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 $$T$$
 * $$\varnothing$$ denotes the empty set;
 * $$\mathbb{U}$$ denotes the universal set;
 * $$\complement$$ denotes set complement.