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

It is given that,, $A = \{ 1,2,3\}$

Now, number of equivalence relations as follows

${R_1} = \{ (1,1),(2,2),(3,3)\}$

${R_2} = \{ (1,1),(2,2),(3,3),(1,2),(2,1)\}$

${R_3} = \{ (1,1),(2,2),(3,3),(1,3),(3,1)\}$

${R_4} = \{ (1,1),(2,2),(3,3),(2,3),(3,2)\}$

${R_5} = \left\{ {(1,2,3) \Leftrightarrow A \times A = {A^2}} \right\}$

$\therefore$ Maximum number of equivalence

relation on the set $A = \{ 1,2,3\} = 5$