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$$\left\{ 1 \right\}$$.

If a set consists of only a finite number of members then it is called a finite set; otherwise it is called an infinite set.