5-4 Inverses, Contrapositives, and Indirect Reasoning

A _____________ is: a statement in If Then form.



The

IF

part is the HYPOTHESIS



The

THEN

part is the conclusion.

is like the opposite of the statement.

Examples:  I like purple.  Negation: I don't like purple.

 It isn’t sunny.  Negation: It is sunny.

is formed by negating the hypothesis and the conclusion.

Example: Conditional: If a man lives in Los Angeles, then he lives in California.

 Inverse: If a man does NOT live in Los Angeles, then he does NOT live in California.

by negating the converse (switch the hypothesis and conclusion and then negate both).

Same example:  Contrapositive:  If a man doesn’t live in California, then he doesn’t live in Los Angeles.

In these examples, the original conditional and the contrapositive are both true.

And the converse and the inverse are both false.

statements that are both true or both false.

Ex. a) Rewrite in if-then form.

b) Identify the hypothesis and conclusion.

c) Write the negation, inverse, and contrapositive.

 Teenagers that are 16 can learn to drive.

a) If a teenager is 16, then he can learn to drive.

b) Hypothesis: a teenager is 16 Conclusion: he can learn to drive

c) Negation of hypothesis : A teenager is not 16.

Negation of conclusion : A teenager cannot learn to drive.

Inverse : If a teenager is not 16, then he can't learn to drive.

Contrapositive : If a teenager is not 16, then he can't learn to drive.

5-4 Continued Indirect Proofs

Until now, the proofs you have written have been direct proofs.

Sometimes it is difficult or even impossible to find a direct proof.

In that case, it may be possible to reason indirectly .

In an indirect proof you begin by assuming temporarily that the desired conclusion is NOT true. Then you reason logically until you reach a contradiction of the hypothesis or a known fact (definition, theorem, etc). Because you’ve reached a contradiction, you know that the temporary assumption is impossible and therefore the desired conclusion must be true.

Writing an Indirect Proof

• Step 1: State as an assumption the opposite (negation) of what you want to prove.

• Step 2: Show that this assumption leads to a contradiction • Step 3: Conclude that the assumption must be false and that what you want to prove must be true.

Step 1: Write the first step of an indirect proof.

a) Assume temporarily that Micah doesn’t love video games.

Micah loves video games.

Assume temp. that geometry isn’t fun.

b) Geometry is fun.

c) d) Assume temporarily that The Wiggles are The Wiggles are not cool.

cool.   90 Assume temporarily that . 90

Step 2: Identify the statements that contradict each other.

I.

Jennifer lives in Orange County.

II. Jennifer is a vegetarian.

III. Jennifer loves to eat In & Out Burgers.

• II and III contradict each other because she cannot be a vegetarian and eat hamburgers at the same time.

Example: Identify the two statements that contradict each other.

I. II. III.

FG



KL

Two segments cannot be parallel and perpendicular at the same time. I and II contradict each other.

Example: Indirect Proof

Read the conditional statement. Think about what is given and what you are to prove. Then give the steps of an indirect proof.

• If Jaelene spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25.

• Given: The cost of two items is more than $50.

• Prove: At least one items costs more than $25.

Step 1

• Assume as true the opposite of what you want to prove – Assume that neither item costs more than $25.

Step 2

• This means that each item costs $25 or less. This, in turn, means that the two items together cost $50 or less. • This

contradicts

the given information that the amount spent is more than $50.

Step 3

• Conclude that the assumption is false.

• One item must cost more than $25 for Jaeleen to spend more than $50 for two items.