## What is relation and types of relations

**What is relation (Mathematics) and types of relations:-**

Relation (mathematics) is same as in real life, it is define as two things connected to each other or it is the property of things to connect in a certain manner.

Example:

- Akbar was the son of Humayun
- 25 is the multiple of 5.
- In a triangle ABC, AB is the base.

**Domain and Range of a Relation:** If R is a relation defined between set A and set B then set of first elements of ordered pair is known as Domain and set of second elements of the ordered pair is known as Range means Domain of R is {a|(a,b) є R} and Range of R is {b|(a,b) є R}.

Example :

If R = { (1,2),(1,4),(1,6),(2,2),(2,4),(2,6)} ( A relation between A and B )

So Domain of R = {1, 2} = A and Range of R = {2,4,6} = B

**Types of Relation:-**

**1.**** Reflexive Relation:** If every element of a set A is related to itself In a Relation R it means that Relations R is reflexive relation.

(a,a) є R for all a є A.

2. **Symmetric Relation:** A relation R defined in a set A, if element ‘a’ is related to ‘b’ and ‘b’ is also related to ‘a’ in the same way then this relation is known as Symmetric Relation.

Relation R is symmetric if (a,b) є R => (b,a) є R for all a,b є A.

**3.** **Anti symmetric Relation:** A relation R defined in a set A. if element ‘a’ is related to ‘b’ and ‘b’ is related to ‘a’, both will be true only when a=b, then this relation is known as anti symmetric Relation.

(a,b) є R and (b,a) є R => a = b, for all a,b, є A

**4.** **Transitive Relation:** A relation R defined in a set A, if element ‘a’ is related to ‘b’ and ‘b’ is related to element ‘c’ from this if ‘a’ has relation to c then this relation is known as Transitive relation.

If (a,b)є R and (b,c)є R => (a,c)єR , for all a,b,c, є A

**5.** **Equivalence Relation: **we call equivalence relation to a Relation R if

R is reflexive

R is Symmetric

R is Transitive

**6.** **Partial order Relation:** we call Partial order relation to a Relation R if

R is reflexive

R is Anti Symmetric

R is Transitive

**7.** **Total order Relation:** we call Partial order relation to a Relation R if

R is Partial Order relation

And for every a,b є A either (a,b) є R or (b,a) є R or a = b are true.