Use truth tables to show that the following statements are logically equivalent. ∼ P ⇔ Q = (P ⇒∼ Q)∧(∼ Q ⇒ P)
Accepted Solution
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Answer: The given logical equivalence is proved below.Step-by-step explanation: We are given to use truth tables to show the following logical equivalence :∼ P ⇔ Q ≡ (P ⇒∼ Q)∧(∼ Q ⇒ P)We know thattwo compound propositions are said to be logically equivalent if they have same corresponding truth values in the truth table.The truth table is as follows :P Q ∼ P ∼Q ∼ P⇔ Q P ⇒∼ Q ∼ Q ⇒ P (P ⇒∼ Q)∧(∼ Q ⇒ P)T T F F F F T FT F F T T T T TF T T F T T T TF F T T F T F FSince the corresponding truth vales for ∼ P ⇔ Q and (P ⇒∼ Q)∧(∼ Q ⇒ P) are same, so the given propositions are logically equivalent.Thus, ∼ P ⇔ Q ≡ (P ⇒∼ Q)∧(∼ Q ⇒ P).