5-4 Inverses, Contrapositives, and Indirect Reasoning.
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Transcript of 5-4 Inverses, Contrapositives, and Indirect Reasoning.
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5-4 Inverses, Contrapositives, and Indirect
Reasoning
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A _____________ ___________ is: a statement in If-Then form.
conditional statement
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The IF part is the HYPOTHESIS
The THEN part is the conclusion.
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The __________ of a statement is like the opposite of the statement.
Examples: I like purple. Negation:It isn’t sunny. Negation:
negation
I don't like purple.
It is sunny.
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The _________ of a conditional is formed by negating the hypothesis and the conclusion.
inverse
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.
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The _______________ is formed by negating the converse (switch the hypothesis and conclusion and then negate both).
contrapositive
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Same example:
Contrapositive:If a man doesn’t live in
California, then he doesn’t live in Los Angeles.
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In these examples, the original conditional and the contrapositive are both true.
And the converse and the inverse are both false.
_________ __________ are statements that are both true or both false.
Equivalent statements
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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.
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b) Hypothesis: a teenager is 16
a) If a teenager is 16, then he can learn to drive.
Conclusion: he can learn to drive
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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.
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5-4 Continued Indirect Proofs
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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.
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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.
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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.
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Step 1: Write the first step of an indirect proof.
a) Micah loves video games.
b) Geometry is fun.
c) The Wiggles are not cool.
d) 90m F 90m F
Assume temporarily that Micah doesn’t love video games.
Assume temp. that geometry isn’t fun.
Assume temporarily that The Wiggles are cool.
Assume temporarily that .
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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.
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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.
FG KL FG KL FG KL
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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.
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•Given: The cost of two items is more than $50.
•Prove: At least one items costs more than $25.
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Step 1
• Assume as true the opposite of what you want to prove – Assume that neither item costs more than $25.
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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.
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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.