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Is there a mathematical symbol for "solution set of"?

ex: "$5 \in \text{[solution set of] } f$", or "$0 \notin \text{[solution set of] } g(x) =x^2+x-2$"

CoilKid
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  • Also, while $x=5$ is true for the first example (assuming it's f(x) and not f(z) or something), it isn't what I'm looking for. – CoilKid Sep 21 '16 at 22:41
  • A useful shortcut for $$0\not\in{x: g(x)=x^2+x-2=0}$$ is simply $g(0)\neq 0$. – Jack D'Aurizio Sep 21 '16 at 22:42
  • @JackD'Aurizio Yes, but if I wanted to write it the long way, do I have to write it in set-builder? – CoilKid Sep 21 '16 at 22:44
  • You may use "for any $f(x)\in\mathbb{R}[x]$, let $$ Z_f={x:f(x)=0}$$" then simply use $Z_f$ to denote the zero set of $f$. I am not aware of any "standard notation". – Jack D'Aurizio Sep 21 '16 at 22:45
  • It isn't uncommon to see $f^{-1}(0)$ or $f^{-1}({0})$ used for this purpose. – Nick Peterson Sep 21 '16 at 22:47
  • Oh, sure, that is true. – Jack D'Aurizio Sep 21 '16 at 22:47
  • I was trying to find an easier way to say "$\text{If } f(x)=x^2+1 \text{ then } 0 \notin \text{[solution set of] f}$. I guess $0 \notin {x:f(x)=x^2+1}$ works, but I was hoping for something that didn't use set builder. – CoilKid Sep 21 '16 at 23:08

1 Answers1

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If $f:X\to Y$ is a function, the set of all points $x\in X$ so that $f(x)=y\in Y$ is commonly denoted $f^{-1}(\{y\})$. So if $5$ is a solution to $f(x)=0$, we could write $$5\in f^{-1}(\{0\}).$$ Some authors will write this as $f^{-1}(y)$, (without brackets), but I don't like this because it could be confused with a reference to the inverse of $f$, which may not even exist.

In fact, if $S\subset Y$, we can write the set of all $x\in X$ so that $f(x)\in S$ as $f^{-1}(S)$. Such a set is called the preimage of $S$.

Plutoro
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