Simplified form of the Boolean function F(P,Q,R,S)= P Q+ PQS+P Q R S+P QR S is

GATE 2024 EE – Question 24

Analog and Digital Electronics · VCOs and timers, combinatorial and sequential logic circuits, multiplexers, demultiplexers · 1 mark · Multiple choice

Simplified form of the Boolean function

$$F(P,Q,R,S)=\bar P\bar Q+\bar PQS+P\bar Q\bar R\bar S+P\bar QR\bar S$$

$\bar PS+\bar Q\bar S$
$\bar P\bar Q+\bar Q\bar S$
$\bar PQ+R\bar S$
$P\bar S+Q\bar R$

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Correct answer: (A) $\bar PS+\bar Q\bar S$

Explanation

Combine the last two terms on $R$: $P\bar Q\bar S(\bar R+R)=P\bar Q\bar S$. Then $F=\bar P\bar Q+\bar PQS+P\bar Q\bar S=\bar P(\bar Q+S)+P\bar Q\bar S$. For $P=0$: $F=\bar Q+S$, and for $P=1$: $F=\bar Q\bar S$. The expression $\bar PS+\bar Q\bar S$ gives $S+\bar Q\bar S=S+\bar Q$ when $P=0$ and $\bar Q\bar S$ when $P=1$, matching exactly.