Answer to Question #273826 in Discrete Mathematics for Anik

Question

In how many ways a relation can be represented? State two different

examples to represent each of them.

Answer 1

There are 4 ways of representing a relation. The 4 ways are,

• Relation as a matrix
• Relation as an arrow diagram
• Relation as directed graph(Digraph)
• Relation as a table

The following examples illustrate how relations can be represented.

$i)$ Relation as a matrix

Let P={1,2,3,4}, Q={a, b, c, d}.

R is given as,

R={(1,a),(1,b),(1,c),(2,b),(2,c),(2,d)}. We can represent this Relation in the matrix below,

$\begin{Bmatrix} (1,a) & (1,b)& (1,c) & (1,d)\\ (2,a) & (2,b) & (2,c)& (2,d)\\ (3,a)&(3,b)&(3,c)&(3,d)\\ (4,a)&(4,b)&(4,c)&(4,d) \end{Bmatrix}$

Let P={1,2,3} and Q={1,2} then its corresponding relation R is given as,

R={(1,1),(1,2),(2,1),(2,2),(3,1),(3,2)}

In matrix form,

$\begin{Bmatrix} (1,1) & (1,2) \\ (2,1) & (2,2)\\ (3,1)&(3,2) \end{Bmatrix}$

$ii)$ Relation as Digraph

Given P={1,2,3,4}, the relation R={(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)}

The digraph for this relation is shown in the image below,

Given A={1,2,3,4}, its relation R is,

R={(1,2),(2,2),(2,4),(3,2),(3,4),(4,1),(4,3)}

The Digraph for this relation is shown in the image below.

 $iii)$ Relation as an arrow diagram.

Let P={1,2,3,4}, Q={a, b, c, d}. The relation R={(1,a),(2,a),(3,a),(1,b),(4,b),(4,c),(4,d)}

The arrow diagram for this relation is shown in the image below.

Let R={(1,1),(2,4),(3,9),(4,16),(5,25)}, then the arrow diagram for this relation is shown in the image below.

$iv)$ Relation as a table

Let P=(1,2,3,4) and Q={x, y, z, k}, the relation R={(1,x),(1,y),(2,z),(3,z),(4,k)}

This relation is represented in the table below,

x y z k

1 * *

2 *

3 *

4 *

The Relation R={(-1,1),(0,-2),(3,1),(4,-1)} is represented using the following table

x y

-1 1

0 -2

3 1

4 -1