# Review of Sets

A set is a collection of distinct and well-defined objects.

In general, capital letters like ** A, B, G, S, T,** etc., are used to denote

**sets**. The words aggregate, class or collection are also often used in place of the word

**set**. But the use of the word set is common. The concept of the word

**set**as described above is considered to be primitive in mathematics.

If $$S$$ denotes a set and $$x$$ is a member (or an element) of it, then we write this fact in notation form as $$x \in S$$. We read $$x \in S$$ as $$x$$ is a member (or an element) of $$S$$, or more precisely “$$x$$ belongs to $$S$$.” When a certain object does not belong to set $$S$$ we write $$x \notin S$$. A set is determined by its members. Therefore, it can either be expressed by writing its various members or by assigning a certain property regarding its members so as all the members of the set are known.

The well defined, distinct and distinguishable objects (or members) in a set may be anything. When a set is described by listing all of its members, for example the set of numbers **1, 5, 17, 257**, we write it by enclosing these members within a pair of curly brackets such as $$\left\{ {1,5,17,257} \right\}$$. The members within the brackets may be written in any order. In case a set is described by assigning a certain property $$P\left( x \right)$$ regarding its members $$x$$, we shall write it as$$\left\{ {x:P\left( x \right)} \right\}$$. For example, $$\left\{ {x:x = {n^8} + 1,{\text{ }}n = 0,2,4,16} \right\}$$ is a set of positive integers **1, 257, 65537, 4294967297**. These are two common ways of describing a set. When the specific reference to members of a set is implied, then a single capital letter may be used to denote the set. The distinctness of the members of a set holds in a conclusive sense. In specifying a set, either by means of listing its members or by assigning some property regarding its members, if a member repeats, then in the set only one representative of the repeated member is ultimately included. Thus,$$\left\{ {1,1} \right\}$$ is actually$$\left\{ 1 \right\}$$.

**finite set**; otherwise it is called an

**infinite set**.