The Boolean function x'y' + xy + x'y is equivalent to

Concept

A sum-of-products Boolean expression can be reduced using three standard identities: the complement law A + A' = 1, the identity law 1·A = A, and the absorption identity A + A'B = A + B (derivable from the distributive form (A + A')(A + B) = A + B). Grouping terms that share a common literal lets a longer expression collapse step by step into an equivalent, simpler form.

Application

1. Group the two terms sharing the factor x': x'y' + x'y = x'(y' + y).

2. Apply the complement law y' + y = 1, so x'(y' + y) = x'·1 = x'.

3. The original expression x'y' + xy + x'y therefore reduces to x' + xy.

4. Apply the absorption identity A + A'B = A + B with A = x', B = y: x' + xy = x' + y.

Cross-check

An independent truth-table check confirms the reduction holds for every input combination:

Both columns match on all four rows, confirming the simplification.

Result: x'y' + xy + x'y is equivalent to x' + y.