Predicates Quantifiers

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PredicatesQuantifiers
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Learning Objectives

• Understand what are predicates
• Understand which quantifiers exist and their
  meaning
• Understand how to translate textual problems into
  predicates

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Predicates and Quantifiers

• Predicate Statement involving variables x gt 5
  x y z
• x greater than 3 Subject the variable x,
  Predicate a property ( greater than 3)
• x y z Subjects x, y and z, Predicate
  the sum of x and y is z.
• P(x), P(x,y,z) Propositional functions.
• P(4) is a proposition, P(4,5,1) is a
  proposition. Note that P(4) T while P(4,5,1)
  F.
• Universe of discourse either specified or
  understood from the context. For example,
  integers, real numbers, people in a well defined
  group etc.

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Predicates and Quantifiers

• Universe of discourse either specified or
  understood from the context. For example,
  integers, real numbers, people in a well defined
  group etc.
• The universal quantifier ? ?xP(x) T if for
  every x in the universe of discourse the
  proposition P(x) is true. ?xP(x) F if there is
  at least one x for which the proposition P(x) is
  false.
• The existential quantifier ? ?xP(x) T if for
  at least one x in the universe of discourse the
  proposition P(x) is true. ?xP(x) F if there is
  no x for which the proposition P(x) is true.

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Predicates and Quantifiers

• Examples
• Let P(x) be the statement x lt 2. Universe of
  discourse N?xP(x) F ?xP(x) T

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Predicates and Quantifiers

• Translating logical statements into English.
• Universe of discourse Students at UWT.
• C(x) x owns a computer F(x,y) x and y
  are friends.
• ?x(C(x) ? ?y(C(y) ? F(x,y))) Every student at
  UWT either owns a computer or has a friend who
  owns a computer.

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Predicates and Quantifiers

• Binding Quantifying a variable or assigning to
  it a value.
• P(x,y) be the statement (xy)2 x2 y2.
  In the statement ?xP(x,1) both x and y are bound.
  Its truth value is F.
• Q(x,y) be the statement x2 y2 2xy. In the
  statement ?x?yQ(x,y) both variables are bound.
  ?x?yQ(x,y) T (let y x).
• Note that ?y?x Q(x,y) F. So the order of
  quantifiers cannot be ignored.

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Predicates and Quantifiers

• Mathematical sentences Let B(x,y) x is ys
  best friend.
• ?x?y?z(B(x,y)?((z ? y) ? ?B(x,z))
• Some student at UWT lives in Seattle and never
  visited any other state of the Union.
• P(x) x lives in Seattle
• Q(x) x is a student at UWT
• U(y) y is a state of the Union
• V(x,y) x visited the state y.
• ?x(P(x) ? Q(x) ?(?y(?(x,y) ? (y Washington)))

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Predicates and Quantifiers

• Negations (De Morgans Laws for quantifiers)
• ? ? x P(x) ? ?x ? P(x)
• ? ? x P(x) ? ? x ? P(x)

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Predicates and Quantifiers

• Other De Morgans Laws (in propositional logic)

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Predicates and Quantifiers

• Examples
• F(x) x is a fleegle
• S(x) x is a snurd
• T(x) x is a thingamabob
• Ufleegles, snurds, thingamabobs
• (Note the equivalent form using the existential
  quantifier is also given)
• Everything is a fleegle
• ? x F( x) ? ? ? x ? F( x)

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Predicates and Quantifiers

• Nothing is a snurd.
• ? x ? S( x) ? ? ? x S( x)
• All fleegles are snurds.
• ? x F(x) --gt S(x)
• ? ? x ? F(x) V S(x)
• ? ? x ? F(x) ? S(x)
• ? ? ? x F(x) ? S(x)

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Predicates and Quantifiers

• Some fleegles are thingamabobs.
• No snurd is a thingamabob.
• If any fleegle is a snurd then it is also a
  thingamabob.