Relations

Topic 1 - Relations & Types of Relations

What is a Relation?

A relation is a rule that connects elements of one set to elements of another set. In mathematics, it is defined as a subset of the Cartesian product of two sets.

If A and B are two sets, then a relation R is written as: R ⊆ A × B
Each element of a relation is an ordered pair (a, b)
Here, 'a' belongs to set A and 'b' belongs to set B

Example:

If A = {1,2} and B = {3,4}, then

R = {(1,3), (2,4)} is a relation from A to B

Ways to Represent a Relation

Roster Form: List of ordered pairs
Set-builder Form: Using a rule or condition
Arrow Diagram: Mapping elements visually
Graph: Plotting points on coordinate plane

Types of Relations

Reflexive Relation

Every element is related to itself.

Condition: (a,a) ∈ R for all a

Example: (1,1), (2,2)

Symmetric Relation

If (a,b) ∈ R then (b,a) ∈ R

Example: (1,2) and (2,1)

Transitive Relation

If (a,b) and (b,c) ∈ R then (a,c) ∈ R

Example: (1,2), (2,3) ⇒ (1,3)

Equivalence Relation

Relation that is reflexive, symmetric, and transitive

Very important for exams

Empty Relation

No elements are related

R = ∅

Universal Relation

All possible pairs are included

R = A × B

Important Notes

Relation is always a subset of Cartesian product
Number of possible relations = 2^(n×m)
Equivalence relation is frequently asked in exams
Relations can be finite or infinite
Understanding ordered pairs is key