Talk:Disjoint union

The foundation article should be thoroughly explained making the thing considerd unknown to the reader clearer. The matter is very tricky in maths.
To my mind, after reading plenty of very well written Wikipedia entries on sets and logic, WE NEED to add examples to make the text a quality information source, no matter how stupid or primitive these explanations may ever seem. Here disjoint union versus exclusive or difference should be covered straight away in the main article.
We do not need to feel ashamed giving simple examples from life, on the contrary.--Capekm (talk) 15:07, 6 April 2012 (UTC)

I don't know what a disjoint union of sets is.

According to this Open University article, "In general, the disjoint union of sets X and Y ... is the set consisting of all items that are either from X or from Y [but not both]", so it's an exclusive-or operation on sets?

Kfor (talk) 20:02, 1 February 2008 (UTC)

It's not the same as an exclusive-or: if X and Y are sets with cardinality a and b repsectively, then their disoint union always has cardinality a+b. The OU article you refer to has two sets that have no elements in common (empty intersection - so they are disjoint sets), and the disjoint union of two sets is the ordinary union if they are already disjoint. The difficulty arises when the two sets are not disjoint - so to form their disjoint union some trick must be played. There are several ways to describe this, and the article does one of those. But I do think it could be clearer. Simplifix (talk) 23:10, 13 February 2008 (UTC)

It appears to me that the OU article may be referring to a Tagged union which is different from this (though related). --WestwoodMatt (talk) 22:58, 25 February 2010 (UTC)

No, but the Symmetric difference seems to be. 83.237.198.221 (talk) 16:59, 14 September 2010 (UTC)

Tagged union seems to be mathematically the same as disjoint union, except in the context of computer science instead of set theory. Symmetric difference is essentially the same as exclusive or, except in the context of set theory instead of logic. The definition "...all items that are either from X or from Y [but not both]" in the above question seems to be the definition of symmetric difference, not disjoint union; i.e. it looks like the wrong definition. 69.91.135.244 (talk) 23:22, 22 February 2012 (UTC)

Example

An example is always useful for explaining things. Is it true that if A_0 = {5, 6} and A_1 = {6, 7} then the disjoint union of A_0 and A_1 is {(5,0), (6,0), (6,1), (7,1)}? If someone can confirm that, could they copy/paste that example into the text? 174.252.55.188 (talk) 19:37, 26 August 2011 (UTC)

Yes, that's correct. The point is that we add the tags 0 or 1 to all the elements in order to guarantee that any duplicated elements (e.g. the two 6's) from the original sets get counted as distinct elements in the disjoint union. 69.91.135.244 (talk) 23:09, 22 February 2012 (UTC)

Union of already disjoint sets

In german wikipedia and several german books I found the union of two disjoint sets to be written as ${\dot {\cup }}$. I find it quite consistent to syntactically differentiate between both concepts. Did anyone notice such a differentiation in english literature? The disjoint union of non-disjoint sets yields a set of tuples - to say that in the case of already disjoint sets the additional "index space" is simply dropped seems kind of "unmathematical" to me... — Preceding unsigned comment added by 77.11.13.179 (talk) 20:55, 30 July 2012 (UTC)

The best mathematical definition of disjoint union is to be a coproduct in the category of sets. As such, the discrete union is defined up to an isomorphism, and the definition with "index space" given in the article is just one realization among others. When the sets are pairwise disjoint, the usual union is another realization. This justifies the second definition of the lead and that of "probability theory" (this section duplicates the lead and I will remove it). What precedes deserve to be expanded in a section "Category theory point of view".

For the notation, care has to be taken that a notation that is widely accepted in set theory is not necessary accepted in other fields of math. As far as I know, when a notation for the disjoint union is needed in a paper that does not belong to set theory, the authors have to define their notation, usually $\bigsqcup$ but also, rather frequently $\coprod$. D.Lazard (talk) 10:32, 31 July 2012 (UTC)

Adjoint?

The adjoint to the product (cartesian product or tensor product) is hom, or the exponential object in the category of sets. See the article on currying for a lurid exposition of this adjunction. Now, the disjunct union is the coproduct, opposite of the product. What can I say about its adjoint, either in some small category or in general? 67.198.37.16 (talk) 07:06, 22 July 2017 (UTC)

Marked union

Disjoint union is also called "marked union". [1] Boris Tsirelson (talk) 06:22, 18 March 2018 (UTC)

Symmetric difference

How’s this any different from Disjunctive union?

Improve along lines of the spanish language article?

I find the introductory section of the spanish language version of this article, the included figure with the labelled polygons, and the boxes surrounding the important definitions, much easier to read.

Would it be possible to copy some of those features into the English language version of the article? Right now I think the spanish language version would be easier to understand (if it was translated into English), especially for someone who has never encountered the topic before.

In particular it also goes through the effort to clearly distinguish between the union of two disjoint subsets of a larger "universal set" and the more abstract construction, acknowledging that both are often referred to as "disjoint union". — Preceding unsigned comment added by 69.143.122.185 (talk) 14:05, 28 January 2022 (UTC)