1. Lecture 1.3: Predicate Logic
CS 250, Discrete Structures, Fall 2015
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren

2. Lecture 1.3 - Predicate Logic
Course Admin
Slides from last lectures all online
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Any questions?
9/14/2019
Lecture Predicate Logic

3. Lecture 1.3 - Predicate Logic
Outline
Predicate Logic (contd.)
9/14/2019
Lecture Predicate Logic

4. Quantifiers – another way to look at them
To simplify, let us say that the universe of discourse is {x1, x2 }
x P(x)  P(x1)  P(x2)
x P(x)  P(x1)  P(x2)
This is very useful in proving equivalences involving propositions that use quantifiers
Let us see some examples
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Lecture Predicate Logic

5. Lecture 1.3 - Predicate Logic
Laws and Quantifiers
Negation or De Morgan’s Law (we saw this last time):
x P(x)  x P(x)
x P(x)  x P(x)
Distributivity:
x (P(x)  Q(x))  x P(x)  x Q(x)
x (P(x)  Q(x))  x P(x)  x Q(x)
Can’t distribute universal quantifier over disjunciton or existential quantifier over conjunction
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Lecture Predicate Logic

6. Predicates – Free and Bound Variables
A variable is bound if it is known or quantified. Otherwise, it is free.
Examples:
P(x) x is free
P(5) x is bound to 5
x P(x) x is bound by quantifier
Reminder: in a proposition, all variables must be bound.
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Lecture Predicate Logic

7. Predicates – Nested Quantifiers
To bind many variables, use many quantifiers!
Example: P(x,y) = “x > y”; universe of discourse is natural numbers
x P(x,y)
xy P(x,y)
xy P(x,y)
x P(x,3) c)
True proposition
False proposition
Not a proposition
No clue b) b) b)
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Lecture Predicate Logic

8. Predicates – Meaning of Nested Quantifiers
xy P(x,y)
xy P(x,y)
xy P(x,y)
xy P(x,y)
P(x,y) true for all x, y pairs.
P(x,y) true for at least one x, y pair.
For every value of x we can find a y so that P(x,y) is true.
There is at least one x for which P(x,y) is always true.
Suppose P(x,y) = “x’s favorite class is y.”
1 and 2 are commutative
3 and 4 are not commutative
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Lecture Predicate Logic

9. Nested Quantifiers – example
N(x,y) = “x is sitting by y”
xy N(x,y)
xy N(x,y)
xy N(x,y)
xy N(x,y)
False
True
True
False
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Lecture Predicate Logic

10. Today’s Reading and Next Lecture
Rosen 1.5
Again, please start solving the exercises at the end of each chapter section!
Please read 1.6 and 1.7 in preparation for the next lecture
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Lecture Predicate Logic