2018 | OriginalPaper | Chapter

# 2. Set Theory

Published in:
The Discrete Math Workbook

## Abstract

One of the fundamental concepts of mathematics is the notion of a set. A set is a collection of objects which we conceive as a whole. These objects are called elements of a set. The belonging of some element a to the set A can be denoted as follows: \(a\in A\). This record reads as follows: “a is an element of the set A” or “element a belongs to the set A.” If a does not belong to A, then we write \(a\notin A\).:

There are several ways to denote which elements belong to a set; the most common are the following:

(1) enumerating the elements \(A=\{a_1,a_2,\dots , a_n\}\). The elements of the set A are enclosed in braces and separated by commas.

(2) using the characteristic predicate \(A=\{x:P(x)\}\).

The characteristic predicate is a statement, allowing one to establish the fact that the object x belongs to the set A. If for some x the predicate P(x) takes a true value, then \(x\in A\); otherwise \(x\notin A\).

Some widely used sets have specific notations:

\(\mathbb {N}=\{1,2,3,\ldots \}\) is the set of natural numbers;

\(\mathbb {Z}=\{0,\pm 1,\pm 2,\pm 3,\ldots \}\) is the set of integers; :

\(\mathbb {Q}=\left\{ p/q:p, q\text { are integers},\;q\ne 0\right\} \) is the set of rational numbers.

A set of real numbers is denoted by \(\mathbb {R}=(-\infty ,+\infty )\) and of complex numbers by \(\mathbb {C}\).