Representation of Relations and Functions

Last Updated : 23 Jul, 2025

A relation R from set A to set B is a subset of the Cartesian product A × B. That is:

R ⊆ A × B

Here, A is called the domain, and B is called the co-domain.

Each pair (a, b) ∈ R means that the element a ∈ A is related to the element b ∈ B by the relation R.

Some of the common representations of relations and functions are:

  • Set of Ordered Pairs
  • Arrow Diagram
  • Matrix Representation
  • Adjacency List
  • Directed Graph
  • Table Representation

Set of Ordered Pairs

A binary relation R between two sets A and B is defined as a subset of the Cartesian product A × B. It can be written as a set of ordered pairs i.e., (a, b) where a ϵ A and b ϵ B.

Example:

  • Sets: A = {1, 2, 3}, B = {4, 5}
  • Relation: R: A → B such that a → b + 3
    • R = {(1, 4), (2, 5)}

Arrow Diagram

An arrow diagram visually shows a relation by drawing arrows from elements of set A to elements of set B that are related.

Example:

  • Sets: A = {-3, -2, -1, 0, 1, 2, 3}, B = {0, 1, 2, 3}
  • Relation: R: A → B such that a and -a maps to the absolute value of a.
    • R = {(-3, 3), (3, 3), (-2, 2), (2, 2), (-1, 1), (1, 1), (0, 0)}
  • Arrow Diagram:
Example-of-Arrow-Diagram

Matrix Representation

A binary relation can be represented using a matrix, especially when A and B are finite. The rows correspond to elements of A, and the columns correspond to elements of B. If (a, b) ∈ R, then the matrix entry is 1; otherwise, it is 0.

Example:

  • Sets: A = {1, 2, 3}, B = {4, 5}
  • Relation: R = {(1, 4), (2, 5)}
  • Matrix: M = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}
    • Row 1 corresponds to 1 ∈ A, and column 1 corresponds to 4 ∈ B.
    • M11 = 1, indicating (1, 4) ∈ R.

Adjacency List Representation

An adjacency list is a compact and efficient way to represent a graph, particularly when the graph is sparse (i.e., has fewer edges relative to the number of vertices).

  • Example:
    • A = {0, 1, 2, 3}
    • Relation R: A → A such that
      • R = {(0, 1), (0, 2), (1, 2), (2, 3)}
    • Adjacency List:
      • 0: {1, 2}
      • 1: {2}
      • 2: {3}
      • 3: {}

Directed Graph

A binary relation can also be represented as a directed graph, where:

  • Elements of A and B are vertices.
  • If (a, b) ∈ R, there is a directed edge from A to B.

Example:

  • Sets: A = {1, 2, 3, 4}
  • Relation: R: A → A such that
    • R = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 1), (3, 1), (4, 1), (3, 2), (4, 2), (4, 3)}
  • Graph:
Example-of-Directed-Graph

Table Representation

A binary relation can be presented as a table where rows correspond to elements of A, and columns correspond to elements of B. A checkmark (✔) or 1 indicates that the pair is in the relation and for those which are not in the relation we can use crossmark (❌) or 0.

Example:

  • Sets: A = {1, 2, 3}, B = {1, 2, 3}
  • Relation: is greater than
    • R = {(2, 1), (3, 1), (3, 2)}
  • Table:
x is greater than y12

3

1❌✔

✔

2❌❌

✔

3❌❌

❌

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