GATE | GATE-CS-2015 (Set 2) | Question 47

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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