Simplify the Boolean expression (X + Y + XY)(X + Z) to its minimal sum form using absorption and distributive properties.

Introduction / Context:Boolean simplification reduces logic cost, delay, and power. This problem reinforces core identities such as absorption and distributivity to reach a minimal or near-minimal expression suitable for implementation.

Given Data / Assumptions:

• Expression: (X + Y + XY)(X + Z).
• Variables X, Y, Z are Boolean (0/1).
• Goal: simplify to a compact sum-of-terms form.

Concept / Approach:Use absorption: A + AB = A. Therefore, Y + XY = Y. Then use the distributive identity: (X + A)(X + B) = X + AB. Apply these stepwise to collapse the expression.

Step-by-Step Solution:

Verification / Alternative check:Construct a quick truth check: when X=1, result is 1; when X=0, result reduces to YZ. This matches the product form behavior, confirming the simplification.

Why Other Options Are Wrong:

• XZ + Y: does not match cases where X=0 and Y=1, Z=0 (gives 1 instead of YZ=0).
• X + Y + Z: over-approximates (too many 1s).
• XY + YZ: misses cases where X alone should drive 1.
• None of the above: incorrect because X + YZ is correct.

Common Pitfalls:Forgetting absorption (Y + XY = Y), or expanding fully then miscombining minterms leading to longer forms.

Final Answer:X + YZ