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**The Logic of Quantified Statements**

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**Definition 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 Quantifier**

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 Statements**

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 Quantifier**

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 Statements**

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.

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**Negations of Quantified Statements**

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 Statements**

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 quantifiers**

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 quantified 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 quantified 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 statements**

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

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**Negating multiply quantified statements**

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

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**The Relation among , , Λ, ν**

Let 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) Λ Q(x_2) Λ … Λ Q(x_n) ; xD, Q(x) is logically equivalent to Q(x_1) ν Q(x_2) ν … ν Q(x_n) .

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**Universal Conditional Statement**

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 conditional 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 Conditions**

“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 Quantified 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: Universal 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: Universal 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

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**Testing 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 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: Converse Error**

Ex: All Math majors are taking Math306; Bill is taking Math306; Bill is a Math major. Math306 class Math majors Bill

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