Answer :
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’.
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