The complement of complement of a variable is always _______.

Question

Accepted Answer

The correct option is (3): Equal to variable.

The correct answer is determined by the Double Complementation Law, also known as the Law of Involution, which is a fundamental axiom in Boolean Algebra. This law states that performing the complement operation twice on a Boolean variable always returns the original value of that variable. Let's represent the variable as A. The complement of A is A', and the complement of the complement is (A')', or A''. According to the law, A'' = A. For instance, if A = 0, its complement A' = 1, and the complement of 1, A'', is 0, which is equal to the original variable A. If A = 1, its complement A' = 0, and the complement of 0, A'', is 1, which is also equal to A. Thus, two successive inversions cancel each other out, restoring the original logical state.

The incorrect option is (1): 0. The result is not always 0. The double complement of a variable, A'', is equal to the variable itself, A. If the original variable A is 1, then A'' must also be 1, not 0. This option would only be true in the specific case where the original variable A happens to have the value 0. Boolean laws must hold true for all possible logical values of the variable.

The incorrect option is (2): 1. The result of the complement of complement operation is not universally 1. If the original variable A has the value 0, then applying the complement operation twice will yield 0, not 1, as per the law A'' = A. This option incorrectly attempts to apply a specific constant as the result of a general law. The result is only 1 when the initial value of the variable A is 1.

The incorrect option is (4): The inverse of the variable. The term "inverse of the variable" is a common synonym for the "complement of the variable" (A'). The question asks for the double complement (A''). Based on the Double Complementation Law, the result is the original variable (A), not its single complement (A'). The double complement operation is equivalent to having no complement operation at all, which is conceptually the opposite of returning the inverse.