Subset Equivalences

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Definitions

In the following:

$S \subseteq T$ denotes that $S$ is a subset of $T$
$S \cup T$ denotes the union of $S$ and $T$
$S \cap T$ denotes the intersection of $S$ and $T$
$S \setminus T$ denotes the set difference between $S$ and $T$
$\varnothing$ denotes the empty set
$\mathbb U$ denotes the universal set
$\complement$ denotes set complement.


Union with Superset is Superset‎

$S \subseteq T \iff S \cup T = T$


Intersection with Subset is Subset‎

$S \subseteq T \iff S \cap T = S$


Set Difference with Superset is Empty Set‎

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


Intersection with Complement is Empty iff Subset

$S \subseteq T \iff S \cap \map \complement T = \O$


Complement Union with Superset is Universe

$S \subseteq T \iff \complement \left({S}\right) \cup T = \mathbb U$


Set Complement inverts Subsets

$S \subseteq T \iff \map \complement T \subseteq \map \complement S$


Sources