Conditional – formed by joining to statements with the words if and then: If p, then q.

Hypothesis – phrase immediately following if

Conclusion – phrase immediately following then

Identify the hypothesis and conclusion of each statement

If you get a 100% on your test, then you get an A.› Hypothesis: you get a 100% on your test› Conclusion: you get an A

The volleyball team will play in the playoffs if they are one of the top 4 teams.› Hypothesis: the volleyball team is one of the

top 4 teams› Conclusion: they will play in the playoffs

Identify the hypothesis and conclusion of the following statements. Then write each statement in if-then form.

A five-sided polygon is a pentagon.› Hypothesis: a polygon has five sides› Conclusion: it is a pentagon› If a polygon has five sides, then it is a

pentagon

An angle formed by perpendicular lines is a right angle.› Hypothesis: an angle is formed by

perpendicular lines› Conclusion: it is a right angle› If an angle is formed by perpendicular

lines, then it is a right angle.

Determine the truth value of the following statement for each set of conditions.

If Yukon rests for 10 days, his ankle will heal.› Yukon rests for 10 days, and he still has a

hurt ankle. The hypothesis is true, but the conclusion is

false. Since the result is not what was expected,

the conditional statement is false

If Yukon rests for 10 days, his ankle will heal.

Yukon rests for 3 days, and he still has a hurt ankle.› The hypothesis is false, and the conclusion

is false. The statement does not say what happens if Yukon only rests for 3 days. His ankle could possibly still heal.

› We cannot say the statement is false. Therefore, the statement is true.

If Yukon rests for 10 days, his ankle will heal. Yukon rests for 10 days, and he does not have

a hurt ankle anymore.› The hypothesis is true since Yukon rested for 10

days, and the conclusion is true because he doesn’t have a hurt ankle.

› Since what was stated is true, the conditional statement is true.

Yukon rests for 7 days, and he does not have a hurt ankle anymore.› The hypothesis is false, and the conclusion is true.

The statement does not say what happens if Yukon only rests for 7 days.

› We cannot say that the statement is false. Therefore, the statement is true.

RULES OF LOGIC:› A conditional is always false when the hypothesis is

true and the conclusion is false.

Conditional – If p, then q. (given hypothesis and conclusion)

Converse – If q, then p. (switching the order of the hypothesis and conclusion)

Inverse – If not p, then not q. (negating both the hypothesis and conclusion)

Contrapositive – If not q, then not p. (negating both the hypothesis and conclusion of the converse)

RULES OF LOGIC:› The truth value of a conditional and its

contrapositive are always the same.› The truth value of a converse and an inverse are

always the same.

Write the converse, inverse, and contrapositive of the statement “All squares are rectangles.” Determine whether each statement is true or false. If the statement is false, give a counterexample.

First, write the conditional in if-then form.› If a shape is a square, then it is a rectangle

True Converse:

› If a shape is a rectangle, then it is a square. False Counterexample: a rectangle with a length of 2

and width of 4 is not a square

Inverse: › If a shape is not a square, then it is not a rectangle. False Counterexample: a 4-sided polygon with lengths 3, 3, 6, and 6 is not a square and is a rectangle.

Contrapositive:› If a shape is not a rectangle, then it is not a square. True

Practice & Word Problem

Practice