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In school life we were taught that $<$ and $>$ are strict inequalities while $\ge$ and $\le$ aren't. We were also taught that $\subset$ was strict containment but. $\subseteq$ wasn't.

My question: Later on, (from my M.Sc. onwards) I noticed that $\subset$ is used for general containment and $\subsetneq$ for strict. The symbol $\subseteq$ wasn't used any longer! We could have simply carried on with the old notations which were analogous to the symbols for inequalities. Why didn't the earlier notations stick on? There has to be a history behind this, I feel. (I could be wrong)

Notations are notations I agree and I am used to the current ones. But I can't reconcile the fact that the earlier notations for subsets (which were more straightforward) were scrapped while $\le$ and $\ge$ continue to be used with the same meaning. So I ask.

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To be honest, I think the usage is dependant on the field of maths you are in - I think I can remember some type of article by Timothy Gowers discussing something similar; I'll try and find it. –  Andrew D Jun 24 '13 at 10:41

2 Answers 2

up vote 3 down vote accepted

I had a professor explain this as "it is much less common to require a proper subset". So much so, that it became customary to use $\subset$ as a (proper or improper) subset, and $\subsetneq$ as a proper subset, just because the former is easier to write on a blackboard.

The order relations $<$, $\le$ between numbers are supposedly used with approximately the same frequency, so the original notation is kept.

I guess when you're first learning about sets, you use $\subset$ and $\subseteq$ because they're similar to $<$ and $\le$, but this indeed seems to not be the common notation "later". (Fortunately, usually it is clear from context what $\subset$ means.)

Edit to add: $\subsetneq$ is also a more complicated relation than $\subset$ (it is the conjunction of $\subset$ and $\neq$), so maybe it deserves a more complicated symbol.

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As @Asaf said in his answer, it depends very much on your field. I’m a set-theoretic topologist, and I consider $\subseteq$ the normal notation; in the mathematics that I read, the use of $\subset$ for $\subseteq$ is not very common. (I myself consider it sloppy and potentially confusing.) –  Brian M. Scott Jun 24 '13 at 20:05
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In my university $\subset$ is used for $\subseteq$ in grad courses in all fields (except, maybe, set theory which I never attended). It might be a local and blackboard-only notation, though. –  Yoni Rozenshein Jun 24 '13 at 21:35

This is very field dependent (and probably depends on the university as well). In my M.Sc. thesis, and in fact anything I write today as a Ph.D. student, I still use $\subseteq$ for inclusion and $\subsetneq$ for proper inclusion. If anything, when teaching freshman intro courses I'll opt for $\subsetneqq$ when talking about proper inclusion.

On the other hand, when I took a basic course in algebraic topology the professor said that we will write $X\setminus x$ when we mean $X\setminus\{x\}$, and promptly apologized to me (the set theory student in the crowd).

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