GATE CS 2013 Q.21 (CS Q.21): Boolean Algebra and Logic Minimization

Which one of the following expressions does NOT represent exclusive NOR of $x$ and $y$ ?

$xy + x' y'$

$x \oplus y'$

$x' \oplus y$

$x' \oplus y'$

Official answer and solution

Solution

Expand each Boolean expression into sum-of-products form to test equivalence with the XNOR function.

The exclusive NOR (XNOR) function of $x$ and $y$ is defined as:

x \odot y = \overline{x \oplus y} = xy + x'y'

Let us evaluate each option:

(A) $xy + x'y'$ : This is the standard algebraic definition of XNOR.
(B) $x \oplus y'$ : $x \oplus y' = x (y')' + x' y' = xy + x'y' = \text{XNOR}$
(C) $x' \oplus y$ : $x' \oplus y = x' y' + (x')' y = x'y' + xy = \text{XNOR}$
(D) $x' \oplus y'$ : $x' \oplus y' = x' (y')' + (x')' y' = x'y + xy' = x \oplus y = \text{XOR}$

Because expression D produces the XOR function rather than XNOR, it does NOT represent XNOR.

Final answer

Option (D)

Common trap

Assuming that negating both inputs inverts the function, whereas $x' \oplus y' = x \oplus y$ .

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