# Relation on a Set

Consider two sets*A*and

*B*. Every subset of the product set

*A*Ã—

*B*is called a relation between

*A*and

*B*. It is denoted by

*R*.

Hence, we can say that .

If (x, y) âˆˆ *R*, then we can write it as x *R *y and is read as â€˜*x* is related to *y*â€™.

If *A* = *B*, then *R* is a relation on *A*.

The set of first elements of the ordered pair that belongs to *R* is known as the *domain* of the relation *R*. It is denoted by *D*.

*D* = {*x*: (*x*, *y*) âˆˆ *R*, for *x* âˆˆ *A*} *i.e.*,

The set of all second elements of the ordered pair that belongs to *R* is known as the *range* of the relation *R*. It is denoted by *E*.

*E* = {*y*: (*x*, *y*) âˆˆ *R*, for *y* âˆˆ *B*} *i.e.*,

*R*be a relation on the set of whole numbers

*W*, defined by

*a*+ 4

*b*= 24. Find

(i) R

(ii) Domain of

*R*

(iii) Range of

*R*

*R*= {(

*a*,

*b*) : a âˆˆ

*W*,

*b*âˆˆ

*W*,

*a*+ 4

*b*= 24} = {(24, 0), (20, 1), (16, 2), (12, 3), (8, 4), (4, 5), (0, 6)}

The above set of ordered pairs is obtained by taking

*b*= 0, 1, 2, 3, 4, 5, 6 in the given relation and finding the value of

*a*. If

*b*> 6, then the value of â€˜

*a*â€™ obtained becomes negative which does not belong to the set

*W*.

Domain of

*R*= {24, 20, 16, 12, 8, 4, 0}

Range of

*R*= {0, 1, 2, 3, 4, 5, 6}

# Different Types of Relation

**Identity relation:***A*relation*R*on a set is said to be an*identity relation,*if both the elements of the ordered pairs are same.*A*is shown as*I*= {(_{A}*x*,*x*) :*x*âˆˆ*A*}**Example:**Let*A*= {1, 2, 3}, then*I*= {(1, 1), (2, 2), (3, 3)}_{A}**Universal relation:***A*Ã—*A*is called the universal relation on a set*A*.**Example:**Let*A*= {2, 4, 6}, then*R*=*A*Ã—*A*= {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)}- Âƒ
**Void relation:**If*a*relation*R*from*A*to*B*is a null set, then*R*is said to be a void relation.**Example:**Let*A*= {1, 3} and*B*= {9, 13}.*R*= {(m, n) :*m*âˆˆ*A*,*n*âˆˆ*B*,*m*Ã—*n*is even}.Since none of the numbers (1 Ã— 9), (1 Ã— 13), (3 Ã— 9), (3 Ã— 13) is even,

*R*is an empty set. - Âƒ
**Reflexive relation:**The relation*R*on a set â€˜*A*â€™ is said to be a reflexive relation, if*R*contains all possible ordered pairs of the type (*x*,*x*) for all*x*âˆˆ*A*.**Example:**Let*A*= {3, 6, 9}*R*= {(3, 3), (3, 6), (6, 6), (9, 9), (9, 6)} is a reflexive relation, since all possible pairs of (*x*,*x*) type are present in*R*.The relation

*R*= {(3, 3), (3, 6), (6, 9), (9, 9), (9, 6)} is a not reflexive relation, since (6, 6) is missing in*R.*

**Note:** ** R** is a reflexive relation, if every element in a set â€˜

**â€™ is related to itself.**

*A***Symmetric relation:**The relation*R*on a set*A*is said to be symmetric, if (*x*,*y*) âˆˆ*R*and (*y*,*x*) âˆˆ*R*.*x*,*y*) if the reverse pair (*y*,*x*) is also there in*R*, then*R*will be symmetric.**Example:**Let*S*be a set of all straight lines in a plane. Then the relation*R*in*S*defined by, â€˜a is perpendicular to*b*â€™. Then the relation*R*is said to be symmetric, since a straight line â€˜*a*â€™ is perpendicular to another line â€˜*bâ€™*, and the line â€˜*b*â€™ is also perpendicular to line â€˜*a*â€™.- Âƒ
**Transitive relation:**The relation*R*on a set*A*is said to be*transitive*if (*x*,*y*) âˆˆ*R*and (*y*,*z*) âˆˆ*R*â‡’ (*x*,*z*) âˆˆ*R***Example:**Let there be a set of straight lines in a plane. Then the relation*R*on the set of lines defined by â€˜*a*is parallel to*b*â€™ is a transitive relation, since if a line*a*is parallel to a line*b*and the line*b*is parallel to another line*c*, then the line*a*is parallel to*c*. - The relation â€˜a is perpendicular to bâ€™ is not a transitive relation. This is because if a line a is perpendicular to a line b and if the line b is perpendicular to a line c, then a and c will be parallel to each other.
**Equivalence relation:**The relation*R*on a set*A*is said to be an*equivalence relation*, if*R*is reflexive, symmetric and transitive.**Example:**The relation*R*= {(*x*,*y*) :*x*âˆˆ*A*,*y*âˆˆ*A*,*x*=*y*} is an equivalence relation.*x*=*x*,*i.e.*, reflexive*x*=*y*â‡’*y*=*x*,*i.e.*, symmetric*x*=*y*,*y*=*z*â‡’*x*=*z*,*i.e.*, transitive- Âƒ
**Inverse relation:**If*R*is a relation from the set*A*to*B*, then the inverse relation*R*^{-1}from*B*to*A*is defined by*R*^{-1}= {(*b*,*a*) : (*a*,*b*) âˆˆ*R*}**Example:**Let*A*= {1, 2},*B*= {a, b} and*R*= {(1, a), (2, a), (1, b), (2, b)} be a relation from*A*to*B*.Then inverse relation of

*R*is*R*^{-1}= {(a, 1), (a, 2), (b, 1), (b, 2)}.

