5-4 Inverses, Contrapositives, and Indirect Reasoning

5-4 Inverses, Contrapositives, and Indirect Reasoning

A _____________ ___________ is: a statement in If-Then form. conditional statement

The IF part is the HYPOTHESIS • The THEN part is the conclusion.

negation The __________ of a statement is like the opposite of the statement. Examples: • I like purple. • Negation: • It isn’t sunny. • Negation: I don't like purple. It is sunny.

inverse The _________ of a conditional is formed by negating the hypothesis and the conclusion. • Inverse: If a man does NOT live in Los Angeles, then he does NOT live in California. Example: Conditional: If a man lives in Los Angeles, then he lives in California.

The _______________ is formed by negating the converse (switch the hypothesis and conclusion and then negate both). contrapositive

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. Equivalent statements _________ __________ are 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.

Assume temporarily that Micah doesn’t love video games. Step 1: Write the first step of an indirect proof. • Micah loves video games. • Geometry is fun. • The Wiggles are not cool. d) Assume temp. that geometry isn’t fun. Assume temporarily that The Wiggles are cool. Assume temporarily that .

Step 2: Identify the statements that contradict each other. • Jennifer lives in Orange County. • Jennifer is a vegetarian. • 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. 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.

5-4 Inverses, Contrapositives, and Indirect Reasoning