( ! ) Warning: preg_match_all(): Compilation failed: missing closing parenthesis at offset 51 in /home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-post-variables.php on line 543 | ||||
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Call Stack | ||||
# | Time | Memory | Function | Location |
1 | 0.0001 | 354376 | {main}( ) | .../index.php:0 |
2 | 0.0001 | 354736 | require( '/home/kbxmjwdb/public_html/wp-blog-header.php ) | .../index.php:17 |
3 | 0.1025 | 10119360 | wp( $query_vars = ??? ) | .../wp-blog-header.php:16 |
4 | 0.1025 | 10119392 | WP->main( $query_args = '' ) | .../functions.php:1336 |
5 | 0.1135 | 10334152 | do_action_ref_array( $hook_name = 'wp', $args = [0 => class WP { public $public_query_vars = [...]; public $private_query_vars = [...]; public $extra_query_vars = [...]; public $query_vars = [...]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE }] ) | .../class-wp.php:830 |
6 | 0.1135 | 10334152 | WP_Hook->do_action( $args = [0 => class WP { public $public_query_vars = [...]; public $private_query_vars = [...]; public $extra_query_vars = [...]; public $query_vars = [...]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE }] ) | .../plugin.php:565 |
7 | 0.1135 | 10334152 | WP_Hook->apply_filters( $value = '', $args = [0 => class WP { public $public_query_vars = [...]; public $private_query_vars = [...]; public $extra_query_vars = [...]; public $query_vars = [...]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE }] ) | .../class-wp-hook.php:348 |
8 | 0.1252 | 10592456 | RankMath\Replace_Variables\Manager->setup( class WP { public $public_query_vars = [0 => 'm', 1 => 'p', 2 => 'posts', 3 => 'w', 4 => 'cat', 5 => 'withcomments', 6 => 'withoutcomments', 7 => 's', 8 => 'search', 9 => 'exact', 10 => 'sentence', 11 => 'calendar', 12 => 'page', 13 => 'paged', 14 => 'more', 15 => 'tb', 16 => 'pb', 17 => 'author', 18 => 'order', 19 => 'orderby', 20 => 'year', 21 => 'monthnum', 22 => 'day', 23 => 'hour', 24 => 'minute', 25 => 'second', 26 => 'name', 27 => 'category_name', 28 => 'tag', 29 => 'feed', 30 => 'author_name', 31 => 'pagename', 32 => 'page_id', 33 => 'error', 34 => 'attachment', 35 => 'attachment_id', 36 => 'subpost', 37 => 'subpost_id', 38 => 'preview', 39 => 'robots', 40 => 'favicon', 41 => 'taxonomy', 42 => 'term', 43 => 'cpage', 44 => 'post_type', 45 => 'embed', 46 => 'post_format', 47 => 'question', 48 => 'question_cat', 49 => 'question_tags', 50 => 'qa_keyword', 51 => 'sitemap', 52 => 'sitemap_n', 53 => 'xsl', 54 => 'rest_route', 55 => 'sitemap-subtype', 56 => 'sitemap-stylesheet', 57 => 'amp', 58 => 'ulpb_post', 59 => 'amazon_auto_links_tag', 60 => 'amazon_auto_links', 61 => 'web-story', 62 => 'web_story_category', 63 => 'web_story_tag', 64 => 'nonamp', 65 => 'nonamphead']; public $private_query_vars = [0 => 'offset', 1 => 'posts_per_page', 2 => 'posts_per_archive_page', 3 => 'showposts', 4 => 'nopaging', 5 => 'post_type', 6 => 'post_status', 7 => 'category__in', 8 => 'category__not_in', 9 => 'category__and', 10 => 'tag__in', 11 => 'tag__not_in', 12 => 'tag__and', 13 => 'tag_slug__in', 14 => 'tag_slug__and', 15 => 'tag_id', 16 => 'post_mime_type', 17 => 'perm', 18 => 'comments_per_page', 19 => 'post__in', 20 => 'post__not_in', 21 => 'post_parent', 22 => 'post_parent__in', 23 => 'post_parent__not_in', 24 => 'title', 25 => 'fields']; public $extra_query_vars = []; public $query_vars = ['name' => 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated', 'amp' => 1]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE } ) | .../class-wp-hook.php:324 |
9 | 0.1260 | 10603952 | RankMath\Replace_Variables\Post_Variables->setup_post_variables( ) | .../class-manager.php:171 |
10 | 0.1261 | 10607608 | RankMath\Replace_Variables\Post_Variables->get_excerpt( ) | .../class-post-variables.php:60 |
11 | 0.1261 | 10607608 | RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $guid = 'https://etsbuy.com/?p=12034'; public $post_type = 'post'; public $comment_count = '1'; public $filter = 'raw' } ) | .../class-post-variables.php:365 |
12 | 0.1326 | 10674936 | preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ) | .../class-post-variables.php:543 |
( ! ) Warning: preg_match_all(): Compilation failed: missing closing parenthesis at offset 51 in /home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-post-variables.php on line 543 | ||||
---|---|---|---|---|
Call Stack | ||||
# | Time | Memory | Function | Location |
1 | 0.0001 | 354376 | {main}( ) | .../index.php:0 |
2 | 0.0001 | 354736 | require( '/home/kbxmjwdb/public_html/wp-blog-header.php ) | .../index.php:17 |
3 | 0.1025 | 10119360 | wp( $query_vars = ??? ) | .../wp-blog-header.php:16 |
4 | 0.1025 | 10119392 | WP->main( $query_args = '' ) | .../functions.php:1336 |
5 | 0.1135 | 10334152 | do_action_ref_array( $hook_name = 'wp', $args = [0 => class WP { public $public_query_vars = [...]; public $private_query_vars = [...]; public $extra_query_vars = [...]; public $query_vars = [...]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE }] ) | .../class-wp.php:830 |
6 | 0.1135 | 10334152 | WP_Hook->do_action( $args = [0 => class WP { public $public_query_vars = [...]; public $private_query_vars = [...]; public $extra_query_vars = [...]; public $query_vars = [...]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE }] ) | .../plugin.php:565 |
7 | 0.1135 | 10334152 | WP_Hook->apply_filters( $value = '', $args = [0 => class WP { public $public_query_vars = [...]; public $private_query_vars = [...]; public $extra_query_vars = [...]; public $query_vars = [...]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE }] ) | .../class-wp-hook.php:348 |
8 | 0.1252 | 10592456 | RankMath\Replace_Variables\Manager->setup( class WP { public $public_query_vars = [0 => 'm', 1 => 'p', 2 => 'posts', 3 => 'w', 4 => 'cat', 5 => 'withcomments', 6 => 'withoutcomments', 7 => 's', 8 => 'search', 9 => 'exact', 10 => 'sentence', 11 => 'calendar', 12 => 'page', 13 => 'paged', 14 => 'more', 15 => 'tb', 16 => 'pb', 17 => 'author', 18 => 'order', 19 => 'orderby', 20 => 'year', 21 => 'monthnum', 22 => 'day', 23 => 'hour', 24 => 'minute', 25 => 'second', 26 => 'name', 27 => 'category_name', 28 => 'tag', 29 => 'feed', 30 => 'author_name', 31 => 'pagename', 32 => 'page_id', 33 => 'error', 34 => 'attachment', 35 => 'attachment_id', 36 => 'subpost', 37 => 'subpost_id', 38 => 'preview', 39 => 'robots', 40 => 'favicon', 41 => 'taxonomy', 42 => 'term', 43 => 'cpage', 44 => 'post_type', 45 => 'embed', 46 => 'post_format', 47 => 'question', 48 => 'question_cat', 49 => 'question_tags', 50 => 'qa_keyword', 51 => 'sitemap', 52 => 'sitemap_n', 53 => 'xsl', 54 => 'rest_route', 55 => 'sitemap-subtype', 56 => 'sitemap-stylesheet', 57 => 'amp', 58 => 'ulpb_post', 59 => 'amazon_auto_links_tag', 60 => 'amazon_auto_links', 61 => 'web-story', 62 => 'web_story_category', 63 => 'web_story_tag', 64 => 'nonamp', 65 => 'nonamphead']; public $private_query_vars = [0 => 'offset', 1 => 'posts_per_page', 2 => 'posts_per_archive_page', 3 => 'showposts', 4 => 'nopaging', 5 => 'post_type', 6 => 'post_status', 7 => 'category__in', 8 => 'category__not_in', 9 => 'category__and', 10 => 'tag__in', 11 => 'tag__not_in', 12 => 'tag__and', 13 => 'tag_slug__in', 14 => 'tag_slug__and', 15 => 'tag_id', 16 => 'post_mime_type', 17 => 'perm', 18 => 'comments_per_page', 19 => 'post__in', 20 => 'post__not_in', 21 => 'post_parent', 22 => 'post_parent__in', 23 => 'post_parent__not_in', 24 => 'title', 25 => 'fields']; public $extra_query_vars = []; public $query_vars = ['name' => 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated', 'amp' => 1]; public $query_string = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&=1'; public $request = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/amp'; public $matched_rule = '([^/]+)/amp(/(.*))?/?$'; public $matched_query = 'name=ncert-solutions-for-class-11-maths-chapter-1-sets-%25e0%25a4%25b8%25e0%25a4%25ae%25e0%25a5%2581%25e0%25a4%259a%25e0%25a5%258d%25e0%25a4%259a%25e0%25a4%25af-updated&='; public $did_permalink = TRUE } ) | .../class-wp-hook.php:324 |
9 | 0.1260 | 10603952 | RankMath\Replace_Variables\Post_Variables->setup_post_variables( ) | .../class-manager.php:171 |
10 | 0.1335 | 10611768 | RankMath\Replace_Variables\Post_Variables->get_excerpt( ) | .../class-post-variables.php:96 |
11 | 0.1335 | 10611768 | RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $guid = 'https://etsbuy.com/?p=12034'; public $post_type = 'post'; public $comment_count = '1'; public $filter = 'raw' } ) | .../class-post-variables.php:365 |
12 | 0.1392 | 10677464 | preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समुच्चय.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समुच्चय).</p>\n\n\nTable of Contents\n\n\n\n\nप्रश्नावली 1.1\nप्रश्नावली 1.2\nप्रश्नावली 1.3\nप्रश्नावली 1.4\nप्रश्नावली 1.5\nप्रश्नावली 1.6\n\n\n\n\nप्रश्नावली 1.1\n<p>प्रश्न 1.निम्नलिखित में कौन से समुच्चय हैं? अपने उत्तर का औचित्य बताइए।(i) '..., $matches = NULL ) | .../class-post-variables.php:543 |
( ! ) Warning: preg_match_all(): Compilation failed: missing closing parenthesis at offset 51 in /home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-post-variables.php on line 543 | ||||
---|---|---|---|---|
Call Stack | ||||
# | Time | Memory | Function | Location |
1 | 0.0001 | 354376 | {main}( ) | .../index.php:0 |
2 | 0.0001 | 354736 | require( '/home/kbxmjwdb/public_html/wp-blog-header.php ) | .../index.php:17 |
3 | 0.1416 | 10689584 | require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ) | .../wp-blog-header.php:19 |
4 | 0.1416 | 10689584 | do_action( $hook_name = 'template_redirect' ) | .../template-loader.php:13 |
5 | 0.1416 | 10689960 | WP_Hook->do_action( $args = [0 => ''] ) | .../plugin.php:517 |
6 | 0.1416 | 10689960 | WP_Hook->apply_filters( $value = '', $args = [0 => ''] ) | .../class-wp-hook.php:348 |
7 | 0.2295 | 18422408 | AMPforWP\AMPVendor\amp_render( '' ) | .../class-wp-hook.php:324 |
8 | 0.4979 | 19366200 | AMPforWP\AMPVendor\AMP_Post_Template->load( ) | .../amp.php:95 |
9 | 0.4979 | 19366200 | AMPforWP\AMPVendor\AMP_Post_Template->load_parts( $templates = [0 => 'single'] ) | .../class-amp-post-template.php:160 |
10 | 0.4979 | 19366520 | AMPforWP\AMPVendor\AMP_Post_Template->verify_and_include( $file = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/single.php', $template_type = 'single' ) | .../class-amp-post-template.php:166 |
11 | 0.4989 | 19368392 | include( '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/single.php ) | .../class-amp-post-template.php:501 |
12 | 0.4989 | 19368392 | amp_header( ) | .../single.php:6 |
13 | 0.4990 | 19368392 | AMPforWP\AMPVendor\AMP_Post_Template->load_parts( $templates = [0 => 'header'] ) | .../components-core.php:425 |
14 | 0.4990 | 19368712 | AMPforWP\AMPVendor\AMP_Post_Template->verify_and_include( $file = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/header.php', $template_type = 'header' ) | .../class-amp-post-template.php:166 |
15 | 0.5041 | 19369224 | include( '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/header.php ) | .../class-amp-post-template.php:501 |
16 | 0.5041 | 19369224 | amp_header_core( ) | .../header.php:6 |
17 | 0.5045 | 19370752 | do_action( $hook_name = 'amp_post_template_head', ...$arg = variadic(class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = ['content_max_width' => 1000, 'document_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated – ETSBUY', 'canonical_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/', 'home_url' => 'https://etsbuy.com', 'blog_name' => 'ETSBUY', 'html_tag_attributes' => [...], 'body_class' => '', 'site_icon_url' => 'https://etsbuy.com/wp-content/uploads/2021/01/cropped-cropped-circle-cropped.v1-150x150.png', 'placeholder_image_url' => 'https://etsbuy.com/wp-content/plugins/accelerated-mobile-pages/images/placeholder-icon.png', 'featured_image' => FALSE, 'comments_link_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/#respond', 'comments_link_text' => 'Leave a Comment', 'amp_runtime_script' => 'https://cdn.ampproject.org/v0.js', 'amp_component_scripts' => [...], 'customizer_settings' => [...], 'font_urls' => [...], 'post_amp_styles' => [...], 'amp_analytics' => [...], 'post_amp_content' => '\n<!-- WP QUADS Content Ad Plugin v. 2.0.83 -->\n<div class="quads-location quads-ad20120 acssefdbc" id="quads-ad20120" data-lazydelay="0">\n<amp-ad width="100vw" height="320" type="adsense" data-ad-client="ca-pub-6245050236237911" data-ad-slot="4229139021" data-auto-format="rspv" data-full-width>\r\n <div overflow></div>\r\n </amp-ad>\n</div>\n\n<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-amp-results-widget"> <div class="rmp-amp-results'..., 'post' => class WP_Post { ... }, 'post_id' => 12034, 'post_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समुच्चय) updated', 'post_publish_timestamp' => 1629533208, 'post_modified_timestamp' => 1629741131, 'post_author' => class WP_User { ... }, 'metadata' => [...], 'ampforwp_amp_content' => FALSE]; public $ID = 12034; public $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समुच्चय).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.1\\u003c/strong\\u003e","text":"प्रश्नावली 1.1","link":"प्रश्नावली-11"},{"level":2,"content":"\\u003cstrong\\u003eप्रश्नावली 1.2\\u003c/strong\\u003e","text":"प्रश्नावली 1.2","link":"प'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } }) ) | .../components-core.php:389 |
18 | 0.5045 | 19371128 | WP_Hook->do_action( $args = [0 => class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = [...]; public $ID = 12034; public $post = class WP_Post { ... } }] ) | .../plugin.php:517 |
19 | 0.5045 | 19371128 | WP_Hook->apply_filters( $value = '', $args = [0 => class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = [...]; public $ID = 12034; public $post = class WP_Post { ... } }] ) | .../class-wp-hook.php:348 |
20 | 0.5066 | 19372320 | ampforwp_rank_math( class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = ['content_max_width' => 1000, 'document_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated – ETSBUY', 'canonical_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/', 'home_url' => 'https://etsbuy.com', 'blog_name' => 'ETSBUY', 'html_tag_attributes' => [...], 'body_class' => '', 'site_icon_url' => 'https://etsbuy.com/wp-content/uploads/2021/01/cropped-cropped-circle-cropped.v1-150x150.png', 'placeholder_image_url' => 'https://etsbuy.com/wp-content/plugins/accelerated-mobile-pages/images/placeholder-icon.png', 'featured_image' => FALSE, 'comments_link_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/#respond', 'comments_link_text' => 'Leave a Comment', 'amp_runtime_script' => 'https://cdn.ampproject.org/v0.js', 'amp_component_scripts' => [...], 'customizer_settings' => [...], 'font_urls' => [...], 'post_amp_styles' => [...], 'amp_analytics' => [...], 'post_amp_content' => '\n<!-- WP QUADS Content Ad Plugin v. 2.0.83 -->\n<div class="quads-location quads-ad20120 acssefdbc" id="quads-ad20120" data-lazydelay="0">\n<amp-ad width="100vw" height="320" type="adsense" data-ad-client="ca-pub-6245050236237911" data-ad-slot="4229139021" data-auto-format="rspv" data-full-width>\r\n <div overflow></div>\r\n </amp-ad>\n</div>\n\n<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-amp-results-widget"> <div class="rmp-amp-results'..., 'post' => class WP_Post { ... }, 'post_id' => 12034, 'post_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated', 'post_publish_timestamp' => 1629533208, 'post_modified_timestamp' => 1629741131, 'post_author' => class WP_User { ... }, 'metadata' => [...], 'ampforwp_amp_content' => FALSE]; public $ID = 12034; public $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } } ) | .../class-wp-hook.php:324 |
21 | 0.5068 | 19370360 | do_action( $hook_name = 'rank_math/head' ) | .../features.php:7450 |
22 | 0.5068 | 19370736 | WP_Hook->do_action( $args = [0 => ''] ) | .../plugin.php:517 |
23 | 0.5068 | 19370736 | WP_Hook->apply_filters( $value = '', $args = [0 => ''] ) | .../class-wp-hook.php:348 |
