Set Difference with Superset is Empty Set

Jump to navigation Jump to search

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

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

$\blacksquare$