Definition and Representation of Set

Definition of Set:

Set is a well-defined collection of distinct objects (i.e. The nature of the object is the same or in other words object in a set may be anything: number, people, place, letters etc.)
These objects are called the elements or members of the sets.

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 comma and enclosed within curly brackets \left\{  {} \right\}.
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 sets.
A = Set of first five natural numbers.
B = Set of positive even integers less 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 sets.

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