### Converse, Inverse, and Contrapositive

Consider the statement: *If the weather is nice, then I’ll wash the car.* We can rewrite this statement using letters to represent the hypothesis and conclusion.

Now the statement is: If , then , which can also be written as .

We can also make the negations, or “nots,” of and . The symbolic version of "not " is .

Using these “nots” and switching the order of and , we can create three new statements.

If the “if-then” statement is true, then the contrapositive is also true. The contrapositive is ** logically equivalent** to the original statement. The converse and inverse may or may not be true. When the original statement and converse are both true then the statement is a

**biconditional statement**. In other words, if is true and is true, then (said “ if and only if ”).

What if you were given a conditional statement like "If I walk to school, then I will be late"? How could you rearrange and/or negate this statement to form new conditional statements?

### Examples

#### Example 1

If , then .

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.

The original statement is true.

#### Example 2

If I am at Disneyland, then I am in California.

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.

The original statement is true.

Notice for the converse and inverse we can use the same counterexample.

#### Example 3

Rewrite as a biconditional statement: Any two points are collinear.

This statement can be rewritten as:

*Two points are on the same line if and only if they are collinear.* Replace the “if-then” with “if and only if” in the middle of the statement.

#### Example 4

Any two points are collinear.

Find the converse, inverse, and contrapositive. Determine if each resulting statement is true or false. If it is false, find a counterexample.

First, change the statement into an “if-then” statement:

*If two points are on the same line, then they are collinear.*

#### Example 5

The following is a true statement:

if and only if is an obtuse angle.

Determine the two true statements within this biconditional.

*Statement 1:* If , then is an obtuse angle.

*Statement 2:* If is an obtuse angle, then .

### Review

For questions 1-4, use the statement:

If and , then is the midpoint of .

- Is this a true statement? If not, what is a counterexample?
- Find the converse of this statement. Is it true?
- Find the inverse of this statement. Is it true?
- Find the contrapositive of this statement. Which statement is it the same as?

Find the converse of each true if-then statement. If the converse is true, write the biconditional statement.

- An acute angle is less than .
- If you are at the beach, then you are sun burnt.
- If , then .

For questions 8-10, determine the two true conditional statements from the given biconditional statements.

- A U.S. citizen can vote if and only if he or she is 18 or more years old.
- A whole number is prime if and only if its factors are 1 and itself.
- if and only if .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.4.