Chapter 1 Section 3-1 Statements and Quantifiers

Chapter 1

Section 3-1Statements and Quantifiers

Statements

A statement is defined as a declarative sentence that is either true or false, but not both simultaneously.

2

Statements

Decide if a Statement

1. The zip code for Folsom is 95630

2. 5 + 8 = 13 or 4 – 3 = 5

3. Where are you going today?

4. Some numbers are positive.

3

Compound Statements

• Compound statement: formed by combining two or more statements.

•Connectives: used to form compound statements; and, or, not, and if…then

4

Example: Compound Statements

Decide whether each statement is compound.a) If Amanda said it, then it must be true.b) The gun was made by Smith and Wesson.c) I read the Sacramento Bee and the Wall

Street Journal.

5

Negations

•Statement: “Max has a valuable card”

• Negation: “Max does not have a valuable card.”

•The negation of a true statement is false and the negation of a false statement is true.

6

Example: Forming Negations

Give a negation of each inequality. Do not use a slash symbol.

a) 3

b) 3 2 12

p

x y

7

Symbols

Connectors

Connective Symbol Type of Statement

and Conjunction

or Disjunction

not ~ Negation

8

Example: Translating from Symbols to Words

p: George Bush is President

q: A rectangle is a 3 sided object.

a) p q d) ~(~q)

b) p q e) ~p q

c) ~q f) ~( p ~q)

Determine if T or F and translate into compound statements

9

Negations of Quantified Statements

Statement Negation

All do. Some do not (not all do)

Some do. None do (all do not)

Some do not All do.

None do. Some do.

10

Existential Quantifiers: Some ---- at least one --- there existsUniversal Quantifiers: None / All

Example: Forming Negations of Quantified Statements

Form the negation of each statement.a) Some cats have fleas.b) Some cats do not have fleas.c) No cats have fleas.

11

Sets of Numbers

Natural (counting) {1, 2, 3, 4, …}Whole numbers {0, 1, 2, 3, 4, …}Integers {…,–3, –2, –1, 0, 1, 2, 3, …} Rational numbers May be written as a terminating decimal, like 0.25, or a repeating decimal like 0.333…

Irrational {x | x is not expressible as a quotient of integers} Decimal representations never terminate and never repeat.Real numbers {x | x can be expressed as a decimal}

and are integers and 0p

p q qq

12

Chapter 1

Section 3-5Analyzing Arguments with Euler

Diagrams

13

Logical Arguments

Made up of……..

premises (assumptions, laws, rules, widely held ideas, or observations)

conclusion

argument: premises together with a conclusion

14

Valid and Invalid Arguments

Valid argument: if the fact that all the premises are true forces the conclusion to be true.

Invalid: Not valid. It is called a fallacy.

15

Euler Diagrams

Several techniques can be used to check the validity of an argument. One of these is a visual technique based on Euler Diagrams.

16

17

Euler Diagrams

• Technique for determining the validity of arguments whose premises contain the words all, some, and no.

Euler Diagrams for Quantified Statements

18

Euler Diagrams and Arguments

1. Make an Euler diagram for the first premise.

2. Make an Euler diagram for the second premise on top of the one for the first premise.

3. The argument is valid if and only if every possible diagram illustrates the conclusion of the argument. If there is even one possible diagram that contradicts the conclusion, this indicates that the conclusion is not true in every case, so the argument is invalid.

Example: Using an Euler Diagram to Determine Validity

Is the following argument valid? All cats are animals.Figgy is a cat.Figgy is an animal.

19

Example: Using an Euler Diagram to Determine Validity

All cats are animals.Figgy is a cat.Figgy is an animal.

Animals

Cats

x represents Figgy.

x

20

Now try Sec 2.5 #1

Example: Using an Euler Diagram to Determine Validity

Is the following argument valid? All sunny days are hot.Today is not hot Today is not sunny.

21

Example: Using an Euler Diagram to Determine Validity

Sunny days

xHot days

x represents today

All sunny days are hot.Today is not hot Today is not sunny.

22

Now try Sec 3.5 #6

Example: Using an Euler Diagram to Determine Validity

Is the following argument valid? All cars have wheels.That vehicle has wheels. That vehicle is a car.

23

Example: Using an Euler Diagram to Determine Validity

All cars have wheels.That vehicle has wheels. That vehicle is a car.

Things that have wheels

Cars

x represents “that vehicle”

x ?

x ?

24

Now try Sec 3.5 #3

Example: Using an Euler Diagram to Determine Validity

Is the following argument valid? Some students drink coffee.I am a student . I drink coffee .

25

Example: Using an Euler Diagram to Determine Validity

Some students drink coffee.I am a student . I drink coffee .

People that drink coffee Students

x ?

x?

26

Now try Sec 3.5 #9

Example: An argument can have a true conclusion yet be invalid

Is the following argument valid? All cars have tires.All tires have rubber . All cars have rubber .

27

This is p. 132 #18

Example: An argument can have a true conclusion yet be invalid

All cars have tires.All tires have rubber . All cars have rubber .

Rubber items

Tires

28

This is p. 132 #18Cars

Example: An argument can have a true conclusion yet be invalid

Is the following argument valid? Quebec is northeast of Ottawa.Quebec is northeast of Toronto . Ottawa is northeast of Toronto .

29

This is similar to Sec 3.5 #21

30

Euler Diagrams and the Quantifier “SOME”All people are mortalSome mortals are students.Therefore, some people are students.Step 1: Make an Euler diagram for the first

premise. All people are mortal.

Step 2: Make an Euler diagram for the second premise on top ofthe one for the first premise.

Some mortals are students.

31

continued

Step 3. The argument is valid if and only if every possible diagram illustrates the conclusion of the argument. The arguments conclusion is:

Some people are students. Can you think of another way to draw the diagram for the second premise?

The diagram does not show the “people” circle and the“students” circle intersecting with a dot in the region of intersection. The conclusion does not follow from thepremises. The argument is invalid.

Chapter 1

Section 3-6Analyzing Arguments with Truth

Tables

33

Truth Tables

In section 3.5 Euler diagrams were used to test the validity of arguments. These work well with simple arguments but may not work well with more complex ones. If the words “all,” “some,” or “no” are not present, it may be better to use a truth table than an Euler diagram to test validity.

34

Example: Truth Tables (Two Premises)

Is the following argument valid? If the door is open, then I must close it.The door is open.I must close it.

35

Example: Truth Tables (Two Premises)

If the door is open, then I must close it.The door is open.I must close it.

Let p represent “the door is open” and q represent “I must close it.”

p q

p

q

Solution

36

Example: Truth Tables (Two Premises)

p q p q

Premise and premise implies conclusion

Now Construct the truth table.

37

Valid Argument Forms

Modus Ponens

Modus Tollens

Disjunctive Syllogism

Reasoning by Transitivity

p q

p

q

~

~

p q

q

p

~

p q

p

q

p q

q r

p r

38

Invalid Argument Forms (Fallacies)

Fallacy of the Converse

Fallacy of the Inverse

p q

q

p

~

~

p q

p

q