Relation

Generally, relation refers to the blood connection between the people. For example, brother and sister, father and son etc.
But in mathematics, if A and B are two sets in which A × B is called the cartesian product of A and B, any subset of A × B is called a relation from A to B. The relation from A to B is denoted by R. The relation from A is called a relation on A.
For examples,
Let, A = {3, 5} and B = {4, 6}
Then, A × B = {3, 5} × {4, 6}
= {(3, 4), (3, 6), (5, 4), (5, 6)}
Let, R = {(3, 4), (5, 6)}
So, R is a subset of A × B.
So, R is a relation from A to B.

Representation of a relation

A relation can be represented in 5 ways as:

• By the set of ordered pairs
• By standard description using a rule of formula
• By table
• By graph
• By arrow diagram
  
  

By the set of ordered pairs
A relation can be represented in the set of ordered pairs in which there are two components. The first component is called x-component and the second component is called y-component.
For example,

• R = {(Sulav, Shree), (Siddartha, Bhagwat), (Asmita, Riya)}
  R = {(1, 1), (2, 2), (3, 3)}

By standard description using a rule or a formula
A relation can be represented by using the formulas and standard description.
For example,

• Let, A = {3, 4, 5} and B = {3, 4, 5}
  Then,
  A × B = {(3, 3), (3, 4), (3, 5), (4, 3), (4, 4), (4, 5), (5, 3), (5, 4), (5, 5)}
  Here, R is a subset of A × B. So, R is a relation from A to B. The relation R can be shown as, R = {(x, y) : x=y}
  
  
• Let, A = (5, 6) and B = (7, 8)
  Then,
  A × B = (5, 6) × (7, 8)
  = {(5, 7), (5, 8), (6, 7), (6, 8)}
  Let, R = {(5, 7), (5, 8), (6, 8)}
  Here,R is a subset of A × B. So R is a relation from A to B. The relationship R can be represented as,R = {(x, y) : x<y}

By Table
A relation can also be represented in table which one as follows: -

• Let, A = {2, 4} and B = {4, 6}
  Then, A × B = {2, 4} × {4, 6}
  = {(2, 4), (2, 6), (4, 2), (4, 6)}
  Then, R is a relation from A to B. This relation can be represented as,
  

  x	2	4
  y	4	6

By Graph
A relation can also be shown in graphs by the use of graph paper. For example,

• Let, A = {4, 5, 6} and {4, 5, 6}
  Then, A × B = {4, 5, 6} × {4, 5, 6}
  = {(4, 4), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)}
  So, R = {(4, 4), (5, 5), (6, 6)}
  Here, R is a relation from A to B which is represented as,
  

Example for graph

By Arrows Diagram

• Let, A = {2, 4} and B = {4, 6}
  Then A × B = {2, 4}×{4, 6}
  = {(2, 4), (2, 6), (4, 4), (4, 6)}
  Let, R = {(2, 4), (4, 6)}
  So, R is a relation from A to B. This relation can be represented as,
  

Example for arrow diagram

Domain and Range of a Relation

Let R be a relation from A to B. Then, the set of all first elements of the ordered pairs of R are called the domain. For example,
Let, A = (1, 2) and B = (3, 4)
Domain of relation = (1, 3)
Similarly, the set of all the second elements of the ordered pair of R is called the range. For example,
Let, A = (1, 2) and B = (3, 4)
Range of a Relation = (2, 4)

Inverse relation

The interchanging of x-component and y-components of each pair of the relation of R is denoted by R^-1.
For example,
Let, A = {1, 2} and B = {3, 4}
Here, A × B = {1, 2} × {3, 4} = {(1, 3) (1, 4) (2, 3) (2, 4)}
Thus, R^-1= {(3, 1), (4, 1), (3, 2), (4, 2)}