by Seth Grief-Albert
A set is a collection of distinct elements.
A is a subset of B if every element of A is an element of B.
$$ A\sube B $$
If A and B are subsets of each other, they are equal.
$$ A=B\implies A\sube B ,\hspace{.1cm}B\sube A $$
If A is a subset of B but is not equal to B, it is a proper subset.
$$ A\sube B,\hspace{.1cm}A\neq B\\\implies A\sub B $$
The set of all elements in both sets
$$ A\cap B $$
If A union B equal the empty set, they are disjoint.
$$ A\cap B=\empty\\\implies disjoint $$
$$ \empty\neq{\{\empty\}} $$