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Is there notation for setting a variable equal to the point where some expressions are equal? E.x. $$ c = (\sqrt{x}=2) $$ which becomes $$ c=4 $$ Because $$ \sqrt{4} = 2 $$

b44ken
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  • It is not common for variables to stand for equations. More importantly, since $4$ is a number, not an equation, I don't think you can say $c=4$. It's $x$ that is the variable that must be $4$. – Mark S. Dec 03 '21 at 23:50
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    This would be useful notation! Given a formula $\phi(x)$, having a notation for "the $x$ such that $\phi(x)$". Alas, I'm not familiar with any inline notation for this. I can imagine using this notation to produce really long and hard to read math! Usually, I just say "let $c$ be such that $\sqrt{c} = 2$. – TomKern Dec 04 '21 at 02:25

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There have been ideas like this before, for example Hilbert’s $\epsilon$ calculus, where you would write $$ c = (\epsilon x. \sqrt x = 2) $$ (with the $\epsilon x$ indicating that you solve the equation for $x$), but none of these are common in current mathematics. Using words as recommended by TomKern in the comments is the usual approach.

You can also use $$ c \in \{ x | \sqrt x = 2 \} $$ which also hints at the “problem” that solutions to equations need not be unique (or exist at all) and you need to specify what is supposed to happen in that case. (The $\epsilon$ operator simply returns some solution if one exists and a default element otherwise, so you need to check $$\sqrt{\epsilon x. \sqrt x = 2}= 2$$ to make sure that $\epsilon x. \sqrt x = 2$ really is a solution.)

Eike Schulte
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There may be some nuance here I am missing, but if you want to say that $c$ is a solution of $\sqrt x=2$, I think what you want to write is just $$\sqrt c=2.$$

If this doesn't fully express your intended meaning, could you leave a comment saying how so?

MJD
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