{"id":234346,"date":"2025-06-14T04:48:44","date_gmt":"2025-06-14T04:48:44","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234346"},"modified":"2025-06-14T04:48:46","modified_gmt":"2025-06-14T04:48:46","slug":"please-prove-that-a-b-a-%e2%88%a9-b-with-words-prove-that-a-b-is-equal-to-a-intersection-b-complement","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/please-prove-that-a-b-a-%e2%88%a9-b-with-words-prove-that-a-b-is-equal-to-a-intersection-b-complement\/","title":{"rendered":"Please prove that A-B = A \u2229 B&#8217; with words Prove that A-B is equal to A intersection B complement."},"content":{"rendered":"\n<p>Please prove that A-B = A \u2229 B&#8217; with words Prove that A-B is equal to A intersection B complement.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To <strong>prove that<\/strong>:A\u2212B=A\u2229B\u2032A &#8211; B = A \\cap B&#8217;A\u2212B=A\u2229B\u2032<\/p>\n\n\n\n<p>we will show that both sets contain exactly the same elements. This is done by proving two inclusions:<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>1. Show that<\/strong> A\u2212B\u2286A\u2229B\u2032A &#8211; B \\subseteq A \\cap B&#8217;A\u2212B\u2286A\u2229B\u2032:<\/h3>\n\n\n\n<p>Let x\u2208A\u2212Bx \\in A &#8211; Bx\u2208A\u2212B.<br>By the <strong>definition of set difference<\/strong>, this means:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2208Ax \\in Ax\u2208A, and<\/li>\n\n\n\n<li>x\u2209Bx \\notin Bx\u2208\/B<\/li>\n<\/ul>\n\n\n\n<p>But if x\u2209Bx \\notin Bx\u2208\/B, then by definition of complement:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2208B\u2032x \\in B&#8217;x\u2208B\u2032<\/li>\n<\/ul>\n\n\n\n<p>So:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2208Ax \\in Ax\u2208A and x\u2208B\u2032x \\in B&#8217;x\u2208B\u2032<\/li>\n<\/ul>\n\n\n\n<p>Hence:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2208A\u2229B\u2032x \\in A \\cap B&#8217;x\u2208A\u2229B\u2032<\/li>\n<\/ul>\n\n\n\n<p>This proves that:A\u2212B\u2286A\u2229B\u2032A &#8211; B \\subseteq A \\cap B&#8217;A\u2212B\u2286A\u2229B\u2032<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>2. Show that<\/strong> A\u2229B\u2032\u2286A\u2212BA \\cap B&#8217; \\subseteq A &#8211; BA\u2229B\u2032\u2286A\u2212B:<\/h3>\n\n\n\n<p>Let x\u2208A\u2229B\u2032x \\in A \\cap B&#8217;x\u2208A\u2229B\u2032.<br>By the <strong>definition of intersection<\/strong>, this means:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2208Ax \\in Ax\u2208A, and<\/li>\n\n\n\n<li>x\u2208B\u2032x \\in B&#8217;x\u2208B\u2032<\/li>\n<\/ul>\n\n\n\n<p>Since x\u2208B\u2032x \\in B&#8217;x\u2208B\u2032, then:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2209Bx \\notin Bx\u2208\/B<\/li>\n<\/ul>\n\n\n\n<p>Thus:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2208Ax \\in Ax\u2208A, and x\u2209Bx \\notin Bx\u2208\/B<\/li>\n<\/ul>\n\n\n\n<p>By the <strong>definition of set difference<\/strong>, this means:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>x\u2208A\u2212Bx \\in A &#8211; Bx\u2208A\u2212B<\/li>\n<\/ul>\n\n\n\n<p>So:A\u2229B\u2032\u2286A\u2212BA \\cap B&#8217; \\subseteq A &#8211; BA\u2229B\u2032\u2286A\u2212B<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Conclusion<\/strong>:<\/h3>\n\n\n\n<p>We have shown both:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A\u2212B\u2286A\u2229B\u2032A &#8211; B \\subseteq A \\cap B&#8217;A\u2212B\u2286A\u2229B\u2032, and<\/li>\n\n\n\n<li>A\u2229B\u2032\u2286A\u2212BA \\cap B&#8217; \\subseteq A &#8211; BA\u2229B\u2032\u2286A\u2212B<\/li>\n<\/ul>\n\n\n\n<p>Therefore:A\u2212B=A\u2229B\u2032A &#8211; B = A \\cap B&#8217;A\u2212B=A\u2229B\u2032<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Explanation (Like in Books)<\/strong>:<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-529.jpeg\" alt=\"\" class=\"wp-image-234347\"\/><\/figure>\n\n\n\n<p>The <strong>set difference<\/strong> A\u2212BA &#8211; BA\u2212B is the set of all elements that are in set AAA but not in set BBB. On the other hand, the <strong>complement of B<\/strong>, denoted by B\u2032B&#8217;B\u2032, includes all elements not in BBB (relative to the universal set). When we take the <strong>intersection<\/strong> A\u2229B\u2032A \\cap B&#8217;A\u2229B\u2032, we are selecting all elements that are both in AAA and not in BBB, which is exactly the same condition as set difference A\u2212BA &#8211; BA\u2212B. Therefore, both expressions describe the same set, and we conclude:A\u2212B=A\u2229B\u2032A &#8211; B = A \\cap B&#8217;A\u2212B=A\u2229B\u2032<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Please prove that A-B = A \u2229 B&#8217; with words Prove that A-B is equal to A intersection B complement. The Correct Answer and Explanation is: To prove that:A\u2212B=A\u2229B\u2032A &#8211; B = A \\cap B&#8217;A\u2212B=A\u2229B\u2032 we will show that both sets contain exactly the same elements. This is done by proving two inclusions: 1. Show [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-234346","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234346","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=234346"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234346\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234346"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234346"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}