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How can I represent a domain of $\mathbb R$ excluding the integer $0$?

Say, for example I have a function $f(x)=-x^2+5\ \{{x=\mathbb R\}}$, and I want every point's tangent line to have a slope not equal to $0$. How could I rewrite the domain to not include zero?

Note: In a previous question, I wrote the domain to be $\{0∉\mathbb R\}$. Is this valid notation?

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CATboardBETA
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    $0\notin\mathbb{R}$ is the statement that $0$ is not a real number, which is false. You want to say "the set of all real numbers except $0$" so you can write this either as $\mathbb{R}\setminus{0}$ or as ${x\in\mathbb{R} | x\ne 0}$ – Snaw Dec 08 '21 at 00:48

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The two standard symbols for "Set minus" are $\setminus$ and $-$ (the first is \setminus in LateX.)

So you could say $\mathbb{R} \setminus \{0\}$, this would probably be the simplest and most common. In setbuilder notation, you would do
$\{x|x\in \mathbb{R}, x\neq 0\}$ or $\{x\in \mathbb{R}|x\neq 0\}$ . If your universe of discourse is already known to be the real numbers (I.e. the only things that exist are real numbers, and all real numbers exist), then you can drop the $\in \mathbb{R}$ and say simply $\{x|x\neq 0\}$

Alan
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  • Why is it $\mathbb R\setminus{0}$ instead of ${\mathbb R\setminus 0}$? – CATboardBETA Dec 08 '21 at 00:53
  • @CATboardBETA Because this is not set-builder notation. The "set minus" notation $A\setminus B$ says whatever elements the set $B$ has, remove them from the set $A$. You don't want to remove the elements of $0$ from $\mathbb R$; you actually want to remove $0$ itself, so you need to subtract a set that contains $0$. That's ${0}$. Moreover, ${\mathbb R} \neq \mathbb R,$ so putting the brackets around everything after the "set minus" makes things worse, not better. – David K Dec 08 '21 at 01:04
  • @DavidK Ok, that makes sense. But for a domain, should I then enclose everything in braces, like this: ${\mathbb R\setminus{0}}$? – CATboardBETA Dec 08 '21 at 01:13
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    @CATboardBETA $\mathbb{R}\backslash{0}$ and ${\mathbb{R}\backslash{0}}$ are two different things. The first is an infinite set of all non-zero real numbers. The second is a set containing only one element - the set of all non-zero real numbers. –  Dec 08 '21 at 01:16
  • Well, yeah. But which one would you want to use in a function defenition? I don't follow. – CATboardBETA Dec 08 '21 at 01:18
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    The first is the one you want. The second is a set containing a set. –  Dec 08 '21 at 01:21