The GATE Grind

GATE 2019 CS – Question 16

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

Which one of the following is NOT a valid identity?

  1. $(x\oplus y)\oplus z=x\oplus(y\oplus z)$
  2. $(x+y)\oplus z=x\oplus(y+z)$
  3. $x\oplus y=x+y$, if $xy=0$
  4. $x\oplus y=(xy+x'y')'$

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Show answer and explanation

Correct answer: (B) $(x+y)\oplus z=x\oplus(y+z)$

Explanation

XOR is associative (A), $x\oplus y=x+y$ when $x$ and $y$ are never both 1 (C), and $x\oplus y=\overline{xy+x'y'}$ (D). For B, take $x=0,y=1,z=1$: $(x+y)\oplus z=1\oplus1=0$ but $x\oplus(y+z)=0\oplus1=1$, so B is not an identity.