Question

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

Answer 1

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).