Proving that exists equivalence relation $r$ in set $A$ such that $ |A \setminus r| = n$

Question

I am trying to show that if $|A| = m$ and $0\neq n \le m $ then exists equivalence relation $r$ in set $A$ such that $ |A \setminus r| = n$. Could someone help me deal with it?

Answer 1

Without loss of generality let $A = \{1, \ldots, m\}$. Choose $A_1 = \{1\}$, $A_2 = \{2\}$, ..., $A_{n-1} = \{n-1\}$, $A_n = \{n, \ldots, m\}$. This defines a partition of $A$ and hence and equivalence relation.