**Relations and Functions** in real life give us the link between any two entities. In our daily life, we come across many patterns and links that characterize relations such as a relation between a father and a son, brother and sister, etc. In mathematics also, we come across many relations between numbers such as a number x is less than y, line l is parallel to line m, etc. Relation and function map elements of one set (domain) to the elements of another set (codomain).

Functions are nothing but special types of relations that define the precise correspondence between one quantity with the other. In this article, we will study how to link pairs of elements from two sets and then define a relation between them, different types of relation and function, and the difference between relations and functions.

### Relation and Function Definition

Relation and function individually are defined as:

**Relations -**A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.**Functions -**A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.

*Z*be the set of integers and

*R*be a relation defined in

*Z*such

*aRb*if (

*a - b*) is divisible by 5. Then number of equivalence classes are

- 2
- 3
- 4
- 5

*R*be a relation defined as

*R*= {

*(x, x), (y, y), (z, z), (x, z)*} in set

*A*= {

*x, y, z*} then relation

*R*is

- reflexive
- symmetric
- transitive
- equivalence

*R*= {(

*x, y*) :

*x*+ 2

*y*= 8} is a relation on

*N*, then range of

*R*is [AI 2014]

- {3}
- {1, 2, 3}
- {1, 2, 3, ....8}
- {1, 2}

*A*= {

*a, b, c*}, then the what is the total numeber of ditinct relations in set

*A*?

*A*= {1, 2, 3} and the relation

*R*= {(1, 2)}, then

*R*is a transitive relation. State true or false and justify your answer.

*R*be a relation in set

*N,*given by

*R*= {(a, b) : a = b - 2, b > 6} then (3, 8) ∈

*R*. State true or false with reason.

*A*= {1, 2, 3}, define a relation

*R*in the set

*A*as follows:

*R*= {(1, 1), (2, 2), (3, ,3), (1, 3)}. Write the ordered pairs to be added to

*R*to make it the smallest equivalence relation. [NCERT Exemplar]

*R*= {(a, a³) : a is a prime number less than 5} be a relation. Find the range of

*R.*[Foriegn 2014]

*A*is called ________ relation, if each element of

*A*is related to itself. [CBSE 2020]

*A*= {1, 2 ,3 }, define relation

*R*on

*A*as

*R*= {(a, b) ∈ A

*X*A : a + b > 6}. Show that

*R*is a universal relation.

*R*in the set

*A*= {5, 6, 7, 8, 9} given by

*R*= {(

*a*,

*b*) : |

*a*-

*b*| is divisible by 2}, is an equivalence realtion. Find all the elements related to the element 6. [Foriegn 2013]

*R*on the set

*A*= {

*x*: 0 ≤

*x*≤ 12} given by

*R*= {(

*a*,

*b*) :

*a*=

*b*} is an equivalence relation, then find the set of all elements related to 1.