Let F be the collection of all functions f : {1, 2, 3} - {1, 2, 3}. If g, f ∈ F, define an equivalence relation ~ by f ...

Question

Let F be the collection of all functions f : {1, 2, 3} -> {1, 2, 3}. If g, f ∈ F, define an equivalence relation ~ by f ~ g if and only if f(3) = g(3).
a. Find the number of equivalence classes defined by ~.
b. Find the number of elements in each equivalence class.

Accepted Answer

a. 3 equivalence classes. b. Each class has 9 elements. Two functions are equivalent if they agree at the single point 3. So the equivalence class of f is determined entirely by f(3), which can be 1, 2, or 3 — giving three classes. Within each class, f(3) is fixed to a particular value (say k), but f(1) and f(2) can each independently be any of {1, 2, 3}. So each class has 3 * 3 = 9 functions.

Let F be the collection of all functions f : {1, 2, 3} -> {1, 2, 3}. If g, f ∈ F, define an equivalence relation ~ by f ~ g if and only if f(3) = g(3). a. Find the number of equivalence classes defined by ~. b. Find the number of elements in each equivalence class.