Chapter 4-3 Notes

Definitions

• Logic

– The study of

• Deductive Reasoning

– When a conclusion is drawn from

– Typically, conclusions are drawn from

about something more specific.

reasoning

facts

general statements

Examples

• Example 1

The following statements are true. What is the logical conclusion?

If Evergreen High School wins today, they will go to the regional tournament.

Evergreen High School won today.

Evergreen will go to the regional tournament

Examples

• Example 2

Here are two true statements. What can you conclude?

Every odd number is the sum of an even and an odd number.

5 is an odd number

5 is a sum of an even and an odd number.

Let’s take example 2 and put it in symbolic form

• p:

• q:

• First Statement:

• Second Statement:

– This is a specific example of p.

• Conclusion:

– This is a specific example of q

• Ultimately, example 2 gives us this:

a number is odd

It is a sum of an even and an odd number

p -> q

p

q

p -> qPq

Definitions

• Law of Detachment

– Suppose that is a true statement and we are given

p. Then you can conclude

p -> q

q

Examples

• Example 3

Here are two true statements. What conclusion can you draw from this?

If ∠A and ∠B are a linear pair, then m∠A + m∠B =

180°.

∠ABC and ∠CBD are a linear pair.

m∠ABC + m∠CBD = 180°

Examples

• Example 4

Here are two statements. What conclusion can you draw?

If ∠A and ∠B are a linear pair, then m∠A + m∠B =

180°.

m∠1 = 90° and m∠2 = 90°

No conclusion can be made

• This is called the because the second statement is the conclusion of the first, like the converse of a statement.

Converse Error

Examples

• Example 5

The following two statements are true. What can you conclude?

If a student is in Geometry, then he/she has passed Algebra 1.

Daniel has not passed Algebra 1.

Daniel is not in Geometry

Definitions

• Law of Contrapositive

– Suppose that is a true statement and we are given

~q. Then, you can conclude

– This works because a conditional statement and its

contrapositive are

p -> q

~p

logically equivalent

Examples

• Example 6

Determine the conclusion from the following true statements below.

Babies wear diapers.

My little brother does not wear diapers.

My little brother is not a baby.

Examples

• Example 7

Determine the conclusion from the true statements below.

If you are not in Chicago, then you can’t be on the L.

Bill is in Chicago.

There is no conclusion you

can make.

• This is an example of the because the second statement is the negation of the hypothesis, like an inverse.

Inverse Error

Examples

• Example 8

Determine the conclusion from the following true statements.

If Pete is late, Mark will be late.

If Mark is late, Karl will be late.

If Pete is late, Karl will be late.

Definitions

• Law of Syllogism

– If and , then p -> r is the logical conclusionp -> q q -> r

Examples

• Example 9

Determine the conclusion from the following true statements.

If I have a dog, then I like animals.

If I like animals, then I like cats.

If I have a dog, then I like cats.

• HW#25, #61-69, 80-83