In my Mathematics textbook, the definition of the power set of a given set is given as follows :
$$P(A) = \{X : X \subseteq A \}$$
Now, this is used to say that the power set of a given set $A$ is the set that contains all sets that are a subset of $A$.
But, the symbol $\subseteq$ is generally used to denote improper subsets. Basically, if $A \subseteq B \iff A = B$. But, if we take that definition of the symbol $\subseteq$, then the power set will only contain the set that is equal to the given set i.e. in that case, $P(A) = A$.
So, that means that the symbol $\subseteq$ is just used as a general subset symbol, which would include both proper and improper subsets of the given set, right? But, sometimes the symbol $\subset$ is also used in place of this. And in the definition of a power set itself too. What I mean is that in another textbook, I saw the definition of a power set as follows :
$$P(A) = \{ X : X \subset A \}$$
So, what I think is that in the definitions of power set that I mentioned above, both $\subset$ and $\subseteq$ are used to just represent a subset (whether it be a proper one, or an improper one).
But, if we use $\subseteq$ as a symbol for improper subsets and $\subset$ as a symbol for proper subsets, would the definition of a power set be :
$$P(A) = \{ X : X \subset A \} \cup \{ X : X \subseteq A \} = \{ X : X \subset A \} \cup \{ A \} \text{ ?}$$
Best Answer
$A\subseteq B$ does NOT mean that $A=B$. It means simply that $A$ is a subset of $B$ and explicitly allows that subset to be $B$ itself. Many people use $A\subset B$ to mean the same thing. Others use $A\subset B$ to mean that $A$ is a proper subset of $B$, i.e., any subset of $B$ except $B$ itself. And some of us, including me, prefer to avoid this ambiguity by writing $A\subsetneqq B$ or $A\subsetneq B$ when we want to specify that $A$ is a proper subset of $B$.
The answer to your final question is yes, though you don’t need the first union: if you use $\subseteq$ for arbitrary subsets and $\subset$ strictly for proper subsets, it’s simply
$$\wp(A)=\{X:X\subseteq A\}=\{X:X\subset A\}\cup\{A\}\;.$$