Types of Relation (Mathematics)

Types of Relation:

(1) Identity Realtion- Relation R on a set A defined as {(a, a) ∀ a ∈ A} is said to be an identity relation on set A and is denoted by I_A.

For example- If A = {1, 2, 3} then I_A = {(1, 1) (2, 2) (3, 3)}

(2) Reflexive Relation- A relation R on a set A is said to be reflexive if every element of A is related to itself i.e. R is reflexive ⇔ (iff) aRa ∀ a ∈ A. Considering the set A = {a, b, c, d}, if any one of the ordered pairs (a, a) (b, b) (c, c) (d, d) is absent in the relation R, then the relation cannot be reflexive.

For example- If A = {1, 2, 3} and R = {(1, 1) (1, 2) (2, 2) (3, 2) (3, 3)} then R is reflexive.

The universal relation on a non-void set is reflexive. The identity relation on a non-void set is always reflexive but a reflexive relation is not necessarily an identity relation.

Let A = set of lines in a plane and then the relation R is defined by a₁Ra₂ iff a₁ || a₂ is reflexive because every line is parallel to itself.

(3) Symmetric Relation- A relation R on a set A is said to be symmetric iff (a, b) ∈ R ⇒ (b, a) ∈ R.

For example- If A = {1, 2, 3} and R = {(1, 1) (1, 2) (2, 1) (3, 3)} then R is symmetric.

Let A = set of lines in a plane, then the relation R is defined by a₁Ra₂ iff a₁ || a₂ is symmetric because if a₁ || a₂ ⇒ a₂ || a₁.

The identity and the universal relations on a non-empty set A are symmetric relations on A.

(4) Transitive Relation- A relation R on a set A is said to be transitive iff (a, b) ∈ R, (b, c) ∈ R ⇒ (a, c) ∈ R. Let A be the set of natural numbers and a relation R on a set A is defined by aRb iff a divides b is transitive because if a divides b and b divides c then a divides c.

The identity and the universal relations on a non-empty set are transitive.

(5) Equivalence Relation- A relation R on a set A is said to be an equivalence relation if-

R is reflexive.
R is symmetric.
R is transitive.

Example- Three relations R₁, R,_2, and R₃ are defined on set A = {-1, 0, 1} as follows:
(i) R₁ = {(-1, -1), (0, 0), (1, 1)}
(ii) R₂ = {(-1, 0), (0, 1), (1, -1), (0, -1), (1, 0), (-1, 1)}
(iii) R₃ = {(0, 0), (1, 1), (0, 1), (1, 0)}
Find which of the relations on A is reflexive, symmetric, or transitive.

Solution- (i) R₁ is reflexive because of a ∈ A ⇒ (a, a) ∈ R.
(ii) R₂ and R₃ are symmetric.
(iii) R₃ is transitive.

Example- Let R be a relation on the set I of all integers defined as (x, y) ∈ R ⇔ x – y is divisible by n. Prove that R is an equivalence relation.

Solution- R is reflexive:
∀ x ∈ I, x – x = 0, which is divisible by n.
∴ (x, x) ∈ R
∴ R is reflexive.

R is symmetric:
Suppose (x, y) ∈ R ⇒ x – y is divisible by n.
⇒ x – y = nλ
⇒ – (y – x) = nλ
⇒ (y – x) = n (-λ)
⇒ (y – x) = nµ
⇒ (y – x) is divisible by n
⇒ (y, x) ∈ R
∴ R is symmetric.

R is transitive:
Suppose (x, y) and (y, z) ∈ R.
⇒ x – y is divisible by n and y – z is divisible by n.
⇒ x – y = nλ and y – z = nµ
⇒ y = x – nλ ……….(i) and y = z + nµ ……….(ii)
From (i) and (ii):
x – nλ = z + nµ
⇒ x – z = nλ + nµ
⇒ x – z = n (λ + µ)
⇒ x – z = np
⇒ x – z is divisble by n
⇒ (x, z) ∈ R
∴ R is transitive.

Thus R is an equivalence relation.

Operation on Sets
Characteristic and Mantissa of a Logarithm
Computation with the Aid of Logarithms
Relation Between the Roots and Coefficients of a Quadratic Equations
Properties of Roots of a Quadratic Equation
Relation (mathematics)– Wikipedia