In Boolean algebra, the expression \(AB + \overline{(A + B)}\) is equivalent to:
2010
In Boolean algebra, the expression \(AB + \overline{(A + B)}\) is equivalent to:
Answer: B. \(A \odot B\) — ConceptDe Morgan’s law gives the complement of a sum as the product of the complements: \(\overline{A+B}=\bar A\bar B\). XNOR is 1 when its two inputs are…
- A.
\(A ⊕ B\)
- B.
\(A \odot B\)
- C.
\((A ⊕ B) \odot A\)
- D.
\((A \odot B) ⊕ B\)
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Correct answer: B
Concept
De Morgan’s law gives the complement of a sum as the product of the complements: \(\overline{A+B}=\bar A\bar B\).
XNOR is 1 when its two inputs are equal. Its sum-of-products form is \(AB+\bar A\bar B\).
Application
Apply De Morgan’s law to the complemented term: \(\overline{(A+B)}=\bar A\bar B\).
Substitute this form into the expression: \(AB+\overline{(A+B)}=AB+\bar A\bar B\).
Recognize the standard XNOR identity: \(AB+\bar A\bar B=A\odot B\).
Cross-check
The truth table shows that both expressions are 1 for equal inputs and 0 for unequal inputs.
A | B | Original expression | \(A\odot B\) |
|---|---|---|---|
0 | 0 | 1 | 1 |
0 | 1 | 0 | 0 |
1 | 0 | 0 | 0 |
1 | 1 | 1 | 1 |
Therefore, the equivalent expression is \(A\odot B\).