In Boolean algebra, A · (A + B) is logically equivalent to:
2023
In Boolean algebra, A · (A + B) is logically equivalent to:
Answer: D. A — Concept: The Absorption Law of Boolean algebra states that for any two variables X and Y, X · (X + Y) = X. It follows from the distributive, idempotent, and…
- A.
B
- B.
A · B
- C.
A + B
- D.
A
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Correct answer: D
Concept: The Absorption Law of Boolean algebra states that for any two variables X and Y, X · (X + Y) = X. It follows from the distributive, idempotent, and identity laws.
Application: Apply these identities to A · (A + B) step by step.
Distributive law: A · (A + B) = A · A + A · B.
Idempotent law: A · A = A, so the expression becomes A + A · B.
Distributive law (factoring A): A + A · B = A · (1 + B).
Annulment law: 1 + B = 1 for any B, so A · (1 + B) = A · 1.
Identity law: A · 1 = A.
Cross-check: Verify with a truth table over all values of A and B. A = 0, B = 0: A + B = 0, so A · (A + B) = 0, which equals A. A = 0, B = 1: A + B = 1, so A · (A + B) = 0, which equals A. A = 1, B = 0: A + B = 1, so A · (A + B) = 1, which equals A. A = 1, B = 1: A + B = 1, so A · (A + B) = 1, which equals A. In every row the expression equals A.
Result: A · (A + B) = A.
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