Can anybody recommend a symbol for ‘equal if defined’ as an asymmetric concept?

In contexts where one might write down notation for an undefined quantity (such as $1/x$ when $x$ might be $0$), sometimes people use the equality symbol to mean that if either side is defined, then so is the other, and then they are equal; and sometimes people use it to mean that if both sides are defined, then they are equal. Sometimes they use this together with a variant symbol with the other meaning. These are not the meanings that I want.

I want a symbol that says that if the left-hand side is defined, then so is the right-hand side, and then they are equal. Or a symbol that means that if the right-hand side is defined, then so is the left-hand side, and then they are equal. Either way, the meaning is asymmetric. Preferably a symbol that is itself left-right asymmetric, so that the reverse symbol has the reverse meaning.

Usage examples: In Algebra, when we write that $(x^2 - 1)/(x^2 - x) = (x + 1)/x$, what we really mean is that if $(x^2 - 1)/(x^2 - x)$ is defined, then $(x + 1)/x$ is also defined and then $(x^2 - 1)/(x^2 - x)$ is equal to $(x + 1)/x$. However, if $(x^2 - 1)/(x^2 - x)$ is undefined, then we're not claiming that it's equal to anything, and we're also not saying whether $(x + 1)/x$ is defined.

In Calculus, when we write that $\lim_{x \to c}(f(x) + g(x)) = \lim(f(x)) + \lim(g(x))$, what we really mean is that if $\lim(f(x)) + \lim(g(x))$ is defined, then $\lim(f(x) + g(x))$ is also defined and then $\lim(f(x) + (x))$ equals $\lim(f(x)) + \lim(g(x))$. However, if $\lim(f(x)) + \lim(g(x))$ is undefined, then we're not claiming that anything's equal to it, and we're also not saying whether $\lim(f(x) + g(x))$ is defined.

Of course, one can always write ‘if the left-hand side is defined’ after the equation, or something like that. But I want a symbol that I can use with a string of conditional equalities (all in the same direction) in the course of an argument to establish an overall conditional equality. Example: $\lim_{x \to 5}(x^2 + 6) = \lim(x^2) + \lim(6) = \lim(x)^2 + 6 = 5^2 + 6 = 31$, where at each stage I use a basic rule of limits in the course of the calculation. Ultimately, I conclude that if $31$ is defined (which it is), then $\lim(x^2 + 6)$ is defined and equals $31$ (which it does).

Of course, I can make up a symbol, but if anybody has already made one up and used it successfully (either in a formal logical context or informally as I was doing above), then I'd like to hear about that.

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    $\begingroup$ People do write things like "if ____ and ____ are defined", often before the equation where the expressions are used. When they don't write this, I think they are relying on the reader to correctly infer what they would have written before the equation. I don't think "=" actually means anything more complicated than "these two things exist and are the same". In your example, to prove that $lim_{x\to5}(x^2+6)$ exists, you could build it up step by step, basically stating your rightmost $=$ first, then the one its left, and so forth. $\endgroup$ – David K Jan 22 '16 at 20:51
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    $\begingroup$ That's true, but I really want to write it from left to right, because I start out not knowing what the limit of $x^2 + 6$ is or even (if I don't think very hard) whether it exists. So I want to write left to right to follow the process of calculation and discovery, but I also want to write things that I know all along are true. $\endgroup$ – Toby Bartels Jan 25 '16 at 1:36
  • $\begingroup$ OK, I can see the merit of always writing true things. For that, perhaps you can make up your own symbol, which only you need to know about, and use it during calculation and discovery; you can still rearrange the order of the derivation that you show to other people so that everything you write is known to be true when they read it, and all the equations are just ordinary "everything is defined on both sides" equations. $\endgroup$ – David K Jan 25 '16 at 2:14
  • $\begingroup$ Keep in mind that sometimes it doesn't end up being equal. For example, I might start by plugging things in, then end up with 0/0, then go back and use L'Hôpital's Rule. The initial calculation wasn't a waste of time, because I needed to see the 0/0 to even know that L'Hôpital's Rule applies, but doing it with ‘=’ makes me write down things that are technically false. And I do want to show the process of calculation and discovery, not skipping over steps. $\endgroup$ – Toby Bartels Jan 26 '16 at 20:03
  • $\begingroup$ Interesting; in the example you just gave, I think you want to say that if the right-hand side is defined, so is the left-hand side and they are equal. $\endgroup$ – David K Jan 27 '16 at 1:39

As far as functions go, note that we already have a symbol for this (coming from set theory): "$\subseteq$"! If $f$ and $g$ are partial functions, identifying them with their graphs means that we can write "$f\subseteq g$" to mean "$f(x)=g(x)$ whenever $f(x)$ is defined."

More generally (or in contexts where imposing set theory might be undesirable), here's a possible recommendation:

First, the symmetric version. I don't know how common this is elsewhere, but in computability theory, we often write $$x\simeq y$$ if $x$ and $y$ are expressions which are either both undefined, or both defined and equal. (Occasionally "$\cong$" is used instead of "$\simeq$," but I prefer "$\simeq$" since (at least within computability theory) it's less often used for isomorphism.)

For the asymmetric version, I would recommend "$\gtrsim$" in analogy with "$\simeq$." But I haven't seen that before, so I don't actually know if it's commonly used; it's just what seems to me the natural choice.

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  • $\begingroup$ In mathematics, these are symbols for ‘is isomorphic to’. $\endgroup$ – Bernard Jan 22 '16 at 20:31
  • $\begingroup$ @Bernard I rarely see "$\simeq$" used for "is isomorphic to". (And to be fair, there's a sense in which it is isomorphism, anyways . . .) $\endgroup$ – Noah Schweber Jan 22 '16 at 20:32
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    $\begingroup$ I've been doing it since I was a student… $\endgroup$ – Bernard Jan 22 '16 at 20:36
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    $\begingroup$ I think $\gtrsim$ sometimes is used to mean "approximately greater than." $\endgroup$ – David K Jan 22 '16 at 20:59
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    $\begingroup$ So I guess computability theory isn't mathematics. Learn something new every day. $\endgroup$ – BrianO Jan 22 '16 at 22:26

Well, I think that the answer seems to be No, that there is no such symbol in standard use in any context. More's the pity! Perhaps we can invent one.

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  • $\begingroup$ This isn't really an answer, but I'm going to try using $\fallingdotseq$ and $\risingdotseq$. The first of these doesn't seem to have any established meaning outside of east Asia (where it means approximate equality), and the latter doesn't seem to have any established meaning anywhere. Although I haven't quite made up my mind which should be which. $\endgroup$ – Toby Bartels Jun 28 at 6:56

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