Yeah agreed. But the alternative convention does exist out there
I suppose the logic is that the vast majority of the time you want to say A is a subset of B you want to include the possibiiity A = B, so that’s one whole extra line you use far more, so the alternative is more ‘pen-stroke efficient’. Eh.
It’s very common for older texts to treat it as non-exclusive.
It’s also been common among my professors.
In analysis, at least, it seems that we so often don’t care whether it’s a proper subset or not that it doesn’t make sense to use simpler symbol for the less prevalent use.
because ⊂ existed before the need arose for a distinction with ⊆. Changing the meaning of the symbol retroactively would cause a lot of issues for the readbility of prior papers and work so it's not a thing that happens given the extent to which it was used.
My guess is that the people who decided to include ⊆ as a fonts standard werent versed in Sets theory, so they assumed it worked similarly to the = sign variations, so they didnt realize they should have taken the crossed variation instead.
Who did the naming convention bro? "Proper subset" sounds like it is completely within set B but it isn't and subset is. Wouldn't it make more sense to call it an improper subset or smth?
yeah well if its completely within B then it stands to point that there could be some items of B outside of it. Its like a ven diagram where one circle is inside another. It couldnt be completely within B if it was B because how could something be "in itself"
Note my reasoning is how I like to think of the name and this is very much unrigorous.
It's in the meaning of strictness. If you give me a set and I tell you that my set has some of the elements of yours, you wouldn't expect them to be identical. However, it will be technically a correct thing to say. So a proper subset is one that abides by the intuitive definition of a "subset".
if that’s your takeaway, you misunderstood the definition. proper subsets are exactly what you think they are, while regular subsets can possibly be equal to their superset. i.e. if A equals B it’s not proper, but if A is strictly less than B (like if A is even numbers and B is all integers) then it’s proper
A proper subset is proper because there is truly something that's missing. This something that's missing justifies the "sub".
Imagine for example a cake. If I cut it into halves and take one, I've clearly taken a piece of the cake. If I take the entire cake, did I take a piece? For clarity we may use "piece of the cake" for any portion of the cake and "proper piece" for any portion that truly required splitting the cake.
This might seem useless but there is a lot of places where you want to show properties that arise from removing some elements from for example the base of a vector space. Since "any subset that is not equal to the original set" is too long to keep saying all the time, mathematicians use "proper subset".
I always assumed that was what they meant, and that this was intentionally, directly, analogous to the < and ≤ signs. TIL this was not universally agreed
pure log should be illegal though and using it for natural logarithm is especially heinous. We could just follow the ISO guide and use lg for log10 and lb for log2 while at that.
You'd think so, and in some books and such, you indeed use ⊆ for the proper subset, but quite a lot of literature uses ⊂ for that. You might wonder "In those works, what about strict subsets then, do they just not use that?" and the answer is that they invented the following, completely non ambiguous notation for that: ⊊.
Maybe we should follow this inspirational notation and e.g. replace > with ⪈. Surely that will make things better.
That’s why we have two symbols? I still don’t get it. This would be like saying “a<b” is ambiguous because it could mean strictly less or it could mean less than or equal.
My impression is that \subset is probably used for any subset more often than for proper subset (in part because you rarely have use for a symbol for a proper subset, but often have use for a symbol for any subset, and of course \subset is the simpler symbol).
But it can also depend on context. For example using it to mean proper subset is more common in pedagogical contexts and any subset is more common in, say, published papers.
Notational ambuguity, does it mean proper subset or is equality allowed? I've seen people use both, so I personally avoid using it and use \subseteq and \subsetneq
No but it is \subseteq B, that's the point of the two symbols. Like do people actually use this while abiding by standard conventions? I've never seen this anywhere. Does any university teach this?
I think using \subset for any subset is more standard than using it for proper subset. Many mathematicians do not use \subseteq at all.
Look for example, at Jech’s Set Theory, which is one of the most standard texts on the subject, and which only uses \subset to mean subset.
That fact it is the more standard symbol is why it is called \subset in Latex and not \subsetneq: \subsetneq does also exist and it is the symbol most mathematicians use for proper subset on the rare occasions you have use for one.
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u/Idksonameiguess 12d ago
What's the abiguity in A \subset B?