As given in the diagram, each branch represents a function in $F$. e.g. $f1 = \left \{ (3,1), (1,1), (2,1) \right \}$ forms a function.
Now, the the relation $\sim$ will be as follows-
$\sim \ = \left \{ {\color {Red}{(f_1, f_1), (f_1, f_2), (f_1, f_3), \dots, (f_1, f_9), (f_2, f_1), (f_2, f_2), (f_2, f_3), \dots, (f_2, f_9), (f_3, f_1), (f_3, f_2), (f_3, f_3), \dots, (f_3, f_9), \dots, (f_9, f_1), (f_9, f_2), (f_9, f_3), \dots, (f_9, f_9),}} \\ {\color {Blue}{(f_{10}, f_{10}), (f_{10}, f_{11}), (f_{10}, f_{12}), \dots, (f_{10}, f_{18}), (f_{11}, f_{10}), (f_{11}, f_{11}), (f_{11}, f_{12}), \dots, (f_{11}, f_{18}), (f_{12}, f_{12}), (f_{12}, f_{11}), (f_{12}, f_{12}), \dots, (f_{12}, f_{18}), \dots, (f_{18}, f_{10}), (f_{18}, f_{11}), (f_{18}, f_{12}), \dots, (f_{18}, f_{18}),}} \\ {\color {Green}{(f_{19}, f_{19}), (f_{19}, f_{20}), (f_{19}, f_{21}), \dots, (f_{19}, f_{27}), (f_{20}, f_{19}), (f_{20}, f_{20}), (f_{20}, f_{21}), \dots, (f_{27}, f_{27}), (f_{21}, f_{21}), (f_{21}, f_{20}), (f_{21}, f_{21}), \dots, (f_{21}, f_{27}), \dots, (f_{27}, f_{19}), (f_{27}, f_{20}), (f_{27}, f_{21}), \dots, (f_{27}, f_{27})}} \right \}$
Therefore, there are 3 equivalence classes which are as follows-
- Class 1 - $[f_1] = [f_2] = [f_3] = \dots = [f_9] = \left \{ f_1, f_2, f_3, \dots, f_9 \right \}$
- Class 2 - $[f_{10}] = [f_{11}] = [f_{12}] = \dots = [f_{18}] = \left \{ f_{10}, f_{11}, f_{12}, \dots, f_{18} \right \}$
- Class 3 - $[f_{19}] = [f_{20}] = [f_{21}] = \dots = [f_{27}] = \left \{ f_{19}, f_{20}, f_{21}, \dots, f_{27 }\right \}$
Therefore, we can see that there are 3 equivalence classes, each having 9 elements.