Discrete Mathematics CS 2610August 24, 2006

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Agenda Last class

Introduction to predicates and quantifiersThis class

Nested quantifiersProofs

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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 PropositionsAn 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)∀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) ∀x LUV(x, Raymond)

Alice likes some carThere is a car that is liked by everyoneEveryone likes some carEveryone loves Raymond

Order matters!!!

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Negation of Nested QuantifiersTo 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 RulesAn Inference Rule:

“∴” means “therefore”

premise 1premise 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)