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1 The Logic of Quantified Statements

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2 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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3 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 x D such that P(x) is true. (denoted {x D | P(x)} ) Ex: P(x) is “5

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4 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 x D, 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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5 Truth and Falsity of Universal Statements Universal statement “ x D, 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. “ x D y D, x+y is even” is true. 2) Let D be the set of all NBA players. “ x D, x has a college degree” is false. Counterexample: Kobe Bryant.

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6 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 “ x D, 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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7 Truth and Falsity of Existential Statements Existential statement “ x D, 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. “ x D, ” is true. 2) Let D = Z. “ x D, x(x-1)(x-2)(x-3)<0” is false. Why? Hint: Use proof by division into cases.

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8 Negations of Quantified Statements The negation of universal statement “ x D, P(x)” is the existential statement “ x D, ~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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9 Negations of Quantified Statements The negation of existential statement “ x D, P(x)” is the universal statement “ x D, ~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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10 Statements containing multiple quantifiers Ex: 1) x R, y Z 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 most2 miles. 3) x Z such that y [3,5], x

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11 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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12 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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13 Negating multiply quantified statements Example: The negation of “for x R, y R s.t. “ is logically equivalent to “ x R s.t. for y R, “. Generally, the negation of x, y s.t. P(x,y) is logically equivalent to x s.t. y, ~P(x,y)

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14 Negating multiply quantified statements Example: The negation of “ x R s.t. y Z, x>y“ is logically equivalent to “ x R y Z 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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15 The Relation among , , Λ, ν Let Q(x) be a predicate; D={x_1, x_2, …, x_n} be the domain of x. Then x D, Q(x) is logically equivalent to Λ Λ … Λ Q(x_1) Λ Q(x_2) Λ … Λ Q(x_n) ; x D, Q(x) is logically equivalent to ν ν … ν Q(x_1) ν Q(x_2) ν … ν Q(x_n).

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16 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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17 Variations of universal conditional statements Variations of x D, if P(x) then Q(x): Contrapositive: x D, if ~Q(x) then ~P(x); Converse: x D, if Q(x) then P(x); Inverse: x D, 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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18 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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19 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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20 Valid Argument Forms: Universal Instantiation x D, P(x); a D; 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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21 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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22 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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23 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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24 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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