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Predicates and Quantified Statements M260 3.1, 3.2
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Predicate Example James is a student at Southwestern College. P(x,y) x is a student at y.
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Predicate Truth Set P(x) is a predicate and x has domain D. The truth set of P(x): {x D P(x)}
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For all For every For arbitrary For any For each Given any
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Universal Statement x D, Q(x) When true When false Counter examples
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There exists We can find a There is at least one For some For at least one
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Existential Statement x D such that Q(x) When true When false
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Translation Examples x , x 2 0 x , x 2 -1 m such that m 2 =m
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Translation Examples If a real number is an integer then it is a rational number All bytes have eight bits No fire trucks are green.
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Formal and Informal For all polygons p, if p is a square, then p is a rectangle. For all squares p, p is a rectangle. There exists a number n such that n is prime and n is even. There exists a prime number n such that n is even.
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Trailing Quantifier For all squares p, p is a rectangle. p is a rectangle for any square p. There exists a prime number n such that n is even. n is even for some prime number n.
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Equivalent forms of and x U if P(x) then Q(x) x D, Q(x) Where D is all x such that P(x)
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Implicit Quantification If n is a number, then it is a rational number If x>2 then x 2 >4. x>2 x 2 >4
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and P(x) Q(x) means that the truth set of P(x) is contained in the truth set of Q(x). P(x) Q(x) means P(x) and Q(x) have identical truth sets.
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Negation of Quantified Statements For all x in D, Q(x) There exists x in D such that ~Q(x). “all are” versus “some are not”
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Negation of Quantified Statements There exists x in D such that Q(x) For all x in D, ~Q(x). “some are” versus “all are not”
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Try These ~(For every prime p, p is odd) ~(There exists a triangle T, such that the sum of the angles of T equals 200 degrees)
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No Politicians are Honest Formal Formal negation Informal negation
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All computer programs are finite. Formal Formal negation Informal negation
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Some dancers are over 40 Formal Formal negation Informal negation
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~( x, P(x) Q(x)) x such that ~(P(x) Q(x)) x such that P(x) ~Q(x)
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More on Universal Conditional x U if P(x) then Q(x) Contrapositive Converse Conditional
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Necessary and Sufficient Conditions, Only If x, r(x) is a sufficient condition for s(x). x, r(x) is a necessary condition for s(x). x, r(x) only if s(x).
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