### 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.

\begin{align*}p = \text{the weather is nice} && q = \text{I'll wash the car}\end{align*}

Now the statement is: If \begin{align*}p\end{align*}, then \begin{align*}q\end{align*}, which can also be written as \begin{align*}p \rightarrow q\end{align*}.

We can also make the negations, or “nots,” of \begin{align*}p\end{align*} and \begin{align*}q\end{align*}. The symbolic version of "not \begin{align*}p\end{align*}" is \begin{align*}\sim p\end{align*}.

\begin{align*}\sim p = \text{the weather is not nice} && \sim q = \text{I won't wash the car}\end{align*}

Using these “nots” and switching the order of \begin{align*}p\end{align*} and \begin{align*}q\end{align*}, we can create three new statements.

\begin{align*}& \text{Converse} && q \rightarrow p && \underbrace{\text{If I wash the car}}_{q}, \ \underbrace{\text{then the weather is nice}}_{p}.\\ & \text{Inverse} && \sim p \rightarrow \sim q && \underbrace{\text{If the weather is not nice}}_{\sim p}, \ \underbrace{\text{then I won't wash the car}}_{\sim q}.\\ & \text{Contrapositive} && \sim q \rightarrow \sim p && \underbrace{\text{If I don't wash the car}}_{\sim q}, \ \underbrace{\text{then the weather is not nice}}_{\sim p}.\end{align*}

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 \begin{align*}p \rightarrow q\end{align*} is true and \begin{align*}q \rightarrow p\end{align*} is true, then \begin{align*}p \leftrightarrow q\end{align*} (said “\begin{align*}p\end{align*} if and only if \begin{align*}q\end{align*}”).

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 \begin{align*}n > 2\end{align*}, then \begin{align*}n^2 > 4\end{align*}.

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.

\begin{align*}& \underline{\text{Converse}}: && \text{If} \ n^2 > 4, \ \text{then} \ n > 2. \\ &&& False. \ \text{If} \ n^2 = 9, n = -3 \ \text{or} \ 3. \ (-3)^2=9\\ & \underline{\text{Inverse}}: && \text{If} \ n \le 2,\ \text{then} \ n^2 \le 4. \\ &&& False. \ \text{If} \ n=-3 , \ \text{then} \ n^2=9.\\ & \underline{\text{Contrapositive}}: && \text{If} \ n^2 \le 4, \ \text{then} \ n \le 2. \\ &&& True. \ \text{The only} \ n^2 \le 4 \ \text{is 1 or 4}. \ \sqrt{1}=\pm 1 \ \text{and}\sqrt{4}=\pm 2, \ \text{which are both less than or equal to 2.} \end{align*}

#### 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.

\begin{align*}& \underline{\text{Converse}}: && \text{If I am in California, then I am at Disneyland.}\\ &&& False. \ \text{I could be in San Francisco.}\\ & \underline{\text{Inverse}}: && \text{If I am not at Disneyland, then I am not in California.}\\ &&& False. \ \text{Again, I could be in San Francisco.}\\ & \underline{\text{Contrapositive}}: && \text{If I am not in California, then I am not at Disneyland.}\\ &&& True. \ \text{If I am not in the state, I couldn't be at Disneyland.}\end{align*}

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.*

\begin{align*}& \underline{\text{Converse}}: && \text{If two points are collinear, then they are on the same line.} \ True.\\ & \underline{\text{Inverse}}: && \text{If two points are not on the same line, then they are not collinear.} \ True.\\ & \underline{\text{Contrapositive}}: && \text{If two points are not collinear, then they do not lie on the same line}. \ True.\end{align*}

#### Example 5

The following is a true statement:

\begin{align*}m\angle ABC > 90^\circ\end{align*} if and only if \begin{align*}\angle ABC\end{align*} is an obtuse angle.

Determine the two true statements within this biconditional.

*Statement 1:* If \begin{align*}m \angle ABC > 90^\circ\end{align*}, then \begin{align*}\angle ABC\end{align*} is an obtuse angle.

*Statement 2:* If \begin{align*}\angle ABC\end{align*} is an obtuse angle, then \begin{align*}m\angle ABC > 90^\circ\end{align*}.

### Review

For questions 1-4, use the statement:

If \begin{align*}AB = 5\end{align*} and \begin{align*}BC = 5\end{align*}, then \begin{align*}B\end{align*} is the midpoint of \begin{align*}\overline{AC}\end{align*}.

- 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 \begin{align*}90^\circ\end{align*}.
- If you are at the beach, then you are sun burnt.
- If \begin{align*}x > 4\end{align*}, then \begin{align*}x+3>7\end{align*}.

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.
- \begin{align*}2x = 18\end{align*} if and only if \begin{align*}x = 9\end{align*}.

### Review (Answers)

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