Define the following terms using mathematical notations. Also provide example for each term- Subset Universal set Power set

Let us define the following terms using mathematical notations and also provide example for each term.

Subset:

We say that a set $A$ is a subset of a set $B$ if each elements $a\in A$ belongs to $B.$ For example, let $A=\{1\}, B=\{1,2\}.$ Since $1\in A$ impllies $1\in B,$ we conclude that $A$ is a subset of $B.$

Universal set:

A universal set $U$ is a set which contains all objects that may appear in some task or class of tasks. For example, if the problem lies in finding the real roots of quadratic equations, then for this task the universal set $U$is the set $\R$ of real numbers.

Power set:

The power set (or powerset) $P(X)$ of a set $X$ is the set of all subsets of $X$, including the empty set and $X$ itself, that is $P(X)=\{A: A\subset X\}.$ For examle for the set $X=\{0,1\}$ its power set $P(X)=\{\emptyset,\{0\},\{1\}, X\},$ where $\emptyset$ is the empty set.