{"id":201755,"date":"2025-03-15T22:16:12","date_gmt":"2025-03-15T22:16:12","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=201755"},"modified":"2025-03-15T22:16:14","modified_gmt":"2025-03-15T22:16:14","slug":"find-the-canonical-sum-of-products-and-product-of-sums-expression-for-the-function","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/15\/find-the-canonical-sum-of-products-and-product-of-sums-expression-for-the-function\/","title":{"rendered":"Find the canonical sum of products and product of sums expression for the function"},"content":{"rendered":"\n<p>Find the canonical sum of products and product of sums expression for the function<\/p>\n\n\n\n<p>F = X1X2X3 + X1X3X4 + X1X2X4.<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s find the <strong>Canonical Sum of Products (SOP)<\/strong> and the <strong>Canonical Product of Sums (POS)<\/strong> for the function:<\/p>\n\n\n\n<p>[<br>F = X_1 X_2 X_3 + X_1 X_3 X_4 + X_1 X_2 X_4.<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. Canonical Sum of Products (SOP)<\/h3>\n\n\n\n<p>In canonical SOP form, each product term (AND operation) must represent a minterm, which corresponds to a unique combination of the variables (X_1, X_2, X_3, X_4). To express (F) as a canonical SOP, we need to write all the minterms for which (F = 1).<\/p>\n\n\n\n<p>Start by identifying the minterms for each term in the sum (X_1 X_2 X_3 + X_1 X_3 X_4 + X_1 X_2 X_4).<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>For (X_1 X_2 X_3):<\/strong> This term is true when (X_1 = 1), (X_2 = 1), and (X_3 = 1), but we need to consider all combinations of (X_4). So the corresponding minterms are:<\/li>\n\n\n\n<li>(X_1 X_2 X_3 \\cdot X_4)<\/li>\n\n\n\n<li>(X_1 X_2 X_3 \\cdot \\overline{X_4})<\/li>\n\n\n\n<li><strong>For (X_1 X_3 X_4):<\/strong> This term is true when (X_1 = 1), (X_3 = 1), and (X_4 = 1), but we also need to consider (X_2). So the corresponding minterms are:<\/li>\n\n\n\n<li>(X_1 X_2 \\cdot X_3 X_4)<\/li>\n\n\n\n<li>(X_1 \\overline{X_2} \\cdot X_3 X_4)<\/li>\n\n\n\n<li><strong>For (X_1 X_2 X_4):<\/strong> This term is true when (X_1 = 1), (X_2 = 1), and (X_4 = 1), but we need to consider (X_3). So the corresponding minterms are:<\/li>\n\n\n\n<li>(X_1 X_2 \\cdot X_3 X_4)<\/li>\n\n\n\n<li>(X_1 X_2 \\cdot \\overline{X_3} X_4)<\/li>\n<\/ul>\n\n\n\n<p>Thus, the <strong>Canonical SOP<\/strong> is:<br>[<br>F = X_1 X_2 X_3 \\cdot X_4 + X_1 X_2 X_3 \\cdot \\overline{X_4} + X_1 X_2 \\cdot \\overline{X_3} X_4 + X_1 \\overline{X_2} \\cdot X_3 X_4.<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. Canonical Product of Sums (POS)<\/h3>\n\n\n\n<p>In canonical POS form, each sum term (OR operation) must represent a maxterm, which corresponds to a unique combination of the variables (X_1, X_2, X_3, X_4). We need to write all the maxterms for which (F = 0).<\/p>\n\n\n\n<p>To find the maxterms, first write the combinations where (F = 0). From the original expression, you can observe where (F = 0) by identifying the combinations of inputs that make (F) false. After deriving those combinations, we can form the maxterms, which are ORed terms involving negations of the variables.<\/p>\n\n\n\n<p>After identifying the zeros in the truth table, we derive the maxterms and write the canonical POS form.<\/p>\n\n\n\n<p>The <strong>Canonical POS<\/strong> form is:<br>[<br>F = (X_1 + X_2 + X_3 + \\overline{X_4})(X_1 + X_2 + \\overline{X_3} + X_4)(X_1 + \\overline{X_2} + X_3 + \\overline{X_4})(\\overline{X_1} + X_2 + X_3 + X_4).<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Summary:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>Canonical SOP<\/strong> is the sum of all the minterms where the function evaluates to 1.<\/li>\n\n\n\n<li>The <strong>Canonical POS<\/strong> is the product of all the maxterms where the function evaluates to 0.<\/li>\n<\/ul>\n\n\n\n<p>These canonical forms are essential in Boolean algebra, especially when designing circuits or simplifying logic expressions. They provide a systematic way to represent any Boolean function using only ANDs, ORs, and NOTs.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find the canonical sum of products and product of sums expression for the function F = X1X2X3 + X1X3X4 + X1X2X4. The correct answer and explanation is : Let&#8217;s find the Canonical Sum of Products (SOP) and the Canonical Product of Sums (POS) for the function: [F = X_1 X_2 X_3 + X_1 X_3 X_4 [&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-201755","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201755","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=201755"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/201755\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=201755"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=201755"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=201755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}