Pictorial Representation of Relation

Definition

A pictorial representation of a relation is the visual display of a relation between sets using diagrams such as arrow diagrams, mapping diagrams, graphs, or coordinate plots, where the related elements are shown by arrows, lines, points, or curves.

If $R$ is a relation from set $A$ to set $B$ , then the pictorial representation shows every ordered pair $(a,b)\in R$ as a visual connection from $a$ in $A$ to $b$ in $B$ .

Example:
If $A=\{1,2\}$ , $B=\{a,b\}$ , and $R=\{(1,a),(2,b)\}$ , then the pictorial representation shows an arrow from 1 to a and another arrow from 2 to b.

Main Content

1. Arrow Diagram Representation

In an arrow diagram, the elements of the first set are written on the left side and the elements of the second set are written on the right side. Arrows are drawn from each element of the first set to the related element(s) of the second set.
This is the simplest and most widely used pictorial form for representing a relation, especially when the sets are finite and small. It clearly shows whether one element is related to many elements, whether some elements are not related at all, and whether a relation is one-to-one, one-to-many, or many-to-many.

Example:
Let $A=\{1,2,3\}$ , $B=\{a,b,c\}$ , and
$R=\{(1,a),(1,b),(2,b),(3,c)\}$

Then the arrow diagram is:

A                     B
1  -----------------> a
|  \
|   ----------------> b
2  -----------------> b
3  -----------------> c

This shows that:

1 is related to a and b
2 is related to b
3 is related to c

Why it is useful:

Easy to construct
Easy to interpret
Useful in examinations for quick visual understanding
Helps in identifying whether the relation is a function or not

2. Mapping and Graphical Representation

A mapping diagram is another visual form similar to the arrow diagram, but it emphasizes the idea of “mapping” or “association” from elements of one set to another. It is especially useful in understanding relations as rules or correspondences.
A graphical representation is used when the relation is between numerical sets and the ordered pairs can be plotted on the Cartesian plane. Each ordered pair $(x,y)$ is represented as a point on the graph. This is very helpful when studying relations in coordinate geometry, algebra, and functions.

Example of graph representation:
If $R=\{(1,2),(2,3),(3,4)\}$ , then these points are plotted on the coordinate plane.

y
4 |           • (3,4)
3 |      • (2,3)
2 | • (1,2)
1 |
  +---------------------- x
    1    2    3    4

Uses of graphical representation:

Shows patterns in relations
Helps identify whether a relation is a function
Useful for studying inequalities, equations, and real-valued relations
Makes it easier to compare multiple relations visually

Difference from arrow diagram:

Arrow diagrams are best for finite sets.
Graphical representation is best for numerical relations and real-number data.

3. Representation Through Ordered Pairs, Tables, and Matrices

A relation can also be shown visually using a table form or matrix form, which is a structured pictorial-style representation. Although not always considered a pure drawing, these forms are still visual and are commonly used together with diagrams.
In a relation table, elements of the first set are listed along rows and elements of the second set along columns. The related pairs are marked with a tick, 1, or some symbol. In a matrix representation, the entry at position $(i,j)$ is 1 if the pair $(a_i,b_j)$ belongs to the relation; otherwise it is 0.

Example:
Let $A=\{1,2\}$ , $B=\{a,b\}$ , and
$R=\{(1,a),(2,b)\}$

Table form:

	a	b
1	1	0
2	0	1

Matrix form:

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Importance of table/matrix form:

Compact representation of relations
Useful for large relations where drawing many arrows becomes confusing
Helps in checking properties like symmetry and transitivity
Useful in computer science and digital logic

Relation to pictorial understanding:
Even though these are not pictures in the usual sense, they provide a visual arrangement of relation data and are often used as part of pictorial learning.

Working / Process

1. Identify the sets and ordered pairs

First, write down the sets involved in the relation and list all the ordered pairs that belong to the relation. This tells you exactly which elements are connected.

2. Choose the suitable visual form

If the sets are finite and small, use an arrow diagram. If the relation involves numbers and coordinates, use a Cartesian graph. If the relation is large or needs compact analysis, use a table or matrix.

3. Draw or plot the relation clearly

Place the elements of the first set and second set in correct positions, then connect related elements using arrows, dots, or marked cells. Finally, verify that every ordered pair is represented correctly and that no unrelated pair is shown.

Advantages / Applications

Makes abstract relations easy to understand and remember
Helps identify whether a relation is a function, one-to-one, many-to-one, or many-to-many
Useful in solving problems involving reflexive, symmetric, and transitive properties
Helps compare relations quickly using diagrams or graphs
Widely used in mathematics, computer science, logic, databases, and network models

Summary

Pictorial representation of relation is a visual way to show how elements of sets are connected. It is commonly done using arrow diagrams, graphs, tables, and matrices. This method makes relations simple, clear, and easy to analyze.