# 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.