Question

Mark the correct alternative in the following question:

The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .

Options

• 1

• 2

• 3

• 5

Answer 1

Consider the relation R₁ = (1, 1)}

It is clearly reflexive, symmetric and transitive

Similarly, R₂ = {(2, 2)} and R₃ = {(3, 3)} are reflexive, symmetric and transitive

Also, R₄ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}

It is reflexive as (a, a) ∈ R₄ for all a ∈ {1, 2, 3}

It is symmetric as (a, b) ∈ R₄ ⇒ (b, a) ∈ R₄ for all a ∈ {1, 2, 3}

Also, it is transitive as (1, 2) ∈ R₄, (2, 1) ∈ R₄ ⇒ (1, 1) ∈ R₄

The relation defined by R₅ = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2)} is reflexive, symmetric and transitive as well.

Thus, the maximum number of equivalence relation on set A = {1, 2, 3} is 5.

Hence, The maximum number of equivalence relations on the set A = {1, 2, 3} is 5.