Set Difference with Empty Set is Self

Theorem
The set difference between a set and the empty set is the set itself:


 * $S \setminus \O = S$

Proof
From Set Difference is Subset:
 * $S \setminus \O \subseteq S$

From the definition of the empty set:
 * $\forall x \in S: x \notin \O$

Let $x \in S$.

Thus:

Thus we have:
 * $S \setminus \O \subseteq S$

and:
 * $S \subseteq S \setminus \O$

So by definition of set equality:
 * $S \setminus \O = S$

Also see

 * Set Difference with Self is Empty Set