Discussion
In Boolean algebra, the expression \( A + \bar{A} = 1 \) holds for any variable A.
Answer: A
Correct answer: True
The statement is true. This is the complement law of Boolean algebra: a variable ORed with its own complement always gives 1. The reasoning is exhaustive. If \( A = 0 \) then \( \bar{A} = 1 \), and \( 0 + 1 = 1 \). If \( A = 1 \) then \( \bar{A} = 0 \), and \( 1 + 0 = 1 \). No third case exists in a two-valued algebra, so the result is 1 whatever A may be. Its dual is equally useful: \( A \cdot \bar{A} = 0 \), since a variable and its complement can never both be 1. The complement laws are among the basic identities used to simplify expressions, alongside the identity laws \( A + 0 = A \) and \( A \cdot 1 = A \), the null laws \( A + 1 = 1 \) and \( A \cdot 0 = 0 \), and the idempotent laws \( A + A = A \) and \( A \cdot A = A \). Note that in Boolean algebra + means OR, not arithmetic addition, which is why \( 1 + 1 = 1 \) rather than 2.
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