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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

Lecture 6

Predicate Calculus

Autumn 2012

CSE 311

### Announcements

• Predicates and Quantifiers

• 1.4, 1.5 7th Edition

• 1.3, 1.4 5th and 6th Edition

CSE 311

### 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

• 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

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 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 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

• “Red cats like tofu”

Cat(x)

Red(x)

LikesTofu(x)

CSE 311

### 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

• 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 Quantifiers

• x (P(x)  Q(x)) vsx P(x) x Q(x)

CSE 311

### 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

CSE 311

### 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

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### 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 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