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GO Classes DPP | GATE CS | Digital Logic | Boolean Algebra Laws

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Simplify the Boolean expression $F=(A+B)(A+\overline{B})(\overline{A}+C)+AC$.

Which of the following is the simplified form?

  1. $A$
     
  2. $AC$
     
  3. $A+C$
     
  4. $A\overline{B}+AC$

2 Answers

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Given expression is $F=(A+B)(A+\overline{B})(\overline{A}+C)+AC$.

First use the Boolean law $(X+Y)(X+Z)=X+YZ$.

Here, $(A+B)(A+\overline{B})=A+B\overline{B}$.

Since $B\overline{B}=0$, we get:

$(A+B)(A+\overline{B})=A+0=A$.

Now substitute this back in the expression.

$F=A(\overline{A}+C)+AC$.

Now distribute $A$.

$F=A\overline{A}+AC+AC$.

Since $A\overline{A}=0$, we get:

$F=0+AC+AC$.

Using idempotent law, $AC+AC=AC$.

Therefore, $F=AC$.

Answer: B.

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