# 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

$S \setminus \O \subseteq S$

From the definition of the empty set:

$\forall x \in S: x \notin \O$

Let $x \in S$.

Thus:

 $\displaystyle x \in S$ $\leadsto$ $\displaystyle x \in S \land x \notin \O$ Rule of Conjunction $\displaystyle$ $\leadsto$ $\displaystyle x \in S \setminus \O$ Definition of Set Difference $\displaystyle$ $\leadsto$ $\displaystyle S \subseteq S \setminus \O$ Definition of Subset

Thus we have:

$S \setminus \O \subseteq S$

and:

$S \subseteq S \setminus \O$

So by definition of set equality:

$S \setminus \O = S$

$\blacksquare$