Logic

• Conclusion based on facts

• “If… then…”

• Conclusions based on observations and patterns

• “Conjecture”• “Counterexample”

will show a conjecture is false

Deductive Reasoning

Inductive Reasoning

“1” if Inductive Reasoning“2” if Deductive Reasoning

For the past two weeks there have always been an even number of birds at your feeder. Today there will be an even number of birds at your feeder.

1

“1” if Inductive Reasoning“2” if Deductive Reasoning

In the last week, whenever you walked out your door and saw your cat in the tree, it rained that afternoon. You saw your cat in the tree this morning. It will rain this afternoon.

1

“1” if Inductive Reasoning“2” if Deductive Reasoning

The sum of two odd numbers is always an even number. The sum of 17 and 21 is even.

2

“1” if Inductive Reasoning“2” if Deductive Reasoning

The last 10 times you pushed an odd-numbered button, a red light flashed. If you push the button “5”, a red light will flash.

1

“1” if Inductive Reasoning“2” if Deductive Reasoning

If yesterday was Thursday, then tomorrow is Saturday.

2

If yesterday was Thursday, then tomorrow is Saturday.

Hypothesis Conclusion

If tomorrow is Saturday, then yesterday was

Thursday Converse

Is the converse true (touch your lips) or false (touch nose)

If a number is divisible by two, then it is an even

number

Is the converse true (touch your lips) or false (touch nose)

If x = 5, then x2 = 25

If false, give a counterexample.

-5

Is the converse true (touch your lips) or false (touch nose)

If a figure is a square, then it has 4 sides.

If false, give a counterexample.

Rectangle

Which is a counterexample to the conjecture

If a>b and c>d, then ac > bd

1234 There is no counterexample. The conjecture is true.

Give a counterexample to the conjecture:

The difference of two rational numbers is always less than at least one of the numbers.

(for example, the difference of 9 and 2 is 7)

8 5 13

8 and -5

If it is not raining, Jack will walk the dog.

If it is not raining, Jerry will go swimming.

If it is not raining, Jerry will go swimming.

Problem Solving Strategy:Use Logical Reasoning

To celebrate getting their drivers' licenses, Liz, Ann, and Phil borrowed their parents cars: a white sedan, a red sports car; and a red compact, (1) Phil's parents own only red cars. (2) Liz borrowed a red car, but her parents wouldn't let her drive their sports car. Who was driving which car?

To celebrate getting their drivers' licenses, Liz, Ann, and Phil borrowed their parents cars: a white sedan, a red sports car; and a red compact, (1) Phil's parents own only red cars. (2) Liz borrowed a red car, but her parents wouldn't let her drive their sports car. Who was driving which car? Sports

(Red)Sedan

(White)Compact

(Red)LizAnn

Phil

To celebrate getting their drivers' licenses, Liz, Ann, and Phil borrowed their parents cars: a white sedan, a red sports car; and a red compact, (1) Phil's parents own only red cars. (2) Liz borrowed a red car, but her parents wouldn't let her drive their sports car. Who was driving which car? Sports

(Red)Sedan

(White)Compact

(Red)LizAnn

Phil X1

To celebrate getting their drivers' licenses, Liz, Ann, and Phil borrowed their parents cars: a white sedan, a red sports car; and a red compact, (1) Phil's parents own only red cars. (2) Liz borrowed a red car, but her parents wouldn't let her drive their sports car. Who was driving which car? Sports

(Red)Sedan

(White)Compact

(Red)LizAnn

Phil X1

X2 X2

X2

X2

The houses on your street are numbered 1 to 120. No numbers are skipped. How many house numbers contain at least one digit 5?

Fundamental Counting Principle 1) How many locker combinations are possible

on a standard lock with digits 0 – 49 using the right/left/right sequence if no number can be used more than once in the combination?

2) How many different phone numbers are possible in the 909 area code starting with 594? 594-__ __ __ __

Assignment:

p. 195 1,3,6,7

We will review Wednesday and the Chapter Test is Friday