Involution (double inversion) rule in Boolean algebra: Does inverting a Boolean variable twice return it to its original value (i.e., (A’’) = A)?

Introduction / Context:
The involution law (double inversion) is one of the simplest and most used identities in Boolean algebra and logic circuit design. It states that if you complement a Boolean variable twice, you get back the original variable. This is fundamental for simplifying expressions and for understanding how active-low signals propagate through cascaded inverters or NAND/NOR structures used as inverters.

Given Data / Assumptions:

• Variables take values in {0, 1}.
• Complement operator (’ or overbar) flips 0 to 1 and 1 to 0.
• Ideal, time-independent Boolean evaluation (no timing hazards).

Concept / Approach:
If A ∈ {0, 1}, then A’ = 1 − A, and applying complement again yields (A’)’ = 1 − (1 − A) = A. Hence (A’’) = A. At the gate level, two inverters in series reproduce the original logic level, a useful trick for buffering or delay but not for changing logic function. This law is frequently used in algebraic reductions and in transforming complements through expressions using De Morgan’s theorems.

Step-by-Step Solution:

Verification / Alternative check:
Truth table check: if A = 0 ⇒ A’ = 1 ⇒ (A’)’ = 0; if A = 1 ⇒ A’ = 0 ⇒ (A’)’ = 1. In both cases, (A’’) = A.

Why Other Options Are Wrong:
It does not depend on canonical forms, input being 1, or propagation delay (the identity is algebraic; delays are implementation concerns).

Common Pitfalls:
Confusing the double negation identity with removing pairs of inversions incorrectly when parentheses or operator precedence matter; always track complements carefully across grouped expressions.

Final Answer:
Correct