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

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).