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.