Set Theory

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 $\mathbb {C}$.