# 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