# Inverses, converses, and contrapositives

### 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.

• Introduction
The inverse, converse, and contrapositive Overview:
a)
Inverse Statements

b)
Converse Statements

c)
Contrapositive Statements

d)
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?