Let $X = \{0,1,2,3,4,5,6,7,8,9\}$ and $ Y = \{0,2,4,6,8,9\}$. In $P(X) =$ power set of $X $ define the following relation:
$$A R B \Leftrightarrow A \setminus Y = B \setminus Y $$
Then, how many elements has the quotient set $P(X)/R$ ? I know that the answer is 16, but I would like know calculate it.
Two sets agree if they agree on the complement of $Y$, so there is a canonical bijection $P(X)/R \to P(X \backslash Y) = P (\{ 1,3,5,7\})$. The bijection identifies $[A] \in P(X)/R$ with $A \cap (X \backslash Y)$. Then you use the fact that $|P(\{x_1,\cdots,x_n\})| = 2^n$.
Hope that helps,