Set Difference with Empty Set is Self

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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:

\(\displaystyle x \in S\) \(\leadsto\) \(\displaystyle x \in S \land x \notin \O\) 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:

$S \setminus \O = S$

$\blacksquare$


Also see


Sources