The number of min-terms after minimizing the following Boolean expression is _________.

      [D′ + AB′ + A′C + AC′D + A′C′D]′

(A) 1
(B) 2
(C) 3
(D) 4


Answer: (A)

Explanation:

Given Boolean expression is: 
              [D′ + AB′ + A′C + AC′D + A′C′D]′

Step 1 : [D′ + AB′ + A′C + C′D ( A + A\')]′ 
( taking C\'D as common )

Step 2 : [D′ + AB′ + A′C + C′D]′ 
( as, A + A\' = 1 )

: [D\' + DC\' + AB\' + A\'C]\' (Rearrange)

Step 3 : [D\' + C\' + AB\' + A\'C]\' 
( Rule of Duality, A + A\'B = A + B )

: [D\' + C\' + CA\' + AB\']\' (Rearrange)

Step 4 : [D\' + C\' + A\' + AB\']\' 
(Rule of Duality)

: [D\' + C\' + A\' + AB\']\' (Rearrange)

Step 5 : [D\' + C\' + A\' + B\']\' 
(Rule of Duality)

:[( D\' + C\' )\'.( A\' + B\')\'] 
(Demorgan\'s law, (A + B)\'=(A\'. B\'))

:[(D\'\'.C\'\').( A\'\'.B\'\')] (Demorgan\'s law)

:[(D.C).(A.B)] (Idempotent law, A\'\' = A)

: ABCD

Hence only 1 minterm after minimization. 

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