The Boolean expression X + X'Y equals

Concept: for any two Boolean variables A and B, the absorption identity A + A'B = A + B always holds. Proof: by the distributive law, A + A'B = (A + A')(A + B); since a variable ORed with its own complement is always 1 (A + A' = 1), this reduces to 1 times (A + B), i.e. A + B.

1. Match the given expression to the identity: take the first term as A and the variable multiplying the complement as B.

2. Given expression X + X'Y has A = X and B = Y.

3. Apply A + A'B = A + B with these substitutions: X + X'Y = X + Y.

Cross-check (independent derivation via the distributive law):

X + X'Y = (X + X')(X + Y) [distributive law] = 1 times (X + Y) [since X + X' = 1] = X + Y -- the same result, confirming the simplification.

Why the other forms don't match:

• Y + XY is not derived from this expression's complemented-variable structure at all -- by the absorption law A + AB = A, Y + XY reduces further to just Y.

• X + XY looks similar but applies the same absorption law A + AB = A to an uncomplemented variable, giving X -- a different case from a term multiplied by a complement.

• X'Y + Y'X is the definition of the exclusive-OR (XOR) function, an entirely different Boolean operation from the OR-based simplification asked here.