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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