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An operation $\ominus$ for the natural numbers is defined as follows:

$$a\ominus 0 = 0\ominus a = a\\S(a)\ominus S(b) = a\ominus b$$

Here $S$ is the successor function. The operation $a\ominus b$ is equivalent in value to $|a-b|$, except negative numbers, absolute value, and subtraction need not be defined. Here are some examples:

$$4\ominus 2 = S(3)\ominus S(1) = 3\ominus 1 = S(2)\ominus S(0) = 2\ominus 0 = 2\\2\ominus 4 = S(1)\ominus S(3) = 1\ominus 3 = S(0)\ominus S(2) = 0\ominus 2 = 2$$

Question: Does this operation have a common name or symbol?

Regret
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4 Answers4

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"Distance" or "Difference" come to mind, as $a\ominus b$ is the difference between $a$ and $b$.

$\Delta a b$ is a notation that's sometimes used to express the difference between $a$ and $b$, particularly in physics.

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    $d(a,b)$ is much more common than $\Delta a b$. – Najib Idrissi Oct 28 '14 at 10:47
  • I have seen $d(a,b)$ mostly in contexts where $d$ is some previously-defined metric (sometimes the absolute value), not as a general synonym for the simple difference between two numbers. –  Oct 28 '14 at 10:49
  • It's an example of a metric, I'm not saying that $d(a,b)$ is always $|a-b|$ where $a,b \in \mathbb{N}$. – Najib Idrissi Oct 28 '14 at 10:52
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    I'd call it the "absolute difference" as "difference" usually refers to the signed quantity $a-b$. – bof Oct 28 '14 at 10:56
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It looks like $a\ominus b=|a-b|$, or the distance between $a$ and $b$.

Akiva Weinberger
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As mentioned by pew I would go for "(absolute) difference" or "distance". I'm sure that there are some conventions of notation, which will probably differ per field of expertise.

Also you are right to say that $\vert a-b\vert$ need not be formally defined on $\mathbb N$. As I don't see the harm in defining your own operator, I would say something like this: $$\ominus : \mathbb N\times \mathbb N\to \mathbb N$$ defined by: $$a\ominus b=\begin{cases}a-b,\quad\text{if }a\geq b\\b-a,\quad\text{if }b>a\end{cases}$$ I don't see what could possibly go wrong like this. (Except maybe mention that you have $0\in \mathbb N$, which is always a subject of discussion.)

gebruiker
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Another way to express gebruiker's answer is $$a\ominus b = \max(a,b) - \min(a,b)$$

lhf
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