Set Difference with Superset is Empty Set

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Theorem

$S \subseteq T \iff S \setminus T = \O$


where:

$S \subseteq T$ denotes that $S$ is a subset of $T$
$S \setminus T$ denotes the set difference between $S$ and $T$
$\O$ denotes the empty set.


Proof

\(\displaystyle \) \(\) \(\displaystyle S \setminus T = \O\)
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \neg \paren {\exists x: x \in S \land x \notin T}\) Definition of Empty Set
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \forall x: \neg \paren {x \in S \land x \notin T}\) De Morgan's Laws (Predicate Logic)
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \forall x: x \notin S \lor x \in T\) De Morgan's Laws: Disjunction of Negations
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \forall x: x \in S \implies x \in T\) Rule of Material Implication
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle S \subseteq T\) Definition of Subset

$\blacksquare$


Sources