## 2. What is relation?

**Definition:** A relation \(R\) from a non-empty set \(A\) to a non-empty set \(B\) is a

subset of the Cartesian product \(A \times B\).

It

*"maps"* elements of one set to another set. The subset is derived by describing a relationship
between the first element and the second element of the ordered pair \(\left( {A \times B} \right)\).

**Domain:** The set of all first elements of the ordered pairs in a relation \(R\) from a set
\(A\) to a set \(B\) is called the

*domain* of the relation \(R\).

**Range:** the set of all the ending points is called the

*range*
**Co-domain:** The whole set B is called the co-domain of the relation R.

**Example**

Let A ={4,5,3} and B ={1,6, 7}

Let R be the Relation "is greater than " from A to B

Then

R= { (4,1),(5,1),(3,1)}

## Algebraic Representation of Relation

A relation can be expressed in Set builder or Roaster form

### Roster forms

In a Roster forms, all the ordered pair in the relation is listed.

Example

R= { (4,1),(5,1),(3,1)}

#### Some Important points

- In roster form, the order in which the elements are listed is immaterial
- while writing the set in roster form an element is not generally repeated

### Set Builder Form

- In set-builder form, all the ordered pair of a relation possess a single common property
which is not possessed by any ordered pair outside the relation. For example, in the relation
\(\left\{ (1,2) ,(2,4) ,(3,6) ,(4,8),(5,10) \right\}\), all the ordered pairs possess a common property, namely, second element in ordered
is doubled of first element . Denoting this set by \(R\), we write

$R = \left\{(x,y) : x,y \in {1,2,3,4,5,6,7,8,9,10},y=2x \right\}$

## Pictorial Representation of Relation

A Relation R from A to B can be depicted pictorially using arrow diagram . In arrow diagram, we write down the elements of two set A and B in two disjoint circle,Then we draw arrow from set A to set B whenever $(a,b) \in R$

Let A={a,b,c,d} and B={x,y,z}

And R ={(a,x),(b,y),(c,z)}

Then this will be represented in arrow diagram as

#### Important Note

The total number of relations that can be defined from a set \(A\) to a set \(B\)
is the number of possible

subsets of \(A \cdot B\). If \(n\left( A \right) = p\) and \(n\left( B \right) = q\), then
\(n\left( {A \cdot B} \right) = pq\) and the total number of relations is \({2^{pq}}\)

**Example:**
Let \(P = \left\{ {1,2,3,.....,18} \right\}\) define a relation \(R\) from \(P\) to \(P\) by \(R = \left\{ {\left( {x,y} \right):2x - y = 0,where \; x,y \in P} \right\}\) Write down its domain, co-domain and range.

Draw the arrow diagram for the relation also

**Solution:**
The relation \(R\) from \(P\) to \(P\) is given as

R = {(x,y):2x-y=0, where x, y ∈ P}

i.e., R = {(x, y): 2x = y, where x, y ∈ P}

Therefore,

\( R = \left\{ {\left( {1,2} \right),\left( {2,4} \right),\left( {3,6} \right),\left( {4,8} \right),\left( {5,10} \right),\left( {6,12} \right),\left( {7,14} \right),\left( {8,16} \right),\left( {9,18} \right)} \right\}\)

The domain of \(R\) is the set of all first elements of the

ordered pairs in the relation.

Therefore,

\(Domain \; of \; R = \left\{ {1,2,3,4,5,6,7,8,9} \right\}\)

The whole set \(P\) is the co-domain of the relation \(R\).

Therefore co-domain of \(R = P = \left\{ {1,2,3, \ldots ,18} \right\}\)

The range of \(R\) is the set of all second elements of the ordered pairs in the relation.

Therefore range of \(R = \left\{ {2,4,6,8,10,12,14,16,18} \right\}\)

Arrow diagram is given below

**Also Read**