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Relations and Functions


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.

Let ‘A’ and ‘B’  be two sets. Then, a relation ‘R’ from set ‘A’ to set ‘B’ is a subset of A x B. Thus, ‘R’ is a relation from A to B $\leftrightarrow$ R is subset A x B. If ‘R’ is a relation from a non void set ‘A’ to a non void set ‘B’ and if (a, b) $\epsilon$ R, then we write aRb, which is read as 'a’ is related to ‘b’ by the relation ‘R'.

Domain: Let ‘R’ be a relation from a set ‘A’ to set ‘B’. Then, the set of all first components or coordinates of the ordered pairs belonging to the domain of ‘R’. Thus, domain of R = (a : (a, b) $\epsilon$ R).

A function assigns exactly one element of one set to each element of the another set. It holds a special relationship between values and mostly represented as f(x), where 'x' is the value user gives.

Domain and Range of a Relation

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Domain is defined as a set of input values or arguments. The set of all the output values is called a range. The range is an interval among the uppermost number to the minor number. In mathematics, the domain and range function is one of the divisions.
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.

Types of Relations

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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. 

Types of Functions

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Different types of functions are explained below:

Even function: A function f is even, if f(-x) = f(x) for all x.
Examples: f(x) = x$^{2}$, f(x) = cos x

Odd function: A function f is said to be odd, if f(-x) = -f(x).
Examples: f(x) = x$^{3}$, f(x) = sin x

Constant function: 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.

Logarithmic function: Logarithmic function is defined as f(x) = log$_{a}(x)$
a > 0, a $\neq$ 1

Greatest integer function: 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.
Examples: The greatest integer less than or equal to the number [8, 5] is [8]

Reciprocal function: Reciprocal function is defined on the set of non zero reals, that sends every real number to its reciprocal.
f(x) = $\frac{1}{x}$, x$\neq$ 0 and the domain is all real numbers

One one function: 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.

Onto 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.

Function Definition

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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,
  1. All elements of set ‘A’ is grouped to elements in set ‘B’.
  2. All elements of set ‘A’ is grouped to unique elements in set ‘B’.

Relations and Functions Examples

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Given below are some of the examples in relations and functions.

Solved Example

Question: Show that F: R $\rightarrow$ R defined by F(x) = x$^{2}$ + 5 is neither one - one nor onto function.
Consider F($x_{1}$) = F($x_{2}$) $\forall$ $x_{1},x_{2}$

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.

Relation Definition

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A relation describes certain properties of things and is a set of ordered pairs. It is a subset of ordered pairs drawn from the set of all possible ordered pairs.

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

Here, Domain is X and Codomain is Y.

Difference Between Relation and Function

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Here, we can see some of the differences between relation and function.

Given below are some of the differences between relation and function:
  1. A function is a particular kind of relation. 
  2. A function is the purpose or action of a certain device, meant to achieve a specific goal. A relation is relative to something else. 
  3. A function is a relation which is one-to-one or one-to-many. 
  4. A relation maybe one-to-one, one-to-many or even many-to-one.