GATE CS 2013 Q21 — Boolean Algebra

The exclusive NOR (XNOR) of two boolean variables $x$ and $y$ is represented as $x \odot y$, which has the boolean expression:

Let's evaluate each option to see if it simplifies to $x \odot y$:

• Option A: $xy + x'y'$
  This is the standard definition of XNOR. Thus, it represents XNOR.
• Option B: $x \oplus y'$
  Using the definition of XOR ($A \oplus B = AB' + A'B$): $$x \oplus y' = x(y')' + x'(y') = xy + x'y' = x \odot y$$ Thus, it represents XNOR.
• Option C: $x' \oplus y$
  Using the definition of XOR: $$x' \oplus y = x'(y)' + (x')'y = x'y' + xy = x \odot y$$ Thus, it represents XNOR.
• Option D: $x' \oplus y'$
  Using the definition of XOR: $$x' \oplus y' = x'(y')' + (x')'y' = x'y + xy' = x \oplus y$$ This represents the exclusive OR (XOR) of $x$ and $y$, which is the complement of XNOR.

Therefore, the expression in Option D does NOT represent the exclusive NOR of $x$ and $y$.