Definition and Representation of Set

Definition of a Set:

A set is a well-defined collection of distinct objects, i.e. the nature of the object is the same, or in other words the objects in a set may be anything: numbers, people, places, letters, etc.
These objects are called the elements or members of the set.

Notation:
A set is usually denoted by capital letters, i.e. A,B,C, \ldots ,X,Y,Z, \ldots etc., and the elements are denoted by small letters, i.e. a,b,c, \ldots ,x,y,z, \ldots etc.

If A is any set and a is the element of set A, then we write a \in A, read as a belongs to A. If A is any set and a is not the element of set A, then we write a \notin A, read as a does not belong to A.

Representation of Sets:
There are three ways to represent a set.

I. Tabular Form:
Listing all the elements of a set, separated by commas and enclosed within curly brackets \left\{ {} \right\}.

Example:
A = \left\{ {1,2,3,4,5} \right\},\,\,B\left\{ {2,4,6, \cdots ,50} \right\},\,\,C\left\{ {1,3,5,7,9, \cdots } \right\}

II. Descriptive Form:
State in words the elements of the set.

Example:
A = Set of first five natural numbers.
B = Set of positive even integers less than or equal to fifty.
C = Set of positive odd integers.

III. Set Builder Form:
Writing in symbolic form the common characteristics shared by all the elements of the set.

Example:
A = \left\{ {x:x \in \mathbb{N} \wedge x \leqslant 5} \right\} N = Natural numbers
A = \left\{ {x:x \in E \wedge 0 < y \leqslant 50} \right\} E = Even numbers
A = \left\{ {x:x \in O \wedge x > 0} \right\} O = Odd numbers