24 | 0.5068 | 19371488 | RankMath\Frontend\Head->metadesc( '' ) | .../class-wp-hook.php:324 |
25 | 0.5068 | 19371488 | RankMath\Paper\Paper->get_description( ) | .../class-head.php:201 |
26 | 0.5068 | 19371488 | RankMath\Paper\Singular->description( ) | .../class-paper.php:193 |
27 | 0.5068 | 19371488 | RankMath\Paper\Singular->get_post_description( $object = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ) | .../class-singular.php:47 |
28 | 0.5068 | 19371584 | RankMath\Paper\Paper::get_from_options( $id = 'pt_post_description', $object = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $default = ??? ) | .../class-singular.php:173 |
29 | 0.5069 | 19371584 | RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $exclude = ??? ) | .../class-paper.php:477 |
30 | 0.5069 | 19371640 | RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' }, $exclude = [] ) | .../class-helper.php:46 |
31 | 0.5069 | 19371680 | RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ) | .../class-replacer.php:104 |
32 | 0.5069 | 19373664 | RankMath\Replace_Variables\Replacer->get_variable_value( $matches = [0 => [0 => '%excerpt%'], 1 => [0 => 'excerpt'], 2 => [0 => ''], 3 => [0 => '']], $index = 0, $id = 'excerpt' ) | .../class-replacer.php:178 |
33 | 0.5069 | 19373664 | RankMath\Replace_Variables\Variable->run_callback( $var_args = [], $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ) | .../class-replacer.php:295 |
34 | 0.5069 | 19373664 | call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = [], class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ) | .../class-variable.php:188 |
35 | 0.5069 | 19373992 | RankMath\Replace_Variables\Post_Variables->get_excerpt( [], class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ) | .../class-variable.php:188 |
36 | 0.5069 | 19373992 | RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = ''; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } ) | .../class-post-variables.php:365 |
37 | 0.5124 | 19439640 | preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समà¥à¤à¥à¤à¤¯.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</p>\n\n\nTable of Contents\n\n\n\n\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.3\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.4\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.5\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.6\n\n\n\n\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\n<p>पà¥à¤°à¤¶à¥à¤¨ 1.निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ à¤à¥à¤¨ सॠसमà¥à¤à¥à¤à¤¯ हà¥à¤? ठपनॠà¤à¤¤à¥à¤¤à¤° à¤à¤¾ à¤à¤à¤¿à¤¤à¥à¤¯ बताà¤à¤à¥¤(i) '..., $matches = NULL ) | .../class-post-variables.php:543 |
( ! ) Warning: preg_match_all(): Compilation failed: missing closing parenthesis at offset 51 in /home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-post-variables.php on line 543 | ||||
---|---|---|---|---|
Call Stack | ||||
# | Time | Memory | Function | Location |
1 | 0.0001 | 354376 | {main}( ) | .../index.php:0 |
2 | 0.0001 | 354736 | require( '/home/kbxmjwdb/public_html/wp-blog-header.php ) | .../index.php:17 |
3 | 0.1416 | 10689584 | require_once( '/home/kbxmjwdb/public_html/wp-includes/template-loader.php ) | .../wp-blog-header.php:19 |
4 | 0.1416 | 10689584 | do_action( $hook_name = 'template_redirect' ) | .../template-loader.php:13 |
5 | 0.1416 | 10689960 | WP_Hook->do_action( $args = [0 => ''] ) | .../plugin.php:517 |
6 | 0.1416 | 10689960 | WP_Hook->apply_filters( $value = '', $args = [0 => ''] ) | .../class-wp-hook.php:348 |
7 | 0.2295 | 18422408 | AMPforWP\AMPVendor\amp_render( '' ) | .../class-wp-hook.php:324 |
8 | 0.4979 | 19366200 | AMPforWP\AMPVendor\AMP_Post_Template->load( ) | .../amp.php:95 |
9 | 0.4979 | 19366200 | AMPforWP\AMPVendor\AMP_Post_Template->load_parts( $templates = [0 => 'single'] ) | .../class-amp-post-template.php:160 |
10 | 0.4979 | 19366520 | AMPforWP\AMPVendor\AMP_Post_Template->verify_and_include( $file = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/single.php', $template_type = 'single' ) | .../class-amp-post-template.php:166 |
11 | 0.4989 | 19368392 | include( '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/single.php ) | .../class-amp-post-template.php:501 |
12 | 0.4989 | 19368392 | amp_header( ) | .../single.php:6 |
13 | 0.4990 | 19368392 | AMPforWP\AMPVendor\AMP_Post_Template->load_parts( $templates = [0 => 'header'] ) | .../components-core.php:425 |
14 | 0.4990 | 19368712 | AMPforWP\AMPVendor\AMP_Post_Template->verify_and_include( $file = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/header.php', $template_type = 'header' ) | .../class-amp-post-template.php:166 |
15 | 0.5041 | 19369224 | include( '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift/header.php ) | .../class-amp-post-template.php:501 |
16 | 0.5041 | 19369224 | amp_header_core( ) | .../header.php:6 |
17 | 0.5045 | 19370752 | do_action( $hook_name = 'amp_post_template_head', ...$arg = variadic(class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = ['content_max_width' => 1000, 'document_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated – ETSBUY', 'canonical_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/', 'home_url' => 'https://etsbuy.com', 'blog_name' => 'ETSBUY', 'html_tag_attributes' => [...], 'body_class' => '', 'site_icon_url' => 'https://etsbuy.com/wp-content/uploads/2021/01/cropped-cropped-circle-cropped.v1-150x150.png', 'placeholder_image_url' => 'https://etsbuy.com/wp-content/plugins/accelerated-mobile-pages/images/placeholder-icon.png', 'featured_image' => FALSE, 'comments_link_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/#respond', 'comments_link_text' => 'Leave a Comment', 'amp_runtime_script' => 'https://cdn.ampproject.org/v0.js', 'amp_component_scripts' => [...], 'customizer_settings' => [...], 'font_urls' => [...], 'post_amp_styles' => [...], 'amp_analytics' => [...], 'post_amp_content' => '\n<!-- WP QUADS Content Ad Plugin v. 2.0.83 -->\n<div class="quads-location quads-ad20120 acssefdbc" id="quads-ad20120" data-lazydelay="0">\n<amp-ad width="100vw" height="320" type="adsense" data-ad-client="ca-pub-6245050236237911" data-ad-slot="4229139021" data-auto-format="rspv" data-full-width>\r\n <div overflow></div>\r\n </amp-ad>\n</div>\n\n<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-amp-results-widget"> <div class="rmp-amp-results'..., 'post' => class WP_Post { ... }, 'post_id' => 12034, 'post_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated', 'post_publish_timestamp' => 1629533208, 'post_modified_timestamp' => 1629741131, 'post_author' => class WP_User { ... }, 'metadata' => [...], 'ampforwp_amp_content' => FALSE]; public $ID = 12034; public $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } }) ) | .../components-core.php:389 |
18 | 0.5045 | 19371128 | WP_Hook->do_action( $args = [0 => class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = [...]; public $ID = 12034; public $post = class WP_Post { ... } }] ) | .../plugin.php:517 |
19 | 0.5045 | 19371128 | WP_Hook->apply_filters( $value = '', $args = [0 => class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = [...]; public $ID = 12034; public $post = class WP_Post { ... } }] ) | .../class-wp-hook.php:348 |
20 | 0.5066 | 19372320 | ampforwp_rank_math( class AMPforWP\AMPVendor\AMP_Post_Template { private $template_dir = '/home/kbxmjwdb/public_html/wp-content/plugins/accelerated-mobile-pages/templates/design-manager/swift'; private $data = ['content_max_width' => 1000, 'document_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated – ETSBUY', 'canonical_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/', 'home_url' => 'https://etsbuy.com', 'blog_name' => 'ETSBUY', 'html_tag_attributes' => [...], 'body_class' => '', 'site_icon_url' => 'https://etsbuy.com/wp-content/uploads/2021/01/cropped-cropped-circle-cropped.v1-150x150.png', 'placeholder_image_url' => 'https://etsbuy.com/wp-content/plugins/accelerated-mobile-pages/images/placeholder-icon.png', 'featured_image' => FALSE, 'comments_link_url' => 'https://etsbuy.com/ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated/#respond', 'comments_link_text' => 'Leave a Comment', 'amp_runtime_script' => 'https://cdn.ampproject.org/v0.js', 'amp_component_scripts' => [...], 'customizer_settings' => [...], 'font_urls' => [...], 'post_amp_styles' => [...], 'amp_analytics' => [...], 'post_amp_content' => '\n<!-- WP QUADS Content Ad Plugin v. 2.0.83 -->\n<div class="quads-location quads-ad20120 acssefdbc" id="quads-ad20120" data-lazydelay="0">\n<amp-ad width="100vw" height="320" type="adsense" data-ad-client="ca-pub-6245050236237911" data-ad-slot="4229139021" data-auto-format="rspv" data-full-width>\r\n <div overflow></div>\r\n </amp-ad>\n</div>\n\n<!-- FeedbackWP Plugin - Results Widget --><div class="rmp-amp-results-widget"> <div class="rmp-amp-results'..., 'post' => class WP_Post { ... }, 'post_id' => 12034, 'post_title' => 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated', 'post_publish_timestamp' => 1629533208, 'post_modified_timestamp' => 1629741131, 'post_author' => class WP_User { ... }, 'metadata' => [...], 'ampforwp_amp_content' => FALSE]; public $ID = 12034; public $post = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw' } } ) | .../class-wp-hook.php:324 |
21 | 0.5068 | 19370360 | do_action( $hook_name = 'rank_math/head' ) | .../features.php:7450 |
22 | 0.5068 | 19370736 | WP_Hook->do_action( $args = [0 => ''] ) | .../plugin.php:517 |
23 | 0.5068 | 19370736 | WP_Hook->apply_filters( $value = '', $args = [0 => ''] ) | .../class-wp-hook.php:348 |
24 | 0.5368 | 19452032 | RankMath\Schema\JsonLD->json_ld( '' ) | .../class-wp-hook.php:324 |
25 | 0.5368 | 19452032 | RankMath\Schema\JsonLD->get_parts( ) | .../class-jsonld.php:140 |
26 | 0.5370 | 19453520 | RankMath\Helper::replace_vars( $content = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://rajboardexam.in/wp-content/uploads/2020/12/up-board-solutions-for-class-11-maths-chapter-1-se-2.png'; public $alttext = 'UP Board Solutions for Class 11 Maths Chapter 1 Sets 16'; public $titletext = '' }, $exclude = ??? ) | .../class-jsonld.php:751 |
27 | 0.5370 | 19453576 | RankMath\Replace_Variables\Replacer->replace( $string = '%excerpt%', $args = class WP_Post { public $ID = 12034; public $post_author = '1'; public $post_date = '2021-08-21 08:06:48'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":[{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1","link":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥-11"},{"level":2,"content":"\\u003cstrong\\u003eपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\\u003c/strong\\u003e","text":"पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2","link":"पà'...; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $post_excerpt = ''; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $filename = 'https://rajboardexam.in/wp-content/uploads/2020/12/up-board-solutions-for-class-11-maths-chapter-1-se-2.png'; public $alttext = 'UP Board Solutions for Class 11 Maths Chapter 1 Sets 16'; public $titletext = '' }, $exclude = [] ) | .../class-helper.php:46 |
28 | 0.5370 | 19453616 | RankMath\Replace_Variables\Replacer->set_up_replacements( $string = '%excerpt%' ) | .../class-replacer.php:104 |
29 | 0.5370 | 19453616 | RankMath\Replace_Variables\Replacer->get_cache( $string = '%excerpt%' ) | .../class-replacer.php:168 |
30 | 0.5370 | 19454056 | RankMath\Replace_Variables\Variable->run_callback( $var_args = '', $args = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://rajboardexam.in/wp-content/uploads/2020/12/up-board-solutions-for-class-11-maths-chapter-1-se-2.png'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = 'UP Board Solutions for Class 11 Maths Chapter 1 Sets 16'; public $titletext = '' } ) | .../class-replacer.php:254 |
31 | 0.5370 | 19454056 | call_user_func:{/home/kbxmjwdb/public_html/wp-content/plugins/seo-by-rank-math/includes/replace-variables/class-variable.php:188}( $callback = [0 => class RankMath\Replace_Variables\Manager { public $post = class WP_Post { ... }; public $args = class stdClass { ... }; private ${RankMath\Replace_Variables\Cache}cache = [...]; protected $counters = [...]; public $is_post_edit = FALSE; public $is_term_edit = FALSE; protected $is_user_edit = FALSE; protected $replacements = [...]; public $remove_non_replaced = TRUE; private $is_setup = TRUE; protected $tmp_args = class stdClass { ... } }, 1 => 'get_excerpt'], $args = '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://rajboardexam.in/wp-content/uploads/2020/12/up-board-solutions-for-class-11-maths-chapter-1-se-2.png'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = 'UP Board Solutions for Class 11 Maths Chapter 1 Sets 16'; public $titletext = '' } ) | .../class-variable.php:188 |
32 | 0.5370 | 19454384 | RankMath\Replace_Variables\Post_Variables->get_excerpt( '', class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://rajboardexam.in/wp-content/uploads/2020/12/up-board-solutions-for-class-11-maths-chapter-1-se-2.png'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = 'UP Board Solutions for Class 11 Maths Chapter 1 Sets 16'; public $titletext = '' } ) | .../class-variable.php:188 |
33 | 0.5370 | 19454384 | RankMath\Replace_Variables\Post_Variables->get_post_content( $object = class stdClass { public $ID = 12034; public $name = ''; public $post_author = '1'; public $post_content = '<!-- wp:paragraph -->\n<p><strong>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</strong></p>\n<!-- /wp:paragraph -->\n\n<!-- wp:table-of-contents-block/table-of-contents-block {"headers":} -->\n<div class="wp-block-table-of-contents-block-table-of-contents-block eb-toc-container" style="border:undefinedpx solid black;background:#fff6f3;box-shadow:0px 0px 0px 0px black;width:100%" data-collapsible="false" data-initial-collapse="false" data-scroll-top="false" data-sticky="false" data-text-color="#707070" data-hide-mobil'...; public $post_date = '2021-08-21 08:06:48'; public $post_excerpt = ''; public $post_modified = '2021-08-23 17:52:11'; public $post_title = 'Ncert Solutions for Class 11 Maths Chapter 1 Sets (समà¥à¤à¥à¤à¤¯) updated'; public $taxonomy = ''; public $term_id = ''; public $term404 = ''; public $filename = 'https://rajboardexam.in/wp-content/uploads/2020/12/up-board-solutions-for-class-11-maths-chapter-1-se-2.png'; public $post_date_gmt = '2021-08-21 02:36:48'; public $post_status = 'publish'; public $comment_status = 'open'; public $ping_status = 'closed'; public $post_password = ''; public $post_name = 'ncert-solutions-for-class-11-maths-chapter-1-sets-%e0%a4%b8%e0%a4%ae%e0%a5%81%e0%a4%9a%e0%a5%8d%e0%a4%9a%e0%a4%af-updated'; public $to_ping = ''; public $pinged = ''; public $post_modified_gmt = '2021-08-23 12:22:11'; public $post_content_filtered = ''; public $post_parent = 0; public $guid = 'https://etsbuy.com/?p=12034'; public $menu_order = 0; public $post_type = 'post'; public $post_mime_type = ''; public $comment_count = '1'; public $filter = 'raw'; public $alttext = 'UP Board Solutions for Class 11 Maths Chapter 1 Sets 16'; public $titletext = '' } ) | .../class-post-variables.php:365 |
34 | 0.5426 | 19520080 | preg_match_all( $pattern = '/<p>(.*Chapter.1.Sets.(समà¥à¤à¥à¤à¤¯.*)<\\/p>/iu', $subject = '<p>Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).</p>\n\n\nTable of Contents\n\n\n\n\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.2\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.3\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.4\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.5\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.6\n\n\n\n\nपà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥ 1.1\n<p>पà¥à¤°à¤¶à¥à¤¨ 1.निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ à¤à¥à¤¨ सॠसमà¥à¤à¥à¤à¤¯ हà¥à¤? ठपनॠà¤à¤¤à¥à¤¤à¤° à¤à¤¾ à¤à¤à¤¿à¤¤à¥à¤¯ बताà¤à¤à¥¤(i) '..., $matches = NULL ) | .../class-post-variables.php:543 |
Chapter 1 Sets (समà¥à¤à¥à¤à¤¯).
