What's the standard notation for summing over all subsets of a set?

Question

I'm reading this book on interpretable AI. In section 8.4 the author uses the following notation for a summation across all subsets of a set:

$$ \sum_{S \subseteq \{1, \dots, p\}} \hat f_S(x_S) $$

Is this standard notation? Wouldn't it better to explicitly state:

$$ \sum_{S \in P(\{1, \dots, p\})} \hat f_S(x_S) $$

where $P(\dots)$ represents the powerset of whatever's contained within the parentheses. Are these two equivalent, and which one is considered clearest?

Answer 1

Assuming that $P$ (as opposed to $\mathbb P$ or $\mathbf P$ or $\wp$ or $2^{X}$ or $\operatorname{Pow}$) has been established for the powerset, then "$S\in P(X)$" is logically equivalent to "$S\subseteq X$".

Relatedly, I would say that the two summation notations are equivalent, and neither is more "explicitly stated" than the other.

If $P(X)$ were used a lot in the context around the summation, I might prefer to use $P(X)$ in the summation notation, for familiarity. But because power set notations vary, and because "subset" is more commonly encountered than "power set", I'd prefer the book's summation notation with $\subseteq$ in most contexts.

Answer 2

I think the first notation is clearly better; subset notation $\subseteq$ is unambiguous and widely understood (it only has this one meaning and no others), and it avoids referring to the powerset.

Notation for the powerset, on the other hand, is not standard, not very commonly used, and collides slightly with other notation (e.g. $P$ is also used to refer to probabilities, or polynomials). If anything I would argue that the use of subset notation is more explicit than the reference to the powerset because it does not require that the reader know that $P$ refers to the powerset as opposed to something else.