Repair and evaluate the idempotent identity: In Boolean algebra, A + A = A (idempotent law for OR). Is this identity valid for all Boolean A?

Introduction / Context:
The original stem was incomplete. Applying the Recovery-First Policy, we minimally repair it to the well-known idempotent identity A + A = A, which expresses that ORing a Boolean variable with itself yields the same variable. Idempotent laws (for both OR and AND) are among the most frequently used rules in algebraic simplification and K-map reasoning.

Given Data / Assumptions:

• A is Boolean (0 or 1).
• + denotes Boolean OR.
• Goal: validate A + A = A for all A.

Concept / Approach:
An operation is idempotent if applying it multiple times does not change the result beyond the first application. For OR, combining A with itself does not introduce any new information beyond A, so the result equals A. There is a dual idempotent law for AND: A * A = A.

Step-by-Step Solution:

Verification / Alternative check:
Truth table enumeration confirms the law instantly. In set-theoretic interpretation (union), X ∪ X = X mirrors the same idempotency, offering further intuition for the identity's correctness.

Why Other Options Are Wrong:

• Incorrect: Contradicted by both truth table and set analogy.
• Ambiguous as stated: After repair, the identity is precise and standard.
• Cannot be determined: No further data is needed to validate an algebraic law.

Common Pitfalls:
Confusing with absorption (A + A*B = A) or domination (A + 1 = 1). While related, idempotency is distinct and should not be interchanged with those identities.

Final Answer:
Correct