Different way to form the power set

Question

The power set can be constructed as a "Cartesian product" as in the following example. Suppose $S=\{a, b, c\}$ and that $\times$ is a modified Cartesian product operator (see following paragraph), then:

$$Pset(S)=\{\emptyset,\{a\}\}\boxtimes \{\emptyset,\{b\}\} \boxtimes \{\emptyset,\{c\}\}$$

The difference being that the pairs of the product are actually unified together to make one set. For example, the pair $(\emptyset, \emptyset)$ yields $\emptyset$, the pair $(\emptyset, \{a\})$ yields $\{a\}$.

Does this operation have a standard name/notation? I want to prove that the resulting set is actually the power set but I need notation or try to introduce appropriate notation.

Note: This construction tells us that the cardinality is $2^{|S|}$ right away which is true.

Answer 1

The set you describe is not the power set of $\{a,b,c\}$. It is a set that has the same cardinality as the power set, but its elements $x$ do not satisfy $x\subseteq S$.

EDIT: after the clarifications and modifications it all makes sense now. I never saw the definition $A\boxtimes B=\{a\cup b\mid a\in A,b\in B\}$, which in the context of axiomatic set theory makes sense. It does not make sense universally in naive set theory though. The equality $\mathcal P(S)=\{\emptyset ,\{s_1\}\}\boxtimes \cdots \{ \emptyset, \{s_n\}\}$ makes sense and is true when $S=\{s_1,\cdots , s_n\}$. With a bit of care it can be extended to infinite sets and infinite box products.