पà¥à¤°à¤¶à¥à¤¨ 1.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ à¤à¥à¤¨ सॠसमà¥à¤à¥à¤à¤¯ हà¥à¤? à¤
पनॠà¤à¤¤à¥à¤¤à¤° à¤à¤¾ à¤à¤à¤¿à¤¤à¥à¤¯ बताà¤à¤à¥¤
(i) j à¤
à¤à¥à¤·à¤° सॠपà¥à¤°à¤¾à¤°à¤®à¥à¤ हà¥à¤¨à¥ वालॠवरà¥à¤· à¤à¥ सà¤à¥ महà¥à¤¨à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(ii) à¤à¤¾à¤°à¤¤ à¤à¥ दस सबसॠà¤
धिठपà¥à¤°à¤¤à¤¿à¤à¤¾à¤¶à¤¾à¤²à¥ लà¥à¤à¤à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(iii) विशà¥à¤µ à¤à¥ सरà¥à¤µà¤¶à¥à¤°à¥à¤·à¥à¤ à¤à¥à¤¯à¤¾à¤°à¤¹ बलà¥à¤²à¤¬à¤¾à¤à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(iv) à¤à¤ª à¤à¥ à¤à¤à¥à¤·à¤¾ à¤à¥ सà¤à¥ बालà¤à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(v) 100 सॠà¤à¤® सà¤à¥ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(vi) लà¥à¤à¤ पà¥à¤°à¥à¤®à¤à¤¨à¥à¤¦ दà¥à¤µà¤¾à¤°à¤¾ लिà¤à¤¿à¤¤ à¤à¤ªà¤¨à¥à¤¯à¤¾à¤¸à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(vii) सरà¥à¤à¥à¤¸à¤® पà¥à¤°à¥à¤£à¤¾à¤à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(viii) à¤à¤¸ à¤
धà¥à¤¯à¤¾à¤¯ मà¥à¤ à¤à¤¨à¥ वालॠपà¥à¤°à¤¶à¥à¤¨à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
(ix) विशà¥à¤µ मà¥à¤ सबसॠà¤
धिठà¤à¤¤à¤°à¤¨à¤¾à¤ à¤à¤¾à¤¨à¤µà¤°à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹à¥¤
हल:
(i) j सॠशà¥à¤°à¥ हà¥à¤¨à¥ वालॠमहà¥à¤¨à¥à¤ à¤à¥ नाम : à¤à¤¨à¤µà¤°à¥, à¤à¥à¤¨ व à¤à¥à¤²à¤¾à¤à¥¤ à¤
तठयह à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(ii) पà¥à¤°à¤¤à¤¿à¤à¤¾à¤¶à¤¾à¤²à¥ लà¥à¤à¤ à¤à¥ परिà¤à¤¾à¤·à¤¿à¤¤ नहà¥à¤ à¤à¤¿à¤¯à¤¾ à¤à¤¾ सà¤à¤¤à¤¾à¥¤ à¤à¤¸à¥à¤²à¤¿à¤ यह à¤à¤ समà¥à¤à¥à¤à¤¯ नहà¥à¤ हà¥à¥¤
(iii) सरà¥à¤µà¤¶à¥à¤°à¥à¤·à¥à¤ बलà¥à¤²à¥à¤¬à¤¾à¤ à¤à¥ परिà¤à¤¾à¤·à¤¿à¤¤ नहà¥à¤ à¤à¤° सà¤à¤¤à¥à¥¤ à¤
तठयह à¤à¤ समà¥à¤à¥à¤à¤¯ नहà¥à¤ हà¥à¥¤
(iv) à¤à¤à¥à¤·à¤¾ à¤à¥ सà¤à¥ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ निशà¥à¤à¤¿à¤¤ हà¥à¤¤à¥ हà¥à¥¤ à¤
तठयह à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(v) 100 सॠà¤à¤® पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾à¤à¤ 1, 2, 3, â¦â¦.. 99 हà¥à¤à¥¤ à¤
तठयह à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(vi) लà¥à¤à¤ पà¥à¤°à¥à¤®à¤à¤¨à¥à¤¦à¥à¤° दà¥à¤µà¤¾à¤°à¤¾ लिà¤à¤¿à¤¤ à¤à¤ªà¤¨à¥à¤¯à¤¾à¤¸à¥à¤ à¤à¤¾ सà¤à¤à¥à¤°à¤¹ à¤à¤¬à¤¨, à¤à¥à¤¦à¤¾à¤¨ à¤à¤¦à¤¿ दà¥à¤µà¤¾à¤°à¤¾ परिà¤à¤¾à¤·à¤¿à¤¤ हà¥à¤à¥¤ à¤
तठयह à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(vii) समपà¥à¤°à¥à¤£à¤¾à¤à¤ {â¦.. -6, -4. 2, 4, 6, â¦.} हà¥à¤à¥¤ à¤à¤¸à¤²à¤¿à¤ यह à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(viii) à¤à¤¸ à¤
धà¥à¤¯à¤¾à¤¯ à¤à¥ पà¥à¤°à¤¶à¥à¤¨ परिà¤à¤¾à¤·à¤¿à¤¤ हà¥à¤à¥¤ à¤
तठयह à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(ix) सà¤à¤¸à¤¾à¤° à¤à¥ सबसॠà¤
धिठà¤à¤¤à¤°à¤¨à¤¾à¤ पशà¥à¤à¤ à¤à¥ सà¤à¤à¥à¤°à¤¹ à¤à¥ परिà¤à¤¾à¤·à¤¿à¤¤ नहॠà¤à¤¿à¤¯à¤¾ à¤à¤¾ सà¤à¤¤à¤¾à¥¤ à¤à¤¸à¤²à¤¿à¤ यह à¤à¤ समà¥à¤à¥à¤à¤¯ नहà¥à¤ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 2.
मान लà¥à¤à¤¿à¤ A = {1, 2, 3, 4, 5, 6}. रिà¤à¥à¤¤ सà¥à¤¥à¤¾à¤¨à¥à¤ मà¥à¤ à¤à¤ªà¤¯à¥à¤à¥à¤¤ पà¥à¤°à¤¤à¥à¤ âà¤
थवा â à¤à¤°à¤¿à¤à¥¤
(i) 5 â¦â¦ A
(ii) 8 â¦â¦.. A
(iii) 0 â¦â¦â¦ A
(iv) 4 â¦â¦. A
(v) 2 â¦â¦.. A
(vi) 10 â¦â¦. A
हल:
(i) 5 â A
(ii) 8 â A
(iii) 0 â 4
(iv) 4 â A
(v) 2 â A
(vi) 10 â A
पà¥à¤°à¤¶à¥à¤¨ 3.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ रà¥à¤¸à¥à¤à¤° रà¥à¤ª मà¥à¤ लिà¤à¤¿à¤:
(i) A = {x : x à¤à¤ पà¥à¤°à¥à¤£à¤¾à¤ हॠà¤à¤° -3 < x < 7}
(ii) B = {x : x सà¤à¤à¥à¤¯à¤¾ 6 सॠà¤à¤® à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(iii) C = {x : x दॠà¤
à¤à¤à¥ à¤à¥ à¤à¤¸à¥ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¤¿à¤¸à¤à¥ à¤
à¤à¤à¥à¤ à¤à¤¾ यà¥à¤à¤«à¤² 8 हà¥à¥¤}
(iv) D = {x : x à¤à¤ à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¥ 60 à¤à¥ à¤à¤¾à¤à¤ हà¥à¥¤
(v) E = TRIGONOMETRY शबà¥à¤¦ à¤à¥ सà¤à¥ à¤
à¤à¥à¤·à¤°à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
(vi) F = BETTER शबà¥à¤¦ à¤à¥ सà¤à¥ à¤
à¤à¥à¤·à¤°à¥à¤ à¤à¤¼à¤¾ समà¥à¤à¥à¤à¥à¤¯
हल:
(i) A = {-2, -1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {1, 2, 3, 4, 5}
(iii) C = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = {2, 3, 5}
(v) E = {T, R, I, G, O, N, M, E, Y}
(vi) F = {B, E, T, R}
पà¥à¤°à¤¶à¥à¤¨ 4.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ निरà¥à¤®à¤¾à¤£ रà¥à¤ª मà¥à¤ वà¥à¤¯à¤à¥à¤¤ à¤à¥à¤à¤¿à¤:
(i) {3, 6, 9, 12}
(i) {2, 4, 8, 16, 32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6, â¦.}
(v) {1, 4, 9, â¦â¦100}
हल:
(i) {x : x = 3n à¤à¤° 1 ⤠n ⤠4}
(ii) {x : x = 2n à¤à¤° 1 ⤠n ⤠5}
(iii) {x : x = 5 à¤à¤° 1 ⤠n ⤠4}
(iv) {x : x à¤à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(v) {x : x = n², 1 ⤠n ⤠10}
पà¥à¤°à¤¶à¥à¤¨ 5.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ सà¤à¥ à¤
वयवà¥à¤ (सदसà¥à¤¯à¥à¤) à¤à¥ सà¥à¤à¥à¤¬à¤¦à¥à¤§ à¤à¥à¤à¤¿à¤à¥¤
(i) A = {x : x à¤à¤ विषम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤
(i) B = x : x à¤à¤ पà¥à¤°à¥à¤£à¤¾à¤ हà¥, Â
पà¥à¤°à¤¶à¥à¤¨ 6.
बाठà¤à¤° रà¥à¤¸à¥à¤à¤° रà¥à¤ª मà¥à¤ लिà¤à¤¿à¤¤ à¤à¤° दाà¤à¤ à¤à¤° समà¥à¤à¥à¤à¤¯ निरà¥à¤®à¤¾à¤£ रà¥à¤ª मà¥à¤ वरà¥à¤£à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¤¾ सहॠमिलान à¤à¥à¤à¤¿à¤à¥¤
(i) {1, 2, 3, 6} (a) {x : x à¤à¤ à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¤° 6 à¤à¥ à¤à¤¾à¤à¤ हà¥à¥¤
(ii) {2, 3} (b) {x : x सà¤à¤à¥à¤¯à¤¾ 10 सॠà¤à¤® à¤à¤ विषम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤
(iii) {M, A, T, H, E, I, C, S} (c) {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¤° 6 à¤à¥ à¤à¤¾à¤à¤ हà¥à¥¤
(iv) {1, 3, 5, 7, 9} (d) {x : x MATHEMATICS शबà¥à¤¦ à¤à¤¾ à¤à¤ à¤
à¤à¥à¤·à¤° हà¥:
हल:
(i) â (c)
(ii) â (a)
(iii) â (d)
(iv) â (b)
पà¥à¤°à¤¶à¥à¤¨ 1.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ सॠà¤à¥à¤¨ सॠरिà¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ à¤à¥ à¤à¤¦à¤¾à¤¹à¤°à¤£ हà¥?
(i) 2 सॠà¤à¤¾à¤à¥à¤¯à¤ विषम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
(ii) सम à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
(iii) {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥, x < 5 à¤à¤° साथ हॠसाथ x > 7}
(iv) {y : y à¤à¤¿à¤¨à¥à¤¹à¥à¤ à¤à¥ दॠसमाà¤à¤¤à¤° रà¥à¤à¤¾à¤à¤ à¤à¤¾ à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ बिनà¥à¤¦à¥ हà¥à¥¤}
हल:
(i) 2 सॠà¤à¤¾à¤à¥à¤¯ à¤à¥à¤ à¤à¥ विषम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾à¤à¤ नहà¥à¤ हà¥à¤à¥¤ à¤
त: यह à¤à¤ रिà¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(ii) सम à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾ à¤à¤¾ समà¥à¤à¥à¤à¤¯ {2} हà¥à¥¤ यह à¤à¤ रिà¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ नहà¥à¤ हà¥à¥¤
(iii) x < 5 à¤à¤° x > 7 à¤à¥à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ नहà¥à¤ हà¥à¥¤ à¤
त: यह à¤à¤ रिà¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(iv) समाà¤à¤¤à¤° रà¥à¤à¤¾à¤à¤ à¤à¤¹à¥à¤ à¤à¥ नहà¥à¤ मिलतॠहà¥à¤à¥¤ à¤
त: यह à¤à¤ रिà¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 2.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¥à¤¯à¥à¤ मà¥à¤ सॠà¤à¥à¤¨ परिमित à¤à¤° à¤à¥à¤¨ à¤
परिमित हà¥à¤?
(i) वरà¥à¤· à¤à¥ महà¥à¤¨à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯à¥¤
(ii) {1, 2, 3, â¦..}
(ii) {1, 2, 3, â¦.. 99, 100}
(iv) 100 सॠबड़ॠधन पà¥à¤°à¥à¤£à¤¾à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
(v) 99 सॠà¤à¥à¤à¥ à¤
à¤à¤¾à¤à¥à¤¯ पà¥à¤°à¥à¤£à¤¾à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
हल:
(i) वरà¥à¤· मà¥à¤ 12 महà¥à¤¨à¥ हà¥à¤¤à¥ हà¥à¤à¥¤
à¤
तठयह à¤à¤ परिमित समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(ii) समà¥à¤à¥à¤à¤¯ {1, 2, 3, â¦â¦} मà¥à¤ à¤
नà¤à¤¤ à¤
वयव हà¥à¤à¥¤
à¤
à¤à¤¤: यह à¤à¤ à¤
परिमित समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(iii) समà¥à¤à¥à¤à¤¯ {1, 2, 3, â¦â¦ 99, 100} मà¥à¤ à¤à¥à¤² 100 à¤
वयव हà¥à¤à¥¤
à¤
त: यह à¤à¤ परिमित समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(iv) 100 सॠबड़ॠपà¥à¤°à¥à¤£à¤¾à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¥à¤¯ {101, 102, 103, â¦.} हॠà¤à¤¿à¤¸à¤®à¥à¤ à¤
नà¤à¤¤ à¤
वयव हà¥à¤à¥¤
à¤
त: यह à¤à¤ à¤
परिमित समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(v) 99 सॠà¤à¥à¤à¥ à¤
à¤à¤¾à¤à¥à¤¯ पà¥à¤°à¥à¤£à¤¾à¤à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ {2, 3, 5, 7, â¦â¦ 97} हॠà¤à¤¿à¤¸à¤®à¥à¤ à¤
वयवà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ निशà¥à¤à¤¿à¤¤ हà¥à¥¤
à¤
त: यह à¤à¤ परिमित समà¥à¤à¥à¤à¤¯ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 3.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ मà¥à¤ सॠपà¥à¤°à¤¤à¥à¤¯à¥à¤ à¤à¥ लिठबताà¤à¤ à¤à¤¿ à¤à¥à¤¨ परिमित हॠà¤à¤° à¤à¥à¤¨ à¤
परिमित ह�
(i) x-à¤
à¤à¥à¤· à¤à¥ समाà¤à¤¤à¤° रà¥à¤à¤¾à¤à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯à¥¤
(ii) à¤
à¤à¤à¥à¤°à¥à¤à¥ वरà¥à¤£à¤®à¤¾à¤²à¤¾ à¤à¥ à¤
à¤à¥à¤·à¤°à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯à¥¤
(iii) à¤à¤¨ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ à¤à¥ 5 à¤à¥ à¤à¥à¤£à¤ हà¥à¤à¥¤
(iv) पà¥à¤¥à¥à¤µà¥ पर रहनॠवालॠà¤à¤¾à¤¨à¤µà¤°à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
(v) मà¥à¤² बिनà¥à¤¦à¥ (0, 0) सॠहà¥à¤à¤° à¤à¤¾à¤¨à¥ वालॠवà¥à¤¤à¥à¤¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯à¥¤
हल:
(i) x-à¤
à¤à¥à¤· à¤à¥ समाà¤à¤¤à¤° à¤
नà¤à¤¤ रà¥à¤à¤¾à¤à¤ à¤à¥à¤à¤à¥ à¤à¤¾ सà¤à¤¤à¥ हà¥à¤à¥¤ à¤
त: यह à¤à¤ à¤
परिमित समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(ii) à¤
à¤à¤à¥à¤°à¥à¤à¥ वरà¥à¤£à¤®à¤¾à¤²à¤¾ मà¥à¤ à¤à¥à¤² 26 à¤
à¤à¥à¤·à¤° हà¥à¤¤à¥ हà¥à¤à¥¤ à¤à¤¨ à¤
à¤à¥à¤·à¤°à¥à¤ सॠबननॠवाला समà¥à¤à¥à¤à¤¯ परिमित हà¥à¤à¤¾à¥¤
(iii) 5 सॠविà¤à¤¾à¤à¤¿à¤¤ हà¥à¤¨à¥ वालॠसà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ {5, 10, 15, 20, â¦.} हà¥, à¤à¤¿à¤¸à¤®à¥à¤ à¤
नà¤à¤¤ à¤
वयव हà¥à¤à¥¤ à¤
तठयह à¤à¤ à¤
परिमित समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(iv) पà¥à¤¥à¥à¤µà¥ पर रहनॠवालॠà¤à¤¾à¤¨à¤µà¤°à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ परिमित हà¥à¤à¤¾à¥¤
(v) मà¥à¤² बिनà¥à¤¦à¥ à¤à¥ à¤à¥à¤¨à¥à¤¦à¥à¤° मानà¤à¤° à¤
ननà¥à¤¤ वà¥à¤¤à¥à¤¤ à¤à¥ à¤à¤¾ सà¤à¤¤à¥ हà¥à¤à¥¤ à¤
त: यह à¤
परिमित हà¥à¤à¤¾à¥¤
पà¥à¤°à¤¶à¥à¤¨ 4.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ बताà¤à¤ à¤à¤¿ A = B हॠà¤
थवा नहà¥à¤ हà¥à¥¤
(i) A = {a, b, c, a}, B = {a, c, b, a}
(ii) A = {4, 8, 12, 16}, B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}, B = {x : x सम धन पà¥à¤°à¥à¤£à¤¾à¤ हॠà¤à¤° x ⤠10}
(iv) A = {x : x सà¤à¤à¥à¤¯à¤¾ 10 à¤à¤¾ à¤à¤ à¤à¥à¤£à¤ हà¥}, B = {10, 15,20, 25, 30, â¦}
हल:
(i) A à¤à¤° B दà¥à¤¨à¥à¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ à¤
वयव a, b, c, d हà¥à¤ à¤
तठA = B.
