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:

$S \setminus \varnothing = S$

First, we have $S \setminus \varnothing \subseteq S$ from Set Difference Subset.

Next, we first note that $\forall x \in S: x \notin \varnothing$ from the definition of the empty set.

Let $x \in S$. Thus:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle x \in S$	$\implies$	$\displaystyle x \in S \land x \notin \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Rule of Conjunction
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle x \in S \setminus \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Set Difference
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\implies$	$\displaystyle S \subseteq S \setminus \varnothing$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of a Subset

Thus we have $S \setminus \varnothing \subseteq S$ and $S \subseteq S \setminus \varnothing$.

So by the definition of set equality, $S \setminus \varnothing = S$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x \in S\)	\(\implies\)	\(\displaystyle x \in S \land x \notin \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rule of Conjunction
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle x \in S \setminus \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Set Difference
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\implies\)	\(\displaystyle S \subseteq S \setminus \varnothing\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of a Subset