COS 150Discrete Structures

Assoc. Prof. Svetla Boytcheva

Fall semester 2014

Lecture № 1

Fundamentals of Logic

Code of Ethics

All course materials are adapted version of the textbook: Susanna S. Epp, Discrete Mathematics with Applications, Fourth Edition, Cengage Learning.

Some images – McGrawHill For materials from other sources, please see the

copyright reverence below each slide.

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Use of Logic

Propositional Logic First Order Logic (Quantifiers)/ Predicate Logic Boolean Algebra

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Outline

Logical Form and Logical Equivalence Statements; Compound Statements; Truth Values;

Evaluating the Truth of More General Compound Statements; Logical Equivalence; Tautologies

and Contradictions; Summary of Logical Equivalences

Conditional Statements Logical Equivalences Involving →; Representation of If-Then As Or; The Negation of a Conditional Statement; The Contrapositive of a Conditional Statement; The Converse and Inverse of a Conditional Statement; Only If and

the Biconditional; Necessary and Sufficient Conditions;

Valid and Invalid Arguments Modus Ponens and Modus Tollens; Additional Valid Argument Forms: Rules of Inference;

Use of Logic

In mathematicsGive a precise meaning of statementsDistinguish between valid and invalid argumentsProvide use of “correct” reasoning

Natural language can be very ambiguousHe ate the cookies on the couchThis is a good soupYou could do with a new automobile. How about a

test drive?

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Use of Logic

Natural language can be very ambiguousThis is a good soupYou could do with a new automobile. How about a

test drive? I shot an elephant in my pajamas.

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Use of Logic

In computingDesign new data/knowledge from existing factDesign of computer circuitsConstruction of computer programsVerification of correctness of programs and circuit

designSpecification

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Statements (propositions)

Propositional logic deals with statements and their truth value

Truth values are TRUE (T or 1) and FALSE (F or 0)

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Example Statements

1+1= 2 (statement, T) The moon is made of cheese (statement, F) Go home! (no statement, imperative) What a beautiful garden (no statement,

exclamation) Alice said: “What a beautiful garden ” (statement,

depends on Alice) Y+1=2 (no statement, uncertain)

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Logic connectives

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Logic connectives

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Compound Statements

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Order of Operations

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Example

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Translating from English to Symbols: But and Neither-Nor

“Jim is tall but he is not heavy.” Shakespeare: “Neither a borrower nor a lender be”

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And, Or, and Inequalities

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Truth Tables

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Truth Table of negation

Unary connective

p: “Today is Wednesday”p: “Today is not Wednesday”

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Binary connective

Example p: “Today is Wednesday” q: “It is raining”p q: “Today is Wednesday and it is raining”

Truth Table of conjunction

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Truth Table of disjunction

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Binary connective

Example p: “Today is Friday” q: “Today is Saturday”p q: “Today is Friday or Saturday”

Binary connective

Example pq “You can follow the rules or be disqualified”

Truth Table of exclusive or

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Truth Table of implication

Binary connective

Example p -> q: “If black is white, then we live in Antarctica”

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Implication as a promise

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MORE READING: CHAPTER 2SUSANNA S. EPP, DISCRETE MATHEMATICS WITH APPLICATIONS

Questions?

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