The Concept of a Set

The basic concepts of set theory were created and developed in the late $19\text{th}$ century by German mathematician Georg Cantor $\left(1845-1918\right).$

According to Cantor's definition, a set is any collection of well defined objects, called the elements or members of the set.

Sets are usually denoted by capital letters $\left(A,B,X,Y,\dots \right).$ The elements of the set are denoted by small letters $\left(a,b,x,y,\dots \right).$

If $X$ is a set and $x$ is an element of $X,$ we write $x\in X.$ The symbol $\in$ was introduced by Italian mathematician Giuseppe Peano $\left(1858-1932\right)$ and is an abbreviation of the Greek word $ϵ\sigma \tau \iota$ - "be".

If $y$ is not an element of $X,$ we write $y\notin X.$

Defining Sets

There are two basic ways of describing sets - the roster method and set builder notation.

Roster Method

In roster notation, we just list the elements of a set. The elements are separated by commas and enclosed in curly braces. For example, $X=\left\{1,5,17,286\right\}$ or the set of natural numbers including zero ${\mathbb{N}}_0=\left\{0,1,2,3,4,\dots \right\}.$

In a listing of the elements of a set, each distinct element is listed only once. The order in which elements are listed does not matter. For example, the following sets are equal:

When writing infinite sets and there is a clear pattern to the elements, an ellipsis (three dots) can be used.

Set Builder Notation

In set builder notation, we define a set by describing the properties of its elements instead of listing them. This method is especially useful when describing infinite sets.

The notation includes on or more set variables and a rule defining which elements belong to the set and which are not. The rule is often expressed in the form of a predicate. The set variable and rule are separated by a colon ":" or vertical slash "|".

Examples:

1. The set of all uppercase letters of the English alphabet.
   $$U=\left\{x|x\text{ is an uppercase letter of the English alphabet}\right\}$$
2. The set of all prime numbers $p$ less than $1000.$
   $$P=\{p|p\text{ is a prime number and }p<1000\}$$
3. The set of all $x$ such that $x$ is a negative real number.
   $$X=\left\{x|x\in \mathbb{R}\text{ and }x<0\right\}\text{ or }X=\left\{x\in \mathbb{R}|x<0\right\}$$
4. The set of all internal points $\left(x,y\right)$ lying within the circle of radius $1$ centered at the origin.
   $$C=\left\{x,y\in \mathbb{R}|x^2+y^2<1\right\}$$

Universal and Empty Sets

A set which contains all the elements under consideration is called the universal set and is denoted as $U.$ The universal set is problem specific. For example, if a question is related to numbers, the universal set can be either all natural numbers $\left(U=\mathbb{N}\right),$ or all integers $\left(U=\mathbb{Z}\right),$ or all rational numbers $\left(U=\mathbb{Q}\right),$ etc. - depending on the context.

There is a special name for the set which contains no elements. It is called the empty set and is denoted by the symbol $\varnothing .$ There is only one empty set.

Subsets

A set $A$ is a subset of the set $B$ if every element of $A$ is also an element of the set $B.$ A subset is denoted by the symbol $\subseteq ,$ so we write $A\subseteq B.$ If a set $C$ is not a subset of $B,$ we write: $C⊈B.$

The empty set $\varnothing$ is a subset of every set, including itself.

The sets $A$ and $B$ are equal if simultaneously $A\subseteq B$ and $B\subseteq A.$ Notation: $A=B$.

A set $A$ is called a proper subset of $B$ if $A\subseteq B$ and $A\ne B.$ In this case, we write $A\subset B.$ Some authors use the symbol $\subset$ to indicate any (proper and improper) subsets.

The power set of any set $A$ is the set of all subsets of $A,$ including the empty set and $A$ itself. It is denoted by $\mathcal{P}\left(A\right)$ or $2^A.$ If the set $A$ contains $n$ elements, then the power set $\mathcal{P}\left(A\right)$ has $2^n$ elements.