**CONTENT LIST**

** Introduction**

In our daily life we come accorss many relations between any two entities like

relation between a father and a son,

relation between a Mother and a son,

relation between a wife and a husband,

relation between a brother and sister,

relation between a student and class,

relation between a student and bag, etc.

In mathematics also, many relations are found between numbers such as

a number x is less than y,

line l is parallel to line m,

x is one more than y,

x is square of y etc.

In this article , we shall learn more about special relations called functions.

Relation and function map elements of one set (domain) to the elements of another set (codomain).Functions is a special types of relations that define the precise correspondence between one quantity with the other. By this artilce Relation and function We shall learn ,how to link pairs of elements from two sets and then define a relation between them, different types of relations and functions, and the difference between relation and function.

** Relation **

A relation R from a non-empty set A to a non empty set B is a subset of the Cartesian product set 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.

The set of all first elements in a relation R, is called the domain of the relation R, and the set of all second elements called images, is called the range of R.

If A and B are two non-empty sets, then the relation R from A to B is a subset of A x B, i.e., R ⊆ A x B.

Ex: Let, A = {1,2,3} and B = {2,3,4}

A X B = {(1,2),(1,3),(1,4);(2,2),(2,3),(2,4);(3,2),(3,3),(3,4)}

R = {(1,2),(2,3),(3,4)} (R:y = x+1)

**Representation of Relation **

A relation may be represented either by the Roster form or by the set builder form, or by an arrow diagram which is a visual representation of a relation.

Consider an example of two sets A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5}.
**The relation is that the elements of A are the square of the elements of B.**

In set-builder form, R = {(x, y): x is the square of y, x ∈ A and y ∈ B}.

In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}.

In arrow diagram form

** Number of Relations **

If n (A) = p, n (B) = q; then the n (A × B) = pq and the total number of possible relations from the set A to set B = (2)^pq

** Functions **

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.

f:A→B show a function from A to B.

For function:: Element of set A has an image in set B.

For function::Element of set A has only image in set B.

b∈B is called image of a∈A under function f .

a∈A is called preimage of b∈B under function f .

Let us take an example Set A={a,b,c,d} Set B={Aarya, Aadhya,Beena,Cema,Eklavya}The above daigram shows relation between set A and B but it is not a function due to following reasons:

1. a ∈ A, has 2 images in set B.

2. There is no image of d ∈ A in set B.

All functions are relation, but all relation are not function

** Example:**

Let A={1,2,3} & B={2,3,4} and f1,f2 & f3 are three subsets of AXB such that f1={(1,2),(2,3),(3,4)} f2={(1,2),(1,3),(2,3),(3,4)} & f3={(1,3),(2,4)} . Identify which relation is showing function.

In relation f1: Each element of set A has uniuqe image in set B. so relation f1 is a function.

In relation f2: An element 1 of set A has two image 2 & 3 in set B. so relation f1 is not a function.

In relation f3: An element 3 of set A has two no image in set B. so relation f3 is not a function.

** Domain Co-domain and Range of function **

If there are two non empty set A and B then a function f is expressed as the set of ordered pairs of AXB in such a way that Each element of set A has uniuqe image in set B & no two pre image of any element of set B in set A .

**Domain of f** = Set of all first elements of memeber of f OR Domain of f={a:(a,b)∈f}

**Range of f **= Set of all second elements of memeber of f OR Range of f={b:(a,b)∈f}

**Co-domain **= all element of set B

** Example: **

Let A={-1,0,1,2,} & B={0,1,2,3,4} Consider rule f(x)=x^2

Under this rule when f(-1)=1,f(0)=0,f(1)=1,f(2)=4 Here we see all element od set A associated with unique element of setB.

Therefore f(x)=x^2 is a function.

**Domain of f(x)**=x^2={-1,0,1,2,}

**Range of f(x)**=x^2={0,1,4}

**Co-domain of f(x)**=x^2={0,1,2,3,4}

**Types of functions**

- Identity function
- Constant function
- Polynomial function
- Rational functions
- Modulus function
- Signum function
- Greatest integer function

**Identity functions**

A function is said to be an identity function when it returns the same value as the output Which was used as its input(argument).

Let us an example of a function of set A = {1, 2, 3, 4, 5} to itself. f: A → A such that, g = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}.

**Identity functions**

A function is said to be an identity function when it returns the same value as the output Which was used as its input(argument).

Let us an example of a function of set A = {1, 2, 3, 4, 5} to itself. f: A → A such that, g = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}.

If we plot a graph of identity function, then it will be a straight line.

Let us draw an identity function f:R→R, f(x)=x for x∈R

When x=1 than y=f(1)=1

When x=0 than y=f(0)=0

When x=-2 than y=f(-2)=-2

**Constant function**

A function is said to be a constant function when it returns constant value as the output for all input(argument).

Constant function f:R→R, f(x)=k for x∈R

If we plot a graph of constant function, then it will be a straight line parallel to x axis .

Let us draw an identity function f:R→R, f(x)=5 for x∈R

When x=1 than y=f(1)=5

When x=0 than y=f(0)=5

When x=-2 than y=f(-2)=5

**Polynomial function**

Polynomial Function is defined as the real valued function f:R→R for x∈R

**Rational functions**

Rational function is the ratio of two polynomial functions where the denominator polynomial is not equal to zero.
where g(x) and t(x) are polynomial functions and t(x) is a non-zero polynomial.
**Modulus function**

The modulus function only gives a positive value as ouput of any variable or input. It is also known as the absolute value function because it gives a non-negative value as out put against any independent variable or input, no matter if it is positive or negative.
The function f:R→R,Modulus is defined by
**Let us draw graph of f(x)=|x|**

When x=-5,then y= f(-5)=|-5|=5

When x=-4,then y= f(-4)=|-4|=4

When x=-3,then y= f(-3)=|-3|=3

When x=-2,then y= f(-2)=|-2|=2

When x=-1,then y= f(-1)=|-1|=2

When x=0,then y= f(0)=|0|=0

When x=1,then y= f(1)=|1|=1

When x=1,then y= f(1)=|1|=1

When x=2,then y= f(2)=|2|=2

When x=3,then y= f(3)=|3|=3

When x=4,then y= f(4)=|4|=4

When x=5,then y= f(5)=|5|=5

**Signum function**

Domain of f = R, Range of f = {1, 0, – 1}
**Greatest integer function**

Greatest integer function: The real function f : R → R defined by

f (x) = [x], x ∈R assumes the value of the greatest integer less than or equal to x, is
called the greatest integer function.

Thus f (x) = [x] = – 1 for – 1 ≤ x < 0

f (x) = [x] = 0 for 0 ≤ x < 1

[x] = 1 for 1 ≤ x < 2

[x] = 2 for 2 ≤ x < 3 and so on

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