Section 3.1 Statements, Negations, and Quantified Statements

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Section 3.1 Statements, Negations, and Quantified
Statements

• Thinking Mathematically
• Math 1001

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Objectives

• Identify English sentences that are statements .
• Express statements using symbols.
• Form the negation of a statement.
• Express negations using symbols.
• Express quantified statements in two ways.
• Write negations of quantified statements.

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Definition of a Statement (page 89)

• A statement is a sentence that is either true or
  false, but not both simultaneously.
• Atlanta is the capital of Georgia.
• Water boils at 98 degrees Fahrenheit.
• Star Wars Episode I is the best movie ever.
• Do your homework.
• What time is it?

True
False
Opinion
Command
Question
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Express Statements Using Symbols

• p Atlanta is the capital of Georgia.
• q Water boils at 98 degrees Fahrenheit.
• Statement p is true.
• Statement q is false.

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Negating Statements

• If Atlanta is the capital of Georgia is
  statement p, then
• p not p
• means the opposite.
• p Atlanta is not the capital of Georgia

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Negation and Truth

• The negation of a true statement is always false.
• The negation of a false statement is always true.

False
True
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Quantified Statements

• All
• All cats are black.
• Some
• Some cats are black.
• None or No
• No cats are black.

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Equivalent Ways of Expressing Quantified
Statements (p 91)

• All A are B.
• All cats are black cats.
• There are no A that are not B.
• There are no cats that are not black cats.

False
False
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Equivalent Ways of Expressing Quantified
Statements (p 91)

• Some A are B.
• Some cats are black cats.
• There exists at least one A that is a B.
• At least one cat is a black cat.

True
True
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Equivalent Ways of Expressing Quantified
Statements (p 91)

• No A are B.
• No cats are black cats.
• All A are not B.
• All cats are not black.

False
False
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Equivalent Ways of Expressing Quantified
Statements (p 91)

• Some A are not B.
• Some cats are not black cats.
• Not all A are B.
• Not all cats are black.

True
True
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Quantified StatementAll, Some, or None

• A quantified statement is one that says
  something about all, some, or none of the
  objects in a collection.
• All students in the college are taking history.
• Some students are taking mathematics.
• No students are taking both mathematics and
  history.

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Negation Diagrams of Quantified Statements

• All students are female.

Is this a true statement?
Females
Females
Students
Students
Is this a true statement?
The negation of a false statement must be true.
Some students are not female.
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Negation Diagrams of Quantified Statements

• Some students are female.

Is this a true statement?
Females
Students
Females
Students
Is this a true statement?
The negation of a true statement must be false.
No students are female.
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Negations of Qualified Statements (Page 92)
Rome
GHC Classes

• Statement
• All A are B.
• All GHC classes are taught in Rome.
• Negation
• Some A are not B
• Some GHC Classes are not taught in Rome.

False
True
Rome
GHC Classes
The negation of a true statement must be false.
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Negations of Qualified Statements (Page 92)
Rome
GHC Classes

• Statement
• Some A are B.
• Some GHC classes are taught in Rome.
• Negation
• No A are B.
• No GHC classes are taught in Rome.

True
False
Rome
GHC Classes
The negation of a true statement must be false.
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Homework 1- 12 Determine whether or not each
statement is a statement.

• Manhattan Island is less than 1 million inches
  long.
• There are 2,500,000 rivets in the Effiel Tower.
• 9 6 16
• 9 6 64
• What is a nice person like you doing in a place
  like this?

Statement
Statement
Statement
Statement
Not a Statement
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In Exercise 13-16 for the negation of each
statement.

• 13. It is raining.
• It is not raining.

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In Exercise 13-16 for the negation of each
statement.

• 15. Macbeth is not a comedy show on television.
• Is to.
• Macbeth is a comedy show on television.

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3.1 Homework

• Page 93
• 1-48 Odd