Consider the following Boolean expressions:
I: $x.y+x'y'$
II: $XOR(x',y')$
III: $XOR(x',y)$
Which of the above expressions represents exclusive NOR operation ?

Question

Consider the following Boolean expressions: 
I: $x.y+x'y'$ 
II: $XOR(x',y')$ 
III: $XOR(x',y)$ 
Which of the above expressions represents exclusive NOR operation ?

Accepted Answer

The correct option is (2): I and III.

The Exclusive NOR (XNOR) operation, often represented as $x \odot y$ or $(x \oplus y)'$, is a core concept in digital logic and Boolean algebra. This operation is defined as producing a TRUE ('1') output only when its two inputs are logically equivalent—that is, both inputs are '0' or both inputs are '1'. Expression I, which is $x.y + x'y'$, is the canonical Sum-of-Products form that directly translates this behavior: $x.y$ covers the case where $x=1$ and $y=1$, while $x'y'$ covers the case where $x=0$ and $y=0$. For Expression III, $XOR(x',y)$, we apply the standard XOR identity, $A \oplus B = A'B + AB'$, by setting $A=x'$ and $B=y$. This results in $(x')'y + x'y'$, which simplifies using the double negation law ($(x')' = x$) to $x.y + x'y'$. Since both Expression I and Expression III simplify to the same standard Boolean equation for XNOR, they both correctly represent the exclusive NOR operation.

The incorrect option is (1): I and II.
While Expression I is a correct representation of the XNOR function, Expression II is not. Expression II is $XOR(x',y')$. Applying the XOR identity ($A \oplus B = A'B + AB'$), this expands to $(x')'y' + x'(y')'$. Using the double negation law, this simplifies to $x y' + x' y$. This resulting expression, $x y' + x' y$, is the standard definition for the **Exclusive OR (XOR)** function, not the Exclusive NOR. Since XNOR is the logical complement of XOR, including Expression II makes the claim that both represent XNOR false.

The incorrect option is (3): II and III.
Expression III is correct because it logically simplifies to the XNOR function, $x.y + x'y'$. However, Expression II is incorrect. As demonstrated, $XOR(x',y')$ simplifies to $x y' + x' y$, which is the Boolean expression for the XOR operation. The question asks for expressions that represent the *exclusive NOR* operation. Since Expression II represents the XOR operation, it is logically inconsistent to include it in a set intended to define the XNOR operation.

The incorrect option is (4): I, II and III.
This option incorrectly asserts that all three expressions represent the XNOR function. Expressions I and III are confirmed to be equivalent to $x.y + x'y'$, the XNOR operation. Conversely, Expression II, which is $XOR(x',y')$, is logically equivalent to the XOR operation, $x \oplus y$. In Boolean algebra, XNOR and XOR are complements, meaning they are distinct logical functions. Therefore, a set that correctly represents XNOR cannot simultaneously include the expression for XOR, making this option fundamentally flawed.