Question

Which of the following are equivalent statements to the implication $p\to q$.

• A. $\sim q\to \sim p$
• B. $p$ only if $q$
• C. $p$ is a sufficient condition for $q$
• D. $q$ is a necessary condition for $p$

Answer 1

Answer: (A), (B), (C), (D)

$q$ is a necessary condition for $p$

Let's analyze the question by looking at the truth table of the implication.

From the truth table its clear that $p\to q$ is equivalent to $\sim q\to \sim p$.

Let's consider option (d). s1: "q is a necessary condition for p” which means for an implication of truth value T, q has to be necessarily 'T' for the value of p to be 'T'. And if it's not so implication assumes value 'F'.

In option (c) s2: "p is a sufficient condition for q”. We can check this by seeing that for implications a truth value of T for p is ensuring a truth value T for q also. If its ensured truth value of implication is T or else F.

Option (b) is only a rephrased version of option (d).