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Converse, Inverse, and Contrapositive

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Converse, Inverse, and Contrapositive

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? After completing this Concept, you'll be able to write the converse, inverse, and contrapositive of a conditional statement like this one.

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CK-12 Converse, Inverse and Contrapositive of a Conditional Statement

James Sousa: Converse, Contrapositive, and Inverse of an If-Then Statement

Guidance

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.

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

Now the statement is: If p , then q , which can also be written as p \rightarrow q .

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

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

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

& \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}.

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 p \rightarrow q is true and q \rightarrow p is true, then p \leftrightarrow q (said “ p if and only if q ”).

Example A

If n > 2 , then n^2 > 4 .

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.

& \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.}

Example B

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.

& \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.}

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

Example C

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.

CK-12 Converse, Inverse and Contrapositive of a Conditional Statement

Guided Practice

1. Any two points are collinear.

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.

2. The following is a true statement:

m\angle ABC > 90^\circ if and only if \angle ABC is an obtuse angle.

Determine the two true statements within this biconditional.

3. p:  x < 10 \quad q:  2x < 50

a) Is p \rightarrow q true? If not, find a counterexample.

b) Is q \rightarrow p true? If not, find a counterexample.

c) Is \sim p \rightarrow \sim q true? If not, find a counterexample.

d) Is \sim q \rightarrow \sim p true? If not, find a counterexample.

Answers

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

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

& \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.

2. Statement 1: If m \angle ABC > 90^\circ , then \angle ABC is an obtuse angle.

Statement 2: If \angle ABC is an obtuse angle, then m\angle ABC > 90^\circ .

3. a) If x < 10 , then 2x < 50 . True .

b) If 2x < 50 , then x < 10 . False , x = 15

c) If x \ge 10 , then 2x \ge 50 . False , x = 15

d) If 2x \ge 50 , then x \ge 10 . True ,  x \ge 25

Practice

For questions 1-4, use the statement:

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

  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 90^\circ .
  2. If you are at the beach, then you are sun burnt.
  3. If x > 4 , then x+3>7 .

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. 2x = 18 if and only if x = 9 .

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