Correct arrow notation for function with natural domain I saw a definition stating that given a function $f:X \to Y$ the natural domain of $f$ is the largest possible subset $D \subseteq X$ for which the rule defining $f$ is valid.
From what I understand, the $X$ in $f:X \to Y$ must be the domain of $X$. Let’s suppose $D$ is a subset of $X$ but is not $X$ itself, then that seems to imply that there is at least one element in $X$ not in $D$. But if $X$ contains an element not in the domain of $f$ then we can’t write $f:X \to Y$. If such a subset $D$ exists and is not $X$, then that implies $f$ is not a function in the first place.
My question is: is this definition flawed? Or, when written in arrow notation, is it acceptable for $X$ to be a set containing the domain; even if it is not the domain of the function itself?
 A: Mathematical notations are mostly a matter of conventions and ultimately the understanding of notations depends very much on the context.
Sometimes people write $f:X\to Y$ while $X$ means a set that contains the domain of $f$ and the author explicitly denotes the domain of $f$ with $D$, which is a proper subset of $X$. Sometimes people make a combination and write something like
$$
f:D(f)\subset \mathcal{H}\to\mathcal{H}
$$
where $D(f)$ denotes the domain of $f$.
Mostly when people write $f:X\to Y$, $X$ means the domain of $f$.

If you assume in the first place that $X$ is the domain of $f$, and write $f:X\to Y$, then it never happens that "$X$ contains an element not in the domain of $f$".
If $B$ is a proper subset of $X$, which is assumed to be the domain of $f$, then it is true that $X$ has an element that is not in $B$. This does not contradict anything about the definition of the domain of $f$: your domain of $f$ is $X$, not $B$.
Think about the following example.
Suppose $f:[0,1]\to\mathbb{R}$ is a function defined on $X=[0,1]$. Then

*

*$B=[0,1)$ is a (proper) subset of $X$.

*$X$ contains an element that is not in $B$, i.e., $x=1$.

A: After thinking way too long about this I'd say scrub the definition as given as you are correct: the notation "$f:X\to Y$ is function" does imply $X$ is the domain and the definition doesn't make sense.
But replace it with this:

If $f$ is a rule that can map some elements of $X$ to $Y$, and then the natural domain is the largest subset $D\subseteq X$ so that $f:D\subseteq X \to Y$ is a valid function. (i.e. in that $D$ is an acceptable domain and all elements of $D$ can be mapped and all elements of $X\setminus D$ can't be.)

That way we avoid saying the incorrect "$f:X\to Y$ is a function" (it isn't... that fails the definition of function) and avoid any weird and awkward "$f:X\to Y$ but not as function maps... we just mean some of the $X$ can be mapped"
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I guess what the book is getting at is if you take a naive idea of function being a "rule" and there is some universal set we are working in then saying things like
$f(x) = \frac {x^2 +7}{x-3}$ or $h(x) = \arcsin x$ or $j(x) = \sqrt x$
and we make the naive and blanket statement that all theses are real functions and map $\mathbb R$ to $\mathbb R$.
Then the "unnatural" super-general-not-much thought put into "domain" of all these functions are $\mathbb R$.
ANd so the natural domain is the one's where the function actually works.  "Natural" domain of $f(x)$ is $\mathbb R\setminus \{3\}$.  Naural domain of $h$ is $[-1,1]$. and natural domain $j(x)$ is $[0,\infty)$.
But the thing is a real "grown-up" mathematician would never say $\mathbb R$ is the domain.  They'd simply say the if $f(x)$ isn't valid then $x$ is simply not in the domain. Period.
Unless.... well if this is practical math text.
Suppose I told you "I have function and it maps reals to reals and the function is $f(x) = \sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}}$ and I need to know what the domain is" well, the technical answer is the domain can be what I want it to be.  If I want the domain to be $\{0,-1, -2,-3\}$ so that $f(x)= \begin{cases}\frac 1{\sqrt {431}}&x=0\\\sqrt{56}+\frac 1{\sqrt{430}}&x=-1\\\sqrt{106}+\frac 1{427}&x=-2\\12 +\frac 1{\sqrt{422}}&x=-3\end{cases}$
well, then I can have $f:\{0,1,2,3\}\to \mathbb R$ via $f(x) =\sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}}$.  But that's not what I meant.
So then I say "C'mon.... you know what I mean.... I want the biggest set of real numbers that can be the domain.... I want the....um, let's call it 'natural domain'" and then .. well that is a legitimate question.
To define the function $f: X \to \mathbb R$ I must specify what the domain $X$ is.
But do I really have to?  Do I really have to say... "well we need $x^3 - 57 \ge 0$ and we need $431 -x^2 > 0$ so the means $x$ must be between...."  Do I even care?
Why shouldn't I just say:  Let $f: NaturalDomain(\sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}})\subset R\to \mathbb R$. and not really worry about knowing what values of $x$ that $ \sqrt{x^3 - 57x} + \frac 1{\sqrt{431- x^2}}$ id defined on?
