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Converse, Inverse, and Contrapositive

Conditional statements drawn from an if-then statement.

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Converse, Inverse, and Contrapositive

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*}.

1. Is this a true statement? If not, what is a counterexample?
2. Find the converse of this statement. Is it true?
3. Find the inverse of this statement. Is it true?
4. 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.

1. An acute angle is less than \begin{align*}90^\circ\end{align*}.
2. If you are at the beach, then you are sun burnt.
3. 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.

1. A U.S. citizen can vote if and only if he or she is 18 or more years old.
2. A whole number is prime if and only if its factors are 1 and itself.
3. \begin{align*}2x = 18\end{align*} if and only if \begin{align*}x = 9\end{align*}.

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

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Vocabulary Language: English

TermDefinition
biconditional statement A statement is biconditional if the original conditional statement and the converse statement are both true.
Conditional Statement A conditional statement (or 'if-then' statement) is a statement with a hypothesis followed by a conclusion.
contrapositive If a conditional statement is $p \rightarrow q$ (if $p$ then $q$), then the contrapositive is $\sim q \rightarrow \sim p$ (if not $q$ then not $p$).
converse If a conditional statement is $p \rightarrow q$ (if $p$, then $q$), then the converse is $q \rightarrow p$ (if $q$, then $p$. Note that the converse of a statement is not true just because the original statement is true.
inverse If a conditional statement is $p \rightarrow q$, then the inverse is $\sim p \rightarrow \sim q$.
Logically Equivalent A statement is logically equivalent if the "if-then" statement and the contrapositive statement are both true.
premise A premise is a starting statement that you use to make logical conclusions.