(ii) A मà¥à¤ à¤
वयव 12 हॠपरनà¥à¤¤à¥ B मà¥à¤ नहà¥à¤ हॠà¤
तठA â B.
(iii) A à¤à¤° B दà¥à¤¨à¥à¤ समà¥à¤à¥à¤à¤¯à¥à¤ मà¥à¤ à¤
वयव 2, 4, 6, 8 à¤à¤° 10 हà¥à¤à¥¤ à¤
तठA = B.
(iv) A = {10, 20, 30, 40, â¦..}, B = {10, 15, 25, 30, â¦.}
10 à¤à¥ à¤à¥à¤£à¤à¥à¤ मà¥à¤ 5, 15, 25 नहà¥à¤ à¤à¤¤à¤¾ हà¥à¥¤ à¤
तठA â B.
पà¥à¤°à¤¶à¥à¤¨ 5.
à¤à¥à¤¯à¤¾ निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯ यà¥à¤à¥à¤® समान हà¥à¤ ? à¤à¤¾à¤°à¤£ सहित बताà¤à¤à¥¤
(i) A = {2, 3}
B = {x : x समà¥à¤à¤°à¤£ x² + 5x + 6 = 0 à¤à¤¾ à¤à¤ हल हà¥à¥¤}
(ii) A = {k : x शबà¥à¤¦ âFOLLOWâ à¤à¤¾ à¤à¤ à¤
à¤à¥à¤·à¤° हà¥à¥¤}
B = {y : y शबà¥à¤¦ âWOLF à¤à¤¾ à¤à¤ à¤
à¤à¥à¤·à¤° हà¥à¥¤}
हल:
(i) A = {2, 3}, B = x : x समà¥à¤à¤°à¤£ x² + 5x + 6 = 0} = {-2, -3}
सà¥à¤ªà¤·à¥à¤ हॠà¤à¤¿ समà¥à¤à¥à¤à¤¯ A à¤à¤° B à¤à¥ à¤
वयव à¤à¤¿à¤¨à¥à¤¨ हà¥à¤à¥¤
à¤
त: A â B.
(ii) A = {F, O, L, W}, B = {W, O, L, F}
समà¥à¤à¥à¤à¥à¤¯ A à¤à¤° B à¤à¥ à¤
वयव समान हà¥à¤à¥¤ à¤
त: A = B.
पà¥à¤°à¤¶à¥à¤¨ 6.
नà¥à¤à¥ दिठà¤à¤ समà¥à¤à¥à¤à¤¯à¥à¤ मà¥à¤ सॠसमान समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¤¾ à¤à¤¯à¤¨ à¤à¥à¤à¤¿à¤:
A = {2, 4, 8, 12}
B = {1, 2, 3, 4}
C = {4, 8, 12, 14}
D = {3, 1, 4, 2}
E = {- 1, 1}
F = {0, a}
G = {1, -1}
H = {0, 1}
हल:
यहाठसमà¥à¤à¥à¤à¤¯ B à¤à¤° D à¤à¥ à¤
वयव 1, 2, 3, 4, हà¥à¤à¥¤
B = D
तथा समà¥à¤à¥à¤à¤¯ E à¤à¤° G मà¥à¤ -1, 1 à¤
वयव समान हà¥à¤à¥¤
E = G
पà¥à¤°à¤¶à¥à¤¨ 1.
रिà¤à¥à¤¤ सà¥à¤¥à¤¾à¤¨à¥à¤ मà¥à¤ पà¥à¤°à¤¤à¥à¤ âया â à¤à¥ à¤à¤° à¤à¤° सहॠà¤à¤¥à¤¨ बनाà¤à¤:
(i) {2, 3, 4} â¦. {1, 2, 3, 4, 5}
(ii) {a, b, c}â¦.. {b, c, d}
(iii) {x : x à¤à¤ªà¤à¥ विदà¥à¤¯à¤¾à¤²à¤¯ à¤à¥ à¤à¤à¥à¤·à¤¾ XI à¤à¤¾ à¤à¤ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ हॠ} â¦. {x : x à¤à¤ªà¤à¥ विदà¥à¤¯à¤¾à¤²à¤¯ à¤à¤¾ à¤à¤ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ हà¥à¥¤}
(iv) {x : x à¤à¤¿à¤¸à¥ समतल मà¥à¤ सà¥à¤¥à¤¿à¤¤ à¤à¤ वà¥à¤¤ हà¥} â¦.. {x : x à¤à¤ समान समतल मà¥à¤ à¤à¤ वà¥à¤¤à¥à¤¤ हà¥à¥¤ à¤à¤¿à¤¸à¤à¥ तà¥à¤°à¤¿à¤à¥à¤¯à¤¾ 1 à¤à¤à¤¾à¤ हà¥à¥¤}
(v) {x : x à¤à¤¿à¤¸à¥ समतल मà¥à¤ सà¥à¤¥à¤¿à¤¤ à¤à¤ तà¥à¤°à¤¿à¤à¥à¤ हà¥} â¦. {x : x à¤à¤¿à¤¸à¥ समतल मà¥à¤ सà¥à¤¥à¤¿à¤¤ à¤à¤ à¤à¤¯à¤¤ हà¥à¥¤}
(vi) {x : x à¤à¤¿à¤¸à¥ सà¤à¤®à¤¤à¤² मà¥à¤ सà¥à¤¥à¤¿à¤¤ à¤à¤ समबाहॠतà¥à¤°à¤¿à¤à¥à¤ हà¥} â¦â¦ {x : x à¤à¤¿à¤¸à¥ समतल मà¥à¤ सà¥à¤¥à¤¿à¤¤ à¤à¤ तà¥à¤°à¤¿à¤à¥à¤ हà¥à¥¤}
(vii) {x : x à¤à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥} â¦â¦. {x : x à¤à¤ पà¥à¤°à¥à¤£à¤¾à¤ हà¥}
हल:
(i) à¤
वयव 2, 3, 4 â {1, 2, 3, 4, 5}
à¤
तठ{2, 3, 4} â {1, 2, 3, 4, 5}
(ii) {a, b, c} à¤à¤¾ à¤
वयव a â {b, c, d}
à¤
त: {a, b, c} â {b, c, d}
(iii) à¤à¥ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ विदà¥à¤¯à¤¾à¤²à¤¯ à¤à¥ à¤à¤à¥à¤·à¤¾ XI मà¥à¤ हà¥à¤ वॠविदà¥à¤¯à¤¾à¤²à¤¯ मà¥à¤ à¤à¥ हà¥à¤à¥¤
à¤
तठ{x : x विदà¥à¤¯à¤¾à¤²à¤¯ à¤à¥ à¤à¤à¥à¤·à¤¾ XI à¤à¤¾ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥} â {x : x à¤à¤ªà¤à¥ विदà¥à¤¯à¤¾à¤²à¤¯ à¤à¤¾ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥}
(iv) समà¥à¤à¥à¤à¤¯ {x : x समतल मà¥à¤ à¤à¤ वà¥à¤¤à¥à¤¤} à¤à¥ à¤à¤ à¤
वयव वà¥à¤¤à¥à¤¤ à¤à¥ तà¥à¤°à¤¿à¤à¥à¤¯à¤¾ 1 सॠà¤à¤¿à¤¨à¥à¤¨ हॠसà¤à¤¤à¥ हà¥à¥¤
à¤
तठ{x : x समतल मà¥à¤ वà¥à¤¤à¥à¤¤} â {x : x वà¥à¤¤à¥à¤¤ à¤à¥ तà¥à¤°à¤¿à¤à¥à¤¯à¤¾ 1 à¤à¤à¤¾à¤ हॠ}
(v) तà¥à¤°à¤¿à¤à¥à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ à¤à¤¯à¤¤à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ सॠबिलà¥à¤à¥à¤² à¤à¤¿à¤¨à¥à¤¨ हà¥à¥¤
à¤
तठ{x : x समतल मà¥à¤ à¤à¤ तà¥à¤°à¤¿à¤à¥à¤} â {x : x समतल मà¥à¤ à¤à¤ à¤à¤¯à¤¤}
(vi) पà¥à¤°à¤¤à¥à¤¯à¥à¤ समबाहॠतà¥à¤°à¤¿à¤à¥à¤ à¤à¤ तà¥à¤°à¤¿à¤à¥à¤ हà¥à¥¤
à¤
तठ{x : x समतल मà¥à¤ à¤à¤ समबाहॠतà¥à¤°à¤¿à¤à¥à¤} â {x : x समतल मà¥à¤ à¤à¤ तà¥à¤°à¤¿à¤à¥à¤}
(vii) पà¥à¤°à¤¤à¥à¤¯à¥à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ à¤à¤ पà¥à¤°à¥à¤£à¤¾à¤ हà¥à¥¤
à¤
तठ{x : x à¤à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾} â {x : x à¤à¤ पà¥à¤°à¥à¤£à¤¾à¤}
पà¥à¤°à¤¶à¥à¤¨ 2.
à¤à¤¾à¤à¤à¤¿à¤ à¤à¤¿ निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¤¥à¤¨ सतà¥à¤¯ हà¥à¤ à¤
थवा à¤
सतà¥à¤¯ हà¥à¤:
(i) {a, b} â {b, c, a}
(ii) {a, e} â {x : x à¤
à¤à¤à¥à¤°à¥à¤à¥ वरà¥à¤£à¤®à¤¾à¤²à¤¾ à¤à¤¾ à¤à¤ सà¥à¤µà¤° हà¥à¥¤
(iii) {1, 2, 3} â {1, 3, 5}
(iv) {a} â {a, b, c}
(v) {a} â {a, b, c}
(vi) {x : x सà¤à¤à¥à¤¯à¤¾ 6 सॠà¤à¤® à¤à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤ â {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥, à¤à¥ सà¤à¤à¥à¤¯à¤¾ 36 à¤à¥ विà¤à¤¾à¤à¤¿à¤¤ à¤à¤°à¤¤à¥ हà¥à¥¤
हल:
(i) समà¥à¤à¥à¤à¤¯ {a, b} à¤à¥ à¤
वयव a, b दासमà¥à¤à¥à¤à¤¯ {b, c, a} मà¥à¤ हà¥à¥¤
{a, b} â {b, c, a}
à¤
तठà¤à¤ªà¤°à¥à¤à¥à¤¤ à¤à¤¥à¤¨ à¤
सतà¥à¤¯ हà¥à¥¤
(ii) a, e दà¥à¤¨à¥à¤ हॠसà¥à¤µà¤° हà¥à¤à¥¤
{a, e} = {x : x, à¤
à¤à¤à¥à¤°à¥à¤à¥ वरà¥à¤£à¤®à¤¾à¤²à¤¾ à¤à¤¾ à¤à¤ सà¥à¤µà¤° हà¥à¥¤
à¤
तठयह à¤à¤¥à¤¨ सतà¥à¤¯ हà¥à¥¤
(iii) समà¥à¤à¥à¤à¤¯ {1, 2, 3} à¤à¤° {1, 3, 5} मà¥à¤ à¤
वयव 2 समà¥à¤à¥à¤à¤¯ {1, 3, 5} नहà¥à¤ हà¥à¥¤
{1, 2, 3} â {1, 3, 5} à¤à¤¥à¤¨à¥ à¤
सतà¥à¤¯ हà¥à¥¤
(iv) a â {a, b, c}
{a} â {a, b, c} यह à¤à¤¥à¤¨ सतà¥à¤¯ हà¥à¥¤
(v) {4} समà¥à¤à¥à¤à¤¯ हà¥, à¤
वयव नहॠहà¥à¥¤
{a} } â {a, b, c} à¤à¤¥à¤¨ à¤
सतà¥à¤¯ हà¥à¥¤
(vi) सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ 2, 4 सà¤à¤à¥à¤¯à¤¾ 6 सॠà¤à¤® हॠतथा 36 à¤à¥ विà¤à¤¾à¤à¤¿à¤¤ à¤à¤°à¤¤à¥ हà¥à¥¤
{x : x à¤à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¥ 6 सॠà¤à¤® हà¥} â {x : x à¤à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ 36 à¤à¥ विà¤à¤¾à¤à¤¿à¤¤ à¤à¤°à¤¤à¥ हà¥à¥¤ à¤
तठयह à¤à¤¥à¤¨ सतà¥à¤¯ हà¥à¥¤}
पà¥à¤°à¤¶à¥à¤¨ 3.
मान लà¥à¤à¤¿à¤ à¤à¤¿ A = {1, 2, 3, 4, 5}, निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ सॠà¤à¥à¤¨ सा à¤à¤¥à¤¨ सहॠनहà¥à¤ हॠà¤à¤° à¤à¥à¤¯à¥à¤?
(i) {3, 4} â A
(i) {3, 4} â A
(iii) {{3, 4}} â A
(iv) 1 â A
(v) 1 â A
(vi) {1, 2, 5} â A
(vii) {1, 2, 5} â A
(viii) {1, 2, 3} â A
(ix) Φ â A
(x) Φ âA
(xi) {Φ} â A
हल:
(i) सहॠनहà¥à¤ हà¥à¥¤ समà¥à¤à¥à¤à¤¯ {3, 4} à¤à¤ à¤
वयव हà¥à¥¤
(ii) सहॠहà¥à¥¤ à¤à¥à¤¯à¥à¤à¤à¤¿ {3, 4} समà¥à¤à¥à¤à¤¯ A à¤à¤¾ à¤à¤ à¤
वयव हà¥à¥¤
(iii) सहॠहà¥à¥¤ A à¤à¥ à¤
वयव {3, 4} à¤à¤¾ à¤à¤ à¤à¤ªà¤¸à¤®à¥à¤à¥à¤à¤¯ हà¥à¥¤
(iv) 1 â A, सहॠहà¥à¥¤
(v) 1 â A सहॠनहà¥à¤ हॠà¤à¥à¤¯à¥à¤à¤à¤¿ 1 à¤à¤ समà¥à¤à¥à¤à¤¯ नहà¥à¤ हà¥à¥¤
(vi) {1, 2, 5} â A सहॠहà¥à¥¤ समà¥à¤à¥à¤à¤¯ {1, 2,5} à¤à¥ à¤
वयव 1, 2, 5 समà¥à¤à¥à¤à¤¯ A मà¥à¤ हà¥à¥¤
(vii) {1, 2, 5} â सहॠनहà¥à¤ हà¥à¥¤ {12, 5} à¤
वयव नहà¥à¤ हà¥à¥¤ यह à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
(viii) {1, 2, 3} â A सहॠनहà¥à¤ हà¥à¥¤ à¤
वयव 3 समà¥à¤à¥à¤à¤¯ मà¥à¤ नहॠहà¥à¥¤
(ix) Φ â A, सहॠनहà¥à¤ हà¥à¥¤ Φ à¤à¤ समà¥à¤à¥à¤à¤¯ हà¥, à¤
वयव नहà¥à¤ हà¥à¥¤
(x) {Φ} â A सहॠहà¥à¥¤ सà¤à¥ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¤¾ à¤à¤ªà¤¸à¤®à¥à¤à¥à¤à¤¯ हà¥à¥¤
(xi) {Φ} â A सहॠनहà¥à¤ हà¥à¥¤ {Φ} समà¥à¤à¥à¤à¤¯ à¤à¤¾ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 4.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ सà¤à¥ à¤à¤ªà¤¸à¤®à¥à¤à¥à¤à¤¯ लिà¤à¤¿à¤à¥¤
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) Φ
हल:
(i) Φ, {a}
(ii) Φ, {a}, {b}, {a, b}
(iii) Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}
(iv) Φ
पà¥à¤°à¤¶à¥à¤¨ 5.
P (A) à¤à¥ à¤à¤¿à¤¤à¤¨à¥ à¤
वयव हà¥à¤, यदि A = Φ
हल:
A = Φ, P(A) = Φ à¤à¤¸ पà¥à¤°à¤à¤¾à¤° P (A) à¤à¥ 2° = 1 à¤
वयव हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 6.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥ à¤
à¤à¤¤à¤°à¤¾à¤² रà¥à¤ª मà¥à¤ लिà¤à¤¿à¤:
(i) {x : x âR, -4 < x ⤠6}
(ii) {x : x âR, -12 < x < -10}
(iii) {x : x â R, 0 ⤠x < 7}
(iv) {x : x â R, 3 ⤠x ⤠4}
हल:
वाà¤à¤à¤¿à¤¤ à¤
à¤à¤¤à¤°à¤¾à¤² à¤à¤¸ पà¥à¤°à¤à¤¾à¤° हà¥à¤à¥¤
(i) (-4, 6]
(ii) (-12, â 10)
(iii) [0, 7)
(iv) [3, 4]
पà¥à¤°à¤¶à¥à¤¨ 7.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤
à¤à¤¤à¤°à¤¾à¤²à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ निरà¥à¤®à¤¾à¤£ रà¥à¤ª मà¥à¤ लिà¤à¤¿à¤:
(i) (-3, 0)
(ii) [6, 12]
(iii) (6, 12]
(iv) [-23, 5]
हल:
(i) (-3, 0) = {x : x â R, -3 < x < 0}
(ii) [6, 12] = {x : x â R, 6 ⤠x ⤠12}
(iii) (6,12] = {x : x â R, 6 < x ⤠12}।
(iv) [-23, 5] = {x : x â R, -23 ⤠x ⤠5}
पà¥à¤°à¤¶à¥à¤¨ 8.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ सॠपà¥à¤°à¤¤à¥à¤¯à¥à¤ à¤à¥ लिठà¤à¤ª à¤à¥à¤¨ सा सारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯ पà¥à¤°à¤¸à¥à¤¤à¤¾à¤µà¤¿à¤¤ à¤à¤°à¥à¤à¤à¥?
