Discrete Mathematics
CS 2610
August 24, 2006
Agenda
Last class

Introduction to predicates and quantifiers
This class


Nested quantifiers
Proofs
2
Overview of last class
A predicate P, or propositional function, is a
function that maps objects in the universe of
discourse to propositions
Predicates can be quantified using the universal
quantifier (“for all”)  or the existential quantifier
(“there exists”) 
Quantified predicates can be negated as follows


x P(x)  x P(x)
x P(x)  x P(x)
Quantified variables are called “bound”
Variables that are not quantified are called “free”
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Predicate Logic and Propositions
An expression with zero free variables is an
actual proposition
Ex. Q(x) : x > 0, R(y): y < 10
 x Q(x)  y R(y)
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Nested Quantifiers
When dealing with polyadic predicates, each argument may be
quantified with its own quantifier.
Each nested quantifier occurs in the scope of another
quantifier.
Examples: (L=likes, UoD(x)=kids, UoD(y)=cars)
 xy L(x,y) reads x(y L(x,y))
 xy L(x,y) reads x(y L(x,y))
 xy L(x,y) reads x(y L(x,y))
 xy L(x,y) reads x(y L(x,y))
Another example

x (P(x)  y R(x,y))
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Examples
If L(x,y) means x likes y, how do you read the
following quantified predicates?
y L(Alice,y)
yx L(x,y)
xy L(x,y)
Alice likes some car
There is a car that is liked by everyone
Everyone likes some car
x LUV(x, Raymond) Everyone loves Raymond
Order matters!!!
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Negation of Nested Quantifiers
To negate a quantifier, move negation to the right,
changing quantifiers as you go.
Example:
xyz P(x,y,z)  x y z P(x,y,z).
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Proofs (or Fun & Games Time)
Assume that the following statements are true:
I have a total score over 96.
If I have a total score over 96,
then I get an A in the class.
What can we claim?
I get an A in the class.
How do we know the claim is true?
Elementary my dear Watson!
Logical Deduction.
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Proofs
• A theorem is a statement that can be proved to be
true.
• A proof is a sequence of statements that form an
argument.
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Proofs: Inference Rules
An Inference Rule:
premise 1
premise 2 …
 conclusion
“” means “therefore”
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Proofs: Modus Ponens
•I have a total score over 96.
•If I have a total score over 96, then I get an A for the class.
 I get an A for this class
p
pq
q
Tautology:
(p  (p  q))  q
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Proofs: Modus Tollens
•If the power supply fails then the lights go out.
•The lights are on.
 The power supply has not failed.
q
pq
 p
Tautology:
(q  (p  q))  p
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Proofs: Addition
•I am a student.
 I am a student or I am a visitor.
p
pq
Tautology:
p  (p  q)
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Proofs: Simplification
•I am a student and I am a soccer player.
 I am a student.
pq
p
Tautology:
(p  q)  p
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Proofs: Conjunction
•I am a student.
•I am a soccer player.
 I am a student and I am a soccer player.
p
q
pq
Tautology:
((p)  (q))  p  q
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Proofs: Disjunctive Syllogism
I am a student or I am a soccer player.
I am a not soccer player.
 I am a student.
pq
q
p
Tautology:
((p  q)  q)  p
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Proofs: Hypothetical Syllogism
If I get a total score over 96, I will get an A in the course.
If I get an A in the course, I will have a 4.0 semester average.
  If I get a total score over 96 then

I will have a 4.0 semester average.
pq
qr
pr
Tautology:
((p  q)  (q  r))  (p  r)
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Proofs: Resolution
I am taking CS1301 or I am taking CS2610.
I am not taking CS1301 or I am taking CS 1302.
 I am taking CS2610 or I am taking CS 1302.
pq
pr
qr
Tautology:
((p  q )  ( p  r))  (q  r)
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Proofs: Proof by Cases
I have taken CS2610 or I have taken CS1301.
If I have taken CS2610 then I can register for CS2720
If I have taken CS1301 then I can register for CS2720
 I can register for CS2720
pq
pr
qr
 r
Tautology:
((p  q )  (p  r)  (q  r))  r
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Fallacy of Affirming the Conclusion
•If you have the flu then you’ll have a sore throat.
•You have a sore throat.
 You must have the flu.
q
pq
p
Abductive reasoning
Fallacy:
(q  (p  q))  p
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Fallacy of Denying the Hypothesis
•If you have the flu then you’ll have a sore throat.
•You do not have the flu.
 You do not have a sore throat.
p
pq
 q
Fallacy:
(p  (p  q))  q
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Inference Rules for Quantified Statements
x P(x)
 P(c)
Universal Instantiation
P(c)___
 x P(x)
Universal Generalization
(for an arbitrary object c from UoD)
(for any arbitrary element c from UoD)
x P(x)
 P(c)
Existential Instantiation
P(c)__
 x P(x)
Existential Generalization
(for some specific object c from UoD)
(for some object c from UoD)
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