{"id":221794,"date":"2025-05-30T14:22:23","date_gmt":"2025-05-30T14:22:23","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=221794"},"modified":"2025-05-30T14:22:25","modified_gmt":"2025-05-30T14:22:25","slug":"convert-the-following-boolean-expression-into-standard-sop-form-and-also-implement-the-expression","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/05\/30\/convert-the-following-boolean-expression-into-standard-sop-form-and-also-implement-the-expression\/","title":{"rendered":"Convert the following Boolean expression into standard SOP form and also implement the expression."},"content":{"rendered":"\n
Q.1.(b)(i) Convert the following Boolean expression into standard SOP form and also implement the expression.\u00c3\u201a\u00c2 ABC + AB + ABCD Q.1.(b)(ii) Convert the following Boolean expression into standard POS form and also implement the expression. (A + B + C)(B + C + D)(A + B + C + D)<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n
Q.1.(b)(i) Convert to Standard SOP Form and Implement:<\/strong><\/h3>\n\n\n\n
Given Boolean Expression:<\/strong> ABC+AB+ABCDABC + AB + ABCD<\/p>\n\n\n\n
Notice that ABC<\/code> is already part of AB<\/code>, and ABCD<\/code> is more specific than ABC<\/code>. So the original expression can be simplified: AB+ABCD=AB(1+CD)=AB\u22c51=ABAB + ABCD = AB(1 + CD) = AB \\cdot 1 = AB<\/p>\n\n\n\n
But we are asked to convert it into Standard SOP<\/strong> (Sum of Products), where each term must include all variables<\/strong> involved (A, B, C, D).<\/p>\n\n\n\n\n\n\n\n
Step 2: Standard SOP Conversion<\/strong><\/h4>\n\n\n\n
We express AB<\/code> in terms of A, B, C, D<\/strong>: AB=AB(C+C\u2032)(D+D\u2032)AB = AB(C + C’)(D + D’)<\/p>\n\n\n\n
OR gate combines the 4 outputs.<\/li>\n<\/ol>\n\n\n\n\n\n\n\n
Q.1.(b)(ii) Convert to Standard POS Form and Implement:<\/strong><\/h3>\n\n\n\n
Given Expression (POS form):<\/strong> (A+B+C)(B+C+D)(A+B+C+D)(A + B + C)(B + C + D)(A + B + C + D)<\/p>\n\n\n\n
This is already in Product of Sums (POS) form. But it is not yet in Standard POS<\/strong>, where each sum term must include all variables<\/strong>.<\/p>\n\n\n\n\n\n\n\n
Step 1: Identify All Variables<\/strong><\/h4>\n\n\n\n
Variables used: A, B, C, D<\/p>\n\n\n\n
\n\n\n\n
Step 2: Expand Each Term to Include All Variables<\/strong><\/h4>\n\n\n\n
\n
(A + B + C) \u2192 Add missing D: (A + B + C + D)(A + B + C + D’)<\/li>\n\n\n\n
(B + C + D) \u2192 Add missing A: (A + B + C + D)(A’ + B + C + D)<\/li>\n\n\n\n
(A + B + C + D) already includes all variables.<\/li>\n<\/ul>\n\n\n\n\n\n\n\n
Standard POS Form:<\/strong><\/h4>\n\n\n\n
(A+B+C+D)(A+B+C+D\u2032)(A\u2032+B+C+D)(A + B + C + D)(A + B + C + D’)(A’ + B + C + D)<\/p>\n\n\n\n
Boolean algebra expressions often require standardization into canonical forms for systematic simplification and implementation in digital circuits. The two primary canonical forms are Sum of Products (SOP)<\/strong> and Product of Sums (POS)<\/strong>.<\/p>\n\n\n\n
In Q.1.(b)(i)<\/strong>, we started with the expression ABC + AB + ABCD<\/code>. Noticing redundancy, we simplified it to AB<\/code>, as it encompasses all original terms. However, SOP form requires each product term to have all variables involved\u2014in this case, A, B, C, and D. We expanded AB<\/code> using identities such as X = X(Y + Y')<\/code> to generate all possible combinations of the missing variables, resulting in: ABCD+ABCD\u2032+ABC\u2032D+ABC\u2032D\u2032ABCD + ABCD’ + ABC’D + ABC’D’<\/p>\n\n\n\n
This SOP form is then implemented using AND gates for each term and a final OR gate to sum the outputs.<\/p>\n\n\n\n
For Q.1.(b)(ii)<\/strong>, the given expression was already in POS format but not standard. A standard POS form mandates that every sum term includes all variables. We augmented each clause by adding missing variables with both complements and original forms using the identity X = (X + Y)(X + Y')<\/code>. The result was: (A+B+C+D)(A+B+C+D\u2032)(A\u2032+B+C+D)(A + B + C + D)(A + B + C + D’)(A’ + B + C + D)<\/p>\n\n\n\n
This form ensures complete coverage of input combinations. It can be implemented using OR gates per clause and an AND gate to combine them, reflecting the POS structure. These standard forms are crucial for minimizing logic and ensuring consistent digital design.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
Q.1.(b)(i) Convert the following Boolean expression into standard SOP form and also implement the expression.\u00c3\u201a\u00c2 ABC + AB + ABCDQ.1.(b)(ii) Convert the following Boolean expression into standard POS form and also implement the expression.(A + B + C)(B + C + D)(A + B + C + D) The Correct Answer and Explanation is: Q.1.(b)(i) Convert […]<\/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-221794","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221794","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=221794"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/221794\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=221794"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=221794"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=221794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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