Question

Given a conditional statement\(p \to q\), find the converse of its inverse, the converse of its converse, and the converse of its contrapositive.

Short answer

The converse of inverse of\(p \to q\)is\(\neg q \to \neg p\).

The converse of converse of \(p \to q\)is\(p \to q\).

The converse of contrapositive of \(p \to q\)is\(\neg p \to \neg q\).

Step-by-step solution

Introduction

If \(p and q\)represent two propositions, and represents their conditional statement then their converse, contrapositive and inverse are as follows, respectively:

• Converse is\(q \to p\)
• Contrapositive is\(\neg q \to \neg p\)
• Inverse is\(\neg p \to \neg q\)

Converse of inverse

Consider the conditional statement\(p \to q\).

The inverse of\(p \to q\)is\(\neg p \to \neg q\).

Therefore, the converse of inverse of\(p \to q\)is\(\neg q \to \neg p\).

Converse of converse

Consider the conditional statement\(p \to q\).

The converse of\(p \to q\)is\(q \to p\).

Therefore, the converse of converse of \(p \to q\)is\(p \to q\).

Converse of contrapositive

Consider the conditional statement\(p \to q\).

The contrapositive of\(p \to q\)is\(\neg q \to \neg p\)

Therefore, the converse of contrapositive of \(p \to q\)is\(\neg p \to \neg q\)