Talk:Complement (set theory)

Old and undated posts

There is also the symmetric complement or symmetric difference, see http://mathworld.wolfram.com/SymmetricDifference.html -- Nichtich 22:10, 12 October 2006 (UTC)

The second picture does not display correctly, or at least it did not display on my computer. Clicking on it, however, brought up the correct image.

Not actually defined

...a concept used in comparisons of sets to refer to the unique values of one set in relation to another

The above could applied to any set in relation to any other. Is a complement not simply that which is not this? Worse still, the term "relative complement" is used early on with out any other definition than ..."absolute" and "relative" complement refer to more specific applications of the concept. These being? --Jubilee♫clipman 00:15, 26 December 2009 (UTC)

I agree that lead paragraph is not ideal. I will attempt a rewrite of this later. As for the use of the term "relative complement" without definition, this probplem came about by an editor switching the order of the sections "relative complement" and absolute complement" — I've now corrected this. I think the article should make more sense now. Thanks for pointing this out. Paul August ☎ 15:37, 27 December 2009 (UTC)

Ah, that does make more sense now. Lead still needs work though, indeed. Thanks Paul! --Jubilee♫clipman 15:54, 27 December 2009 (UTC)

Ok, I've now rewritten the lead, I hope it is a bit better now. Paul August ☎ 16:02, 27 December 2009 (UTC)

Yes: that actually explains what it specifically is now! Good work. --Jubilee♫clipman 16:06, 27 December 2009 (UTC)

Proposition 2 given before Proposition 1!

This should obviously not be the case. Either

• Rename the propositions, or
• Reverse the order of the sections "Relative complement" and "Absolute complement".

Jameshfisher (talk) 10:38, 13 April 2011 (UTC)

Thanks for pointing this out. I've eliminated using the term "propositions" altogether. Paul August ☎ 14:35, 13 April 2011 (UTC)

A and B swapped in programming language examples?

Aren't A and B swapped in the programming language examples? The initial definition states that the relative complement of A with respect to B are the elements of B not present in A. However, the java example does a.removeAll(b), which leaves the elements of a not present in b. — Preceding unsigned comment added by 212.62.245.98 (talk) 10:31, 2 November 2011 (UTC)

Or is it the other way around? Anyway, it seems like one or the other is wrong.. — Preceding unsigned comment added by Kayskull (talk • contribs) 13:40, 4 November 2011 (UTC)

Ā

This is often times denoted as Ā would be great to add to article. --ben_b (talk) 10:06, 30 April 2012 (UTC)

Etymology

I'm curious. Why is nonoccurrence called a "complement"? How does the lack of occurrence of an event make the description of the probability of an event occurring "perfect"? 68.37.254.48 (talk) 23:11, 28 July 2012 (UTC)

Complements in various programming languages

It's not clear that we need this long list of the names of operators in various languages: and it looks a lot like original research. To justify it's inclusion, we should have a reference to an independent secondary reliable source listing languages with a "set" type and relative complement or set difference operators. In particular, Unix shell does not have a set type, and using files as set types requires a lot more in the way of sourcing than a reference to a man page. Deltahedron (talk) 19:47, 17 August 2014 (UTC)

MINUS is not standard ANSI SQL, it's Oracle specific. Changed to EXCEPT

This list seems only tangentially related to the subject of the article, considering that the article is about the mathematical concept of a set complement. Everything in the list can also easily be found in the documentation of the languages, so I think it should be removed. If this really needs to exist, it should be moved to some other place, for example Comparison of programming languages. Clmul (talk) 22:02, 31 December 2017 (UTC)

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Notation

In the literature I've never seen the use of such a big C in the notation

$A^{\complement }$

Instead one sees:

$A^{c}$

Maybe this symbol is meant for use in

$\complement A$

(forgot to login)Madyno (talk) 17:23, 25 September 2018 (UTC)

I wouldn't really call that different notation. It's more like different typesetting of the same notation. I kind of agree that the stretched-out C is something I haven't seen that much of, but maybe that's just the choice made by the Wikimedia LaTeX-rendering engine? --Trovatore (talk) 05:50, 26 September 2018 (UTC)

I think the task of Wikipedia is not introducing new typesetting (way of notation). Madyno (talk) 08:46, 26 September 2018 (UTC)

This is certainly not a choice of the Wikimedia LaTeX-rendering engine. But LaTeX command \complement is certainly not intended to be used in exponent (otherwise, its use should be \complement{A}). I have often encountered the notation $\complement A,$ which has been introduced by Bourbaki (see [1] for an early use, and [2] for a previous Bourbaki's text that uses a different notation). In the literature influenced by Bourbaki, the notation $\complement _{B}A$ is also commonly used for $B\setminus A,$ at least when $A\subseteq B.$

Thus, the notations of this article must be changed and sourced. D.Lazard (talk) 09:37, 26 September 2018 (UTC)

