# 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
$\land$ and (conjunction)
$\lor$ or (nonexclusive disjunction)
$\lnot$ not (negation)
$\to$ if then/implies
$\leftrightarrow$ if and only if

## Truth Table

Truth tables are used to analyze formulae of propositional logic.

### Example

Truth table for $p\to (q\to p)$ $p\,\!$ $q\,\!$ $q\to p$ $p\to (q\to p)$ T T T T
T F T T
F T F T
F F T T

## Tautology

### Definition

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

### Examples

Truth table for $(p\to \lnot p)\to \lnot p=\theta$ $p\,\!$ $\lnot p$ $p\to \lnot p$ $\theta \,\!$ T F F T
F T T T

Truth table for $(p\to (q\leftrightarrow \lnot q))\to \lnot p=\theta$ $p\,\!$ $q\,\!$ $\lnot q$ $q\leftrightarrow \lnot q$ $p\to (q\leftrightarrow \lnot q)$ $\lnot p$ $\theta \,\!$ 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 $(\lnot p\vee q)\to (p\to q)=\theta$ $p\,\!$ $q\,\!$ $\lnot p$ $\lnot p\vee q$ $p\to q$ $\theta \,\!$ 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 $\varphi \,\!$ and $\theta \,\!$ are tautologically equivalent if $\varphi \leftrightarrow \theta$ is a tautology.

### Examples

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

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

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

## Related Resources

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