# Functions

Let*A*and

*B*be two non-empty sets. Then, if each element

*x*of

*A*is related with a unique element

*f*(

*x*) of

*B*, it is called

*function*from

*A*to

*B*and it is denoted by

*f*:

*A*â†’

*B*.

The element *f *(*x*) is called the *image* of *x*, while *x* is called the *pre-image* of *f *(*x*).

Let *f *: *A *â†’ *B*, then *A* is called the *domain* of *f*, while *B* is called the *co-domain* of *f*.

The set *f *(*A*) = { *f *(*x*) : *x *âˆˆ *A*} is called the *range* of *f*.

**Example : **Let *A* = {1, 2, 3, 4}, *B* = {4, 9, 16, 25}

Let *f *(*x*) = *x*^{2}

Then, *f* is not a function, since no element of *B* is assigned to the element 1 âˆˆ *A*.

**Example:** Let *A* = {1, 2, 3, 4, 5}, *B* = {2, 4, 6, 8, 10, 12, 14}

Let *f*(*x*) = 2*x*

Then *f*(1) = 2

*f*(2) = 4

*f*(3) = 6

*f*(4) = 8

*f*(5) = 10

We can see that each element of *A* has a unique image in *B*.

So *f* : *A *â†’ *B* is a function from *A* to *B*.

# Various Types of Functions

**Oneâ€“one function:**If distinct elements in*A*have distinct images in*B*,*i.e.*,*f*is â€˜one-oneâ€™ if*f*(*x*_{1}) =*f*(*x*_{2}) â‡’*x*_{1}=*x*_{2}. It is also called as*injective function.***Example:**Let*f*:*A*â†’*B*be defined by*f*(*x*) =*x*^{2}.*We know that, 1 â‰ -1, but**f*(1) =*f*(-1). So â€˜*f*â€™ is not oneâ€“one.*Let**f*:*A*â†’*B*be defined by*f*(*x*) =*x*^{3}.*Here,**f*is oneâ€“one as*f*(*x*_{1}) =*f*(*x*_{2}) â‡’*x*_{1}=*x*_{2}.- Âƒ
**Manyâ€“one function:**If two or more than two elements of set*A*have the same image in*B*, then it is called â€˜manyâ€“oneâ€™ function.**Example:**Let*f*:*A*â†’*B*be defined by*f*(*x*) =*x*^{2}.*f*(-1) = (-1)^{2}= 1 and also*f*(1) = (1)^{2}= 1.*A*have the same image in set*B*. Hence â€˜*f*â€™ is a manyâ€”one function. -
**Onto function:**A function f : A â†’ B is called onto or surjective if f(A) = B, i.e., for all b âˆˆ B there at least one a âˆˆ A with f(a) = bIt is also called as surjective function. In order words, no element in B should be without a pre-image. For onto functions, range = co-domain.**Example:**Let*A*= {1, 2, 3},*B*= {a, b} and let*f*= {(1, a), (2, a), (3, b)}.Here, the domain is

*A*and no two components of*f*have the same first element. Hence it is a function. Also to every element y in*B*, there is an element in*A*and hence*f*is an onto function. **Into function:**If at least one element in*B*has no pre-image in*A*, it is called*into function*.**Example:**Let*A*= {1, 2, 3},*B*= {a, b, c, d*f*= {(1, a), (2, b), (3, c)}*f*is a function from*A*to*B*.

Here the element d in*B*does not have a pre-image in*A*.Hence

*f*is an into function.**Bijective function:**A oneâ€“one and onto function is said to be*bijective function*.**Equal functions:**If two functions*f*and*g*have the same domain and satisfy the condition f(*x*) =*g*(*x*), then*f*and*g*are said to be equal functions.**Composite function:**Let*f*:*A*â†’*B*and*g*:*B*â†’*C*be two functions. The function from*A*to*C*which maps an element*x*âˆˆ*A*into*g*(*f*(*x*)) âˆˆ*C*is called a*composite function*of â€˜*f*â€™ and â€˜*g*â€™. It is represented as â€˜*g**o**f*â€™.**Example:**Let*A*= {1, 2, 3},*B*= {w, x, y, z},*C*= {2, 4, 6, 8, 10}*f*= {(1, w), (2, x), (3, y)} and*g*= {(w, 2), (x, 4), (y, 6), (z, 8)}Then gof is the function {(1, 2), (2, 4), (3, 6)}

**Inverse function:**Let*f*be a bijective function from*A*to*B*and*f*(*x*) =*y*.*f*^{-1}such that*f*^{-1}(*y*) =*x*.*f*^{-1}(*y*) is called the*inverse*of*f*.

**Note:** A function has an inverse function if and only if it is bijective.

*f*(

*x*) = 2 +

*x*, then

*f*

^{-1}(

*x*) = ?

*y*=

*f*(

*x*)