The Logic of Quantified Statements

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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**Truth Set of a Predicate**• IfP(x) is a predicate and x has domain D, thenthe 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}**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”.**Truth and Falsity of Universal Statements**• Universal statement “xD, P(x)” is true iffP(x) is true for every x in D; is falseiff P(x) is false for at least one x. (that x is called counterexample) • Ex: 1) Let D be the set of even integers. “xDyD, 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.**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”.**Truth and Falsity of Existential Statements**• Existential statement “xD, P(x)” is true iffP(x) is true for at least one x in D; is falseiff 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.**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”.**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”.**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.**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.**Truth values of multiply quantified statements**Ex(cont.): Determine truth values of the following statements: 1) a student S s.t. language L, S speaksL. 2) a student S s.t.for language group G L in Gs.t. S speaksL. 3) a language group Gs.t. for student SL in Gs.t. S speaks L.**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) is logically equivalent to x s.t. y, ~P(x,y)**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) is logically equivalent to x y s.t. ~P(x,y)**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) ΛQ(x_2) Λ…ΛQ(x_n) ; xD, Q(x)is logically equivalent to Q(x_1) νQ(x_2) ν…νQ(x_n) .**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.**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.**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)”**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.**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.**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 **Rational numbers**Integers 5 Testing validity by diagrams • Ex: All integers are rational numbers; 5 is an integer; 5 is a rational number.**Mathematicians**Logicians Testing validity by diagrams • Ex: All logicians are mathematicians; John is not a mathematician; John is not a logician. John**Math306 class**Math majors Testing validity by diagrams:Converse Error • Ex: All Math majors are taking Math306; Bill is taking Math306; Bill is a Math major. Bill