Master

Converse, Inverse, Contrapositive

Master converse, inverse, contrapositive with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Converse, Inverse, Contrapositive

Study Guide

Key Definition

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. We can create three related statements from a conditional statement: the converse, inverse, and contrapositive.

Important Notes

The converse of a statement interchanges the hypothesis and the conclusion.
The inverse of a statement takes the negation of both the hypothesis and the conclusion.
The contrapositive of a statement interchanges the hypothesis and the conclusion of the inverse statement.

Mathematical Notation

$p \rightarrow q$ represents 'If p, then q'

$q \rightarrow p$ represents the converse

$\neg p \rightarrow \neg q$ represents the inverse

$\neg q \rightarrow \neg p$ represents the contrapositive

Remember to use proper notation when solving problems

Why It Works

The truth of a conditional statement and its contrapositive are always the same, as are the truth of the converse and the inverse. This helps in deductive reasoning and proofs.

Remember

If a statement is true, its contrapositive is also true. If the converse is true, the inverse is also true.

Quick Reference

Statement:$p \rightarrow q$

Converse:$q \rightarrow p$

Inverse:$\neg p \rightarrow \neg q$

Contrapositive:$\neg q \rightarrow \neg p$

Understanding Converse, Inverse, Contrapositive

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

We start with a simple 'if-then' statement, known as a conditional statement. From this, we can form the converse, inverse, and contrapositive.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Given the statement $p \rightarrow q$ (If p, then q), which of the following represents its contrapositive?

Real-World Problem

Question Exercise

Intermediate

Cinema Scenario

If a movie is rated PG-13, then the viewer must be at least 13 years old. Formulate the converse, inverse, and contrapositive of this statement.

Advanced Challenge

Thinking Exercise

Advanced

Think About This

If a statement and its contrapositive have different truth values, what can you conclude?

Recap

Watch & Learn

Review key concepts and takeaways