let $(X_i)_{i \in I}$ be an infinite family of sets with $|X_i| \geq 2$. we define an equivalence relation on $X = \prod_{i \in I} X_i$ by $x \sim y \Leftrightarrow \{i : x_i \neq y_i\}$ is finite. what is the cardinality of $X/\sim$? we may endow the $X_i$ with group structures and write this set as $\prod_{i \in I} X_i / \oplus_{i \in I} X_i$.

it is easy to see that $\min_i |X_i| \leq |X/\sim|$ and $|X| \leq |X/\sim| * \max_i |X_i|$. in particular, if $|X_i|$ is constant, $|X/\sim| = |X_i|$.

if all the $X_i$ are finite, it can be shown $|X/\sim|=2^{|I|}$. the same equation holds, if every $X_i$ satisfies $\aleph_0 \leq |X_i| \leq |I|$ (I'll add the proofs if needed).

the general case struggles me.