The minimum number of gates required to implement the Boolean expression AB +…

Concept: Boolean simplification uses three identities here — the complement law A + A' = 1 (and its dual, A·A' = 0), the factoring/distributive law used as A·B + A·B' = A·(B + B'), and the absorption identity A + A'C = A + C (obtained by distributing: A + A'C = (A + A')·(A + C) = 1·(A + C) = A + C).

Application: Reduce AB + AB' + A'C step by step.

1. Factor the first two terms on A: AB + AB' = A(B + B').

2. Apply the complement law B + B' = 1, so A(B + B') = A·1 = A. The expression is now A + A'C.

3. Apply the absorption identity A + A'C = A + C (shown above via the distributive law), leaving the two-literal expression A + C.

A + C uses only the literals A and C in their true (uncomplemented) form combined by a single logical sum, so it can be realized with exactly one 2-input OR gate and nothing else — no AND gate is needed (no product term survives) and no NOT gate is needed (neither surviving literal is complemented).

Cross-check: List all eight input combinations of A, B, C and evaluate the original expression against A + C independently of the algebra above (an exhaustive check, not just a couple of sample rows).

All eight rows match, confirming the simplification exhaustively rather than only algebraically.

Hence the minimum number of gates required is 1 (a single OR gate combining A and C).