Prove the following expression using Boolean algebra : (X + Y)(X + Y̅)(X̅ + Z)…

Concept

For Boolean variables, the complement law gives AĀ = 0, and the identity law gives A + 0 = A.

The distributive identity (A + B)(A + C) = A + BC lets us simplify two sums that share A before multiplying the remaining factor.

Application

1. Apply the distributive identity to the first two factors: (X + Y)(X + Y̅) = X + YY̅.

2. By the complement law, YY̅ = 0; hence X + YY̅ = X.

3. Substitute this result into the full left-hand side: (X + Y)(X + Y̅)(X̅ + Z) = X(X̅ + Z).

4. Distribute X over the remaining sum: X(X̅ + Z) = XX̅ + XZ.

5. Again using the complement and identity laws, XX̅ + XZ = 0 + XZ = XZ.

Cross-check

• If X = 0, then (X + Y)(X + Y̅) = YY̅ = 0, so both sides are 0.

• If X = 1, then the first two factors are 1 and the left-hand side becomes X̅ + Z = Z, while XZ = Z. Therefore the identity holds for both values of X.