Introduction to set theory/Lecture 1

Introduction

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We will start the course by introducing Propositional Logic. Even though this is a set theory class and not a logic course, most of notations from the logic courses can be used in set theory. Furthermore, logic is important in various proofs we will encounter in this course.

Notations

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Here are the notations and what they mean:

Symbols Meaning
  and (conjunction)
  or (nonexclusive disjunction)
  not (negation)
  if then/implies
  if and only if

Truth Table

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Truth tables are used to analyze formulae of propositional logic.

Example

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Truth table for  

       
T T T T
T F T T
F T F T
F F T T

Tautology

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Definition

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A formula   of propositional logic is a tautology if only T's occur in the   column of the truth table.

Examples

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Truth table for  

       
T F F T
F T T T

Truth table for  

             
T T F F F F T
T F T F F F T
F T F F T T T
F F T F T T T

Truth table for  

           
T T F T T T
T F F F F T
F T T T T T
F F T T T T

Tautological Equivalence

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Definition

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The proposition formulas   and   are tautologically equivalent if   is a tautology.

Examples

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Contraposition:   is tautologically equivalent to  .

             
T T F F T T T
T F T F F F T
F T F T T T T
F F T T T T T

de Morgan's Law I:   is tautologically equivalent to  .

               
T T F F T F F T
T F F T T F F T
F T T F T F F T
F F T T F T T T

de Morgan's Law II:   is tautologically equivalent to  . Truth table for Assignment #1

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The materials in this course overlap with Introductory Discrete Mathematics for Computer Science, particularly Lesson 1.