Question

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

Short answer

The inverse of its inverse is\(p \to q\)

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

The inverse of its contrapositive is\(q \to p \)

Step-by-step solution

Introduction

Given conditional statement is\(p \to q\)

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

Inverse of inverse

The inverse of its inverse is\(\neg \left( {\neg p} \right) \to \neg \left( {\neg q} \right)\)i.e. \(p \to q\)

Inverse of converse

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

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

Inverse of contrapositive

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

Therefore the inverse of its contrapositive is\(\neg \left( {\neg q} \right) \to \neg \left( {\neg p } \right)\)i.e. \(q \to p \)