In propositional language π β π is equivalent to (where βΌ denotes NOT) :
2015
In propositional languageΒ π β πΒ is equivalent to (whereΒ βΌΒ denotes NOT) :
- A.
βΌ(πβ¨π)β§βΌ(πβ¨π)
- B.
(βΌπβ¨π)β§(βΌπβ¨π)
- C.
(πβ¨π)β§(πβ¨π)
- D.
βΌ(πβ¨π)ββΌ(πβ¨π)
Attempted by 181 students.
Show answer & explanation
Correct answer: B
Derivation: Start from the biconditional and rewrite implications.
PβQ β‘ (PβQ) β§ (QβP).
Replace each implication using PβQ β‘ Β¬P β¨ Q and QβP β‘ Β¬Q β¨ P to get:
(Β¬P β¨ Q) β§ (Β¬Q β¨ P)
Therefore the biconditional is equivalent to (Β¬P β¨ Q) β§ (Β¬Q β¨ P).
Why the other expressions are not equivalent:
The expression Β¬(Pβ¨Q) β§ Β¬(Qβ¨P) simplifies to Β¬P β§ Β¬Q, which is true only when both P and Q are false.
The expression (Pβ¨Q) β§ (Qβ¨P) simplifies to Pβ¨Q, which is true when at least one of P or Q is true; it does not require P and Q to have the same truth value.
The implication Β¬(Pβ¨Q) β Β¬(Qβ¨P) has identical antecedent and consequent, so it is always true (a tautology); the biconditional is not a tautology.