Define X\#Y=X+Y on Boolean variables. S1: (P#Q)#R=P#(Q#R). S2: Q#R=R#Q. Which hold?

GATE 2015 CS (CS3) – Question 16

Digital Logic · Boolean Algebra and Minimization · 1 mark · Multiple choice

Define $X\#Y=X\prime+Y\prime$ on Boolean variables. S1: (P#Q)#R=P#(Q#R). S2: Q#R=R#Q. Which hold?

S1 only
S2 only
Both
Neither

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Show answer and explanation

Correct answer: (B) S2 only

Explanation

The operator is NAND by De Morgan’s law, so it is commutative, proving S2. It is not associative: with P=0,Q=1,R=1, the left side is (0 NAND 1) NAND 1=0, while the right is 0 NAND (1 NAND 1)=1. Thus only S2, B.

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