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