Relations and functions play a very important role in mathematics. Relation can be represented in the roster form and tabular form. For example, a relation ‘R’ on Set A = {1, 2, 3, 4, 5} is defined by R = {(a, b): b = a + 2}. It can also be expressed as aRb, if and only if b = a + 2. |

For any function y = f(x), domain is the set of all the possible values of x, where x being the independent variable.

Range of a function is the set of all the corresponding values of y, for every value of x. It is the set of all second component values, that can result from using values in the domain.

Different types of relations are explained below:

**Universal Relation:**Universal relation consists of all ordered pairs of the cartesian product "B x B".

R = B x B

Consider a set B = {a, b, c}

Universal set is R = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}

__Symmetric Relation:__A relation R is said to be symmetric, if (a, b) $\epsilon$ R

$\Rightarrow$ (b, a) $\epsilon$ R

__Transitive Relation:__A relation R is said to be transitive, if (a, b) $\epsilon$ R, (b, c) $\epsilon$ R

$\Rightarrow$ (a, c) $\epsilon$ R

__Identity Relation:__In an identity relation "R", every element of the set “A” is related to itself only.

For a set A = {1, 2, 3}, the identity relation is R = {(1, 1), (2, 2), (3, 3)}

**Binary Relation:**A binary relation on sets A and B is a subset of A x B. Any R $\subseteq$ A x B is a binary relation.

If (a, b) $\epsilon$ R, we can write it as aRb.

A binary relation R on a set S is said to be complete, if and only if [aRb or bRa] for all distinct a, b $\epsilon$ S.

Different types of functions are explained below:

**A function f is even, if f(-x) = f(x) for all x.**

__Even function:__**Examples:**f(x) = x$^{2}$, f(x) = cos x

**A function f is said to be odd, if f(-x) = -f(x).**

__Odd function:__**Examples:**f(x) = x$^{3}$, f(x) = sin x

**Constant function is a linear function of the form y = b, where b is a constant. It is written as f(x) = b. The graph of a constant function is a horizontal line.**

__Constant function:__**Logarithmic function is defined as f(x) = log$_{a}(x)$**

__Logarithmic function:__a > 0, a $\neq$ 1

**Greatest Integer Function is a step function written as f(x) = [x], where f(x) is the greatest integer less than or equal to x.**

__Greatest integer function:__**Examples:**The greatest integer less than or equal to the number [8, 5] is [8]

**Reciprocal function is defined on the set of non zero reals, that sends every real number to its reciprocal.**

__Reciprocal function:__f(x) = $\frac{1}{x}$, x$\neq$ 0 and the domain is all real numbers

**A one-to-one (injective) function f from set X to set Y is a function such that, each x in X is related to a different y in Y.**

__One one function:__**A function f: X -> Y is said to be onto (surjective), if for every y in Y, there is an x in X such that f(x) = y.**

__Onto function:__If f: A --> B is a function, then ‘f’ associates all elements of set ‘A’ to elements in set ‘B’, such that an element of set ‘A’ is associated to a unique element of set ‘B’. Let ‘A’ and ‘B’ be two non empty sets, then a function ‘f’ from set ‘A’ to set ‘B’ is a rule which groups elements of set ‘A’ to elements of set ‘B’ such that,

- All elements of set ‘A’ is grouped to elements in set ‘B’.
- All elements of set ‘A’ is grouped to unique elements in set ‘B’.

### Solved Example

**Question:**Show that F: R $\rightarrow$ R defined by F(x) = x$^{2}$ + 5 is neither one - one nor onto function.

**Solution:**

x$_{1}^{2}$ + 5 = x$_{2}^{2}$ + 5

$\Rightarrow$ $x_{1}^{2}$ = $x_{2}^{2}$

$\Rightarrow$ $x_{1}^{2}$ - $x_{2}^{2}$ = 0

$x_{1}$ = $x_{2}$

Therefore, F is not one-one function.

Let y = F(x) where x $\epsilon$ domain, y $\epsilon$ codomain

y = x$^{2}$ + 5 $\Rightarrow$ x = $\pm$$\sqrt{y-5}$

There is no value of x in the domain $\forall$ y < 5

Therefore, F is not an onto function.

R is a relation if R $\subseteq$ ($\frac{(x,y)}{x\epsilon X, y\epsilon Y}$)

Here, Domain is X and Codomain is Y. Here, we can see some of the differences between relation and function.

Given below are some of the differences between relation and function:

- A function is a particular kind of relation.
- A function is the purpose or action of a certain device, meant to achieve a specific goal. A relation is relative to something else.
- A function is a relation which is one-to-one or one-to-many.
- A relation maybe one-to-one, one-to-many or even many-to-one.