Is $\{a \pm b\}$ correct notation? Is $\{a \pm b\}$ correct notation? I want to mean a set $\{a-b, a+b\}$ by this, but I need some shorthand notation to fit an equation to double-column paper.
 A: TL;DR
The notation is probably fine, but might be considered ambiguous by some.  I would prefer to find an alternative, but the decision is ultimately up to you and your publisher / instructor / editor / whoever.
Longer Answer
I am going to address this question a little differently than the other answers.  In particular, I want to specifically address the question of whether or not the notation $\{a\pm b\}$ is "correct".  This, in my opinion, is not the question you should be asking.  Rather, you should be asking if the notation is clear, and whether or not it will be unambiguously understood.  Remember that the goal of mathematical writing is to clearly communicate ideas.  Any notation which helps that goal is "correct", whereas any notation which hinders that goal is "incorrect".
In the case of the notation proposed, I think that
$$ \{a\pm b\} $$
is okay, though not necessarily ideal, notation.  Personally, I would prefer to avoid it, and to be more explicit.  On the other hand, while there is room for ambiguity, I think that most readers would see that notation and understand that it means
$$ \{a\pm b\} = \{ a+b, a-b\}. $$
A reader might understand the notation to indicate a singleton set consisting of either $a+b$ or $a-b$, but I would think that this would be an unusual interpretation.
A quick way to continue using this notation and also reduce ambiguity would be to add a couple of words.  For example, suppose that this set represents the two roots of a quadratic polynomial.  Then one could say, for example, "The roots are the two elements of the set $\{a\pm b\}$."  That is, point out, in plain English, that the set has two elements.  This additional language virtually eliminates any ambiguity.
Other possible solutions are:


*

*If you are using this notation a lot, then it might be reasonable to explain it near the beginning of your document.  For example, when you use the notation for the first time, say something to the effect of "The notation $\{a\pm b\}$ will always denote the two element set $\{a+b, a-b\}$."

*Be more explicit about the notation, and write
$$ \{a+b, a-b\} $$
in a display environment (as is done here).  If the problem is that the text $\{a+b,a-b\}$ does not fit into a two-column layout, or causes strange linebreaks, then put it in a display.  It gets its own space, and there is no ambiguity.

*Of course, maybe $a$ and $b$ are both very large expressions.  So perhaps something like the following would work:
\begin{align}
&\biggl\{ \frac{\text{long expression}}{\text{another long expression}} + \frac{\text{frightfully long expression}}{\text{oh my god, why?!}} \\
&\qquad\qquad\qquad \frac{\text{long expression}}{\text{another long expression}}- \frac{\text{frightfully long expression}}{\text{oh my god, why?!}} \biggr\}.
\end{align}
Finally, as I think that this is a matter of style, you should take it up with the person for whom you are writing.  If this is for a journal article, ask your editor / publisher / reviewers if they have a preference (or just use whatever notation you like, but change it if you are asked to do so).  If this is part of a homework assignment, ask your instructor what their preference is.  If this is for your own notes, do whatever you like.
A: This notation can be used. Similiarly I have seen $\{0,\pm1,\pm2,\dots\}$ for denoting $\Bbb Z$. Anyway, for a two element set there is not much of an advantage for using $\{a\pm b\}$ over $\{a+b,a-b\}$.
A: I think your should stick with the latter: $\{a-b, a+b\}$.  Most might understand what you mean, but the tiny effort saved in key strokes doesn't justify, IMHO, risking ambiguity.
Note that $\pm x$ reads: plus OR minus x, not plus and minus x.
I have often seen solutions to the quadratic formula as $x= \pm (\text{ foo })$.  But when using set notation, for example, $x \in \{\pm(\text{foo})\},$ that's less standard when only two elements are involved.
