# Inverses, converses, and contrapositives

##### Intros

###### Lessons

##### Examples

###### Lessons

**Finding the inverse, converse and contrapositive**

Given the statements, write the inverse, converse, and contrapositive:- For each statement, write the inverse, converse, and contrapositive in symbolic form:
**Truth value of inverse, converse and contrapositive**

Write the converse, and find the truth value of the converse:- Assume that the conditional statement is true. Write the inverse and state whether the inverse is always true, sometimes true, or never true:
**Logical Equivalents**Find the truth value of the following conditionals. Then write the contrapositive and find the truth value of the contrapositive. Are the truth values the same?

**Finding truth values of original statements**Find possible truth values for $p$ and $q$ where:

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###### Topic Notes

Let $p$ be the hypothesis and $q$ be the conclusion. Then:

An **inverse** statement is formed by negating both the hypothesis and conclusion of the conditional. In symbolic form it would be:

~$p$ $\to$ ~$q$

A **converse** statement is formed by switching the hypothesis and the conclusion of the conditional. In symbolic form it would be:

$q\ \to p$

A **contrapositive** statement is formed by negating both the hypothesis and conclusion, AND switching them. In symbolic form it would be:

~$q$ $\to$ ~$p$

Statements which always have the same truth values are **logical equivalents**.

Conditionals and contrapositives are logical equivalents, and inverses and converses are logical equivalents.

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