Pictures

In my opinion of the two pictures the second does not contribute to the understanding of the term complement. One picture with red=A and white=its complement is more than sufficient. Madyno (talk) 08:43, 26 September 2018 (UTC)

I disagree with that opinion which is why I reverted your change. For those of us who are comfortable with set notation and its visualizations, I would agree, one picture is sufficient, but for those readers coming to this page without that level of understanding, the two picture version is more informative. Another point to keep in mind is that all the Venn diagrams on this page have the subset being described in red, a uniformity that your edit destroyed. While this has nothing to do with the math, these subtle visual clues are often helpful to the novice. --Bill Cherowitzo (talk) 18:55, 26 September 2018 (UTC)

Well, the pictures lacks this clarification of the use of the color red. Moreover a set and its complement appear in the same situation ("picture"). Also the second picture does not mention that A now is in white. To me it's a mess. It would be much clearer if in just one picture the symbols for A and A^c where inserted. Madyno (talk) 08:25, 28 September 2018 (UTC)

I'd appreciate the comment of others. Madyno (talk) 12:09, 12 October 2018 (UTC)

I also completely do not understand the role of the picture about the absolute complement. It shows two circles, one of which is absolute(!) meaningless. Madyno (talk) 13:15, 16 December 2018 (UTC)

Note that when you hover over a link to this page, the image shown has the disk in red, not its complement--I suppose because that is the first image on the page. But that is unfortunate in the light of the uniformity across all these pages in which red is the region of interest. — Preceding unsigned comment added by 24.56.247.67 (talk) 19:56, 21 December 2023 (UTC)

Sets in programming languages

I have removed the long list of the names of the function "set complement" in various languages. Several reasons for that:

• Not really encyclopedic
• Some of these languages do not have any builtin data structure modeling finite mathematical sets. So the list is confusing
• This is an article about mathematics, and this list is out of scope here.
• This list breaks the guidelines of MOS:CODE and MOS:MATH#Algorithms. In particular, the lattes says "Articles should not include multiple implementations of the same algorithm in different programming languages unless there is encyclopedic interest in each implementation."

However, it is worth to have in this article an explanation of the relationship between the mathematical operation of set complement and the programming operation of the same name. I have written such an explanation, which deserves probably to be expanded. D.Lazard (talk) 12:29, 7 January 2019 (UTC)

"Universe" vs "Universal set"

The current article defines the complement of a set using the concept of a "universe".

"if U is the universe that contains all the elements under study, and there is no need to mention it because it is obvious and unique, then the absolute complement of A is the relative complement of A in U:[1]"

Following the link in the article to "Universe", we find that a "Universe" is different than the "Universal set". Yet in the article, the symbol "U" is used as if it represents a set.

The current Wikipedia article on "Universal set" says:

"In set theory as usually formulated, the conception of a universal set leads to Russell's paradox and is consequently not allowed."

Tashiro~enwiki (talk) 00:33, 22 September 2019 (UTC)

 Fixed. I agree that the link is confusing and unnecessary. In fact, the absolute complement is not a concept, but simply an abbreviation, as expressed in the footnote. I have rewritten the sentence for fixing this. However, the section should be completely rewritten, for putting the footnote in front of the section. D.Lazard (talk) 07:52, 22 September 2019 (UTC)

There remains the question of whether the notation for an absolute complement $A^{C}$ denotes a particular set. I see two distinct interpretations of the articles definition of $U$ as the "a set that contains all elements under study".

I think the intent of the article is to say that a writer who uses the notation "$A^{C}$" must have established that he uses that notation to mean $U\setminus A$ where $U$ is a particular set within the collection of sets treated by set theory. In particular, $U$ does not denote the collection of all things that are elements.

A possible misinterpretation of the article is that when set theory is applied to a particular situation, we use $U$ to denote the universal set and such a universal set exists within set theory.

The concept of the universal set is useful in thinking intuitively about set theory. However, a technical article should make it clear that $U$ does not represent the universal set.

Tashiro~enwiki (talk) 18:47, 22 September 2019 (UTC)

I

Maps to logic operators

We have that AND maps to INTERSECTION, and OR maps to UNION.

Does anyone have any references for the maps to RELATIVE COMPLEMENT (tt:0100) and also we have MATERIAL IMPLICATION maps to ABSOLUTE RELATIVE COMPLEMENT (tt:1011), and for the other logic operators?

Darcourse (talk) 04:00, 14 February 2021 (UTC)

See Indicator function for the general relationship between subsets and functions with values in {0, 1}. For your case, you have to consider a bit string as a function that maps the position of a bit in a string to the value of the bit. The remainder is a simple computation for translating the definition of a logical operator in terms of AND, OR and NOT in term of operations on sets. D.Lazard (talk) 09:05, 14 February 2021 (UTC)