Equality between sets. Subsets and Supersets

We will say that two sets $$A$$ and $$B$$ are equal, written as $$A = B$$ if they have the same elements. That is, if, and only if, every element of $$A$$ is contained also in $$B$$ and every element of $$B$$ is contained in $$A$$. In symbols: $$$x\in A \Leftrightarrow x\in B$$$

We say that a set $$A$$ is a subset of another set $$B$$,if every element of $$A$$ is also an element of $$B$$, that is, when the following is verified: $$$x\in A \Rightarrow x\in B$$$ whatever the element $$x$$ is. In this case, it is written $$$A\subseteq B$$$

Note that by definition, the possibility that if $$A\subseteq B$$, then $$A = B$$ is not excluded. If $$B$$ has at least one element not belonging to $$A$$, but if every element of d$$A$$ is an element of $$B$$, then we say that $$A$$ is a proper subset of $$B$$, which is represented as $$A\subset B$$.

Thus, the empty set is a proper subset of every set (except of itself), and any set $$A$$ is an improper subset of itself.

If $$A$$ is a subset of $$B$$, we can also say that $$B$$ is a superset of $$A$$, written $$B\supseteq A$$ and say that $$B$$ is a proper superset of $$A$$ if $$B \supset A$$.

By principle of identity, it is always true that $$$x\in A \Rightarrow x\in A $$$ for every element $$x$$, so, every set is a subset and a superset of itself.