# Introduction to set theory/Lecture 1

## Introduction

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

Here are the notations and what they mean:

Symbols Meaning
${\displaystyle \land }$ and (conjunction)
${\displaystyle \lor }$ or (nonexclusive disjunction)
${\displaystyle \lnot }$ not (negation)
${\displaystyle \to }$ if then/implies
${\displaystyle \leftrightarrow }$ if and only if

## Truth Table

Truth tables are used to analyze formulae of propositional logic.

### Example

Truth table for ${\displaystyle p\to (q\to p)}$

${\displaystyle p\,\!}$ ${\displaystyle q\,\!}$ ${\displaystyle q\to p}$ ${\displaystyle p\to (q\to p)}$
T T T T
T F T T
F T F T
F F T T

## Tautology

### Definition

A formula ${\displaystyle \theta \,\!}$ of propositional logic is a tautology if only T's occur in the ${\displaystyle \theta \,\!}$ column of the truth table.

### Examples

Truth table for ${\displaystyle (p\to \lnot p)\to \lnot p=\theta }$

${\displaystyle p\,\!}$ ${\displaystyle \lnot p}$ ${\displaystyle p\to \lnot p}$ ${\displaystyle \theta \,\!}$
T F F T
F T T T

Truth table for ${\displaystyle (p\to (q\leftrightarrow \lnot q))\to \lnot p=\theta }$

${\displaystyle p\,\!}$ ${\displaystyle q\,\!}$ ${\displaystyle \lnot q}$ ${\displaystyle q\leftrightarrow \lnot q}$ ${\displaystyle p\to (q\leftrightarrow \lnot q)}$ ${\displaystyle \lnot p}$ ${\displaystyle \theta \,\!}$
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 ${\displaystyle (\lnot p\vee q)\to (p\to q)=\theta }$

${\displaystyle p\,\!}$ ${\displaystyle q\,\!}$ ${\displaystyle \lnot p}$ ${\displaystyle \lnot p\vee q}$ ${\displaystyle p\to q}$ ${\displaystyle \theta \,\!}$
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

### Definition

The proposition formulas ${\displaystyle \varphi \,\!}$ and ${\displaystyle \theta \,\!}$ are tautologically equivalent if ${\displaystyle \varphi \leftrightarrow \theta }$ is a tautology.

### Examples

Contraposition: ${\displaystyle p\to q=\theta }$ is tautologically equivalent to ${\displaystyle \lnot q\to \lnot p=\varphi }$.

${\displaystyle p\,\!}$ ${\displaystyle q\,\!}$ ${\displaystyle \lnot q}$ ${\displaystyle \lnot p}$ ${\displaystyle \theta \,\!}$ ${\displaystyle \varphi \,\!}$ ${\displaystyle \theta \leftrightarrow \varphi }$
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: ${\displaystyle \lnot (p\lor q)=\theta }$ is tautologically equivalent to ${\displaystyle \lnot p\land \lnot q=\varphi }$.

${\displaystyle p\,\!}$ ${\displaystyle q\,\!}$ ${\displaystyle \lnot p}$ ${\displaystyle \lnot q}$ ${\displaystyle p\lor q}$ ${\displaystyle \theta \,\!}$ ${\displaystyle \varphi \,\!}$ ${\displaystyle \theta \leftrightarrow \varphi }$
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: ${\displaystyle \lnot (p\land q)=\theta }$ is tautologically equivalent to ${\displaystyle \lnot p\lor \lnot q=\varphi }$. Truth table for Assignment #1

## Related Resources

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