(i) समà¤à¥à¤£ तà¥à¤°à¤¿à¤à¥à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
(ii) समदà¥à¤µà¤¿à¤¬à¤¾à¤¹à¥ तà¥à¤°à¤¿à¤à¥à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯
हल:
दà¥à¤¨à¥à¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ लिठसारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯ :
{x : x समतल मà¥à¤ सà¥à¤¥à¤¿à¤¤ à¤à¤ तà¥à¤°à¤¿à¤à¥à¤}
पà¥à¤°à¤¶à¥à¤¨ 9.
समà¥à¤à¥à¤à¤¯ A = {1, 3, 5}, B = {2, 4, 6} à¤à¤° C = {0, 2, 4, 6, 8} पà¥à¤°à¤¦à¤¤à¥à¤¤ हà¥à¤à¥¤ à¤à¤¨ तà¥à¤¨à¥à¤ समà¥à¤à¥à¤à¤¯à¥à¤ A, B à¤à¤° C à¤à¥ लिठनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ सॠà¤à¥à¤¨ सा (सà¥) सारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯ लिठà¤à¤¾ सà¤à¤¤à¥ हà¥à¤?
(i) {0,1, 2, 3, 4, 5, 6}
(ii) Φ
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
हल:
समà¥à¤à¥à¤à¤¯ (iii),
तà¥à¤¨à¥à¤ समà¥à¤à¥à¤à¤¯ A, B, C à¤à¥ लिठ{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} सारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯ हà¥à¤à¥¤
पà¥à¤°à¤¶à¥à¤¨ 1.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ सॠपà¥à¤°à¤¤à¥à¤¯à¥à¤ समà¥à¤à¥à¤à¤¯ यà¥à¤à¥à¤® à¤à¤¾ समà¥à¤®à¤¿à¤²à¤¨ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) X = {1, 3, 5}, Y = {1, 2, 3}
(i) A = {a, e, i, 0, u}, B = {a, b, c}
(iii) A = {x : à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¤° 3 à¤à¤¾ à¤à¥à¤£à¤ हà¥à¥¤
B = {x : x सà¤à¤à¥à¤¯à¤¾ 6 सॠà¤à¤® à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤
(iv) A = {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¤° 1 < x ⤠6}
B = {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¤° 6 < x < 10}
(v) A = {1, 2, 3}, B = Φ
हल:
(i) X ⪠Y = {1, 3, 5} ⪠{1, 2, 3} = {1, 2, 3, 5}
(ii) A ⪠B = {a, e, i, 0, u} ⪠{a, b, c} = {a, b, c, e, i, 0, u}
(iii) A ⪠B = {3, 6, 9â¦.} ⪠{1, 2, 3, 4, 5} = {1, 2, 4, 5 या सà¤à¤à¥à¤¯à¤¾ 3 à¤à¤¾ à¤à¥à¤£à¤}
(iv) A = {2, 3, 4, 5, 6}, B = {7, 8, 9}
A ⪠B = {2, 3, 4, 5, 6, 7, 8, 9} à¤
रà¥à¤¥à¤¾à¤¤à¥ {x : 1 < x < 10, x â N}
(v) A âªB = {1, 2, 3} ⪠Φ = {1, 2, 3}
पà¥à¤°à¤¶à¥à¤¨ 2.
मान लà¥à¤à¤¿à¤ à¤à¤¿ A = {a, b}, B = {a, b, c} à¤à¥à¤¯à¤¾ A â B? A ⪠B à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤à¥¤
हल:
A = {a, b}, B = {a, b, c}। समà¥à¤à¥à¤à¤¯ A à¤à¥ à¤
वयव a, b समà¥à¤à¥à¤à¤¯ B मà¥à¤ à¤à¥ हà¥à¥¤ A â B = A ⪠B = B à¤à¤° A ⪠B = {a, b} ⪠{a, b, c} = {a, b, c}
पà¥à¤°à¤¶à¥à¤¨ 3.
यदि A à¤à¤° B दॠà¤à¤¸à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤¿ A â B, तॠA ⪠B à¤à¥à¤¯à¤¾ हà¥?
हल:
A â B समà¥à¤à¥à¤à¤¯ A à¤à¥ सà¤à¥ à¤
वयव समà¥à¤à¥à¤à¤¯ B मà¥à¤ हà¥à¤à¥¤ A â B = B.
पà¥à¤°à¤¶à¥à¤¨ 4.
यदि A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} à¤à¤° D = {7, 8, 9, 10}, तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) A ⪠B
(ii) A ⪠C
(ii) B ⪠C
(iv) B ⪠D
(v) A ⪠B ⪠C
(vi) A ⪠B âªD
(vii) B ⪠C ⪠D
हल:
(i) A ⪠B = {1, 2, 3, 4} ⪠{3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}
(ii) A ⪠C = {1, 2, 3, 4} ⪠{5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ⪠C = {3, 4, 5, 6} ⪠{5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}
(iv) B ⪠D = {3, 4, 5, 6} ⪠{7, 8, 9, 10} = {3,4, 5, 6, 7, 8, 9, 10}
(v) A ⪠B âªC = ({1, 2, 3, 4} {3, 4, 5, 6}) ⪠{5, 6, 7, 8} = {1, 2, 3, 4, 5, 6} ⪠{5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}.
(vi) A ⪠B ⪠D = ({1, 2, 3, 4} ⪠{3, 4, 5, 6}) ⪠{7, 8, 9, 10} = {1, 2, 3, 4, 5, 6} ⪠{7, 8, 9, 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ⪠C ⪠D = ({3, 4, 5, 6} ⪠{5, 6, 7, 8}) ⪠{7, 8, 9, 10} = {3, 4, 5, 6, 7, 8} ⪠{7, 8, 9, 10) = {3, 4, 5, 6, 7, 8, 9, 10}
पà¥à¤°à¤¶à¥à¤¨ 5.
पà¥à¤°à¤¶à¥à¤¨ 1 मà¥à¤ दिठपà¥à¤°à¤¤à¥à¤¯à¥à¤ समà¥à¤à¥à¤à¤¯ यà¥à¤à¥à¤® à¤à¤¾ सरà¥à¤µà¤¨à¤¿à¤·à¥à¤ समà¥à¤à¥à¤à¤¯ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
हल:
(i) X â© Y= {1, 3, 5} â© {1, 2, 3} = {1, 3}
(ii) A â© B = {a, e, i, o, u} â© {a,b,c} = {a}.
(iii) A â© B = {3, 6, 9 â¦..} â© {1, 2, 3, 4, 5} = {3}.
(iv) A â© B = {2, 3, 4, 5, 6} â©{7, 8, 9} = Φ
(v) A ⩠B = {1, 2, 3} ⩠Φ = 0
पà¥à¤°à¤¶à¥à¤¨ 6.
यदि A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} à¤à¤° D = {15, 17}; तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) A â© B
(ii) B â©C
(iii) A â© C â©D
(iv) A â©C
(v) B â© D
(vi) à ⩠(B ⪠C)
(vii) A â© D
(viii) A â©(B âªD)
(ix) (A ⩠B) ⩠(B ⪠C)
(x) (A ⪠D) ⩠(B ⪠C)
हल:
(i) A â© B = {3, 5, 7, 9, 11} â© {7, 9, 11, 13} = {7, 9, 11}
(ii) B â© C = {7, 9, 11, 13} â© {11, 13, 15} = {11, 13}
(iii) à ⩠C â© D = ({3, 5, 7, 9, 11} â© {11, 13, 15}) â© {15, 17} = {11} â©{15, 17} = Φ
(iv) A â© C = {3, 5, 7, 9, 11} â© {11, 13, 15} = {11}.
(v) B ⩠D = {7, 9, 11, 13} ⩠{15, 17} = Φ
(vi) A ⩠(B ⪠C) = {3, 5, 7, 9, 11} ⩠({7, 9, 11, 13} ⪠{11, 13, 15}) = {3, 5, 7, 9, 11} ⩠{7, 9, 11, 13, 15} = {7, 9, 11}.
(vii) A ⩠D = {3, 5, 7, 9, 11} ⩠{15, 17} = Φ
(viii) A â©(B âªD) = {3, 5, 7, 9, 11} â©{7, 9, 11, 13} âª{15, 17}) = {3, 5, 7, 9, 11} â© {7, 9, 11, 13, 15, 17} = {7, 9, 11).
(ix) A â© B = {3, 5, 7, 9, 11} â© {7, 9, 11, 13} = {7, 9, 11}
B ⪠C = {7, 9, 11, 13} ⪠{11, 13, 15) = {7, 9, 11, 13, 15).
(A ⩠B) ⩠(B ⪠C) = {7, 9, 11} ⩠{7, 9, 11, 13, 15} = {7, 9, 11}.
(x) A ⪠D = {3, 5, 7, 9, 11} ⪠{15, 17} = {3, 5, 7, 9, 11, 15, 17}
B ⪠C = {7, 9, 11, 13} ⪠{11, 13, 15} = {7, 9, 11, 13, 15}
(A ⪠D) ⩠(B ⪠C) = {3, 5, 7, 9, 11, 15, 17} ⩠{7, 9, 11, 13, 15} = {7, 9, 11, 15}
पà¥à¤°à¤¶à¥à¤¨ 7.
यदि A = {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥}, B = {x : x à¤à¤ सम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥} C = {x : x à¤à¤ विषम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥}, D = {x : à¤à¤ à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हà¥} तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) A â© B
(ii) A â©C
(iii) A â© D
(iv) B â©C
(v) B â©D
(vi) C â© D
हल:
A = {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥} = {1, 2, 3, 4â¦â¦} B = {x : x à¤à¤ समठपà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥} = {2, 4, 6, 8â¦} C = {x : x à¤à¤ विषम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥} = {1, 3, 5, 7â¦} D = {x : x à¤à¤ à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हà¥} = {2, 3, 5, 7, 11â¦.}
(i) A â© B = {1, 2, 3, 4â¦.} â©{2, 4, 6, 8â¦.} = {2, 4, 6, 8â¦.} = B
(ii) A â©C = {1, 2, 3, 4â¦â¦} â© {1, 3, 5, 7â¦.} = {1, 3, 5, 7â¦.} = C
(iii) A â©D = {1, 2, 3, 4â¦} â© {2, 3, 5, 7â¦..} = {2, 3, 5, 7â¦â¦} = D
(iv) B â© C = {2, 4, 6, 8â¦} â© {1, 3, 5, 7â¦â¦} = Φ
(v) B â©D = {2, 4, 6, 8â¦..} â© {2, 3, 5, 7â¦..} = {2}
(vi) C â©D = {1, 3, 5, 7â¦..} â©{2, 3, 5, 7, 11â¦â¦.} = {3, 5, 7, 11, 13â¦â¦} = {x : x à¤à¤ विषम à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾}
पà¥à¤°à¤¶à¥à¤¨ 8.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯ यà¥à¤à¥à¤®à¥à¤ मà¥à¤ सॠà¤à¥à¤¨ सॠयà¥à¤à¥à¤® à¤
सà¤à¤¯à¥à¤à¥à¤¤ हà¥à¤?
(i) {1, 2, 3, 4} तथा {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हॠà¤à¤° 4 ⤠x ⤠6}
(ii) {a, e, i, 0, u} तथा {c, d, e, f}
(iii) {x : x à¤à¤ सम पà¥à¤°à¥à¤£à¤¾à¤à¤ हà¥à¥¤ à¤à¤° {x : x à¤à¤ विषम पà¥à¤°à¥à¤£à¤¾à¤ हà¥à¥¤
हल:
(i) मान लà¥à¤à¤¿à¤ E = {1, 2, 3, 4} F = {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ à¤à¤° 4 ⤠x ⤠6} = {4, 5, 6} à¤
वयव 4, E à¤à¤° F दà¥à¤¨à¥à¤ समà¥à¤à¥à¤à¤¯à¥à¤ मà¥à¤ हà¥à¥¤ à¤
त: दà¥à¤¨à¥à¤ यà¥à¤à¥à¤® à¤
सà¤à¤¯à¥à¤à¥à¤¤ नहà¥à¤ हà¥à¤à¥¤
(ii) दियॠहà¥à¤ समà¥à¤à¥à¤à¤¯à¥à¤ मà¥à¤ à¤
वयव à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ हà¥à¥¤ à¤
त: यह à¤
सà¤à¤¯à¥à¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ नहà¥à¤ हà¥à¥¤
(iii) मान लà¥à¤à¤¿à¤ A = {x : x à¤à¤ सम पà¥à¤°à¥à¤£à¤¾à¤à¤ हà¥à¤à¥¤ = {â¦.-4, -2, 0, 2, 4â¦} B = {x : x à¤à¤ विषम पà¥à¤°à¥à¤£à¤¾à¤à¤ हà¥} = {â¦.-5, -3, -1, 1, 3, 5â¦..} A à¤à¤° B समà¥à¤à¥à¤à¤¯à¥à¤ मà¥à¤ à¤à¥à¤ à¤à¥ à¤
वयव à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ नहà¥à¤ हà¥à¥¤ à¤
त: यह समà¥à¤à¥à¤à¤¯ à¤
सà¤à¤¯à¥à¤à¥à¤¤ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 9.
यदि A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) A â B
(ii) A â C
(iii) A â D
(iv) B â A
(v) C â A
(vi) D â A
(vii) B â C
(viii) B â D
(ix) C â B
(x) D â B
(xi) C â D
(xii) D â C
हल:
(i) A â B = {3, 6, 9, 12, 15, 18, 21} â {4, 8, 12, 16, 20} = {3, 6, 9, 15, 18, 21}
(ii) A â C = {3, 6, 9, 12, 15, 18, 21} â {2, 4, 6, 8, 10, 12, 14, 16} = {3, 9, 15, 18, 21}
(iii) A â D = {3, 6, 9, 12, 15, 18, 21} â {5, 10, 15, 20} = {3, 6, 9, 12, 18, 21}
(iv) B â A = {4, 8, 12, 16, 20} â {3, 6, 9, 12, 15, 18, 21} = {4, 8, 16, 20}
(v) C â A = {2, 4, 6, 8, 10, 12, 14, 16} â {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}
(vi) D â A = {5, 10, 15, 20} â {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}
(vii) B â C = {4, 8, 12, 16, 20} â {2, 4, 6, 8, 10, 12, 14, 16} = {20}
(viii) B â D = {4, 8, 12, 16, 20} â {5, 10, 15, 20} = {4, 8, 12, 16}
(ix) C â B = {2, 4, 6, 8, 10, 12, 14, 16} â {4, 8, 12, 16, 20} = {2, 6, 10, 14}
(x) D â B = {5, 10, 15, 20} â {4, 8, 12, 16, 20} = {5, 10, 15}
(xi) C â D = {2, 4, 6, 8, 10, 12, 14, 16} â {5, 10, 15, 20} = {2, 4, 6, 8, 12, 14, 16}
(xii) D â C = {5, 10, 15, 20} â {2, 4, 6, 8, 10, 12, 14, 16} = {5, 15, 20}
पà¥à¤°à¤¶à¥à¤¨ 10.
यदि X = {a, b, c, d} à¤à¤°à¥ Y = {f, b, d, g} तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) X â Y
(ii) Y â X
(iii) X â© Y
हल:
(i) X â Y= {a, b, c, d} â {f, b, d, g} = {a, c}
(ii) Y â X = {f, b, d, g} â {a, b, c, d} = {f, g}
(iii) X â©Y= {a, b, c, d} â© {f, b, d, g} = {b, d}
पà¥à¤°à¤¶à¥à¤¨ 11.
यदि R वासà¥à¤¤à¤µà¤¿à¤ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¤° Q परिमà¥à¤¯ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤, तॠR â Q à¤à¥à¤¯à¤¾ हà¥à¤à¤¾?
हल:
R = {x : x à¤à¤ वासà¥à¤¤à¤µà¤¿à¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤} p= {x : x à¤à¤ परिमà¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
R â Q = {x : x à¤à¤ à¤
परिमà¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤} à¤
त: यह à¤
परिमà¥à¤¯ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 12.
