What is the best notation for the "image" or "codomain" of the function that maps a subset $A \subseteq X$ to it's cardinal number, referencing the set $X$ directly?
Once a set $X$ has been chosen, there is a function $card$ that with domain $\mathcal{P}(X)$ that maps subsets to cardinalities. However it is presently unclear to me what do label the codomain of this function.
Typically $|X|$ is used to denote the cardinality of a set $X$. But it is unclear how to label the proper of all cardinalities of subsets of a given set $X$, in a way that references $X$ . Or in other words, the proper set of cardinalities which are less than or equal to $|X|$, the cardinality of the set $X$.
Is it best to simply write:
- Define $card: \mathcal{P}(X) \rightarrow |\mathcal{P}(X)|$ by $card(A) = |A|$ for each $A \subseteq X$.
?
Note the choice of the notation $|\mathcal{P}(X)|$ as the codomain of $card$. Is the powerset notation $|\mathcal{P}(X)|$ a good choice for this codomain?
Consider the cases where $X = \mathbb{N}$, $X = \mathbb{R}$, or more generically where $X$ is an algebraic structure such as a group. Is aleph notation here best? Is it avoidable?
Edit:
So, focusing on the case where $X = \mathbb{R}$ may I write:
Define $card_\mathbb{R}: \mathcal{P}(\mathbb{R}) \rightarrow \vert\mathbb{R}\vert + 1$ by $card_{\mathbb{R}}(A) = |A|$ for each $A \subseteq \mathbb{R}$
so that
$card_\mathbb{R}(\emptyset) = 0$,
$card_\mathbb{R}(\{1,7, 42\}) = 3$,
$card_\mathbb{R}(\mathbb{Z}) = \aleph_0$, and
$card_\mathbb{R}(\mathbb{R}) = \mathfrak{c}$
?
There are a few colliding notations, the vertical bar notation: $\vert \cdot \vert$ (commonly seen in abstract algebra) and the $card()$ notation (as in Real Analysis by Gerald B Folland, where it is is not defined alone, but only as part of an expression such as $card(X) = card(Y)$). I wish to define a function as a restriction of "function-class" [as in Notes on Logic and Set Theory by P.T. Johnstone] $card: V \rightarrow V$ by restricting its domain to to $\mathcal{P}(\mathbb{R})$, which should give a proper function in the sense of set theory, with a domain and a codomain and, hopefully, a notation for each which clearly distinguishes the function from it's codomain.
Counting subsets of $\mathbb{R}$ arises naturally in the formalization of simple ideas such as the "vertical line test" (see below). In the textbooks I've refered to (Dummit & Foote, etc) the question of the codomain here is often glossed over even though in most other situations the codomain of a function is explicitly given a notation, or at least there is one available in needed.
The purpose of this notation is for this context, in defining $v_a$: Given a relation, how is the set of values for which the" vertical line test" fails usually described and what is the notation?.
Best Answer
It's the set of cardinals less than or equal to $|X|$ — that is, the set of all cardinals less than $|X|^+$, the cardinal successor of $|X|$.
In case that's not clear: Of course $|A| \leq |X|$ for any $A \subseteq X$. Conversely, for every cardinal $\lambda \leq |X|$ there's a subset $A$ of $X$ with $|A| = \lambda$ — namely, $f^{-1}[\lambda]$ where $f\colon X \to |X|$ is any bijection.
The range of $A\mapsto |A|$ on $\mathcal{P}(X)$ is $\mathrm{Card} \cap |X|^+$, and if you call it that you'll be understood. If you need to refer to this set frequently, you might define a shorthand for it. For example, if you declare that $\mathrm{Card}_{\leq \kappa} = \mathrm{Card} \cap \kappa^+$ by definition, then the set you want to denote would be $\mathrm{Card}_{\leq |X|}$.