# Use of ternary operator in formal writing

I would like to write $$f(x) = \begin{cases}1&\mbox{if }x = 1\\ 0&\mbox{otherwise.}\end{cases}$$ However, this eats up a lot of vertical space for a very simple statement. Is there agreed upon or common notation to inline this into a single normal-height line in technical writing? As a computer scientist I am tempted to write $$f(x) = (x=1\,?\,1 : 0),$$ which is the notation of the common programming notation for the ternary operator. If you were reviewing a paper that did this (saying that it used the ternary operator notation of C++) would you declare it an abomination? Is there an alternative that you would consider?

• I'd use the first version. The ternary operator is not common notation in math papers and I at least would be confused by it. I suppose it might depend on the field though. – Denis Nardin May 17 '16 at 15:05
• $f(x) = \delta_{x,1}$ or $f(x) = 1_{\{1\}}(x)$ – Steve Huntsman May 17 '16 at 15:06
• I would write a sentence: "Define $f(x)=1$ if $x=1$ and $f(x)=0$ otherwise." – Tom Goodwillie May 17 '16 at 15:27
• I think it's perfectly fine to use the ternary operator provided it's clearly defined (and provided it's used often enough to make the gain of space worth while). A compromise might be to write $f(x) = (\mathtt{if~}x=1\mathtt{~then~}1\mathtt{~else~}0)$. – Gro-Tsen May 17 '16 at 15:58
• I use the ternary operator quite often in notes that I write for myself, and I would be happy if it were more widely accepted in the mathematical literature. I suggest that you make a LaTeX macro for it, so you can change back to more standard notation fairly easily if the referee complains. – Neil Strickland May 17 '16 at 16:47

The ternary operator doesn't seem to be used in mathematical papers, with the vertical brace notation being the most popular option. If there is a strong reason for not using the vertical brace (such as it being embedded in a more complicated expression), I would suggest using the Iverson bracket:

https://en.wikipedia.org/wiki/Iverson_bracket

in which case you would write $f(x) = [x = 1]$. Even then, the Iverson bracket might not necessarily be known to your reader, in which case you could use the vertical brace the first time you introduce the Iverson bracket:

$$f(x) = [x = 1] := \begin{cases}1&\mbox{if }x = 1\\ 0&\mbox{otherwise.}\end{cases}$$

Then, any subsequent time you use the Iverson bracket, its meaning should be understood without the need for a clarifying brace:

$$\delta_{ij} = [i = j]$$

• I also immediately thought of the Iverson bracket, and frequently find occasion to use it. However, perhaps just out of mathematical conservatism, I think that @TomGoodwillie's comment mathoverflow.net/questions/239098/… is better; if space is at such a premium, then prefer a written explanation to the use of what will for many readers be an ad hoc symbolism. (Of course, the same objection could have been urged against Recorde's use of '=' in place of 'equals', and I think it's good that it wasn't. :-) ) – LSpice May 17 '16 at 17:51

Even the Iverson bracket is used only in certain parts of mathematics. If the "cases" notation is absolutely ruled out (perhaps because it is to be used a few hundred times), then my inclination is to follow Steve Hartsman and use the indicator function (a.k.a. characteristic function) $$f = \mathbb{1}_{\{1\}} \\ f = \chi_{\{1\}}$$ or similar.

• I know that you explicitly don't advocate them, but I think that both of those have problems. As a representation theorist, I prefer to reserve $\chi$ for characters, rather than characteristic functions. I don't have any particularly principled objection to $\mathbb 1$ for characteristic functions; I just find it ugly. A modification of the Iverson bracket mentioned by @AdamP.Goucher, and which I have heard attributed to Kottwitz, uses brackets: $[\{1\}]$. Of course, a big problem with all of these is that they give no idea what the domain is! – LSpice May 17 '16 at 22:13