According to De Morgan's Law, which expression correctly equals (A · B)′?
2025
According to De Morgan's Law, which expression correctly equals (A · B)′?
Answer: B. (A · B)′ = A′ + B′ — De Morgan's Laws describe how negation distributes across AND and OR in Boolean algebra. For any two variables X and Y: the complement of a product (AND)…
- A.
(A + B)′ = A′ · B′ - B.
(A · B)′ = A′ + B′ - C.
(A · B)′ = A′ + B - D.
(A · B)′ = A + B′
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Correct answer: B
De Morgan's Laws describe how negation distributes across AND and OR in Boolean algebra. For any two variables X and Y: the complement of a product (AND) equals the sum (OR) of the individual complements — (X · Y)′ = X′ + Y′ — and the complement of a sum (OR) equals the product (AND) of the individual complements — (X + Y)′ = X′ · Y′. In each law, the operator switches (AND becomes OR, or OR becomes AND) as the complement moves inside.
Applying the product form to A and B, and checking it against every input combination:
A | B | A · B | (A · B)′ | A′ | B′ | A′ + B′ |
|---|---|---|---|---|---|---|
0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 | 1 | 1 |
1 | 1 | 1 | 0 | 0 | 0 | 0 |
In every row of the table, (A · B)′ matches A′ + B′, confirming the identity (A · B)′ = A′ + B′ for the complement of a product.
The identity (A + B)′ = A′ · B′ is also a true Boolean law, but it is the identity for the complement of a sum, not a product — notice it swaps AND for OR on the other side of the equation. Because this question specifically asks for the complement of the product term A · B, only the product-form identity answers it; the sum-form identity is a different, equally valid, but separate law.
So the complement of the product (A · B) is given by the sum of the individual complements: (A · B)′ = A′ + B′.