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Converse, Inverse, and Contrapositive

Conditional statements drawn from an if-then statement.

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Converse, Inverse, and Contrapositive

What if your sister told you "if you do the dishes, then I will help you with your homework"? What's a statement that is logically equivalent to what your sister said? 

Converse, Inverse, and Contrapositive 

Consider the statement: If the weather is nice, then I will wash the car. This can be rewritten using letters to represent the hypothesis and conclusion:

\begin{align*}\text{If} \ p, \text{then} \ q && \text{where} && p = \text{the weather is nice and} && q = \text{I will wash the car.} && \text{Or}, p \rightarrow q.\end{align*}If p,then qwherep=the weather is nice andq=I will wash the car.Or,pq.

In addition to these positives, we can also write the negations, or “not”s of \begin{align*}p\end{align*}p and \begin{align*}q\end{align*}q. The symbolic version of not \begin{align*}p\end{align*}p, is \begin{align*}\sim p\end{align*}p.

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

Using these negations and switching the order of \begin{align*}p\end{align*}p and \begin{align*}q\end{align*}q, we can create three more conditional 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 we accept “If the weather is nice, then I’ll wash the car” as true, then the converse and inverse are not necessarily true. However, if we take the original statement to be true, then the contrapositive is also true. We say that the contrapositive is logically equivalent to the original if-then statement. It is sometimes the case that a statement and its converse will both be true. These types of statements are called biconditional statements. So, \begin{align*}p \rightarrow q\end{align*} is true and \begin{align*}q \rightarrow p\end{align*} is true. It is written \begin{align*}p \leftrightarrow q\end{align*}, with a double arrow to indicate that it does not matter if \begin{align*}p\end{align*} or \begin{align*}q\end{align*} is first. It is said, “\begin{align*}p\end{align*} if and only if \begin{align*}q\end{align*}”. Replace the “if-then” with “if and only if” in the middle of the statement. “If and only if” can be abbreviated “iff.”

 

Finding the Converse, Inverse, and Contrapositivive 

1. Use the statement: If \begin{align*}n > 2\end{align*}, then \begin{align*}n^2 > 4\end{align*}.

a) Find the converse, inverse, and contrapositive.

b) Determine if the statements from part a are true or false. If they are false, find a counterexample.

The original statement is true.

\begin{align*}& \underline{\text{Converse}}: && \text{If} \ n^2>4, \ \text{then} \ n > 2. && False. \ n \ \text{could be} \ -3, \ \text{making} \ n^2=9.\\ & \underline{\text{Inverse}}: && \text{If} \ n < 2, \text{then} \ n^2 < 4. && False. \ \text{Again, if} \ n = -3, \ \text{then} \ n^2=9.\\ & \underline{\text{Contrapositive}}: && \text{If} \ n^2<4, \text{then} \ n < 2. && True, \ \text{the only square number less than}\\ & && && \text{4 is 1, which has square roots of 1 or -1, both}\\ & && && \text{less than 2.}\end{align*}

2. Use the statement: If I am at Disneyland, then I am in California.

a) Find the converse, inverse, and contrapositive.

b) Determine if the statements from part a are true or false. If they are 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 inverse and converse we can use the same counterexample. This is because the inverse and converse are also logically equivalent.

 

Determining True Statements within a Biconditional Statement 

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*}.

You should recognize this as the definition of an obtuse angle. All geometric definitions are biconditional statements.

 

Earlier Problem Revisited

The following information answers the question asked at the beginning of this Section:

Your sister presented you with the if-then statement, "If you do the dishes, then I will help you with your homework." If we take the original statement to be true, then the contrapositive is also true. The following contrapositive statement is logically equivalent to the original if-then statement:

"If I do not help you with your homework, then you will not do the dishes."

Examples

Example 1 

Use the statement: Any two points are collinear.

a) Find the converse, inverse, and contrapositive, and determine if the statements are true or false. If they are false, find a counterexamples

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 2

2. \begin{align*}p: x < 10 \qquad q: 2x < 50\end{align*}

Is \begin{align*}p \rightarrow q\end{align*} true? If not, find a counterexample. Is \begin{align*}q \rightarrow p\end{align*} true? If not, find a counterexample. Is \begin{align*}\sim p \rightarrow \sim q\end{align*} true? If not, find a counterexample. Is \begin{align*}\sim q \rightarrow \sim p\end{align*} true? If not, find a counterexample.

 

\begin{align*}& p \rightarrow q: \qquad \quad \ \text{If}\ x < 10, \ \text{then}\ 2x < 50. \quad \ True.\\ & q \rightarrow p: \qquad \quad \ \text{If}\ 2x < 50, \ \text{then} \ x < 10. \quad \ False, \ x = 15 \ \text{would be a counterexample}.\\ &\sim p \rightarrow \sim q: \quad \ \text{If}\ x > 10, \ \text{then}\ 2x > 50. \quad \ False, \ x = 15 \ \text{would also work here.}\\ &\sim q \rightarrow \sim p: \quad \ \text{If}\ 2x > 50, \ \text{then} \ x > 10. \quad \ True. \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. If this is the converse, what is the original statement? Is it true?
  2. If this is the original statement, what is the inverse? Is it true?
  3. Find a counterexample of the statement.
  4. Find the contrapositive of the original statement from #1.
  5. What is the inverse of the inverse of \begin{align*}p \rightarrow q\end{align*}? HINT: Two wrongs make a right in math!
  6. What is the one-word name for the converse of the inverse of an if-then statement?
  7. What is the one-word name for the inverse of the converse of an if-then statement?
  8. What is the contrapositive of the contrapositive of an if-then statement?

For questions 9-12, 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 it has exactly two distinct factors.
  3. Points are collinear if and only if there is a line that contains the points.
  4. \begin{align*}2x = 18\end{align*} if and only if \begin{align*}x = 9\end{align*}.
  5. \begin{align*}p: x = 4 \quad q: x^2=16\end{align*}
    1. Is \begin{align*}p \rightarrow q\end{align*} true? If not, find a counterexample.
    2. Is \begin{align*}q \rightarrow p\end{align*} true? If not, find a counterexample.
    3. Is \begin{align*}\sim p \rightarrow \sim q\end{align*} true? If not, find a counterexample.
    4. Is \begin{align*}\sim q \rightarrow \sim p\end{align*} true? If not, find a counterexample.
  6. \begin{align*}p:x=-2 \quad q:-x+3=5\end{align*}
    1. Is \begin{align*}p \rightarrow q\end{align*} true? If not, find a counterexample.
    2. Is \begin{align*}q \rightarrow p\end{align*} true? If not, find a counterexample.
    3. Is \begin{align*}\sim p \rightarrow \sim q\end{align*} true? If not, find a counterexample.
    4. Is \begin{align*}\sim q \rightarrow \sim p\end{align*} true? If not, find a counterexample.
  7. \begin{align*}p:\end{align*} the measure of \begin{align*}\angle ABC=90^\circ \ q: \angle ABC\end{align*}is a right angle
    1. Is \begin{align*}p \rightarrow q\end{align*} true? If not, find a counterexample.
    2. Is \begin{align*}q \rightarrow p\end{align*} true? If not, find a counterexample.
    3. Is \begin{align*}\sim p \rightarrow \sim q\end{align*} true? If not, find a counterexample.
    4. Is \begin{align*}\sim q \rightarrow \sim p\end{align*} true? If not, find a counterexample.
  8. \begin{align*}p:\end{align*} the measure of \begin{align*}\angle ABC=45^\circ \ q: \angle ABC\end{align*}is an acute angle
    1. Is \begin{align*}p \rightarrow q\end{align*} true? If not, find a counterexample.
    2. Is \begin{align*}q \rightarrow p\end{align*} true? If not, find a counterexample.
    3. Is \begin{align*}\sim p \rightarrow \sim q\end{align*} true? If not, find a counterexample.
    4. Is \begin{align*}\sim q \rightarrow \sim p\end{align*} true? If not, find a counterexample.
  9. Write a conditional statement. Write the converse, inverse and contrapositive of your statement. Are they true or false? If they are false, write a counterexample.
  10. Write a true biconditional statement. Separate it into the two true conditional statements.

Review (Answers)

To view the Review answers, open this PDF file and look for section 2.3. 

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Vocabulary

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.

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.

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