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Chapter 2

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Chapter 2

Logic and Proofs

- Conditional statements are just that – statements that contain a condition.
- If p then q
- p is the Hypothesis and q is the Conclusion
- Example: If an animal is a robin, then the animal is a bird.

- A conditional tells us that if we have one certain condition present, then some other condition MUST be present.
- Ex.: If you want an “A” on a test, then you must study.
- This conditional tells us that in order for us to satisfy some desire (getting an A) we will need to participate in some specific type of behavior (studying).

- Example: Vertical angles share a vertex.
- Identify the hypothesis and the conclusion.
- Write the conditional. (if _____,then_____)
- If two angles are vertical, then they share a vertex.

- If we determine that a statement has a “false” truth value, then we must provide a counterexample.
- A counterexample is something that provides evidence that the statement is false.

For something to be true, it must always be true. One case when it is not true, makes the statement false! Look for counterexamples.

- In the following conditional, 1) identify the hypothesis 2) identify the conclusion 3) give its truth value4) if false, give a counterexample:
- If you live in Alpharetta, then you live in Georgia.
- If x+1=4, then x=3
- If it is Monday, we have math class.
- If a number is divisible by 3, it is odd.

- Converse – the converse of a conditional statement is formed by switching the hypothesis and the conclusion.
- Remember, the “if” and the “then” are not parts of the hypothesis or the conclusion, so we do not switch them.

- In the following conditional, 1) identify the hypothesis 2) identify the conclusion 3) give its truth value4) provide the converse5) give the truth value of the converse:
- If today is Tuesday, then tomorrow is Wednesday.
- If AB + BC = AC, then point B lies on segment AC.
- If Y is the midpoint of segment XZ, then XY=YZ.

- If XY=YZ, then Y is the midpoint of segment XZ.
- We said this is false, so what is a counterexample?
- If you spend a lot of money, then you will be broke.
- What is a counterexample?

- An inverse is the negation of the original hypothesis and conclusion.
- The negation of a statement is the opposite.
- Original conditional: If there is lightning tomorrow night, then the swim meet will be cancelled.
- Inverse: If there is NO lightning, then the swim meet will NOT be cancelled.
- The original is true, but what about the inverse?

- A contrapositive statement is the inverse of the converse.
- This means that we take the converse, then we negate both parts
(switch p and q and put “not” in front of both of then).

- Original conditional: If there is lightning tomorrow night, then the swim meet will be cancelled.
- Contrapositive: If the swim meet is not cancelled, then there is no lightning.

- p.85 #6, 8, 10, 11, 14, 33-41 odd
- p.93-94 #5-7, 9, 12, 14, 16, 18, 20, 21, 35, 38
- (24 problems)