{"id":198967,"date":"2025-03-10T15:35:40","date_gmt":"2025-03-10T15:35:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=198967"},"modified":"2025-03-10T15:35:43","modified_gmt":"2025-03-10T15:35:43","slug":"express-the-following-function-in-a-sum-of-minterms-and-a-product-of-maxterms","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/10\/express-the-following-function-in-a-sum-of-minterms-and-a-product-of-maxterms\/","title":{"rendered":"Express the following function in a sum of minterms and a product of maxterms"},"content":{"rendered":"\n

Express the following function in a sum of minterms and a product of maxterms:<\/p>\n\n\n\n

F(A,B,C,D) = D(CA’) + B’D<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To express the function ( F(A,B,C,D) = D(CA’) + B’D ) in both a sum of minterms<\/strong> (SOP) and a product of maxterms<\/strong> (POS), follow these steps:<\/p>\n\n\n\n

Step 1: Expand the Given Function<\/strong><\/h3>\n\n\n\n

The given Boolean function is:<\/p>\n\n\n\n

[
F(A,B,C,D) = D(CA’) + B’D
]<\/p>\n\n\n\n

Distribute Terms<\/strong><\/h4>\n\n\n\n

Expanding using the distributive property:<\/p>\n\n\n\n

[
F(A,B,C,D) = DCA’ + B’D
]<\/p>\n\n\n\n

Factor (D) in the first term:<\/p>\n\n\n\n

[
F(A,B,C,D) = D(A’C + B’)
]<\/p>\n\n\n\n

Step 2: Find the Sum of Minterms (SOP)<\/strong><\/h3>\n\n\n\n

A minterm is a product (AND) of all variables in either complemented or uncomplemented form.<\/p>\n\n\n\n

Identify Minterms<\/strong><\/h4>\n\n\n\n

We analyze each product term:<\/p>\n\n\n\n

    \n
  1. (DCA’)<\/strong>:<\/li>\n<\/ol>\n\n\n\n
      \n
    • ( A = 0, B = -) (does not appear), ( C = 1, D = 1 )<\/li>\n\n\n\n
    • Possible values for ( B ): 0 or 1<\/li>\n\n\n\n
    • Corresponding minterms: ( (0,0,1,1) = m3 ) and ( (0,1,1,1) = m7 )<\/li>\n<\/ul>\n\n\n\n
        \n
      1. (B’D)<\/strong>:<\/li>\n<\/ol>\n\n\n\n
          \n
        • ( B = 0, D = 1 ) (A and C are free)<\/li>\n\n\n\n
        • Possible values: ( A,C = 00, 01, 10, 11 )<\/li>\n\n\n\n
        • Corresponding minterms: ( m1, m3, m5, m7 )<\/li>\n<\/ul>\n\n\n\n

          Combine the minterms<\/strong><\/h4>\n\n\n\n

          [
          F(A,B,C,D) = \\sum m(1,3,5,7)
          ]<\/p>\n\n\n\n

          Step 3: Find the Product of Maxterms (POS)<\/strong><\/h3>\n\n\n\n

          A maxterm is a sum (OR) of all variables in either complemented or uncomplemented form.<\/p>\n\n\n\n

          Find the Missing Minterms<\/strong><\/h4>\n\n\n\n

          The function covers minterms ( 1,3,5,7 ), so the missing minterms (for a 4-variable function) are:<\/p>\n\n\n\n

          [
          M_0, M_2, M_4, M_6, M_8, M_9, M_{10}, M_{11}, M_{12}, M_{13}, M_{14}, M_{15}
          ]<\/p>\n\n\n\n

          Convert to Maxterms<\/strong><\/h4>\n\n\n\n

          [
          F(A,B,C,D) = \\prod M(0,2,4,6,8,9,10,11,12,13,14,15)
          ]<\/p>\n\n\n\n

          Final Answer<\/strong><\/h3>\n\n\n\n
            \n
          1. Sum of minterms (SOP):<\/strong>
            [
            F(A,B,C,D) = \\sum m(1,3,5,7)
            ]<\/li>\n\n\n\n
          2. Product of maxterms (POS):<\/strong>
            [
            F(A,B,C,D) = \\prod M(0,2,4,6,8,9,10,11,12,13,14,15)
            ]<\/li>\n<\/ol>\n\n\n\n

            Explanation<\/strong><\/h3>\n\n\n\n
              \n
            • The sum of minterms<\/strong> form represents the function as an OR of all the minterms where ( F = 1 ).<\/li>\n\n\n\n
            • The product of maxterms<\/strong> form represents the function as an AND of all the maxterms where ( F = 0 ).<\/li>\n\n\n\n
            • The process involves expanding the function into product terms, identifying the minterms it covers, and finding the missing maxterms to construct the complementary representation.<\/li>\n<\/ul>\n\n\n\n

              This method ensures the function is correctly expressed in both forms.<\/p>\n","protected":false},"excerpt":{"rendered":"

              Express the following function in a sum of minterms and a product of maxterms: F(A,B,C,D) = D(CA’) + B’D The correct answer and explanation is : To express the function ( F(A,B,C,D) = D(CA’) + B’D ) in both a sum of minterms (SOP) and a product of maxterms (POS), follow these steps: Step 1: […]<\/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-198967","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/198967","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=198967"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/198967\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=198967"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=198967"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=198967"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}