# Use of the $\subset$ and $\subseteq$ symbols in the definition of a power set and re-defining the power set with these symbols.

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 : $$P(A) = \{ X : X \subset A \} \cup \{ X : X \subseteq A \} = \{ X : X \subset A \} \cup \{ A \} \text{ ?}$$

$$A\subseteq B$$ 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
$$\wp(A)=\{X:X\subseteq A\}=\{X:X\subset A\}\cup\{A\}\;.$$
• Thanks for the answer! I don't understand what you mean by It simply means that A is a subset of B and explicitly allows that subset to be B itself. I'd be glad if you can clear that to me. And, the last part was actually a mistake. I interchange $\cap$ and $\cup$ a lot by mistake. Shall I edit it or not? – Rajdeep Sindhu Jun 3 at 20:42
• @Rajdeep: The notation $A\subseteq B$ says that $A$ is any subset of $B$; the inclusion of the underscore in the symbol is simply a way of saying explicitly that $A$ could be $B$ as well as any proper subset of $B$. Yes, by all means edit the question to fix the mistake; when you do, I’ll add a little at the end of mine to comment on the corrected version. – Brian M. Scott Jun 3 at 20:47