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.