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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?
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DefinitionDefinition
• Hypothesis
The if part of a conditional statement
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DefintionDefintion
• Conclusion
The then part of a conditional statement
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DefinitionDefinition
• 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
Conditional StatementsConditional Statements
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”
Conditional StatementsConditional Statements
Example: (Original) I breathe when I sleep
(Conditional) If I am sleeping, then I am breathing.
Lesson 2-1 Conditional Statements 14
Conditional StatementsConditional Statements
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……
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DefinitionDefinition
• 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.
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ExamplesExamples
• If you are 13 years old, then you are a teenager.
• Hypothesis:– You are 13 years old
• Conclusion:– You are a teenager
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If a car is a Corvette, then it is a Chevrolet
Hypothesis Conclusion
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Euler Diagram (Venn Diagram)
Euler Diagram (Venn Diagram)
CarsChevys
Corvettes
Euler Diagram (Venn Diagram)
Euler Diagram (Venn Diagram)
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)
Conditional StatementsConditional Statements
• 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.
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DefinitionDefinition
• 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
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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
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DefinitionDefinition
• 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?
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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.
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DefinitionDefinition
• Deductive Reasoning
The process of drawing logically certain conclusions by using an argument
Euler Diagram (Venn Diagram)
Euler Diagram (Venn Diagram)
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
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DefinitionDefinition
• 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
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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
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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.
1. Identify the underlined portion of the conditional statement.
A. hypothesisB. ConclusionC. neither
2. Identify the underlined portion of the conditional statement.
2. Identify the underlined portion of the conditional statement.
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.
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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.