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I belong the the group of people who still write (not necessarily proper) subset as $\subseteq$ to avoid any confusion with proper subset, which I notate $\subsetneq$; I usually do not use $\subset$ at all. But now that I think about it, I have not seen anyone use $\subset$ for proper subset in any post-1960s textbooks. Is it still used for that at all, or can that use of the $\subset$ symbol be considered archaic?

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    $\begingroup$ When I was a student at the university of Copenhagen 11 years ago, only very few people there didn't use $\subset$ for proper subset. But I suppose it depends a lot on what you do. $\endgroup$ Commented Jan 22, 2015 at 11:56
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    $\begingroup$ I currently study in Aarhus; it is strange how the conventions apparently vary from university to university. At Aarhus, it appears that all topologists use $\subset$ for subset, while all analysts use $\subseteq$. It varies among the algebraists. $\endgroup$
    – Gaussler
    Commented Jan 22, 2015 at 12:00
  • $\begingroup$ I have seen people use both $\subset$ and $⊆$ for subset but never seen anyone use $\subset$ for proper subset only. At the very least not in any papers or books that were written fairly recently or any conferences/lectures I have been to. $\endgroup$
    – Jack Yoon
    Commented Jan 22, 2015 at 12:06
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    $\begingroup$ I think the use of $\subset$ and $\subseteq$ was originally intended to reflect the use of $<$ and $\le$. However, since the early days of set theory, it has become increasingly obvious that subset is the more important notion, and that it is used much more than proper subset. So I do indeed see the idea of letting the most simple symbol stand for "subset" (even if I do not use it for that myself). $\endgroup$
    – Gaussler
    Commented Jan 22, 2015 at 12:17
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    $\begingroup$ I only use $\subset$. I don't like the way $\subseteq$ looks. If I needed to stress that $A$ was a proper subset of $B$ then I would write $A\subset B$, $A\neq B$. Just personal preference. $\endgroup$
    – Math1000
    Commented Jan 22, 2015 at 12:36

1 Answer 1

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It is hard to say what is common or archaic use of a symbol, because it would require studying a large number of publications.

The use of $\subset$ for proper subset and $\subseteq$ for subset is obviously motivated by analogy with $<$ and $\le$. On the other hand, using $\subsetneq$ for proper subset and $\subset$ for subset is motivated by the fact that proper subset is a concept which is seldom needed in actual mathematical context. (I cannot easily create a meaningful example where proper subset is required.)

At my university (Budapest), only $\subset$ was ever used and always meant subset. No one ever used $\subseteq$ (nor $\subsetneq$), because the concept proper subset was seldom used, and the simpler form, $\subset$ had a better use for the more common meaning subset (a kind of economical choice of symbol). In fact, these symbols had a certain "childish" or at least "high schoolish" look. On the rare occasion that proper subset was needed, one would write $X \subset Y, X \ne Y$, in order to emphasize this constraint.

Below I collected some of the global resources that may be useful in deciding one way or other:

ISO

ISO has a standard for math symbols, but it is targeted towards natural sciences and technology rather than mathematics itself.

The standard is behind a paywall, so only the Wikipedia page can be linked here: ISO 80000-2:2009. An eralier, obsolete standard (which nevertheless is very similar) is available on Wikipeda: ISO 31-11:1992.

  • Item 2.5.7 defines $B \subseteq A$ as "$B$ is included in $A$, $B$ is a subset of $A$".
  • Item 2.5.8 defines $B \subset A$ as "$B$ is properly included in $A$, $B$ is a proper subset of $A$".

However; even the ISO document contains a remark for 2.5.7 and 2.5.8 that defines an alternative: $\subset$ is subset and then $\subsetneq$ must be used for proper subset.

Wikipedia

The Wikipedia article on math symbols takes the same approach, preferring the pair $\subseteq$ / $\subset$, but mentioning $\subset$ / $\subsetneq$ as well.

Unicode character names

Unicode defines these operators in the Mathematical Operators block and uses the following names:

  • U+2282 $\subset$ SUBSET OF
  • U+2286 $\subseteq$ SUBSET OF OR EQUAL TO
  • U+228A $\subsetneq$ SUBSET OF WITH NOT EQUAL TO

This seems to prefer the $\subset$ / $\subsetneq$ pair, but is not decisive (ambiguity is very common with Unicode names, unfortunately).

HTML entities

In the official HTML entity list (renamed to "named character reference" since HTML 5):

  • $\subset$ is &subset;
  • $\subseteq$ is &sube;
  • $\subsetneq$ is &subne;

Again, this seems to favor the $\subset$ / $\subsetneq$ pair.

Summary

Where I come from, professional mathematicians prefer $\subset$ for subset. Proper subset is virtually not used. If a single character were still required for that case, it would be $\subsetneq$. But better avoid that as well.

In education and applied sciences one can see $\subseteq$ for subset more often.

There are also some mentions of $\subset$ meaning proper subset, but I have never seen an actual example where it was used like that. In order to avoid confusion, one could (in fact, should) use $\subsetneq$ for proper subset even if $\subseteq$ is used for subset.

Bottom line: use of $\subset$ meaning proper subset is ill-advised in any case because it creates confusion. Use either

  • $\subset$ for subset and (if absolutely necessary) $\subsetneq$ for proper subset (my preference), or
  • $\subseteq$ for subset and $\subsetneq$ for proper subset (totally unambiguous choice).
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    $\begingroup$ I can think of a few times when proper subsets come into play. Perhaps the most important is that a maximal ideal is defined to be a proper subset of a ring. $\endgroup$ Commented Mar 23, 2018 at 13:55
  • $\begingroup$ "Professional mathematicians use $\subset$ only" is totally wrong; both conventions are common among professional mathematicians. $\endgroup$ Commented Jan 31, 2019 at 16:38
  • $\begingroup$ @EricWofsey: thank you for the comment. I updated the answer to reflect it and reworded to a less biased form. I have seen some more papers recently that use $\subseteq$ for subset . Yet as to the original question, which was about $\subset$ meaning proper subset I still believe that is a confusing use of the symbol. $\endgroup$
    – balage
    Commented May 11, 2019 at 13:51
  • $\begingroup$ @CarlMummert: thank you for the comment. I changed the wording somewhat. $\endgroup$
    – balage
    Commented May 11, 2019 at 13:51

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