<p>My question is about the following notation:</p>

<p>I have noticed in several places (for example <a href="https://rads.stackoverflow.com/amzn/click/1107458110" rel="nofollow noreferrer">here</a> (page 15)  and <a href="http://www.columbia.edu/~md3405/DT_Choice_15.pdf" rel="nofollow noreferrer">here</a> (page 1)) that different authors use different notations for <em>choice correspondences</em>. Given a set $X$ of <em>objects that might be chosen</em> and a collection of non-empty subsets $\mathcal{A}$ of the power set of $X$, some use </p>

<ol>
<li>$c:\mathcal{A} \rightarrow 2^{X}\backslash \emptyset$ (as in the second link)</li>
<li>$c:\mathcal{A} \rightarrow 2^{X}\backslash \{ \emptyset\}$ (as in the first link)</li>
</ol>

<p>They both want to express that the co-domain of the choice correspondence must include all the <em>non-empty subsets</em> of the power set of X, i.e from any menu of options, you <em>must</em> choose something. </p>

<p>My question is, which one is the correct way to define it? Clearly, since $\emptyset \neq \{\emptyset\}$, both definitions cannot be equivalent. </p>

<p>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:</p>

<p>$$
2^{X}=\Bigg\{\emptyset, \{p \},  \ \{c \}, \{s \},     \{p,c \}, \{p,s \} , \{c,s \}, \{ p,c, s\} \Bigg\}
$$</p>

<p>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 </p>

<p>$$
c:\mathcal{A} \rightarrow 2^{X}\backslash \emptyset
$$</p>

<p>Thank you for any answers!</p>

<choice-theory>
Choice correspondence notation
thekiciminister
# Choice correspondence notation

(Asked by: username_0 on 2018-01-16)

My question is about the following notation:

I have noticed in several places (for example [here](https://rads.stackoverflow.com/amzn/click/1107458110) (page 15) and [here](http://www.columbia.edu/~md3405/DT_Choice_15.pdf) (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

1. $c:\mathcal{A} \rightarrow 2^{X}\backslash \emptyset$ (as in the second link)
2. $c:\mathcal{A} \rightarrow 2^{X}\backslash \{ \emptyset\}$ (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!

**Tags:** choice-theory

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## 0 answers

