GATE CS 2024 Set 2 Q20 — Boolean Algebra

We need to identify which of the given Boolean algebraic statements are FALSE for a Boolean variable $x \in \{0, 1\}$.

Let's evaluate each option:

• Option A: $x \cdot 1 = x$
  This is the Identity Law.
  • If $x = 0$, then $0 \cdot 1 = 0$.
  • If $x = 1$, then $1 \cdot 1 = 1$.
  Thus, this statement is TRUE.


• Option B: $x + 1 = x$
  According to the Annulment Law, $x + 1 = 1$.
  • If $x = 0$, then $0 + 1 = 1 \neq x$.
  Thus, this statement is FALSE.


• Option C: $x \cdot x = 0$
  According to the Idempotent Law, $x \cdot x = x$.
  • If $x = 1$, then $1 \cdot 1 = 1 \neq 0$.
  Thus, this statement is FALSE.


• Option D: $x + \bar{x} = 1$
  This is the Complementarity Law.
  • If $x = 0$, then $0 + \bar{0} = 0 + 1 = 1$.
  • If $x = 1$, then $1 + \bar{1} = 1 + 0 = 1$.
  Thus, this statement is TRUE.

Since the question asks for the FALSE statements, the correct options are B and C.