Types of Relation in Maths
Last Updated :
16 Jan, 2025
A relation is like a rule that tells us how two things are connected.
- For example:
- In a classroom, the relation can be "is the friend of."
- In numbers, the relation can be "is greater than" or "is equal to."
Mathematically, a relation links elements from one set (group) to another.
Given two sets A and B, a relation R from A to B is a subset of the Cartesian product A × B. Mathematically:
R ⊆ A × B
Here, A × B represents the set of all ordered pairs (a, b), where a ∈ A and b ∈ B.
Some of the common types of relations in maths are:
Universal Relation
A relation R on a set A is called a universal relation if every element in A × A is related.
- Example: For set A = {1, 2}, R = A × A = {(1, 1), (1, 2), (2, 1), (2, 2)}.
Empty Relation
A relation R is empty if it contains no ordered pairs.
- Example: For A = {1, 2}, R = ∅.
Identity Relation
A relation R on set A is called an identity relation if every element is related to itself and no other elements.
- Example: For A = {1, 2, 3}, R = {(1, 1), (2, 2), (3, 3)}.
Reflexive Relation
R is reflexive if (a, a) ∈ R for all a ∈ A.
- Example: For A = {1, 2}, R = {(1, 1), (2, 2), (1, 2)}.
Symmetric Relation
R is symmetric if (a, b) ∈ R ⇒ (b, a) ∈ R.
- Example: If R = {(1, 2), (2, 1)}, R is symmetric.
Anti-Symmetric Relation
R is anti-symmetric if (a, b) ∈ R and (b, a) ∈ R ⇒ a = b.
- Example: If R = {(1, 2)}, R is anti-symmetric.
Transitive Relation
R is transitive if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R.
- Example: If R = {(1, 2), (2, 3), (1, 3)}, R is transitive.
Equivalence Relation
R is an equivalence relation if it is reflexive, symmetric, and transitive.
- Example: Relation R defined by "is of the same age group" is an equivalence relation.
Partial Order Relation
R is a partial order if it is reflexive, anti-symmetric, and transitive.
- Example: ≤ on the set of integers.
Total Order Relation
R is a total order if it is a partial order and every pair of elements is comparable.
Inverse Relation
The inverse of a relation R is defined as R−1 = {(b, a): (a, b) ∈ R}.
- Example: If R = {(1, 2), (3, 4)}, then R−1 = {(2, 1), (4, 3)}.
Complement of a Relation
The complement of R is defined as Rc = (A × A) − R.
- Example: For A = {1, 2}, R = {(1, 1)}, Rc = {(1, 2), (2, 1), (2, 2)}.
Void Relation
A relation with no pairs, R = ∅, is called a void relation.
- Example: For any set A, R = ∅ is void.
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