Is there a symbol for ‘equal if defined’

Question

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 $\lim(f(x) + g(x))$ 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.

Answer 1

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."

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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.

Answer 2

Motivated by Toby's answer and his accompanying comments (especially the second one), perhaps a little modification like the following can make clearer which side the "if defined" applies to. We can write $$L \mathop{{}^\downarrow{=}} R$$ to mean "if $L$ is defined then $L=R$". For example, $$\frac{x^2−1}{x−1} \mathop{{}^\downarrow{=}} x+1$$

Similarly, we can write $$L \mathop{{=}^\downarrow} R$$ to mean "if $R$ is defined then $L=R$". For example, $$\lim (f(x)+g(x)) \mathop{{=}^\downarrow} \lim f(x) + \lim g(x)$$

The choice of using $\downarrow$ is from this comment from another question about the symbol for "defined". Another choice would be $\vartriangle$ (that is "$L \mathop{{}^\vartriangle{=}} R$" and "$L \mathop{{=}^\vartriangle} R$") which perhaps work better for those already familiar with $\triangleq$.

Personally I like $\downarrow$ more though, since it is still visible but takes up less space.

Answer 3

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!

I've tried using $ \fallingdotseq $ and $ \risingdotseq $. These exist in Unicode (as U+2252 and U+2253) and as relations in the amssymb LaTeX package (as \fallingdotseq and \risingdotseq) and so also in MathJax. But the first one doesn't seem to have any established meaning outside of east Asia (where it means approximate equality, that is $ \approx $), and the second doesn't seem to have any established meaning at all (although the Unicode name suggests ‘image of’).

I've been using these so that the top dot is on the side that is hypothesized to be defined. So $ u \fallingdotseq v $ means that $ u = v $ if $ u $ is defined, while $u \risingdotseq v $ means that $ u = v $ if $ v $ is defined. For example, $ ( x ^ 2 − 1 ) / ( x − 1 ) \fallingdotseq x + 1 $, and $ \lim ( f ( x ) + g ( x ) ) \risingdotseq \lim f ( x ) + \lim g ( x ) $.