Chapter 2 Reasoning in Geometry

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Chapter 2 Reasoning in Geometry

• 2.2 Introduction to Logic

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Introduction

• 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!

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Introduction

• Do you recognize the following conversation?

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"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'!"
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"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.
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Charles 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.

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Conditional Statements

• In order to analyze statements, we will
  translate them into a logic statement called a
  conditional statement.
• (You will be taking notes now)

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Essential Question

• How do I recognize and analyze a conditional
  statement?

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Definition

• Hypothesis
• The if part of a conditional statement

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Defintion

• Conclusion
• The then part of a conditional statement

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Definition

• 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

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Conditional Statements
conditional statement

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

if-then

• 2. A conditional statement has _________.
• The __________ is the ____ part.
• The __________ is the ______ part.

two parts
hypothesis
if
conclusion
then
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Conditional Statements

• Example

(Original) I breathe when I sleep (Conditional)
If I am sleeping, then I am
breathing.
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Conditional 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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Definition

• 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 feet smell, nose runs q built upside down
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Rewrite 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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Examples

• 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)
Cars
Chevys
Corvettes
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Euler Diagram (Venn Diagram)

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

Chevrolets
(Conclusion then part)
Corvettes
(Hypothesis If part)
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Example Euler Diagram

• What is the conditional statement?
• If two angles form a linear pair, then the angles
  are supplementary angles

Supplementary angles
(Conclusion then part)
Linear pairs
(Hypothesis If part)
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Conditional Statements

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

converse
(Conditional) If I am sleeping, then I am
breathing.
(Converse) If I am breathing, then I am
sleeping.
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Definition

• 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 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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Definition

• 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 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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Definition

• Deductive Reasoning
• The process of drawing logically certain
  conclusions by using an argument

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Euler Diagram (Venn Diagram)

• Susans car is a Corvette
• If a car is a Corvette, then it is a Chevrolet
• Susans car is a Corvette
• Therefore the conclusion is Susan's car is a
  Chevrolet.

Chevrolets

• Corvettes
• Susans car

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Definition

• 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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Example

• 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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Logical Chain (Transitive property)
If cats freak, then mice frisk If sirens
shriek, then dogs howl If dogs howl, then cats
freak
Using the provided statements to prove the
following conclusion If sirens shriek, then
mice frisk
Look for the conditional that begins with the
then statement and write it down under the first
Keep repeating until you get a conclusion that
matches the one youre looking for
First, find the hypothesis of the conditional
you are trying to prove
Second, write down the conditional with that
hypothesis
If sirens shriek, then dogs howl
Conclusion If sirens shriek, then mice frisk
If dogs howl, then cats freak
If cats freak, then mice frisk
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1. Identify the underlined portion of the
conditional statement.

1. hypothesis
2. Conclusion
3. neither

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2. Identify the underlined portion of the
conditional statement.

1. hypothesis
2. Conclusion
3. neither

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4. Identify the converse for the given
conditional.

1. If you do not like tennis, then you do not play
   on the tennis team.
2. If you play on the tennis team, then you like
   tennis.
3. If you do not play on the tennis team, then you
   do not like tennis.
4. You play tennis only if you like tennis.

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Assignment

• 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.