Converse, Inverse, and
Contrapositive
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: November 15, 2015
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
Chapter 1. Converse, Inverse, and Contrapositive
C HAPTER
1
Converse, Inverse, and
Contrapositive
Here you’ll learn how to write the converse, inverse, and contrapositive of any conditional statement. You’ll also
learn what a biconditional statement is.
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.
Watch This
MEDIA
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URL: http://www.ck12.org/flx/render/embeddedobject/136611
CK-12 Converse, Inverse and Contrapositive of a Conditional Statement
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/1377
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 = the weather is nice
q = I’ll wash the car
Now the statement is: If p, then q, which can also be written as p → q.
We can also make the negations, or “nots,” of p and q. The symbolic version of "not p" is ∼ p.
∼ p = the weather is not nice
∼ q = I won’t wash the car
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Using these “nots” and switching the order of p and q, we can create three new statements.
q→ p
Converse
If
the weather
is nice} .
| I wash
{z the car}, then
|
{z
q
∼ p →∼ q
Inverse
p
If
I won’t{zwash the car} .
| the weather
{z is not nice}, then
|
∼p
Contrapositive
∼ q →∼ p
∼q
If
the weather
| I don’t wash
{z the car}, then
|
{z is not nice} .
∼q
∼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 → q is true and q → p is true, then
p ↔ q (said “p if and only if q”).
Example A
If n > 2, then n2 > 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.
Converse :
If n2 > 4, then n > 2.
False. If n2 = 9, n = −3 or 3. (−3)2 = 9
Inverse :
If n ≤ 2, then n2 ≤ 4.
False. If n = −3, then n2 = 9.
Contrapositive : If n2 ≤ 4, then n ≤ 2.
True. The only n2 ≤ 4 is 1 or 4.
√
√
1 = ±1 and 4 = ±2, 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.
Converse :
If I am in California, then I am at Disneyland.
False. I could be in San Francisco.
Inverse :
If I am not at Disneyland, then I am not in California.
False. Again, I could be in San Francisco.
Contrapositive :
If I am not in California, then I am not at Disneyland.
True. 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.
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Chapter 1. Converse, Inverse, and Contrapositive
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.
MEDIA
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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:
m6 ABC > 90◦ if and only if 6 ABC is an obtuse angle.
Determine the two true statements within this biconditional.
3. p : x < 10 q : 2x < 50
a) Is p → q true? If not, find a counterexample.
b) Is q → p true? If not, find a counterexample.
c) Is ∼ p →∼ q true? If not, find a counterexample.
d) Is ∼ q →∼ 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.
Converse :
If two points are collinear, then they are on the same line. True.
Inverse :
If two points are not on the same line, then they are not collinear. True.
Contrapositive :
If two points are not collinear, then they do not lie on the same line. True.
2. Statement 1: If m6 ABC > 90◦ , then 6 ABC is an obtuse angle.
Statement 2: If 6 ABC is an obtuse angle, then m6 ABC > 90◦ .
3. a) If x < 10, then 2x < 50. True.
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b) If 2x < 50, then x < 10. False, x = 15
c) If x ≥ 10, then 2x ≥ 50. False, x = 15
d) If 2x ≥ 50, then x ≥ 10. True, x ≥ 25
Explore More
For questions 1-4, use the statement:
If AB = 5 and BC = 5, then B is the midpoint of AC.
1.
2.
3.
4.
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.
5. An acute angle is less than 90◦ .
6. If you are at the beach, then you are sun burnt.
7. If x > 4, then x + 3 > 7.
For questions 8-10, determine the two true conditional statements from the given biconditional statements.
8. A U.S. citizen can vote if and only if he or she is 18 or more years old.
9. A whole number is prime if and only if its factors are 1 and itself.
10. 2x = 18 if and only if x = 9.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 2.4.
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