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Is there a notation that replaces the "union" operator $A\cup B$ and emphasizes that the outcome should be considered a multi-set rather than a set? For example, if $A = \{1,2,3\}$ and $B =\{3,4,5\}$, then:

$$A ? B = \{1,2,3,3,4,5\}$$

since the $3$ is counted twice. What operator can I used instead of the "$?$"?

Erel Segal-Halevi
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  • Maybe disjoint union is what you are looking for. See the Wikipedia article https://en.wikipedia.org/wiki/Disjoint_union – Crostul Aug 18 '20 at 15:59

2 Answers2

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This is sometimes called “additive union”. (For example, see Blizard, W. “Multiset Theory”.) There is no truly standard notation for multiset operations, but $$A\uplus B$$ is often used for additive union. The MathJax code is \uplus.

Martin Sleziak
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MJD
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    Just to add, there are various terms used for this multiset operation, including 'addition', 'sum' and 'merge'. [See Definition 7, Page 78 of Singh, D. et.al. (2007). An overview of the applications of multisets. Novi Sad J. Math. 37(2), 73-92]. I have a preference for merge. Not only does merge have a natural alignment to essence of this operation, it also provides multisets with a uniquely different term from classical sets. – Mari153 Sep 23 '20 at 12:01
  • If I need to additively union together a large/varying/arbitrary amount of these, would \biguplus be an appropriate notation?$$\biguplus\left{\left[1, 1\right], \left[1, 2\right], \left[2, 3\right]\right} = \left[1, 1, 1, 2, 2, 3\right]$$ – JamesTheAwesomeDude Sep 28 '23 at 21:59
  • Any notation is appropriate as long as you state clearly what it means. For something like this, don't assume that anyone will understand without an explanation. – MJD Sep 29 '23 at 19:15
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For multiset union (and intersection, etc) just use the usual symbols. Just make clear you are talking multisets, not sets.

vonbrand
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    Even in the context of multisets, the writer will want to make clear that $\cup$ represents additive union $A+B$ rather than max-union $\max(A, B)$. – MJD Aug 18 '20 at 16:06