# 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.

Notation:
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 $\left\{ {} \right\}$.
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 sets.
Example:
$A =$ 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.

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

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