Image of Set Difference under Relation

Theorem
Let $\mathcal R \subseteq S \times T$ be a relation. Let $A$ and $B$ be subsets of $S$.

Then:
 * $\mathcal R \left({A}\right) \setminus \mathcal R \left({B}\right) \subseteq \mathcal R \left({A \setminus B}\right)$

where $\setminus$ denotes set difference.

Note
Note that equality does not hold in general.

See the note on Mapping Image of Set Difference for an example of a mapping (which is of course a relation) for which it does not.