# Does anyone use $\subset$ for proper subset anymore?

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?

• 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. Commented Jan 22, 2015 at 11:56
• 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. Commented Jan 22, 2015 at 12:00
• 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. Commented Jan 22, 2015 at 12:06
• 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). Commented Jan 22, 2015 at 12:17
• 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. Commented Jan 22, 2015 at 12:36

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).
• 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. Commented Mar 23, 2018 at 13:55
• "Professional mathematicians use $\subset$ only" is totally wrong; both conventions are common among professional mathematicians. Commented Jan 31, 2019 at 16:38
• @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. Commented May 11, 2019 at 13:51
• @CarlMummert: thank you for the comment. I changed the wording somewhat. Commented May 11, 2019 at 13:51