Better notation for $A\mapsto\chi_A$ and a question about the elements of $\{0,1\}^X$

Question

I have two questions that come from the same exercise so I put together in this question. I have an exercise that say

$f:\mathcal P(X)\to \{0,1\}^X$ and $A\mapsto\chi_A$. Show that $f$ is bijective.

First question: I understand what the exercise is asking but the notation used, specifically the part $A\mapsto\chi_A$, seems terrible to my eyes.

I understand that a characteristic function $x\mapsto \chi_G$ map $1$ or $0$ depending if $x$ is an element of $G$ or not. But then we have that $\chi_A(A)=0$ because $A\notin A$.

I understand that the notation $A\mapsto\chi_A$ want the sequence of zeros and ones of some subset $A\subseteq X$ through the function $\chi_A(x)$ for all $x\in X$ (we can infer this because any $f(A)\in\{0,1\}^X$).

Then my question is, there is a better notation for this situation?

Second question: if $X$ is countable I understand that we can interpret it elements as a sequence, if $X$ is finite we can interpret the elements as vectors or $|X|$-tuples.

But, how we must interpret the elements of $\{0,1\}^X$ if $X$ is uncountable? Thank you in advance.

Answer 1

I'm not sure you do understand the notation; you write "$x\mapsto \chi_G$" and "$\chi_A(A)=0$", but neither of these really make sense.

• The function $f:\mathcal{P}(X)\to \{0,1\}^X$ takes elements of $\mathcal{P}(X)$ and produces elements of $\{0,1\}^X$.
• Elements of $\mathcal{P}(X)$ are subsets of $X$, and elements of $\{0,1\}^X$ are functions from $X$ to $\{0,1\}$.
• For a subset $A\subseteq X$, recall that the element $\chi_A\in\{0,1\}^X$ is the function $\chi_A:X\to\{0,1\}$ defined by $$\chi_A(x)=\begin{cases} 1&\text{ if }x\in A\\ 0&\text{ if }x\notin A \end{cases}$$
• The exercise states that the function $f$ sends an element $A\in\mathcal{P}(X)$ to the element $\chi_A\in\{0,1\}^X$. In other words, $f(A)=\chi_A$.

Personally, the only thing I would change about the notation in the exercise is that I would write "$f(A)=\chi_A$" instead of "$A\mapsto \chi_A$", but other than that I think the notation used is both standard and intelligible.

As for your second question, the elements of $\{0,1\}^X$ are just the functions from $X$ to $\{0,1\}$, and I'm not sure I see a particular need to interpret them any other way.

Answer 2

The set $\{ 0, 1 \}^{X}$ denotes the class of all maps from $X$ to $\{ 0, 1\}$; so the map $A \mapsto \chi_{A}$ defined on the class of all subsets of $X$ lies in $\{ 0, 1 \}^{X}$.

For each $A \subset X$, we can assign exactly one characteristic function, namely $\chi_{A}: X \to \{ 0, 1\}$; hence we may define the function $f: A \mapsto \chi_{A}: X \to \{ 0, 1 \}^{X}$.

Can you now clear up your questions?