GATE CS 2024 Set 2 Q.30: Boolean Algebra and Logic Minimization

For a Boolean variable $x$ , which of the following statements is/are FALSE?

$x . 1 = x$

$x + 1 = x$

$x . x = 0$

$x + \bar{x} = 1$

Official answer and solution

Solution

Check each Boolean identity for $x \in \{0, 1\}$ and select the FALSE ones.

Checking each Boolean equation:

(A) $x \cdot 1 = x$ is TRUE (Identity law for AND).
(B) $x + 1 = x$ is FALSE because $x + 1 = 1$ (Annulment/Dominance law for OR; when $x = 0$ , $0 + 1 = 1 \neq 0$ ).
(C) $x \cdot x = 0$ is FALSE because $x \cdot x = x$ (Idempotent law for AND; when $x = 1$ , $1 \cdot 1 = 1 \neq 0$ ).
(D) $x + \bar{x} = 1$ is TRUE (Complement law for OR).

Thus (B) and (C) are FALSE.

Final answer

Options (B), (C)

Common trap

Selecting the TRUE identities (A) and (D) instead of the FALSE ones.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

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