बताà¤à¤ à¤à¤¿ निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¤¥à¤¨à¥à¤ मà¥à¤ सॠपà¥à¤°à¤¤à¥à¤¯à¥à¤ सतà¥à¤¯ हॠया à¤
सतà¥à¤¯? à¤
पनॠà¤à¤¤à¥à¤¤à¤° à¤à¤¾ à¤à¤à¤¿à¤¤à¥à¤¯ à¤à¥ बताà¤à¤à¥¤
(i) {2, 3, 4, 5} तथा {3, 6} à¤
सà¤à¤¯à¥à¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¤à¥¤
(ii) {a, e, i, 0, u} तथा {a, b, c, 4} à¤
सà¤à¤¯à¥à¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¤à¥¤
(iii) {2, 6, 10, 14} तथा {3, 7, 11, 15} à¤
सà¤à¤¯à¥à¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¤à¥¤
(iv) {2, 6, 10} तथा {3, 7, 11} à¤
सà¤à¤¯à¥à¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¤à¥¤
हल:
(i) यह à¤à¤¥à¤¨ सतà¥à¤¯ नहà¥à¤ हॠà¤à¥à¤¯à¥à¤à¤à¤¿ समà¥à¤à¥à¤à¤¯ {2, 3, 4, 5} à¤à¤° {3, 6} मà¥à¤ à¤
वयव 3 à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ हà¥à¥¤
(ii) यह à¤à¤¥à¤¨ सतà¥à¤¯ नहà¥à¤ हॠà¤à¥à¤¯à¥à¤à¤à¤¿ समà¥à¤à¥à¤à¤¯ {a, e, i, o, u} à¤à¤° {a, b, c, d} मà¥à¤ à¤
वयव a à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ हà¥à¥¤
(iii) यह à¤à¤¥à¤¨ सतà¥à¤¯ हॠà¤à¥à¤¯à¥à¤à¤à¤¿ समà¥à¤à¥à¤à¤¯ {2, 6, 10, 14} à¤à¤° {3, 7, 11, 15} मà¥à¤ à¤à¥à¤ à¤
वयव à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ नहà¥à¤ हà¥à¥¤ à¤
त: यह समà¥à¤à¥à¤à¤¯ à¤
सà¤à¤¯à¥à¤à¥à¤¤ हà¥à¥¤
(iv) यह à¤à¤¥à¤¨ सतà¥à¤¯ हॠà¤à¥à¤¯à¥à¤à¤à¤¿ समà¥à¤à¥à¤à¤¯ {2, 6, 10} à¤à¤° {3, 7, 11} मà¥à¤ à¤à¥à¤ à¤
वयव à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ नहà¥à¤ हà¥à¥¤ à¤
तठयह समà¥à¤à¥à¤à¤¯ à¤
सà¤à¤¯à¥à¤à¥à¤¤ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 1.
मान लà¥à¤à¤¿à¤ à¤à¤¿ U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} à¤à¤° C = {3, 4, 5, 6} तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) Aâ
(ii) Bâ
(iii) (A ⪠C)â
(iv) (A ⪠B)â
(v) (Aâ)â
(vi) (B â C)â
हल:
(i) Aâ = U â A = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {1, 2, 3, 4} = {5, 6, 7, 8, 9}
(ii) Bâ = U â B = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {2, 4, 6, 8} = {1, 3, 5, 7, 9)
(iii) A ⪠C = {1, 2, 3, 4} ⪠{3, 4, 5, 6} = {1, 2, 3, 4, 5, 6} (A ⪠C)â = U â (A ⪠C) = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {1, 2, 3, 4, 5, 6} = {7, 8, 9}
(iv) A ⪠B = {1, 2, 3, 4} ⪠{2, 4, 6, 8} = {1, 2, 3, 4, 6, 8} (A ⪠B)â = U â (A âªB) = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {1, 2, 3, 4, 6, 8} = {5, 7, 9} (v) (A)â = U â A = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {1, 2, 3, 4} = {5, 6, 7, 8, 9} (Aâ)â = U- Aâ = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {5, 6, 7, 8, 9} = {1, 2, 3, 4}
(vi) B â C = {2, 4, 6, 8} â {3, 4, 5, 6} = {2, 8} (B â C)â = U â (B â C) = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {2,8} = {1, 3, 4, 5, 6, 7, 9}
पà¥à¤°à¤¶à¥à¤¨ 2.
यदि U = {a, b, c, d, e, f, g, h}, तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ पà¥à¤°à¤ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤¯à¥:
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}
हल:
(i) Aâ = U â A = {a, b, c, d, e, f, g, h} â {a, b,c} = {d, e, f, g, h}
(ii) Bâ = U â B = {a, b, c, d, e, f, g, h} â {d, e, f, g} = {a, b, c, h}
(iii) C = U â C = {a, b, c, d, e, f, g, h} â {a, c, e, g} = {b, d, f, h}
(iv) Dâ = U â D = {a, b, c, d, e, f, g, h} â {f, g, h, a} = {b, c, d, e}.
पà¥à¤°à¤¶à¥à¤¨ 3.
पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾à¤à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ à¤à¥ सारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯ मानतॠहà¥à¤, निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ पà¥à¤°à¤ लिà¤à¤¿à¤:
(i) {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ सम सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(ii) {x : x à¤à¤ पà¥à¤°à¤¾à¤à¥à¤¤ विषम सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(iii) {x : x सà¤à¤à¥à¤¯à¤¾ 3 à¤à¥ à¤à¤ धन à¤à¥à¤£à¤ हà¥à¥¤}
(iv) {x : x à¤à¤ à¤
à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(v) {x : x, 3 à¤à¤° 5 सॠविà¤à¤¾à¤à¤¿à¤¤ हà¥à¤¨à¥ वालॠà¤à¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(vi) {x : x à¤à¤ पà¥à¤°à¥à¤£ वरà¥à¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(vii) {x : x à¤à¤ पà¥à¤°à¥à¤£ à¤à¤¨ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(viii){x : x + 5 = 8}
(ix) {x : 2x + 5 = 9}
(x) {x : x ⥠7}
(xi) {x : x â N à¤à¤° 2x + 1 > 10}
हल:
(i) {x : x à¤à¤ विषम पà¥à¤°à¤¾à¤à¥à¤¤ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(ii) {x : à¤à¤ सम सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(iii) {x : x â N à¤à¤° x सà¤à¤à¥à¤¯à¤¾ 3 à¤à¤¾ धन à¤à¥à¤£à¤ नहà¥à¤ हà¥à¥¤}
(iv) {x : x = 1 à¤à¤° x à¤à¤ धन à¤à¤¾à¤à¥à¤¯ सà¤à¤à¥à¤¯à¤¾ हà¥à¥¤}
(v) {x : x â N à¤à¤° x, सà¤à¤à¥à¤¯à¤¾ 3 व 5 à¤à¤¿à¤¸à¥ सॠà¤à¥ विà¤à¤¾à¤à¤¿à¤¤ नहà¥à¤ हà¥à¤¤à¥à¥¤}
(vi) {x : x â N तथा x à¤à¤ पà¥à¤£ वरà¥à¤ सà¤à¤à¥à¤¯à¤¾ नहà¥à¤ हà¥à¥¤}
(vii) {x : x â N तथा x à¤à¤ पà¥à¤°à¥à¤£ वरà¥à¤ à¤à¤¨ सà¤à¤à¥à¤¯à¤¾ नहà¥à¤ हà¥à¥¤}
(viii) {x : x â N तथा x â 3}
(ix) {x : x â N तथा x â 2}
(x) {x : x â N तथा x < 7}
(xi) {x : x â N तथा x < Â
पà¥à¤°à¤¶à¥à¤¨ 4.
यदि U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} à¤à¤° B = {2, 3, 5, 7}, तॠसतà¥à¤¯à¤¾à¤ªà¤¿à¤¤ à¤à¥à¤à¤¿à¤ à¤à¤¿:
(i) (A ⪠B)â = Aâ â© Bâ
(ii) (A â© B)â = Aâ ⪠Bâ
हल:
(i) A ⪠B = {2, 4, 6, 8} âª{2, 3, 5, 7} = {2, 3, 4, 5, 6, 7, 8}
बायाठपà¤à¥à¤· = (A âªB)â = U â (A ⪠B) = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {2, 3, 4, 5, 6, 7, 8} = {1, 9}
Aâ = U â A = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {2, 4, 6, 8} = {1, 3, 5, 7, 9}
Bâ = U â B = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {2, 3, 5, 7} = {1, 4, 6, 8, 9}
दायाठपà¤à¥à¤· = Aâ â© Bâ = {1, 3, 5, 7, 9} â© {1, 4, 6, 8, 9} = {1, 9}
à¤
तठ(A ⪠B)â = Aâ â© Bâ.
(ii) बायाठपà¤à¥à¤· = (A â©B)â
(A â©B) = {2, 4, 6, 8} â© {2, 3, 5, 7} = {2}
(A â© B)â = {1, 2, 3, 4, 5, 6, 7, 8, 9} â {2} = {1, 3, 4, 5, 6, 7, 8, 9}
दायाठपà¤à¥à¤· : Aâ ⪠Bâ = {1, 3, 5, 7, 9} ⪠{1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}
à¤
त: (A â© B)â = Aâ ⪠Bâ.
पà¥à¤°à¤¶à¥à¤¨ 5.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ सॠपà¥à¤°à¤¤à¥à¤¯à¥à¤ à¤à¥ लिठà¤à¤ªà¤¯à¥à¤à¥à¤¤ वà¥à¤¨ à¤à¤°à¥à¤ à¤à¥à¤à¤à¤¿à¤à¥¤
(i) (A ⪠B)â
(ii) Aâ â© Bâ
(iii) (A â©B)â
(iv) (Aâ ⪠Bâ)
हल:
à¤à¤¾à¤¯à¤¾à¤à¤à¤¿à¤¤ à¤à¥à¤·à¥à¤¤à¥à¤° à¤à¥ निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ दà¥à¤µà¤¾à¤°à¤¾ दरà¥à¤¶à¤¾à¤¤à¥ हà¥à¤:
पà¥à¤°à¤¶à¥à¤¨ 6.
मान लà¥à¤à¤¿à¤ à¤à¤¿ à¤à¤¿à¤¸à¥ समतल मà¥à¤ सà¥à¤¥à¤¿à¤¤ सà¤à¥ तà¥à¤°à¤¿à¤à¥à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ सारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯ U हà¥à¥¤ यदि A à¤à¤¨ सà¤à¥ तà¥à¤°à¤¿à¤à¥à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤¿à¤¨à¤®à¥à¤ à¤à¤® सॠà¤à¤® à¤à¤ à¤à¥à¤£ 60° सॠà¤à¤¿à¤¨à¥à¤¨ हà¥, तॠAâ à¤à¥à¤¯à¤¾ हà¥?
हल:
U = {x : समतल मà¥à¤ à¤à¤ तà¥à¤°à¤¿à¤à¥à¤ हà¥à¥¤}
A = {x : x à¤à¤ तà¥à¤°à¤¿à¤à¥à¤ à¤à¤¿à¤¸à¤à¤¾ à¤à¤® सॠà¤à¤® à¤à¤ à¤à¥à¤£ 60° à¤à¤¾ न हà¥à¥¤}
Aâ = {सà¤à¥ समबाहॠतà¥à¤°à¤¿à¤à¥à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ हà¥à¥¤}
पà¥à¤°à¤¶à¥à¤¨ 7.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¤¥à¤¨à¥à¤ à¤à¥ सतà¥à¤¯ बनानॠà¤à¥ लिठरिà¤à¥à¤¤ सà¥à¤¥à¤¾à¤¨ à¤à¤°à¤¿à¤:
(i) A ⪠Aâ = â¦â¦â¦..
(ii) Φâ â© A = â¦â¦â¦..
(iii) A â©Aâ = â¦â¦â¦â¦.
(iv) Uâ â© A = â¦â¦â¦â¦
हल:
(i) A ⪠Aâ = U
(ii) Φâ â© A = U â© A = A
(iii) A â©Aâ = Φ
(iv) Uâ â© A = Φ â© A = Φ
पà¥à¤°à¤¶à¥à¤¨ 1.
यदि X à¤à¤° Y दॠà¤à¤¸à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤¿ n(X) = 17, n(Y) = 23 तथा n(X ⪠Y) = 38, तॠn(X â©Y) à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤à¥¤
हल:
दिया हà¥à¥¤ n(X) = 17, n(Y) = 23
n(X ⪠Y) = 38
n(X ⪠Y) = n(X) + n(Y) â n(X â© Y)
38 = 17 + 23 â n(K â© Y) = 40 â n (X â© Y).
n(X â© Y) = 40 â 38 = 2.
पà¥à¤°à¤¶à¥à¤¨ 2.
यदि X à¤à¤° Yदॠà¤à¤¸à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤¿ X ⪠Y मà¥à¤ 18, X मà¥à¤ 8 à¤à¤° Y मà¥à¤ 15 à¤
वयवॠहà¥à¤ तॠX â©Y मà¥à¤ à¤à¤¿à¤¤à¤¨à¥ à¤
वयव हà¥à¤à¤à¥?
हलà¤
n(X) = 8, n(Y) = 15, n(X ⪠Y) = 18
हम à¤à¤¾à¤¨à¤¤à¥ हà¥à¤ à¤à¤¿,
n(X ⪠Y) = n(X) + n(Y) â n (X â© Y)
18 = 8 + 15 â n(X â© Y) = 23 â n(X â© Y)
n(X â© Y) = 23 â 18 = 5.
पà¥à¤°à¤¶à¥à¤¨ 3.
400 वà¥à¤¯à¤à¥à¤¤à¤¿à¤¯à¥à¤ à¤à¥ समà¥à¤¹ मà¥à¤, 250 हिनà¥à¤¦à¥ तथा 200 à¤
à¤à¤à¥à¤°à¥à¤à¥ बà¥à¤² सà¤à¤¤à¥ हà¥à¤à¥¤ à¤à¤¿à¤¤à¤¨à¥ वà¥à¤¯à¤à¥à¤¤à¤¿ हिनà¥à¤¦à¥ तथा à¤
à¤à¤à¥à¤°à¥à¤à¥ दà¥à¤¨à¥à¤ बà¥à¤² सà¤à¤¤à¥ हà¥à¤?
हल:
मान लà¥à¤à¤¿à¤ à¤à¤¿ H à¤à¤° E à¤à¥à¤°à¤®à¤¶à¤ हिनà¥à¤¦à¥ व à¤
à¤à¤à¥à¤°à¥à¤à¥ बà¥à¤²à¤¨à¥ वालà¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤, तब
n(H) = 250, n(E) = 200 à¤à¤°
n(H ⪠E) = 400
n(H âªE) = n(H) + n(E) â n(H â©E)
400 = 250 + 200 â n(H â©E) = 450 â n(H â©E)
n (H â©E) = 450 â 400 = 50.
पà¥à¤°à¤¶à¥à¤¨ 4.
यदि S à¤à¤° T दॠà¤à¤¸à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤¿ S मà¥à¤ 21, T मà¥à¤ 32 à¤à¤° S â© T मà¥à¤ 11 à¤
वयव हà¥à¤ तॠS âªT मà¥à¤ à¤à¤¿à¤¤à¤¨à¥ à¤
वयव हà¥à¤à¤à¥?
हल:
यहाठn(S) = 21, n (T) = 32, n(S â©T) = 11
n(SâªT) = n(S) + n(T) â n(S â© T) = 21 + 32 â 11 = 53 â 11 = 42.
पà¥à¤°à¤¶à¥à¤¨ 5.
यदि X à¤à¤° दॠà¤à¤¸à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤¿ X मà¥à¤ 40, X âªY मà¥à¤ 60, à¤à¤° X â© Y मà¥à¤ 10 à¤
वयव हà¥à¤, तॠ? मà¥à¤ à¤à¤¿à¤¤à¤¨à¥ à¤
वयव हà¥à¤à¤à¥?
हल:
n(X) = 40, n(X âªY) = 60, n(X â© Y) = 10, n(Y) = ?
n(X ⪠Y) = n(X) + n(Y) â n (X â©Y)
60 = 40 + n (Y) â 10
n(Y) = 60 â 40 + 10 = 30.
पà¥à¤°à¤¶à¥à¤¨ 6.
70 वà¥à¤¯à¤à¥à¤¤à¤¿à¤¯à¥à¤ à¤à¥ समà¥à¤¹ मà¥à¤ 37 à¤à¥à¤«à¥, 52 à¤à¤¾à¤¯ पसà¤à¤¦ à¤à¤°à¤¤à¥ हà¥à¤ à¤à¤° पà¥à¤°à¤¤à¥à¤¯à¥à¤ वà¥à¤¯à¤à¥à¤¤à¤¿ दà¥à¤¨à¥à¤ मà¥à¤ सॠà¤à¤® सॠà¤à¤® à¤à¤ पà¥à¤¯ पसà¤à¤¦ à¤à¤°à¤¤à¤¾ हà¥, तॠà¤à¤¿à¤¤à¤¨à¥ वà¥à¤¯à¤à¥à¤¤à¤¿ à¤à¥à¤«à¥ à¤à¤° à¤à¤¾à¤¯ दà¥à¤¨à¥à¤ पसà¤à¤¦ à¤à¤°à¤¤à¥ हà¥à¤?
हल:
मान लिया C, à¤à¥à¤«à¥ पà¥à¤¨à¥ वालॠलà¥à¤à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ à¤à¥ à¤à¤° T, à¤à¤¾à¤¯ पà¥à¤¨à¥ वालॠलà¥à¤à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤, तब
n(C âªT) = 70, n(C) = 37, n(T) = 52
n(C âªT) = n (C) +n(T) â n(C â© T)
70 = 37 + 52 â n(C â©T)
n(C â© T) = 37 + 52 -70 = 89 â 70 = 19.
पà¥à¤°à¤¶à¥à¤¨ 7.
