Discrete Mathematics CS 2610 August 24, 2006. 2 Agenda Last class Introduction to predicates and...

Discrete Mathematics CS 2610

August 24, 2006

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Agenda Last class Introduction to predicates and quantifiers

This class Nested quantifiers Proofs

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Overview of last classA predicate P, or propositional function, is a

function that maps objects in the universe of discourse to propositionsPredicates 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 QuantifiersWhen 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) xy L(x,y) reads x(y L(x,y)) xy L(x,y) reads x(y L(x,y)) xy L(x,y) reads x(y L(x,y)) xy 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)yx L(x,y) xy L(x,y)

x LUV(x, Raymond)

Alice likes some carThere is a car that is liked by everyone

Everyone likes some car

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:

xyz 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 RulesAn Inference Rule:

“” means “therefore”

premise 1 premise 2 … conclusion

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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.

Tautology:

(q (p q)) p

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 CS2720If 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.

Fallacy:

(q (p q)) p

q

p q

p

Abductive reasoning

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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.

Fallacy:

(p (p q)) q

p

p q

q

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Inference Rules for Quantified Statements

x P(x) P(c)

x P(x) P(c)

P(c)___ x P(x)

Universal Instantiation

(for an arbitrary object c from UoD)

Universal Generalization

(for any arbitrary element c from UoD)

Existential Instantiation

(for some specific object c from UoD)

P(c)__ x P(x)

Existential Generalization

(for some object c from UoD)