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CSE 311 Foundations of Computing I

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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  1. CSE 311 Foundations of Computing I Lecture 6 Predicate Calculus Autumn 2012 CSE 311

  2. Announcements • Reading assignments • Predicates and Quantifiers • 1.4, 1.5 7th Edition • 1.3, 1.4 5th and 6th Edition CSE 311

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. English to Predicate Calculus • “Red cats like tofu” Cat(x) Red(x) LikesTofu(x) CSE 311

  9. 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

  10. 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

  11. Scope of Quantifiers • x (P(x)  Q(x)) vsx P(x) x Q(x) CSE 311

  12. 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

  13. Quantification with two variables CSE 311

  14. 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

  15. De Morgan’s Laws for Quantifiers x P(x) x P(x)  x P(x) x P(x) CSE 311

  16. 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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