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The Logic of The Logic of Quantified StatementsQuantified Statements

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Definition of PredicateDefinition of Predicate Predicate is a sentence that

contains finite number of variables; becomes a statement when

specific values are substituted for the variables.

Ex: let predicate P(x,y) be “x>2 and x+y=8” when x=5 and y=3,

P(5,3) is “5>2 and 5+3=8”

Domain of a predicate variable is the set of all possible values of the variable.

Ex (cont.): D(x)= ; D(y)=R

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Truth Set of a Predicate• If P(x) is a predicate and

x has domain D,

then the truth set of P(x) is

all xD such that P(x) is true.

(denoted {xD | P(x)} )

• Ex: P(x) is “5<x<9” and D(x)=Z.

Then {xD | P(x)} ={6, 7, 8}

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Universal Statement and Universal Statement and QuantifierQuantifier

Let P(x) be “x should take Math306”;

D={Math majors} be the domain of x.

Then “all Math majors take Math306”

is denoted xD, P(x)

and is called universal statement. is called universal quantifier;

expressions for : “for all”, “for arbitrary”,

“for any”, “for each”.

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Truth and Falsity of Universal Truth and Falsity of Universal StatementsStatements

Universal statement “xD, P(x)” is true iff P(x) is true for every x in D; is false iff P(x) is false for at least one x.

(that x is called counterexample)

Ex: 1) Let D be the set of even integers. “xD yD, x+y is even” is true. 2) Let D be the set of all NBA players. “xD, x has a college degree” is false. Counterexample: Kobe Bryant.

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Existential Statement and Existential Statement and QuantifierQuantifier

Let P(x) be “x(x+2)=24”; D =Z be the domain of x. Then ”there is an integer x such that x(x+2)=24”

is denoted “xD, P(x)” and is called existential statement.

is called existential quantifier; expressions for : “there exists”, “there is a”,

“there is at least one”, “we can find a”.

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Truth and Falsity of Existential Truth and Falsity of Existential StatementsStatements

Existential statement “ xD, P(x)”

is true iff P(x) is true for at least one x in D;

is false iff P(x) is false for all x in D.

Ex: 1) Let D be the set of rational numbers.

“ xD, ” is true.

2) Let D = Z.

“ xD, x(x-1)(x-2)(x-3)<0” is false.

Why? Hint: Use proof by division into cases.

0122 xx

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Negations of Quantified Negations of Quantified StatementsStatements

The negation of universal statement “xD, P(x)” is

the existential statement “xD, ~P(x)” Example: The negation of “All NBA players have college degree” is “There is a NBA player who doesn’t have college degree”.

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Negations of Quantified Negations of Quantified StatementsStatements

The negation of existential statement “ xD, P(x)” is

the universal statement “ xD, ~P(x)”

Example: The negation of

“ x Z such that x(x+1)<0”

is “ x Z, x(x+1) ≥ 0”.

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Statements containing multiple Statements containing multiple quantifiersquantifiers

Ex: 1) xR, yZ such that |x-y|<1.

2) For any building x in the city

there is a fire-station y such that the distance between x and y

is at most 2 miles.

3) xZ such that y[3,5], x<y.

4) There is a student who solved all the problems of the exam correctly.

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Truth values of multiply Truth values of multiply quantified statementsquantified statements

Ex: Students = {Joe, Ann, Bob, Dave} 2 groups of languages:Asian languages={Japanese,Chinese,Korean};European languages={French, German, Italian, Spanish}. Joe speaks Italian and French;Ann speaks German, French and Japanese;Bob speaks Spanish, Italian and Chinese;Dave speaks Japanese and Korean.

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Truth values of multiply Truth values of multiply quantified statementsquantified statements

Ex(cont.): Determine truth values of the following statements:

1) a student S s.t. language L,

S speaks L.

2) a student S s.t. for language group G L in G s.t. S speaks L.

3) a language group G s.t. for student S L in G s.t. S speaks L.

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Negating multiply quantified Negating multiply quantified statementsstatements

Example: The negation of “for xR, yR s.t. “

is logically equivalent to “xR s.t. for yR,

“.Generally,

the negation of x, y s.t. P(x,y) is logically equivalent to

x s.t. y, ~P(x,y)

xy 2

xy 2

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Negating multiply quantified Negating multiply quantified statementsstatements

Example: The negation of “ xR s.t. yZ, x>y“

is logically equivalent to “xR yZ s.t. x≤y“.

Generally,the negation of x s.t. y, P(x,y)

is logically equivalent to x y s.t. ~P(x,y)

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The Relation among The Relation among , , , , ΛΛ, ,

ννLet Q(x) be a predicate;

D={x_1, x_2, …, x_n} be the domain of x.

Then xD, Q(x) is logically equivalent to

Q(x_1) ΛΛ Q(x_2) ΛΛ …… ΛΛ Q(x_n) ; xD, Q(x) is logically equivalent to

Q(x_1) νν Q(x_2) νν …… νν Q(x_n) .

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Universal Conditional Universal Conditional StatementStatement

Definition: x, if P(x) then Q(x) . Example: undergrad S,

if S takes CS300,

then S has taken CS240.

Negation of universal conditional statement:

x such that P(x) and ~Q(x) Ex(cont.): undergrad who takes CS300

but hasn’t taken CS240.

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Variations of universal Variations of universal conditional statementsconditional statements

Variations of xD, if P(x) then Q(x):

• Contrapositive: xD, if ~Q(x) then ~P(x);

• Converse: xD, if Q(x) then P(x);

• Inverse: xD, if ~P(x) then ~Q(x).

The original statement is logically equivalent to its contrapositive.

Converse is logically equivalent to inverse.

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Necessary and Sufficient Necessary and Sufficient ConditionsConditions

• “x, P(x) is a sufficient condition for Q(x)”

means “x, if P(x) then Q(x)”

• “x, P(x) is a necessary condition for Q(x)”

means “x, if Q(x) then P(x)”

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Validity of Arguments with Validity of Arguments with Quantified StatementsQuantified Statements

Argument form is valid means that

for any substitution of the predicates,

if the premises are true,

then the conclusion is also true.

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Valid Argument Forms: Valid Argument Forms: Universal InstantiationUniversal Instantiation

x D, P(x);

aD;

P(a).

• If some property is true

for everything in a domain,

then it is true

for any particular thing in the domain.

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Valid Argument Forms: Valid Argument Forms: Universal InstantiationUniversal Instantiation

Ex: 1) All Italians are good cooks;

Tony is an Italian;

Tony is a good cook.

2) For x,y R,

74.5, 73.5 R

))((22 yxyxyx

)5.735.74)(5.735.74(5.735.74 22

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Testing validity by diagramsTesting validity by diagrams

• Ex: All integers are rational numbers;

5 is an integer;

5 is a rational number.

Rational numbers

Integers

5

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Testing validity by diagramsTesting validity by diagrams

• Ex: All logicians are mathematicians;

John is not a mathematician;

John is not a logician.

Mathematicians

Logicians

John

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Testing validity by diagrams:Testing validity by diagrams:Converse ErrorConverse Error

• Ex: All Math majors are taking Math306;

Bill is taking Math306;

Bill is a Math major.

Math306 class

Math majors

Bill