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

An equivalence relation is one which is reflexive, symmetric and transitive.

Given that, A = {1, 2, 3}

We can define equivalence relation on A as follows.

R₁ = A × A = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3),

(3,1),(3,2),(3,3)}

R₁ is reflexive ∵ (1,1),(2,2),(3,3) ∈ R

R₁ is symmetric ∵ (1,2),(1,3),(2,3) ∈ R ⇒ (2,1),(3,1),(3,2) ∈ R

R₁ is Transitive ∵ (1,2) ∈ R and (2,3) ∈ R ⇒ (1,3) ∈ R

Similarly,

R₂ = {(1,1),(2,2),(3,3),(1,2),(2,1)}

R₃ = {(1,1),(2,2),(3,3),(1,3),(3,1)}

R₄ = {(1,1),(2,2),(3,3),(2,3),(3,2)}

R₅ = {(1,1),(2,2),(3,3)}

∴ maximum number of equivalence relation on A is ‘5’.