A logical binary relation □ ,is defined as follows: Let ~ be the unary…
2006
A logical binary relation □ ,is defined as follows:
Let ~ be the unary negation (NOT) operator, with higher precedence than □.
Which one of the following is equivalent to A∧B ?
- A.
(~A □ B)
- B.
~(A □ ~B)
- C.
~(~A □ ~B)
- D.
~(~A □ B)
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Correct answer: D
Key observation: From the truth table, the binary operator □ is true except when A is false and B is true, so for any X and Y we have X □ Y = X ∨ ¬Y.
Therefore, A □ B = A ∨ ¬B.
Compute ~A □ B: ~A □ B = ¬A ∨ ¬B, which equals ¬(A ∧ B) by De Morgan.
Negate that: ~(~A □ B) = ¬(¬A ∨ ¬B) = A ∧ B (by De Morgan). Thus ~(~A □ B) is equivalent to A∧B.
Quick checks of the other candidate expressions show they simplify to different forms: (~A □ B) = ¬(A ∧ B), ~(A □ ~B) = ¬A ∧ ¬B, and ~(~A □ ~B) = A ∧ ¬B.
Conclusion: The expression ~(~A □ B) is equivalent to A∧B.
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