What is the number of equivalence classes formed by a merge I have this question related to the number of equivalence classes of equivalence relations. If $R_1$ and $R_2$ are two equivalence relations on a set $A$ with number of equivalence classes of $R_1 = n_1$ and number of equivalence classes of $R_2= n_2$. What is the number of equivalence classes formed by $R_1 \cap  R_2\,$?

It is less than or equal to $max\{n1,n2\}$.
The equivalent class formed by is $R_1 \cap R_2$ is $span(R_1 \cap R_2)$. However,
$span(R_1 \cap R_2)\subseteq R_1$ and $span(R_1 \cap R_2)\subseteq R_2$. 
Because $R_1$ and $R_2$ are all equivalent classes, the span cannot have more elements than $R_1$ and $R_2$ at the same time.