65 वà¥à¤¯à¤à¥à¤¤à¤¿à¤¯à¥à¤ à¤à¥ समà¥à¤¹ मà¥à¤, 40 वà¥à¤¯à¤à¥à¤¤à¤¿ à¤à¥à¤°à¤¿à¤à¥à¤ à¤à¤° 10 वà¥à¤¯à¤à¥à¤¤à¤¿ à¤à¥à¤°à¤¿à¤à¥à¤ तथा à¤à¥à¤¨à¤¿à¤¸ दà¥à¤¨à¥à¤ à¤à¥ पà¤à¤¸à¤¦ à¤à¤°à¤¤à¥ हà¥à¤, तॠà¤à¤¿à¤¤à¤¨à¥ वà¥à¤¯à¤à¥à¤¤à¤¿ à¤à¥à¤µà¤² à¤à¥à¤¨à¤¿à¤¸ à¤à¥ पà¤à¤¸à¤¦ à¤à¤°à¤¤à¥ हà¥à¤ à¤à¤¿à¤à¤¤à¥ à¤à¥à¤°à¤¿à¤à¥à¤ à¤à¥ नहà¥à¤? à¤à¤¿à¤¤à¤¨à¥ वà¥à¤¯à¤à¥à¤¤à¤¿ à¤à¥à¤¨à¤¿à¤¸ à¤à¥ पà¤à¤¸à¤¦ à¤à¤°à¤¤à¥ हà¥à¤?
हल:
मान लà¥à¤à¤¿à¤ C, à¤à¥à¤°à¤¿à¤à¥à¤ पà¤à¤¸à¤¦ à¤à¤°à¤¨à¥ वालॠलà¥à¤à¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ हॠà¤à¤° T à¤à¥à¤¨à¤¿à¤¸ पà¤à¤¸à¤¦ à¤à¤°à¤¨à¥ वालà¥à¤ à¤à¤¾ समà¥à¤à¥à¤à¤¯ हà¥, तब
n(C âªT) = 65, n(C) = 40, n(C â©T) = 10
हम à¤à¤¾à¤¨à¤¤à¥ हà¥à¤ à¤à¤¿
n(C ⪠T) = n(C) + n(T) â n(C â© T)
65 = 40 + n(T) â 10 = 30 + n(T)
n(T) = 65 â 30 = 35
à¤à¥à¤µà¤² à¤à¥à¤¨à¤¿à¤¸ पà¤à¤¸à¤¦ à¤à¤°à¤¨à¥ वालॠà¤à¥ सà¤à¤à¥à¤¯à¤¾ = n(T) â n(C â©T) = 35 â 10 = 25.
à¤à¤¸ पà¥à¤°à¤à¤¾à¤° à¤à¥à¤¨à¤¿à¤¸ पà¤à¤¸à¤¦ à¤à¤°à¤¨à¥ वालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ à¤à¥ à¤à¥à¤°à¤¿à¤à¥à¤ पà¤à¤¸à¤¦ नहà¥à¤ à¤à¤°à¤¤à¥ = 25
à¤
त: à¤à¥à¤¨à¤¿à¤¸ पà¤à¤¸à¤¦ à¤à¤°à¤¨à¥ वालॠलà¥à¤à¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 35.
पà¥à¤°à¤¶à¥à¤¨ 8.
à¤à¤ à¤à¤®à¥à¤à¥ मà¥à¤, 50 वà¥à¤¯à¤à¥à¤¤à¤¿ फà¥à¤°à¥à¤à¤ 20 वà¥à¤¯à¤à¥à¤¤à¤¿ सà¥à¤ªà¥à¤¨à¤¿à¤¶ à¤à¤° 10 वà¥à¤¯à¤à¥à¤¤à¤¿ सà¥à¤ªà¥à¤¨à¤¿à¤¶ à¤à¤° फà¥à¤°à¥à¤à¤ दà¥à¤¨à¥à¤ हॠà¤à¤¾à¤·à¤¾à¤à¤ à¤à¥ बà¥à¤² सà¤à¤¤à¥ हà¥à¤à¥¤ à¤à¤¿à¤¤à¤¨à¥ वà¥à¤¯à¤à¥à¤¤à¤¿ à¤à¤¨ दà¥à¤¨à¥à¤ हॠà¤à¤¾à¤·à¤¾à¤à¤ मà¥à¤ सॠà¤à¤® सॠà¤à¤® à¤à¤ à¤à¤¾à¤·à¤¾ बà¥à¤² सà¤à¤¤à¥ हà¥à¤?
हल:
मान लà¥à¤à¤¿à¤ फà¥à¤°à¤¾à¤à¤¸à¥à¤¸à¥ बà¥à¤²à¤¨à¥ वालॠलà¥à¤à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ à¤à¥ F सॠतथा सà¥à¤ªà¥à¤¨à¤¿à¤¶ बà¥à¤²à¤¨à¥ वालॠलà¥à¤à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ à¤à¤¾ S सॠनिरà¥à¤ªà¤¿à¤¤ à¤à¤¿à¤¯à¤¾ हà¥, तब
n(F) = 50, n(S) = 20, n(F â©S) = 10
à¤
ब, n(F âªS) = n(F) + n (S) â n (F â©S) = 50 + 20 â 10 = 60
à¤à¤® सॠà¤à¤® à¤à¤ à¤à¤¾à¤·à¤¾ बà¥à¤²à¤¨à¥ वालॠलà¥à¤à¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 60.
ठधà¥à¤¯à¤¾à¤¯ 1 पर विविध पà¥à¤°à¤¶à¥à¤¨à¤¾à¤µà¤²à¥
पà¥à¤°à¤¶à¥à¤¨ 1.
निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ समà¥à¤à¥à¤à¤¯à¥à¤ मà¥à¤ सॠà¤à¥à¤¨ à¤à¤¿à¤¸à¤à¤¾ à¤à¤ªà¤¸à¤®à¥à¤à¥à¤à¤¯ हà¥, à¤à¤¸à¤à¤¾ निरà¥à¤£à¤¯ à¤à¥à¤à¤¿à¤:
A = x : x âR तथा x² â 8x + 12 = 0 à¤à¥ सà¤à¤¤à¥à¤·à¥à¤ à¤à¤°à¤¨à¥ वालॠसà¤à¥ वासà¥à¤¤à¤µà¤¿à¤ सà¤à¤à¥à¤¯à¤¾à¤à¤ = x}, B= {2, 4, 6}, C = {2, 4, 6, 8â¦.}, D = {6}.
हल:
A = {x : x â R, x समà¥à¤à¤°à¤£ x² â 8x + 12 = 0 à¤à¥ सà¤à¤¤à¥à¤·à¥à¤ à¤à¤°à¤¤à¤¾ हà¥à¥¤ à¤
रà¥à¤¥à¤¾à¤¤à¥
A = {2, 6}
B = {2, 4, 6} ।
C = {2, 4, 6, 8â¦.}
D = {6}
(i) समà¥à¤à¥à¤à¤¯ A à¤à¥ à¤
वयव 2, 6 समà¥à¤à¥à¤à¤¯ B मà¥à¤ à¤à¥ हà¥à¤à¥¤
A â B.
(ii) à¤à¤¸ पà¥à¤°à¤à¤¾à¤° समà¥à¤à¥à¤à¤¯ A à¤à¥ à¤
वयव 2, 6 समà¥à¤à¥à¤à¤¯ C मà¥à¤ à¤à¥ हà¥à¥¤
A â C.
(iii) समà¥à¤à¥à¤à¤¯ B à¤à¥ à¤
वयव 2, 4, 6 समà¥à¤à¥à¤à¤¯ C मà¥à¤ हà¥à¤à¥¤
B âC.
(iv) समà¥à¤à¥à¤à¤¯ D à¤à¤¾ à¤
वयव 6, समà¥à¤à¥à¤à¤¯ A, B à¤à¤° C तà¥à¤¨à¥à¤ मà¥à¤ हà¥à¤,
D â A, D â B, D âC.
पà¥à¤°à¤¶à¥à¤¨ 2.
à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤ à¤à¤¿ निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ मà¥à¤ सॠपà¥à¤°à¤¤à¥à¤¯à¥à¤ à¤à¤¥à¤¨ सतà¥à¤¯ हॠया à¤
सतà¥à¤¯ हà¥à¥¤ यदि सतà¥à¤¯ हà¥, तॠà¤à¤¸à¥ सिदà¥à¤§ à¤à¥à¤à¤¿à¤à¥¤ यदि à¤
सतà¥à¤¯ हॠतॠà¤à¤ à¤à¤¦à¤¾à¤¹à¤°à¤£ दà¥à¤à¤¿à¤à¥¤
(i) यदि x â A तथा A â B, तॠx â B
(ii) यदि A âB तथा B â C, तॠA â C
(iii) यदि A â B तथा B âC, तॠA âC
(iv) यदि A â B B â C, तॠA â C
(v) यदि x â A तथा A â B, तॠx â B
(vi) यदि A â B तथा x â B, तॠx â A
हल:
(i) à¤
सतà¥à¤¯ : मान लà¥à¤à¤¿à¤ A = {1}, B = {{1}, 2}
सà¥à¤ªà¤·à¥à¤ हॠà¤à¤¿ 1 â A, A â B परà¤à¤¤à¥ 1 â समà¥à¤à¥à¤à¤¯ B à¤à¥à¤¯à¥à¤à¤à¤¿ 1 B मà¥à¤ नहà¥à¤ हà¥à¥¤ à¤à¤¸ पà¥à¤°à¤à¤¾à¤° दिया हà¥à¤ à¤à¤¥à¤¨ सतà¥à¤¯ नहà¥à¤ हà¥
(ii) à¤
सतà¥à¤¯ : मान लà¥à¤à¤¿à¤ A = {1}, B = {1, 2} à¤à¤° C = {{1, 2}, 3}
समà¥à¤à¥à¤à¤¯ A à¤à¤¾ à¤
वयव समà¥à¤à¥à¤à¤¯ B मà¥à¤ हà¥à¤ A â B
à¤
वयव {1, 2} समà¥à¤à¥à¤à¤¯ C मà¥à¤ हà¥à¤ B â C
पà¤à¤°à¤¤à¥ A = {1} समà¥à¤à¥à¤à¤¯ C मà¥à¤ नहà¥à¤ हà¥à¥¤
à¤à¤¥à¤¨ A â C सतà¥à¤¯ नहà¥à¤ हà¥à¥¤
(iii) सतà¥à¤¯ : A â B â यदि x â A तथा x â B
परà¤à¤¤à¥ B â C â यदि x â B तब x â C
यदि x â A तब x â A तब x â C â A â C
(iv) à¤
सतà¥à¤¯ : मान लà¥à¤à¤¿à¤ A = {1, 2}, B = {2, 3}, C = {1, 2, 5}
समà¥à¤à¥à¤à¤¯ A à¤à¥ सà¤à¥ à¤
वयव 1, 2 समà¥à¤à¥à¤à¤¯ B मà¥à¤ नहà¥à¤ हà¥à¤à¥¤
A â D
समà¥à¤à¥à¤à¤¯ B à¤à¥ सà¤à¥ à¤
वयव 2, 3 समà¥à¤à¥à¤à¤¯ C मà¥à¤ नहà¥à¤ हà¥à¤à¥¤
A â C
पà¤à¤°à¤¤à¥ समà¥à¤à¥à¤à¤¯ A à¤à¥ सà¤à¥ à¤
वयव 1, 2 समà¥à¤à¥à¤à¤¯ C मà¥à¤ हà¥à¤à¥¤
A â C
à¤à¤¸ पà¥à¤°à¤à¤¾à¤° दिया à¤à¤¥à¤¨ सतà¥à¤¯ नहà¥à¤ हà¥à¥¤
(v) समà¥à¤à¥à¤à¤¯ A = {1, 2}, B = {2, 3, 4, 5}
समà¥à¤à¥à¤à¤¯ A à¤à¤¾ à¤
वयव 1, 2 समà¥à¤à¥à¤à¤¯ B मà¥à¤ नहà¥à¤ हॠ।
A â B
समà¥à¤à¥à¤à¤¯ A à¤à¤¾ à¤
वयव 1 समà¥à¤à¥à¤à¤¯ B मà¥à¤ नहà¥à¤ हà¥à¤à¥¤
x â B
à¤à¤¸ पà¥à¤°à¤à¤¾à¤° दिया à¤à¤¯à¤¾ à¤à¤¥à¤¨ सतà¥à¤¯ नहà¥à¤ हà¥à¥¤
(vi) सतà¥à¤¯ : A â B = यदि x â A तब x â B यदि x â B तथा x â A
à¤à¤¸ पà¥à¤°à¤à¤¾à¤° à¤à¤¥à¤¨ A â B, x â B तब x â A सतà¥à¤¯ हà¥à¤à¥¤
पà¥à¤°à¤¶à¥à¤¨ 3.
मान लà¥à¤à¤¿à¤ A, B à¤à¤° Cà¤à¤¸à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤¿ A ⪠B = A ⪠C तथा A â© B = A â© C, तॠदरà¥à¤¶à¤¾à¤à¤ à¤à¤¿ B = C
हल:
दिया हà¥:
A ⪠B = A ⪠C
(A ⪠B) ⩠C = (A ⪠C) ⩠C = C [(A ⪠C) ⩠C = C ]
(A â© C) â© (B â© C) = C
(A â© B) ⪠(B â© C) = C â¦.(i) [A â© C = A â© B = दिया हà¥à¥¤]
A ⪠B = A ⪠C
(A ⪠B) ⩠B = (A ⪠C) ⩠B
B = (A âªC) â© B = (A â© B) ⪠(C â© B)
या (A â© ) ⪠(B â© C) = B â¦â¦(ii)
(i) à¤à¤° (ii) सॠB = C पà¥à¤°à¤¾à¤ªà¥à¤¤ हà¥à¤¤à¤¾ हà¥à¥¤
पà¥à¤°à¤¶à¥à¤¨ 4.
दिà¤à¤¾à¤à¤ à¤à¤¿ निमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¤¾à¤° पà¥à¤°à¤¤à¤¿à¤¬à¤¨à¥à¤§ तà¥à¤²à¥à¤¯ हà¥à¤:
(i) A â B
(ii) A â B = Φ
(iii) A ⪠B = B
हल:
(i) A â B à¤
रà¥à¤¥à¤¾à¤¤à¥ समà¥à¤à¥à¤à¤¯ A à¤à¥ सà¤à¥ à¤
वयव B मà¥à¤ हà¥à¤à¥¤
A â B = Φ à¤
रà¥à¤¥à¤¾à¤¤ (i) â (ii)
(ii) A â B = Φ â समà¥à¤à¥à¤à¤¯ A à¤à¥ सà¤à¥ à¤
वयव B मà¥à¤ हà¥à¤à¥¤
A ⪠B = B
à¤
रà¥à¤¥à¤¾à¤¤ (ii) â (iii)
(iii) A ⪠B = B â समà¥à¤à¥à¤à¤¯ A à¤à¥ सà¤à¥ à¤
वयव B मà¥à¤ हà¥à¥¤
समà¥à¤à¥à¤à¤¯ A à¤à¤° B मॠसमà¥à¤à¥à¤à¤¯ A à¤à¥ à¤
वयव à¤à¤à¤¯à¤¨à¤¿à¤·à¥à¤ हà¥à¥¤
A â©B = A
à¤à¤¸à¤¸à¥ सà¥à¤ªà¤·à¥à¤ हॠसà¤à¥ à¤à¤¥à¤¨ समान हà¥à¤à¥¤
पà¥à¤°à¤¶à¥à¤¨ 5.
दिà¤à¤¾à¤à¤ à¤à¤¿ यदि A â B तॠC â B â C â A.
हल:
मान लà¥à¤à¤¿à¤ x â C â B â x â C पà¤à¤°à¤¤à¥ x â B
दिया हà¥: A â B â यदि x â B â x â A
à¤
रà¥à¤¥à¤¾à¤¤, x â C à¤à¤° x â A â x â C â A
यहाठहम पातॠहà¥à¤ à¤à¤¿
यदि x â C â B तब x â C â A
â C â B â C â A.
पà¥à¤°à¤¶à¥à¤¨ 6.
मान लà¥à¤à¤¿à¤ à¤à¤¿ P(A) = P(B), सिदà¥à¤§ à¤à¥à¤à¤¿à¤ à¤à¤¿ A = B.
हल:
मान लà¥à¤à¤¿à¤ x, समà¥à¤à¥à¤à¤¯ A à¤à¤¾ à¤à¥à¤ à¤
वयव हà¥à¥¤
तब à¤à¤ à¤à¤ªà¤¸à¤®à¥à¤à¥à¤à¤¯ X (मान लà¥) à¤à¤¸à¤¾ हà¥à¤à¤¾ à¤à¤¿à¤¸à¤®à¥ x â A à¤à¤¿à¤¸à¤à¥ à¤
नà¥à¤¸à¤¾à¤°
X â A â X â P(A)
X â P(B) [P(A) = P(B)]
X â B या x â B
à¤
रà¥à¤¥à¤¾à¤¤ यदि
x â A तब x â B â A â B â¦..(i)
y समà¥à¤à¥à¤à¤¯ B à¤à¤¾ à¤à¥à¤ à¤
वयव हà¥, तब
समà¥à¤à¥à¤à¤¯ B à¤à¤¾ à¤à¥à¤ à¤à¤ªà¤¸à¤®à¥à¤à¥à¤à¤¯ Y (मान लà¥) हà¥à¤à¤¾ à¤à¤¿à¤¸à¤¸à¥ y â Y
Y â B â Y âP(B)
Y âP(A) [P(A) = P(B)]
Y â A यदि y â B तब y â A
B â A â¦â¦â¦(ii)
समà¥à¤à¤°à¤£ (i) à¤à¤° (ii) सà¥, हम पातॠहà¥à¤à¥¤
A = B.
