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Conditional Statements. Chapter 2-3. Analyze statements in if-then form. Write the converse, inverse, and contrapositive of if-then statements. conditional statement. inverse. if-then statement hypothesis conclusion related conditionals converse. contrapositive logically equivalent. - PowerPoint PPT Presentation
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Conditional Statements
Chapter 2-3
• conditional statement
• if-then statement
• hypothesis
• conclusion
• related conditionals
• converse
• Analyze statements in if-then form.
• Write the converse, inverse, and contrapositive of if-then statements.
• inverse
• contrapositive
• logically equivalent
Conditional Statement• Usually written in if-then form.• The if part is called the hypothesis.• The then part is called the conclusion.• Hypothesis and conclusions do not include
the words if or then.
• Example: If it is raining, then it is cloudy.
Hypothesis Conclusion
Identify Hypothesis and Conclusion
A. Identify the hypothesis and conclusion of the following statement.
Answer: Hypothesis: a polygon has 6 sidesConclusion: it is a hexagon
If a polygon has 6 sides, then it is a hexagon.
If a polygon has 6 sides, then it is a hexagon.
hypothesis conclusion
A. A
B. B
C. C
D. D
A. Hypothesis: you will cryConclusion: you are a baby
B. Hypothesis: you are a babyConclusion: you will cry
C. Hypothesis: babies cryConclusion: you are a baby
D. none of the above
A. Which of the choices correctly identifies the hypothesis and conclusion of the given conditional?
If you are a baby, then you will cry.
A. A
B. B
C. C
D. D
A. Hypothesis: you want to find the distance between 2 pointsConclusion: you can use the distance formula
B. Hypothesis: you are taking geometryConclusion: you learned the distance formula
C. Hypothesis: you used the distance formulaConclusion: you found the distance between 2 points
D. none of the above
B. Which of the choices correctly identifies the hypothesis and conclusion of the given conditional?
To find the distance between two points, you can use the Distance Formula.
Rewriting a Conditional in if-then form
• You may go outside if you clean your room.
If you clean your room, then you may go outside.
• Two points are collinear if they lie on the same line.
If two points lie on the same line, then they are collinear.
• All cheerleaders are dumb.
If you are a cheerleader, then you are dumb.
Write a Conditional in If-Then Form
A. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form.
Distance is positive.
Answer: Hypothesis: a distance is measuredConclusion: it is positiveIf a distance is measured, then it is positive.
Sometimes you must add information to a statement. Here you know that distance is measured or determined.
Write a Conditional in If-Then Form
B. Identify the hypothesis and conclusion of the following statement. Then write the statement in the if-then form.
A five-sided polygon is a pentagon.
Answer: Hypothesis: a polygon has five sidesConclusion: it is a pentagonIf a polygon has five sides, then it is a pentagon.
1. A
2. B
3. C
4. D
A. If an octagon has 8 sides, then it is a polygon.
B. If a polygon has 8 sides, then it is an octagon.
C. If a polygon is an octagon, then it has 8 sides.
D. none of the above
A. Which of the following is the correct if-then form of the given statement?
A polygon with 8 sides is an octagon.
1. A
2. B
3. C
4. D
A. If an angle is acute, then it measures less than 90°.
B. If an angle is not obtuse, then it is acute.
C. If an angle measures 45°, then it is an acute angle.
D. If an angle is acute, then it measures 45°.
B. Which of the following is the correct if-then form of the given statement?
An angle that measures 45° is an acute angle.
• A counterexample must make the hypothesis true and the conclusion false.
• If x2 = 16, then x = 4.
• Counterexample: x = -4
x2 = 16, but x 4
Writing a Counterexample
True False
A. Determine the truth value of the following statement for each set of conditions.
If Yukon rests for 10 days, his ankle will heal.
Answer: Since the result is not what was expected, the conditional statement is false.
Truth Values of Conditionals
The hypothesis is true, but the conclusion is false.
