Representation of Relations and Functions

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:

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:
  • 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:

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:

Read More,

• Relation and Functions
• Relations in Maths
• Types of Relations
• Functions in Maths
• Types of Functions