Subset Equivalences

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 $T$
 * $\O$ denotes the empty set
 * $\mathbb U$ denotes the universal set
 * $\complement$ denotes set complement.