Question

Which of the following statement are NOT logically equivalent?

• A. $\sim (p\vee \sim q)$ and $(\sim p\wedge q)$
• B. $\sim (p\to q)$ and $(p\wedge \sim q)$
• C. $(p\to q)$ and $(\sim q\to \sim p)$
• D. $(p\to q)$ and $(\sim p\wedge q)$

Answer 1

Answer: (D)

$(p\to q)$ and $(\sim p\wedge q)$

We make an option wise check for this.

Option A: $\sim (p\vee \sim q)and(\sim p\wedge q)$

By application of Demorgan's Law on $\sim (p\vee \sim q)$ we get, $\sim (p\wedge q)$

So this option is logically equivalent.

Option B: $\sim (p⟶q)and(p\wedge \sim q)$

Again by application Conditional Disjunction rule, we see that this option is also logically equivalent.

Option C: $(p⟶q)and(\sim q⟶\sim p)$

This is again true by Contrapositive tautology.

Option D:$(p⟶q)and(\sim p\wedge q)$

This is not logically equivalent.