Recovery (clarified stem): Absorption law equivalence — are the Boolean expressions “A + A'B” and “A + B” logically equivalent for all possible values of A and B?
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AEquivalent (Correct)
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BNot equivalent (Incorrect)
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CEquivalent only when A = 0
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DEquivalent only when B = 1
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EEquivalent only if A and B are complements
Answer
Correct Answer: Equivalent (Correct)
Explanation
Introduction / Context:Incomplete original wording has been repaired using the Recovery-First Policy. A classic identity in Boolean algebra is the absorption law, which enables significant simplification of logic by removing redundant terms. One common form is A + A'B = A + B, which this question asks you to validate.
Given Data / Assumptions:
- A and B are Boolean variables taking values 0 or 1.
- Operator “+” denotes OR; juxtaposition or “·” denotes AND; prime (′) denotes NOT.
- We test equivalence across all input combinations.
Concept / Approach:Using Boolean algebra: A + A'B = (A + A') (A + B) by distributive expansion. Since A + A' = 1 (complementarity), the expression reduces to 1 · (A + B) = A + B, proving equivalence. Intuitively, if A is 1, both sides are 1 regardless of B; if A is 0, the left side reduces to A'B = B, and the right side is 0 + B = B — again identical.
Step-by-Step Solution:
Start with A + A'B.Apply distributive law: X + YZ = (X + Y)(X + Z) with X=A, Y=A', Z=B.Compute (A + A') = 1 via complementarity.Reduce: 1 * (A + B) = A + B. Equivalence shown.Verification / Alternative check:Create a 4-row truth table for (A, B) ∈ {(0,0), (0,1), (1,0), (1,1)}. Evaluate both expressions; results match in all rows, confirming equivalence.
Why Other Options Are Wrong:
“Not equivalent”: contradicts algebraic proof and truth table.Conditional statements (only when A=0, B=1, or A= B′): unnecessary constraints; the identity holds universally.Common Pitfalls:Misapplying distributive law or misreading A'B as AB; forgetting that A + A' = 1 and A · A' = 0 are fundamental complementarity relations.
Final Answer:Equivalent (Correct)