Propositional Function - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Discrete Mathematics

CS 2610

August 21, 2008

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Agenda

Nested quantifiers Rules of inference Proofs

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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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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 and inference

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? Logical Deduction.

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Proofs: Inference Rules

An 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: 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: 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: Proof by Cases

I have taken CS2610 or I have taken CS1301. If I have taken CS2610 then I can register for CS If I have taken CS1301 then I can register for CS

∴ I can register for CS

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, rather than deductive reasoning!