Set Equation: Union

Theorem
Let $A$ and $B$ be sets.

Consider the set equation:
 * $A \cup X = B$

The solution set of this is:
 * $\varnothing$ if $A \nsubseteq B$


 * $\left\{ {\left({B \setminus A}\right) \cup Y: Y \subseteq A}\right\}$ otherwise.

Proof
In the first case $A$ is a not a subset of $B$.

So there exists an $x \in A$ such that $x \notin B$.

By the definition of union:


 * $\forall X: x \in A \implies x \in A \cup X$

Hence the solution set is empty.