Question

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

Answer 1

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

There is a maximum of $5$ equivalence relations on the set $A=\{1,2,3\}$

They are

$$\begin{gathered} R_1=\left\{\right(1,1),(2,2),(3,3\left)\right\} \\ {R}_2=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right\} \\ {R}_3=\left\{\right(1,1),(2,2),(3,3),(1,3),(3,1\left)\right\} \\ {R_4=\left\{\right(1,1),(2,2),(3,3),(2,3),(3,2\left)\right\}}_{} \\ {R}_5=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2\left)\right\} \end{gathered}$$

Hence,the Maximum number of equivalence relation's is $\text{'}5\text{'}$