Number of Equivalence Classes

Question

Let $M=\{1,2,\ldots,20\}$ and define a function $f:M\to \mathbf{Z}$ by $f(x)=\min(x,3)$. Define an equivalence relation on M by letting two element $m$ and $n$ be equivalent if $f(m)=f(n)$. 1) How many equivalence classes are there? 2) How many elements are part of the biggest equivalence class?

Answer 1

If both $m,n \geq 3$, then they are in the same equivalence class since then $$f(m) = \min(m,3) = 3 = \min(n,3) = f(n).$$ If $m,n\leq 2$, then $\min(m,3) = \min(n,3)$ if and only if $m = n$. So the equivalence classes are $[1] = \{1\}, [2] = \{2\}, [3] = \{3,\ldots,20\}$. That is, there are 3 distinct equivalence classes and the largest one is $[3]$ with 18 elements. The graph idea mentioned above by @julien is also a good way to see this visually.