Many day-to-day occurrences involve two objects that are connected with each other by some rule of correspondence.

**Relations**

Many day-to-day
occurrences involve two objects that are connected with each other by some rule
of correspondence. We say that the two objects are related under the specified
rule. How shall we represent it? Here are some examples,

How are New
Delhi and India related? We may
expect the response, “New Delhi is the
capital of India”. But there are several ways in which ‘New Delhi’ and ‘India’
are related. Here are some possible answers.

·
New Delhi is the capital of India.

·
New Delhi is in the northern part of India.

·
New Delhi is one of the largest cities of India etc.

So, when we wish to
specify a particular relation, providing only one ordered pair

(New Delhi, India) it
may not be practically helpful. If we ask the relation in the following set of
ordered pairs,

{(New
Delhi, India), (Washington, USA), (Beijing, China), (London, U.K.), (Kathmandu,
Nepal)} then specifying the relation is easy

**Illustration 4**

Let us define a
relation between heights of corresponding students. (Fig.1.7)

*R *= {(heights,
students)}

* R *= {(4.5, *S*_{1}* *), (4.5, *S _{4}*), (4.7,

Let *A *and *B *be
any two non-empty sets. A ‘relation’ R from *A *to *B *is a subset of *A* ×*B* satisfying some specified
conditions. If *x* ∈ *A* is related to *y* ∈ *B* through R , then we write it as *x* R*y*. *x* R*y*
if and only if (*x* , *y*) ∈ R .

The domain of the relation R = {*x
*∈ *A *| *x *R *y*, for some *y *∈ *B*}

The co-domain of the relation R is *B*

The range of the relation R = {*y
*∈ *B *| *x *R*y*, for some *x *∈ *A*}

From these
definitions, we note that domain of R ⊆ *A *, co-domain
of R = *B *and range of R ⊆ *B *.

**Illustration 5**

Let *A *=
{1,2,3,4,5} and *B *= {Mathi, Arul, John}

A relation R between
the above sets *A *and *B *can be represented by an arrow diagram
(Fig. 1.8).

Then, domain of R =
{1,2,3,4}

range of R= {Mathi,
Arul, John} = co-domain of R . Note that domain of R is a proper subset of *A.*

Let *A *=
{1,3,5,7} and *B *= {4,8}. If R is a relation defined by “is less
than” from *A *to *B*, then 1R4 (since 1 is less than 4). Similarly,
it is observed that 1R8, 3R4, 3R8, 5R8, 7R8

Equivalently R=
{(1,4), (1,8), (3,4), (3,8), (5,8), (7,8)}

In the above
illustration *A*×*B *= {(1,4), (1,8), (3,4), (3,8), (5,4), (5,8),
(7,4),(7,8)}

R = {(1,4), (1,8),
(3,4), (3,8), (5,8), (7,8)} We see that R is a subset of *A*×*B*

In a particular area
of a town, let us consider ten families *A*, *B*, *C*, *D*,
*E*, *F*, *G*, *H*, *I *and *J *with two
children. Among these, families *B*, *F*, *I *have two girls; *D*,
*G*, *J *have one boy and one girl; the remaining have two boys. Let
us define a relation R by *x*R*y*, where *x *denote the number
of boys and *y *denote the family with *x *number of boys.
Represent this situation as a relation through ordered pairs and arrow diagram.

Since the domain of
the relation R is concerned about the number of boys, and we are considering
families with twochildren, the domain of R will consist of three elements given
by {0,1,2}, where 0, 1, 2 represent the number of boys say no, one, two boys
respectively. We note that families with two girls are the ones with no boys.
Hence the relation R is given by

R = {(0, *B*),(0,
*F*),(0, *I *),(1, *D*),(1,*G*),(1,*J *),(2, *A*),(2,*C
*),(2, *E*),(2, *H *)}

This relation is shown
in an arrow diagram (Fig.1.9).

Let *A *=
{3,4,7,8} and *B *= {1,7,10}. Which of the following sets are relations
from *A *to *B*?

(i) R_{1}
={(3,7), (4,7), (7,10), (8,1)}

(ii) R_{2}=
{(3,1), (4,12)}

(iii) R_{3}=
{(3,7), (4,10), (7,7), (7,8), (8,11), (8,7), (8,10)}

*Solution **A*×*B *= {(3,1),
(3,7), (3,10), (4,1), (4,7), (4,10), (7,1), (7,7), (7,10), (8,1), (8,7),
(8,10)}

(i) We note that, R_{1}
⊆ *A*× *B *. Thus, R_{1} is a relation from *A
*to *B*.

(ii) Here, (4,12) ∈ R_{2} , but (4,12) ∉ *A*×*B *.
So, R_{2} is not a relation from *A *to *B*.

(iii) Here, (7, 8) ∈ R_{3} , but (7, 8) ∉ *A*×*B *.
So, R_{3} is not a relation from *A *to *B*.

A relation may be
represented algebraically either by the roster method or by the set builder
method.

An arrow diagram is a
visual representation of a relation.

**Example 1.5 **The arrow diagram
shows (Fig.1.10) a relationship between the sets *P *and *Q*. Write
the relation in (i) Set builder form (ii) Roster form (iii) What is the domain
andrange of .

*Solution*

(i) Set builder form
of R = {(*x*,*y*)| *y *= *x *− 2, *x *∈ *P*,*y *∈ *Q*}

(ii) Roster form
R = {(5, 3),(6, 4),(7, 5)}

(iii) Domain of R =
{5,6,7} and range of R = {3, 4, 5}

‘**Null relation**’

Let us
consider the following examples. Suppose *A
*= {–3,–2,–1} and* B *= {1,2,3,4}. A
relation from* A *to *B *is defined as* a *−* b *=* *8* *i.e., there is no pair (*a*,*b*) such that *a *−* b *=* *8* *. Thus* *R* *contain no element and so R = *ɸ* .

A
relation which contains no element is called a “Null
relation”.

Tags : Definition, Illustration, Example, Solution | Mathematics Definition, Illustration, Example, Solution | Mathematics

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