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

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

`      [Dâ€² + ABâ€² + Aâ€²C + ACâ€²D + Aâ€²Câ€²D]â€²`

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

Answer: (A)

Explanation:

```Given Boolean expression is:
[Dâ€² + ABâ€² + Aâ€²C + ACâ€²D + Aâ€²Câ€²D]â€²

Step 1 : [Dâ€² + ABâ€² + Aâ€²C + Câ€²D ( A + A')]â€²
( taking C'D as common )

Step 2 : [Dâ€² + ABâ€² + Aâ€²C + Câ€²D]â€²
( 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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