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