Chapter 2 Reasoning in Geometry

Chapter 2 Reasoning in Geometry

Chapter 2 Reasoning in Geometry

2.2 Introduction to Logic

IntroductionIntroduction

In chapter 2 section 2, we will discuss how we use logic to develop mathematical proofs.

When writing proofs, It is important to use exact and correct mathematical language. We must say what we mean!

IntroductionIntroduction

Do you recognize the following conversation?

"Then you should say what you mean." the March Hare went on.

"I do," Alice hastily replied; "at least -- at least I mean what I say -- that's the same thing, you know. "

"Not the same thing a bit!" said the Hatter, "Why, you might just as well say that 'I see what I eat' is the

same thing as 'I eat what I see'!"

"You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!“

"You might just as well say," added the Dormouse, who seemed to be talking in his sleep, "that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!“

"It is the same thing with you," said the Hatter, and here the conversation dropped, and the party sat silent for a minute.

Charles DodgsonCharles Dodgson

Charles Dodgson lived from 1832 to 1898

Dodgson was a mathematics lecturer and author of mathematics books who is better known by the pseudonym Lewis Carroll. He is known especially for Alice's Adventures in Wonderland.

Conditional StatementsConditional Statements

In order to analyze statements, we will translate them into a logic statement called a conditional statement.

(You will be taking notes now)

Essential Question:Essential Question:

How do I recognize and analyze a conditional

statement?

04/20/23 Free PowerPoint Template from www.brainybetty.com 9

DefinitionDefinition

• Hypothesis

The if part of a conditional statement


DefintionDefintion

• Conclusion

The then part of a conditional statement



• Conditional

IF something, THEN something else

If a car is a Corvette, then it is a Chevy

If you are in this room right now, then you are in Geometry


1. A _________________ is a statement that can be expressed in ________form.

conditional conditional statementstatement ““if-if-

then”then”

2.2. A conditional statement has A conditional statement has __________________..

The The ____________________ is the is the ________ part. part.The The ____________________ is the is the ____________ part. part.

hypothesihypothesiss

two partstwo parts““if”if”

conclusioconclusionn

““thethen”n”


Example: (Original) I breathe when I sleep

(Conditional) If I am sleeping, then I am breathing.

Lesson 2-1 Conditional Statements 14


Definition: A conditional statement is a statement that can be written in if-then form.“If _____________, then ______________.”

Example: If your feet smell and your nose runs, then you're built upside down.

Continued……



• Conditional

If / then statements are conditional. The then part of the statement is depends on (is conditional to) the if part.

In shorthand, the statement is “if p then q”

In symbol form, p qp = feet smell, nose runs

q = built upside down

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Rewrite in the if-then formRewrite in the if-then form

• All mammals breathe oxygen– If an animal is a mammal, then it

breathes oxygen.

• A number divisible by 9 is also divisible by 3– If a number s divisible by 9, then it is

divisible by 3.


ExamplesExamples

• If you are 13 years old, then you are a teenager.

• Hypothesis:– You are 13 years old

• Conclusion:– You are a teenager


If a car is a Corvette, then it is a Chevrolet

Hypothesis Conclusion


Euler Diagram (Venn Diagram)


CarsChevys

Corvettes



If a car is a Corvette, then it is a Chevrolet

Chevrolets

Corvettes

(Conclusion: then part)

(Hypothesis: If part)

Example: Euler DiagramExample: Euler DiagramWhat is the conditional statement?•If two angles form a linear pair, then the angles are supplementary angles

Supplementary angles

Linear pairs

(Conclusion: then part)

(Hypothesis: If part)


• The ________ of a conditional statement is formed by switching the hypothesis and the conclusion.

• Example:

converseconverse

(Conditional)(Conditional) If If I am sleepingI am sleeping, then , then I amI am breathingbreathing..

(Converse)(Converse) If If I am breathingI am breathing, then , then I I am am sleeping. sleeping.



• Converse

Changing the if and the then around

• Conditional: If a car is a Corvette, then it is a Chevrolet

• Converse: If a car is a Chevrolet, then it is a Corvette


Determine the ConverseDetermine the Converse

If you are wearing a skirt, then you are a female

If you are a female, then you are wearing a skirt



• Counterexample

An example that proves a statement false

Consider the conditional statement:

If you are a female, then you are wearing a skirt

Is there any females in the room that are not wearing a skirt?


Writing a CounterexampleWriting a Counterexample

• Write a counterexample to show that the following conditional statement is false– If x2 = 16, then x = 4.– As a counterexample, let x = -4.

• The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false.



• Deductive Reasoning

The process of drawing logically certain conclusions by using an argument



Susan’s car is a Corvette1.If a car is a Corvette, then it is a Chevrolet2. Susan’s car is a Corvette3.Therefore the conclusion is: Susan's car is a Chevrolet.

Chevrolets

Corvettes• Susan’s car



• If-Then Transitive Property

• If A then B• If B then C

• You can conclude: If A then C

• Also known as a logic chain


ExampleExample

Consider the following conditionals - If cats freak, then mice frisk– If sirens shriek, then dogs howl– If dogs howl, then cats freak

Prove the following: If sirens shriek, then mice frisk


If cats freak, then mice friskIf sirens shriek, then dogs howlIf dogs howl, then cats freak

First, find the hypothesis of the conditional you are trying to prove

Using the provided statements to prove the following conclusion: If sirens shriek, then mice frisk

If sirens shriek, then dogs howl

Second, write down the conditional with that hypothesisLook for the conditional that begins with the then statement and write it down under the first

If dogs howl, then cats freak

Keep repeating until you get a conclusion that matches the one you’re looking for

If cats freak, then mice frisk

Conclusion: If sirens shriek, then mice frisk

Logical Chain (Transitive property)

1. Identify the underlined portion of the conditional statement.


A. hypothesisB. ConclusionC. neither



A. hypothesisB. ConclusionC. neither

4. Identify the converse for the given conditional.4. Identify the converse for the given conditional.

A. If you do not like tennis, then you do not play on the tennis team.

B. If you play on the tennis team, then you like tennis.

C. If you do not play on the tennis team, then you do not like tennis.

D. You play tennis only if you like tennis.


AssignmentAssignment

• Read pages 90-93, Ch2 Sec 2 Complete problems on Page 95 #9-34 Due Friday Oct. 15.

• This is an involved set of problems and will take some time to complete. You will be making a big mistake if you wait until Thursday evening to begin this assignment.

• Suggestion: Break into small parts, complete 6 to 10 problems per day/night.

Chapter 2 Reasoning in Geometry

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