पà¥à¤°à¤¶à¥à¤¨ 7.
à¤à¤¿à¤¨à¥à¤¹à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯à¥à¤ A तठB à¤à¥ लिठà¤à¥à¤¯à¤¾ यह सतà¥à¤¯ हॠà¤à¤¿ P(A) ⪠P(B) = P(A ⪠B) ? à¤
पनॠà¤à¤¤à¥à¤¤à¤° à¤à¤¾ à¤à¤à¤¿à¤¤à¥à¤¯ बताà¤à¤à¥¤
हल:
मान लà¥à¤à¤¿à¤à¥¤
A = {a}, B = {b}, à¤à¤° A ⪠B = {a, b}
P(A) = {Φ, {a}}, P(B) = {Φ, {b}}
P(A) ⪠P(B) = {Φ, {4}, {5}} â¦(i)
à¤
ब A ⪠B = {a, b}
P(A ⪠B) = {Φ, {a}, {b}, {a, b}}
समà¥. (i) à¤à¤° (ii) सॠहम दà¥à¤à¤¤à¥ हà¥à¤ à¤à¤¿
à¤
तठP(A) ⪠P(B) â P(A ⪠B)
पà¥à¤°à¤¶à¥à¤¨ 8.
à¤à¤¿à¤¨à¥à¤¹à¥à¤ दॠसमà¥à¤à¥à¤à¤¯à¥à¤ A तथा B à¤à¥ लिठसिदà¥à¤§ à¤à¥à¤à¤¿à¤ à¤à¤¿
A = (A â© B) ⪠(A â B) à¤à¤° A ⪠(B â A) = A ⪠B.
हल:
(i) दायाठपà¤à¥à¤· = (A â© B) ⪠(A â B)
= (A â© B) ⪠(A â B) [A â B = A â© Bâ]
= (A â© (B ⪠Bâ) (वितरण à¤à¥à¤£ सà¥)
= A â© U (यहाठU सारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯)
= A
à¤
तठ(A â© B) ⪠(A â B) = A.
(ii) बायाठपà¤à¥à¤· = A ⪠(B â A)
= A ⪠(B â© Aâ) [B â A = B â© Aâ]
= (A ⪠B) â© (A ⪠Aâ) (वितरण à¤à¥à¤£ सà¥)
= (A ⪠B) â© U [यहाठU सारà¥à¤µà¤¤à¥à¤°à¤¿à¤ समà¥à¤à¥à¤à¤¯]
= A ⪠B
à¤
तठ: A ⪠(B â A) = A ⪠B
पà¥à¤°à¤¶à¥à¤¨ 9.
समà¥à¤à¥à¤à¤¯à¥à¤ à¤à¥ à¤à¥à¤£à¤§à¤°à¥à¤®à¥à¤ à¤à¤¾ पà¥à¤°à¤¯à¥à¤ à¤à¤°à¤à¥ सिदà¥à¤§ à¤à¥à¤à¤¿à¤ à¤à¤¿
(i) A ⪠(A ⩠B) = A
(ii) A ⩠(A ⪠B) = A.
हल:
(i) बायाठपà¤à¥à¤· = A ⪠(A â© B)
= (A ⪠A) â© (A ⪠B) (वितरण à¤à¥à¤£ सà¥)
= A ⩠(A ⪠B) (A ⪠A = A)
= A [A â A ⪠B]
A ⪠(A ⩠B) = A.
(ii) बायाठपà¤à¥à¤·à¥ = A â© (A ⪠B)
= (A â© A) ⪠(A â© B) [वितरण à¤à¥à¤£ सà¥]
= A ⪠(A ⩠B) [A ⩠A = A]
= A [A â© B â A]
à¤
तठA â©(A ⪠B) = A.
पà¥à¤°à¤¶à¥à¤¨ 10.
दिà¤à¤²à¤¾à¤à¤ à¤à¤¿ A â© B = A â© C à¤à¤¾ तातà¥à¤ªà¤°à¥à¤¯ B = C à¤à¤µà¤¶à¥à¤¯à¤ रà¥à¤ª सॠनहà¥à¤ हà¥à¤¤à¤¾à¥¤
हल:
मान लà¥à¤à¤¿à¤ A = {1, 2}, B = {1, 7} तथा C = {1, 4} हà¥, तब
A â© B = {1, 2} â© {1, 7} = {1}
A â© C = {1, 2} â© {1, 4} = {1}
A â© B = A â© C
B â C
यदि A â© B = A â© C तॠà¤à¤µà¤¶à¥à¤¯à¤ नà¥à¤¹ हॠà¤à¤¿ B = C.
पà¥à¤°à¤¶à¥à¤¨ 11.
मान लà¥à¤à¤¿à¤ à¤à¤¿ A à¤à¤° B समà¥à¤à¥à¤à¤¯ हà¥à¤à¥¤ यदि à¤à¤¿à¤¸à¥ समà¥à¤à¥à¤à¤¯ X à¤à¥ लिठA⪠X = B ⪠X = Φ तथा A ⪠X = B ⪠X तॠसिदà¥à¤§ à¤à¥à¤à¤¿à¤ à¤à¤¿ A = B.
हल:
दिया हॠA ⪠X = B ⪠X, à¤à¤¬ à¤à¤¿ X à¤à¥à¤ समà¥à¤à¥à¤à¤¯ हà¥à¥¤
A â© (A ⪠X) = A â© (B ⪠X) [A â A âªX, A â© (A ⪠X) = A]
A = A ⩠(B ⪠X)
= (A â© B) ⪠(A â© X) [वितरण à¤à¥à¤£ सà¥]
= (A â© B) ⪠Φ (दिया हà¥, A â© X = Φ
= A â© B
A â B â¦â¦(i)
A ⪠X = B ⪠X
B â©(A ⪠X) = B â© (B ⪠X)
B â©(A ⪠X) = B [B â B ⪠X]
(B â© A) ⪠(B â© X) = B [वितरण à¤à¥à¤£ सà¥]
(B â© A) ⪠Φ = B [दिया हà¥: B â© X = Φ]
(B â© A) = B
B â A â¦..(ii)
समà¥. (i) à¤à¤° (ii) सà¥, हम पातॠहà¥à¤ à¤à¤¿ A = B.
पà¥à¤°à¤¶à¥à¤¨ 12.
à¤à¤¸à¥ समà¥à¤à¥à¤à¤¯ A, B à¤à¤° C à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤ ताà¤à¤¿ A â© B, B â© C तथा A â© C à¤à¤°à¤¿à¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ हà¥à¤ à¤à¤° A â© B â© C = Φ.
हल:
मान लà¥à¤à¤¿à¤à¥¤ A = {1, 2}, B = {2, 3}, C = {1, 3}
A â© B = {1, 2} â© {2, 3} = {2},
B â© C = {2, 3} â© {1, 3} = {3}
C â© A = {1, 3} â© {1, 2} = {1}
à¤
तठA â© B, B â© C, C â© A रिà¤à¥à¤¤ समà¥à¤à¥à¤à¤¯ नहà¥à¤ हà¥à¤à¥¤
A ⩠B ⩠C = (A ⩠B) ⩠C = {2} ⩠{1, 3} = Φ
à¤à¤¤à¤¿ सिदà¥à¤§à¤®à¥
पà¥à¤°à¤¶à¥à¤¨ 13.
à¤à¤¿à¤¸à¥ विदà¥à¤¯à¤¾à¤²à¤¯ à¤à¥ 600 विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ सरà¥à¤µà¥à¤à¥à¤·à¤£ सॠà¤à¥à¤à¤¾à¤¤ हà¥à¤ à¤à¤¿ 150 विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ à¤à¤¾à¤¯, 225 विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ à¤à¥à¤«à¥ तथा 100 विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ à¤à¤¾à¤¯ à¤à¤° à¤à¥à¤«à¥ दà¥à¤¨à¥à¤ पà¥à¤¤à¥ हà¥à¤à¥¤ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤ à¤à¤¿ à¤à¤¿à¤¤à¤¨à¥ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ न तॠà¤à¤¾à¤¯ पà¥à¤¤à¥ हà¥à¤ à¤à¤° न à¤à¥à¤«à¥ पà¥à¤¤à¥ हà¥à¤à¥¤
हल:
मान लà¥à¤à¤¿à¤ 7 à¤à¤° C à¤à¤¾à¤¯ तथा à¤à¥à¤«à¥ पà¥à¤¨à¥ वालॠविदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤, तब
n(T) = 150, n(C) = 225, n(T â© C) = 100
n(T ⪠C) = n(T) + n(C) â n(T â© C) = 150 + 225 â 100 = 275
= à¤à¤¨ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ à¤à¥ à¤à¤¾à¤¯ या à¤à¥à¤«à¥ पà¥à¤¤à¥ हà¥à¤ या à¤à¤¾à¤¯ à¤à¤° à¤à¥à¤«à¥ दà¥à¤¨à¥à¤ पà¥à¤¤à¥ हà¥à¤à¥¤
विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ à¤à¥à¤² सà¤à¤à¥à¤¯à¤¾ = 600
à¤à¤¨ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ à¤à¥ à¤à¤¾à¤¯ या à¤à¥à¤«à¥ à¤à¥à¤ à¤à¥ नहà¥à¤ पà¥à¤¤à¥ = 600 â 275 = 325.
पà¥à¤°à¤¶à¥à¤¨ 14.
विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ समà¥à¤¹ मà¥à¤, 100 विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ हिनà¥à¤¦à¥, 50 विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ à¤
à¤à¤à¥à¤°à¥à¤à¥ तथा 25 विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ दà¥à¤¨à¥à¤ à¤à¤¾à¤·à¤¾à¤à¤ à¤à¥ à¤à¤¾à¤¨à¤¤à¥ हà¥à¤à¥¤ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ मà¥à¤ सॠपà¥à¤°à¤¤à¥à¤¯à¥à¤ या तॠहिनà¥à¤¦à¥ या à¤
à¤à¤à¥à¤°à¥à¤à¥ à¤à¤¾à¤¨à¤¤à¤¾ हà¥à¥¤ समà¥à¤¹ मà¥à¤ à¤à¥à¤² à¤à¤¿à¤¤à¤¨à¥ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¥ हà¥à¤?
हल:
पाना तथा à¤à¥à¤°à¤®à¤¶à¤ हिनà¥à¤¦à¥ à¤à¤° à¤
à¤à¤à¥à¤°à¥à¤à¥ à¤à¤¾à¤¨à¤¨à¥ वालà¥à¤ à¤à¥ समà¥à¤à¥à¤à¤¯ हà¥à¤, तब
n(H) = 100, n(E) = 50, n(H â© E) = 25
n(H ⪠E) = n(H) + n(E) â n(H â©E) = 100 + 50 â 25 = 125
à¤à¤¨ विदà¥à¤¯à¤¾à¤°à¥à¤¥à¤¿à¤¯à¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ à¤à¥ हिनà¥à¤¦à¥ या à¤
à¤à¤à¥à¤°à¥à¤à¥ à¤à¤¾à¤¨à¤¤à¥ हà¥à¤ = 125.
पà¥à¤°à¤¶à¥à¤¨ 15.
60 लà¥à¤à¥à¤ à¤à¥ सरà¥à¤µà¥à¤à¥à¤·à¤£ मà¥à¤ पाया à¤à¤¯à¤¾ à¤à¤¿ 25 लà¥à¤ समाà¤à¤¾à¤° पतà¥à¤° H, 26 लà¥à¤ समाà¤à¤¾à¤° पतà¥à¤° T, 26 लà¥à¤ समाà¤à¤¾à¤° पतà¥à¤° I, 9 लà¥à¤ H तथा I दà¥à¤¨à¥à¤, 11 लà¥à¤ H तथा T दà¥à¤¨à¥à¤, 8 लà¥à¤ T तथा I दà¥à¤¨à¥à¤ à¤à¤° 3 लà¥à¤ तà¥à¤¨à¥à¤ हॠसमाà¤à¤¾à¤° पतà¥à¤° पढ़तॠहà¥à¤, तॠनिमà¥à¤¨à¤²à¤¿à¤à¤¿à¤¤ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤:
(i) à¤à¤® सॠà¤à¤® à¤à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾à¥¤
(ii) ठà¥à¤ ठà¥à¤ à¤à¥à¤µà¤² à¤à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾à¥¤
हल:
à¤à¥à¤² लà¥à¤à¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ à¤à¤¿à¤¨à¤à¤¾ सरà¥à¤µà¥à¤à¥à¤·à¤£ à¤à¤¿à¤¯à¤¾ à¤à¤¯à¤¾ = 60
H समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾, n (H) = 25
T समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾, n (T) = 26
I समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾, n (I) = 26
H à¤à¤° I समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾, n (H â© I) = 9
H à¤à¤° I समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾, n (H â© T) = 11
T à¤à¤° I समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾, n (T â© I) = 8
तà¥à¤¨à¥à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾, n (H â© T â© I) = 3
H à¤à¤° I समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालॠतथा T समाà¤à¤¾à¤° पतà¥à¤° न पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 9 â 3 = 6
H à¤à¤° T समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालॠतथा I समाà¤à¤¾à¤° पतà¥à¤° न पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 11 â 3 = 8
T à¤à¤° I समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालॠतथा H समाà¤à¤¾à¤° पतà¥à¤° न पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 8 â 3 = 5
à¤à¥à¤µà¤² H समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 25 â 8 â 6 â 3 = 8
à¤à¥à¤µà¤² T समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 26 â 8 â 3 â 5 = 10
à¤à¥à¤µà¤² I समाà¤à¤¾à¤° पतà¥à¤°à¤ पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 26 â 6 â 3 â 5 = 12
à¤à¤® सॠà¤à¤® à¤à¤à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = à¤à¥à¤µà¥à¤² à¤à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ + à¤à¥à¤µà¤² दॠसमाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ + तà¥à¤¨à¥à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = (8 + 10 + 12) + (8 + 6 + 5) + 3 = 30 + 19 + 3 = 52
वà¥à¤à¤²à¥à¤ªà¤¿à¤ विधि :
n(H ⪠T ⪠I) = n(H) + n(T) + n(I) â n(H â© T) â n(T â© I) â n(Hâ©I) + n(H â© T â© I)
= 25 + 26 + 26 â 11 â 8 â 9 + 3 = 77 â 28 + 3 = 80 â 28 = 52
(ii) à¤à¥à¤µà¤² H à¤à¤° T समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 11 â 3 = 8
à¤à¥à¤µà¤² T à¤à¤° 1 समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 8 â 3 = 5
à¤à¥à¤µà¤² 1 à¤à¤° H समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 9 â 3 = 6
तà¥à¤¨à¥à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 3
à¤à¥à¤µà¤² à¤à¤ समाà¤à¤¾à¤° पतà¥à¤° पढ़नॠवालà¥à¤ à¤à¥ सà¤à¤à¥à¤¯à¤¾ = 52 â (8 + 5 + 6 + 3) = 52 â 22 = 30.
पà¥à¤°à¤¶à¥à¤¨ 16.
à¤à¤ सरà¥à¤µà¥à¤à¥à¤·à¤£ मà¥à¤ पाया à¤à¤¯à¤¾ à¤à¤¿ 21 लà¥à¤ à¤à¤¤à¥à¤ªà¤¾à¤¦ A, 26 लà¥à¤ à¤à¤¤à¥à¤ªà¤¾à¤¦ B, 29 लà¥à¤ à¤à¤¤à¥à¤ªà¤¾à¤¦ C पसà¤à¤¦ à¤à¤°à¤¤à¥ हà¥à¤à¥¤ यदि 14 लà¥à¤ à¤à¤¤à¥à¤ªà¤¾à¤¦ A तथा B, 12 लà¥à¤ à¤à¤¤à¥à¤ªà¤¾à¤¦ C तथा A, 14 लà¥à¤ à¤à¤¤à¥à¤ªà¤¾à¤¦ B तथा C à¤à¤° 8 लà¥à¤ तà¥à¤¨à¥à¤ हॠà¤à¤¤à¥à¤ªà¤¾à¤¦à¥à¤ à¤à¥ पसà¤à¤¦ à¤à¤°à¤¤à¥ हà¥à¤à¥¤ à¤à¥à¤à¤¾à¤¤ à¤à¥à¤à¤¿à¤ à¤à¤¿ à¤à¤¿à¤¤à¤¨à¥ लà¥à¤ à¤à¥à¤µà¤² à¤à¤¤à¥à¤ªà¤¾à¤¦ C à¤à¥ पसà¤à¤¦ à¤à¤°à¤¤à¥ हà¥à¤?
हल:
दिया हà¥:
n(A) = 21,
n(B) = 26,
n(C) = 29
n(A â© B) = 14,
n(A â© C) = 12
n(B â© C) = 14,
n(A â© B â© C) = 8
n(A â© C) = 12,
P(A â© B â© C) = 8
n(à¤à¥à¤µà¤² A à¤à¤° C) = 12 â 8 = 4
n(à¤à¥à¤µà¤² B à¤à¤° C) = 14 â 8 = 6
n(à¤à¥à¤µà¤² C) = n(C) â n (à¤à¥à¤µà¤² A à¤à¤° C) â n (à¤à¥à¤µà¤² B à¤à¤° C) â n(A â© B â© C) = 29 â 4 â 6 â 8 = 29 â 18 = 11
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