Cse 311 foundations of computing i
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CSE 311 Foundations of Computing I. Lecture 6 Predicate Calculus Autumn 2012. Announcements. Reading assignments Predicates and Quantifiers 1.4, 1.5 7 th Edition 1.3, 1.4 5 th and 6 th Edition. Predicate Calculus. Predicate or Propositional Function

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CSE 311 Foundations of Computing I

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Cse 311 foundations of computing i

CSE 311 Foundations of Computing I

Lecture 6

Predicate Calculus

Autumn 2012

CSE 311


Announcements

Announcements

  • Reading assignments

    • Predicates and Quantifiers

      • 1.4, 1.5 7th Edition

      • 1.3, 1.4 5th and 6th Edition

CSE 311


Predicate calculus

Predicate Calculus

  • Predicate or Propositional Function

    • A function that returns a truth value

  • “x is a cat”

  • “x is prime”

  • “student x has taken course y”

  • “x > y”

  • “x + y = z” or Sum(x, y, z)

    NOTE: We will only use predicates with variables or constants as arguments.

Prime(65353)

CSE 311


Quantifiers

Quantifiers

  • x P(x) : P(x) is true for every x in the domain

  • x P(x) : There is an x in the domain for which P(x) is true

Relate  and  to  and 

CSE 311


Statements with quantifiers

Statements with quantifiers

Domain:

Positive Integers

  • x Even(x)

  • x Odd(x)

  • x (Even(x)  Odd(x))

  • x (Even(x)  Odd(x))

  • x Greater(x+1, x)

  • x (Even(x)  Prime(x))

Even(x)

Odd(x)

Prime(x)

Greater(x,y)

Equal(x,y)

CSE 311


Statements with quantifiers1

Statements with quantifiers

Domain:

Positive Integers

  • xy Greater (y, x)

  • xy Greater (x, y)

  • xy (Greater(y, x)  Prime(y))

  • x (Prime(x)  (Equal(x, 2)  Odd(x))

  • xy(Sum(x, 2, y)  Prime(x)  Prime(y))

Even(x)

Odd(x)

Prime(x)

Greater(x,y)

Equal(x,y)

Sum(x,y,z)

CSE 311


Statements with quantifiers2

Statements with quantifiers

Domain:

Positive Integers

Even(x)

Odd(x)

Prime(x)

Greater(x,y)

Sum(x,y,z)

  • “There is an odd prime”

  • “If x is greater than two, x is not an even prime”

  • xyz((Sum(x, y, z)  Odd(x)  Odd(y)) Even(z))

  • “There exists an odd integer that is the sum of two primes”

CSE 311


English to predicate calculus

English to Predicate Calculus

  • “Red cats like tofu”

Cat(x)

Red(x)

LikesTofu(x)

CSE 311


Goldbach s conjecture

Goldbach’s Conjecture

  • Every even integer greater than two can be expressed as the sum of two primes

Even(x)

Odd(x)

Prime(x)

Greater(x,y)

Equal(x,y)

  • xyz ((Greater(x, 2)  Even(x))  (Equal(x, y+z)  Prime(y)  Prime(z))

Domain:

Positive Integers

CSE 311


Scope of quantifiers

Scope of Quantifiers

  • Notlargest(x) y Greater (y, x) z Greater (z, x)

    • Value doesn’t depend on y or z “bound variables”

    • Value does depend on x “free variable”

  • Quantifiers only act on free variables of the formula they quantify

  •  x ( y (P(x,y)  x Q(y, x)))

  • CSE 311


    Scope of quantifiers1

    Scope of Quantifiers

    • x (P(x)  Q(x)) vsx P(x) x Q(x)

    CSE 311


    Nested quantifiers

    Nested Quantifiers

    • Bound variable name doesn’t matter

      •  x  y P(x, y)  a  b P(a, b)

    • Positions of quantifiers can change

      •  x (Q(x)  y P(x, y))  x  y (Q(x)  P(x, y))

    • BUT: Order is important...

    CSE 311


    Quantification with two variables

    Quantification with two variables

    CSE 311


    Negations of quantifiers

    Negations of Quantifiers

    • Not every positive integer is prime

    • Some positive integer is not prime

    • Prime numbers do not exist

    • Every positive integer is not prime

    CSE 311


    De morgan s laws for quantifiers

    De Morgan’s Laws for Quantifiers

    x P(x) x P(x)

     x P(x) x P(x)

    CSE 311


    De morgan s laws for quantifiers1

    De Morgan’s Laws for Quantifiers

    x P(x) x P(x)

     x P(x) x P(x)

    “There is no largest integer”

    •   x  y ( x ≥ y)

    • x  y ( x ≥ y)

    • x  y  ( x ≥ y)

    • x  y (y > x)

    “For every integer there is a larger integer”

    CSE 311


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