Yukon rests for 10 days, and he still has a hurt ankle.
B. Determine the truth value of the following statement for each set of conditions.
If Yukon rests for 10 days, his ankle will heal.
Answer: In this case, we cannot say that the statement is false. Thus, the statement is true.
Truth Values of Conditionals
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.
Yukon rests for 3 days, and he still has a hurt ankle.
C. Determine the truth value of the following statement for each set of conditions.
If Yukon rests for 10 days, his ankle will heal.
Answer: Since what was stated is true, the conditional statement is true.
Truth Values of Conditionals
The hypothesis is true since Yukon rested for 10 days, and the conclusion is true because he does not have a hurt ankle.
Yukon rests for 10 days, and he does not have a hurt ankle anymore.
D. Determine the truth value of the following statement for each set of conditions.
If Yukon rests for 10 days, his ankle will heal.
Answer: In this case, we cannot say that the statement is false. Thus, the statement is true.
Truth Values of Conditionals
The hypothesis is false, and the conclusion is true. The statement does not say what happens if Yukon only rests for 7 days.
Yukon rests for 7 days, and he does not have a hurt ankle anymore.
1. A.
2. B.
A. Determine the truth value of the following statement for each set of conditions.
If it rains today, then Michael will not go skiing.
It does not rain today; Michael does not go skiing.
A. true
B. false
1. A.
2. B.
B. Determine the truth value of the following statement for each set of conditions.
If it rains today, then Michael will not go skiing.
It rains today; Michael does not go skiing.
A. true
B. false
1. A.
2. B.
C. Determine the truth value of the following statement for each set of conditions.
If it rains today, then Michael will not go skiing.
It snows today; Michael does not go skiing.
A. true
B. false
1. A.
2. B.
D. Determine the truth value of the following statement for each set of conditions.
If it rains today, then Michael will not go skiing.
It rains today; Michael goes skiing.
A. true
B. false
Symbolic Notation• Given a conditional: if p, then q.
• Conditional p q
• Converse q p
• Inverse ~ p ~ q
~ means “not”
• Contrapositive ~ q ~ p
Converse• The converse of a conditional switches the
hypothesis and the conclusion.
• Original:
If you own a black car, then you drive fast.
• Converse:
If you drive fast, then you own a black car.
Inverse• The negation of the hypothesis and the
conclusion of a conditional.
• Conditional:
If mA = 30o, then A is acute.
• Inverse:
If mA 30o, then A is not acute.
Contrapositive• The negation of the hypothesis and the
conclusion of the converse of a conditional.
• Conditional:
If you are tall, then you are smart.
• Converse:
If you are smart, then you are tall.
• Contrapositive:
If you are not smart, then you are not tall.
Write the converse, inverse, and contrapositive of the statement All squares are rectanglesAll squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample.
Conditional: If a shape is a square, then it is a rectangle.
The conditional statement is true.Converse: If a shape is a rectangle, then it is a square.
The converse is false.
A rectangle with l = 2 and w = 4 is not a square. Inverse: If a shape is not a square, then it is not a rectangle.
The inverse is false.
A 4-sided polygon with sides 2, 2, 4, and 4 is not a square.
Contrapositive:If a shape is not a rectangle, then it is not a square. The contrapositive is true.
Logically Equivalent Statements• Two statements that are both true or both false.
• A conditional is equivalent to its contrapositive.
• The inverse is equivalent to the converse.
A. A
B. B
C. C
D. D
A. All 4 statements are true.
B. Only the conditional and contrapositive are true.
C. Only the converse and inverse are true.
D. All 4 statements are false.
Write the converse, inverse, and contrapositive of the statement The sum of the measures of two complementary angles is 90. Which of the following correctly describes the truth values of the four statements?
Homework
Chapter 2-3
• Pg 94
11-23 odd, 27 – 36 all, 37 – 41 odd,
67-69 all, 73-75 all
Animation:Conditional Statements
