Which of the following is not a valid expression of Boolean algebra ?

Question

Accepted Answer

The correct option is (2):
The expression $x \cdot \overline{y} + \overline{x} \cdot y + \overline{x} \cdot \overline{y} = 1$ is not a valid or universally true identity in Boolean algebra. We can simplify the Left Hand Side (LHS) to prove this. By factoring out $\overline{x}$ from the last two terms, we get $x \cdot \overline{y} + \overline{x} \cdot (y + \overline{y})$. Since the Complementary Law states that $y + \overline{y} = 1$, the expression simplifies to $x \cdot \overline{y} + \overline{x} \cdot 1$, or simply $x \cdot \overline{y} + \overline{x}$. Applying the Distributive and Complementary Laws again, this further simplifies to $\overline{x} + \overline{y}$. Since $\overline{x} + \overline{y}$ is not always equal to 1 (it is 0 when $x=1$ and $y=1$), this equality is false for that specific case. Therefore, it is the expression that is *not* a valid Boolean algebra identity.

The incorrect option is (1):
The expression $\overline{x} + (y \cdot \overline{z}) + x + \overline{y} = 1$ is a valid Boolean identity due to the property of the Dominance Law. The LHS can be regrouped using the Commutative and Associative Laws as $(\overline{x} + x) + \overline{y} + (y \cdot \overline{z})$. According to the Complementary Law, $\overline{x} + x$ is always equal to 1. Once this term is reduced to 1, the entire expression becomes $1 + \overline{y} + (y \cdot \overline{z})$. Since any OR operation with 1 always results in 1 (Dominance Law: $1 + A = 1$), the LHS is universally true and equals 1.

The incorrect option is (3):
The expression $\overline{(x + y + z)} = \overline{x} \cdot \overline{y} \cdot \overline{z}$ is a direct application of De Morgan's Law. De Morgan's Law defines how the complement (NOT operator) distributes over the OR and AND operators. Specifically, it states that the complement of a sum (OR operation) is the product (AND operation) of the complements. This rule is a fundamental theorem in Boolean algebra and holds true for any number of variables, including $x$, $y$, and $z$.

The incorrect option is (4):
The expression $x \cdot \overline{y} + \overline{x} \cdot y + x \cdot y = x + y$ is a valid identity that demonstrates the simplification of a minterm expansion. We can simplify the Left Hand Side (LHS) by factoring the term $y$ from the last two components: $LHS = x \cdot \overline{y} + y \cdot (\overline{x} + x)$. Since $\overline{x} + x$ is equal to 1 (Complementary Law), the expression reduces to $x \cdot \overline{y} + y$. Using the Absorption Law variant ($A + \overline{A} B = A + B$), where $A=y$ and $B=x$, we can simplify $y + \overline{y}x$ to $y + x$, which is equivalent to $x + y$.