# Inverses, converses, and contrapositives

- Intro Lesson: a8:26
- Intro Lesson: b5:01
- Intro Lesson: c6:47
- Intro Lesson: d5:17
- Lesson: 1a6:45
- Lesson: 1b4:47
- Lesson: 1c3:17
- Lesson: 1d4:10
- Lesson: 2a2:50
- Lesson: 2b5:27
- Lesson: 2c3:58
- Lesson: 2d4:30
- Lesson: 3a5:57
- Lesson: 3b3:02
- Lesson: 3c2:20
- Lesson: 4a8:11
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- Lesson: 5a7:12
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- Lesson: 5c4:44
- Lesson: 6a5:47
- Lesson: 6b5:13
- Lesson: 6c3:26
- Lesson: 6d6:03

### Inverses, converses, and contrapositives

#### Lessons

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.

- IntroductionThe inverse, converse, and contrapositive Overview:a)Inverse Statementsb)Converse Statementsc)Contrapositive Statementsd)Logical equivalents
- 1.
**Finding the inverse, converse and contrapositive**

Given the statements, write the inverse, converse, and contrapositive:a)Two intersecting lines create an angle.b)If today is Monday, then Kevin will play soccer.c)If $1+2=3$, then $1^2+2^2=3^2$.d)If the polygon is a triangle, then it has 3 sides. - 2.For each statement, write the inverse, converse, and contrapositive in symbolic form:a)$r \to s$b)$p$ $to$ ~$q$c)~$m$ $\to$ $n$d)~$u$ $\to$ ~$v$
- 3.
**Truth value of inverse, converse and contrapositive**

Write the converse, and find the truth value of the converse:a)If $x+7=13$, then $x=6$.b)If $3$ is odd, then $3+1$ is even.c)If $2$ is an integer, then $2$ is a whole number. - 4.Assume that the conditional statement is true. Write the inverse and state whether the inverse is always true, sometimes true, or never true:a)$p \to q$b)$q\ to p$c)~$q$ $\to$ ~$p$d)$p$ $\to$ ~$q$
- 5.
**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?

a)If (2+3)×4=20, then $2+3=5$.b)If $4$ is even, then $4+2$ is even.c)If $3$ is an integer, then $3$ is not a whole number. - 6.
**Finding truth values of original statements**Find possible truth values for $p$ and $q$ where:

a)$p \to q$ and ~$q$ $\to$ ~$p$ is both false?b)$p$ $\to$ $q$ and $q$ $\to$ $p$ is both true?c)$p \to q$ and $q \to p$ is both false?d)$q \to p$ and ~$q$ $\to$ ~$p$ is both true?