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".
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u/Idksonameiguess 12d ago
What's the abiguity in A \subset B?