Is it true that $\frac{A\cup B}{C\cup D}=\frac{A}{C}\cup \frac{B}{D}$? Suppose $A, B$ are sets and $C\subset A$, $D\subset B$ subsets such that $C\cap D=\phi$. Is it true that
$$\frac{A\cup B}{C\cup D}=\frac{A}{C}\cup \frac{B}{D}.$$
Recall that if $X$ is a set and $A\subset X$ is a subset then $\displaystyle\frac{X}{A}$ is the quotient set of $X$ modulo the equivalence relation: $x\sim y$ if $x, y\in A$.
 A: After the edits, I get the question. This statement is false. The simplest example I can come up with is the following:
Say that $A = B = \{1,2\}$ and $C = \{1\}$ and $D = \{2\}$. Then the left hand side has one element because $A\cup B = C\cup D$ and the right hand side has two elements. In fact, the set $\frac{A}{C}$ has two elements since here $1\not\sim 2$ because $1,2$ are not both elements in $C$.
A: This is certainly not true, since the union $C\cup D$ is not sensitive to whether elements came from $C$ or $D$, but the expression on the right side is sensitive to this sort of rearrangement. (Thanks to @Thomas for pointing out that "re-ordering" is not the right word.) Here's an intuitive angle: $C$ and $D$ are merged to become a single element in $\frac{A \cup B}{C \cup D}$. On the other hand, $C$ becomes a single element in $A/C$ and $D$ becomes a single element in $B/D$, but there's no reason that these two elements have to be linked together in $\frac{A}{C} \cup \frac{B}{D}$.
Example: Take $A=B=\{1,2\}$. Now let $C=\{1\}$ and $D=\{2\}$, $C'=\{1,2\}$ and $D'=\emptyset$.  Then $C \cup D = C' \cup D'$, so 
\begin{equation}\tag{$*$}
\frac{A \cup B}{C \cup D}=\frac{A \cup B}{C' \cup D'}.
\end{equation} But $$\frac{A}{C} \cup \frac{B}{D} = \frac{\{1,2\}}{\{1\}} \cup \frac{\{1,2\}}{\{2\}}=\{1,2\} \cup\{1,2\}=\{1,2\}$$
and
$$\frac{A}{C'} \cup \frac{B}{D'} = \frac{\{1,2\}}{\{1,2\}} \cup \frac{\{1,2\}}{\emptyset}=\{[1,2]\} \cup\{1,2\}=\{[1,2]\}.$$
One of these sets contains two elements and the other contains three elements. Hence they cannot be equal, contradicting ($*$). (Note that this example can be generalized to larger sets and decompositions in which no subset is empty.)
