3.6 The Inverse and the Contrapositive
F Math 12
of Conditional Statements p. 208
Name
Date
Goal: Understand and interpret the contrapositive and inverse of a conditional
statement.
1. inverse: A statement that is formed by negating both the hypothesis and the conclusion of
a conditional statement; for example, for the statement “If a number is even, then it is
divisible by 2,” the inverse is “If a number is not even, then it is not divisible by 2.”
2. contrapositive: A statement that is formed by negating both the hypothesis and the
conclusion of the converse
of a conditional statement; for example, for the statement “If
a number is even, then it is divisible by 2,” the contrapositive is “If a number is not divisible
by 2, then it is not even.”
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INVESTIGATE the Math
Puneet’s math teacher said, “If a polygon is a triangle, then it has three sides.”
Puneet’s geography teacher said, “If you live in Saskatoon, then you live in Saskatchewan.”
She wonders what other statements she can write using this information.
What other variations can Puneet write, and are they true?
A. Begin with the math teacher’s statement. Is it true? Explain.
B. Write the converse of the math teacher’s statement. Is it true? Explain.
C. Write the inverse of the math teacher’s statement. Is it true? Explain.
D. Write the contraposative of the math teacher’s statement. Is it true? Explain.
E. Repeat parts A to D using the geography teacher’s statement.
Reflecting
F. If you are given a conditional statement that you know is true, can you predict whether
i.
the converse is true?
ii.
the inverse is true?
iii.
the contrapositive is true?
G. Test your conjectures from part F using this true statement: “If a quadrilateral is a
rectangle, then it is a parallelogram.”
H. Examine the statements you wrote for the inverse and converse for each of the
conditional statements given by Puneet’s teachers. What do you notice?
Example 1: Verifying the inverse and contrapositive of a conditional Statement (p. 209)
Consider the following conditional statement: “If today is February 29, then this year is a leap
year.”
a) Verify the statement, or disprove it with a counterexample.
b) Verify the inverse, or disprove it with a counterexample.
c) Verify the contrapositive, or disprove it with a counterexample.
Example 2: Examining the relationship between a conditional statement and its contraposative
(p. 210)
Consider the following conditional statement: “If a number is a multiple of 10, then it is a
multiple of 5.”
a) Write the contrapositive of this statement.
b) Verify that the conditional and contrapositive statements are both true.
Example 3: Examining the relationship between the converse and inverse of a conditional
statement (p. 212)
Arizona is studying the colour wheel in art
class. She observes the following: “If a
colour is red, yellow, or blue, then it is a
primary colour.”
a) Write the converse of this
statement.
b) Write the inverse of this statement.
c) Verify that the converse and the
inverse are both true.
d) Is Arizona’s statement
biconditional? Explain.
HW: 3.6 p. 214-216 #1, 5, 6, 7, 8, 9 & 12