Which of the following is NOT a correct identity in Boolean Algebra?
2018
Which of the following is NOT a correct identity in Boolean Algebra?
Answer: B. A(A+B) = B — Concept In Boolean Algebra, four core laws govern simplification: the Absorption Law (X + XY = X), the Distributive Law (X(Y+Z) = XY + XZ), the Idempotent Law…
- A.
A + AB = A
- B.
A(A+B) = B - C.
AB + AB' = A
- D.
(A+B)(A+B') = A
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Correct answer: B
Concept
In Boolean Algebra, four core laws govern simplification: the Absorption Law (X + XY = X), the Distributive Law (X(Y+Z) = XY + XZ), the Idempotent Law (XX = X), and Complementation (X + X' = 1, XX' = 0). An identity is correct only when both sides evaluate to the same expression for every value of the variables.
Applying it to this identity
Take the expression A(A+B) and apply the Distributive Law: A(A+B) = AA + AB.
Apply the Idempotent Law: AA = A, so the expression becomes A + AB.
Apply the Absorption Law to A + AB: this simplifies to A.
So A(A+B) always equals A, not B — the claimed equation A(A+B) = B does not hold in general, making it the incorrect identity.
Cross-check the other three identities
A + AB = A holds directly by the Absorption Law.
AB + AB' = A(B + B') = A · 1 = A, using the Distributive Law and Complementation.
(A+B)(A+B') expands, via the Distributive Law, to AA + AB' + AB + BB' = A + AB' + AB + 0 = A(1 + B' + B) = A, so it also holds.
Only one equation fails to hold — A(A+B) = B — because the left side always simplifies to A, not B. That is the identity that is not correct.