Chapter 1 Logics Discrete Mathematics I on 13 March 2012

Applications in Computer ScienceDesign of computer circuits • Construction of computer programs • Verification of the correctness of programs • Constructing proofs automatically • Artificial intelligence

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Contents Propositional Logic

Chapter 1

Logics

Discrete Mathematics I on 13 March 2012

Tran Vinh Tan

Tran Vinh Tan

Contents Propositional LogicContents

1 Propositional Logic

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Contents Propositional Logic

Logic

Definition (Averroes)

Definition (Penguin Encyclopedia)

Definition (Discrete Mathematics - Rosen)

Rules of logic are used to distinguish between valid and invalid

mathematical arguments

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Applications in Computer Science

• Artificial intelligence

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Propositional Logic

Definition

but not both

Examples

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Contents Propositional Logic

Examples

Examples (Which of these are propositions?)

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Contents Propositional LogicNotations

• Propositions are denoted byp, q,

• Thetruth value(”chân trị”) istrue(T) orfalse(F)

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Contents Propositional Logic

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Operators

Exclusive OR- Tuyển loại : p ⊕ q

“p or q (but not both)”

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More Expressions for Implication p → q

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Contents Propositional LogicConditional Statements From p → q

p → q

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Exercise

conditional statement

“If he plays online games too much, his girlfriend leaves him.”

games too much

does not play online games too much

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• “p isnecessary and sufficientfor q”

• “if p then q, andconversely”

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Translating Natural Sentences

Exercise

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Contents Propositional LogicTranslating Natural Sentences

Exercise

He will not run the red light if he sees the police unless he is too

risky

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Construct Truth Table

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• type “dai hoc bach khoa” in Google

• means “daiANDhocANDbachANDkhoa”

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Applications (cont.)

• Logic puzzles

• There are two kinds of inhabitants on an island,knights, who

always tell the truth, and their opposites,knaves, who always

lie You encounter two people A and B What are A and B if

A says“ B is a knight” and B says”The two of us are

opposite types”?

• Bit operations

• 101010011is a bit string of length nine

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Tautology and Contradiction

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Contents Propositional Logic

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Contents Propositional Logic

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Contents Propositional Logic

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Contents Propositional Logic

Constructing New Logical Equivalences

Example

developing a series of logical equivalences

Solution

¬(p ∨ (¬p ∧ q)) ≡ ¬p ∧ ¬(¬p ∧ q) by the second De Morgan law

≡ ¬p ∧ [¬(¬p) ∨ ¬q] by the first De Morgan law

≡ ¬p ∧ (p ∨ ¬q) by the double negation law

≡ (¬p ∧ p) ∨ (¬p ∧ ¬q) by the second distributive law

≡ F ∨ (¬p ∧ ¬q) because ¬p ∧ p ≡ F

≡ ¬p ∧ ¬q by the identity law for F