My question is about the following notation:
I have noticed in several places (for example here (page 15) and here (page 1)) that different authors use different notations for choice correspondences. Given a set $X$ of objects that might be chosen and a collection of non-empty subsets $\mathcal{A}$ of the power set of $X$, some use
- $c:\mathcal{A} \rightarrow 2^{X}\backslash \emptyset$ (as in the second link)
- $c:\mathcal{A} \rightarrow 2^{X}\backslash \{ \emptyset\}$ (as in the first link)
They both want to express that the co-domain of the choice correspondence must include all the non-empty subsets of the power set of X, i.e from any menu of options, you must choose something.
My question is, which one is the correct way to define it? Clearly, since $\emptyset \neq \{\emptyset\}$, both definitions cannot be equivalent.
My intuition would tell me that from the list above, (1) is the correct one. For example, if $X=\{\text{pasta (p)}, \text{chicken (c)}, \text{sushi (s)}\}$, then the power set of $X$ is:
$$ 2^{X}=\Bigg\{\emptyset, \{p \}, \ \{c \}, \{s \}, \{p,c \}, \{p,s \} , \{c,s \}, \{ p,c, s\} \Bigg\} $$
Since I want to “take out” the empty set from $2^{X}$ so that the choice from any non-empty subset of $X$ is itself non-empty, and since $\{\emptyset\} \notin 2^{X}$, it should be
$$ c:\mathcal{A} \rightarrow 2^{X}\backslash \emptyset $$
Thank you for any answers!