Suppose X = {1, 2, 3, 4} and R is a relation on X.

If R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}, then which one of the following is correct?

Concept:

For any two non-empty sets A and B, a relation R from A to B is a subset of the cartesian product A X B.

Reflexive:

Let R be a relation on a non-empty set A, if every element of A is related to itself then R is said to be a reflexive relation.

Thus, R is reflexive if and only if (a, a) ∈ R, ∀ a ∈ A.

Symmetric:

Let R be a relation on a non-empty set A, then the relation R is said to be symmetric relation if and only if (a, b) ∈ R ⇒ (b, a) ∈ R

Transitive:

Let R be a relation on a non-empty set A, then the relation R is said to be transitive relation if and only if (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R

Equivalence:

Let R be a relation on a non-empty set A, then the relation R is said to be equivalence relation if R is reflexive, symmetric and transitive.

Calculation:

Given: X = {1, 2, 3, 4} and R is a relation on X given by R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2), (1, 3), (3, 1)}

Reflexive:

The given relation R on X is not reflexive ∵ (a, a) ∉ R, ∀ a ∈ X i.e (4, 4) ∉ R.

Symmetric:

The given relation R on X is symmetric ∵ for all (a, b) ∈ R ⇒ (b, a) ∈ R 

Transitive: