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18 May, 00:19

Use truth tables to show that the following statements are logically equivalent. ∼ P ⇔ Q = (P ⇒∼ Q) ∧ (∼ Q ⇒ P)

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  1. 18 May, 00:33
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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 that

    two 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 F

    T F F T T T T T

    F T T F T T T T

    F F T T F T F F

    Since 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).
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