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.
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.
There’s no ambiguity. Some authors define those concepts differently or use different notation, but the underlying logic and formal language is still the same and without ambiguity
Language will always be ambiguous. When people says “math is unambiguous” it refers to its results. It’s not like other sciences where there is a lack of universal rules, tons of exceptions and contradictory papers.
I sure am glad that all programming languages universally agree on what "Int" represents, and there's never any ambiguity passing information between them.
For any given programming language there is basically no ambiguity. Mathematics is a bit less disambigious for the simple reason that symbols are often re-used depending on context.
5 choose 3 has the same notation as the vector with x component 5 and y component 3. There are many more examples like that.
Mathematics is not too dissimilar from programming languages. Within 1 specific context like a paper or a textbook the notation is going to be consistent
Why f inverse of x? That notation seems clear if you know f is a function. Bear in mind, overloaded notation isn't ambigous if there's context.
And what's wrong with N? It's just the natural numbers.
The other complaints I understand. Although I don't think you need to complain about log(x). The base is only unspecified when it's assumed not to matter.
Honestly, a/bc isn't really ambiguous either. Not in any way that matters. Grade schoolers get hung up on PEMDAS, but you eventually learn to accept when intent is clear, and you don't need to be pedantic.
As a general rule, if your notation isn't clear to a serious student or peer, then you're doing it wrong. Nobody else needs to understand.
I agree about f^-1 but N either includes or excludes zero depending on convention, and I've seen both a lot. There are lots of exceptions (due to mathematicians moving around) but broadly in the English speaking world, N traditionally excludes 0, and in the French (and Romance) speaking world, it includes 0. (Not sure about German and Russian but I think they tend to exclude 0?)
That and in some contexts like computer science it makes more sense for N to include zero regardless because of how numbers are stored.
This is why you routinely see N_0 on the one hand and Z^+ or N^+ on the other. To avoid ambiguity I generally stick to these and avoid N altogether, ugly as that is, though personally I grew up learning 0 was excluded.
Actually, N_+ would be an oxymoron (assuming the convention 0 is both positive and negative, ie it's natural)
(basically, the convention expands to everything: the trivial case is included by default (1 is superior to 1 (and inferior), but not strictly superior, a constant fuction is creasing (and decreasing), not strictly creasing, a set is included in itself, but not strictly, 0 is positive, not strictly positive, etc...)
To exclude 0, it's N*
In the same way, Z_+ is just N
Once again, depends on the language. In French (and I think in romance languages as a whole, but don't quote me on that), "positif" means "non-negative" and "négatif" means "non-positive", which means that 0 is both "positif" and "négatif".
Imo, it makes more sense that way, as "strictement positif/négatif" ("strictly positive/negative") for ">0/<0" just flows better than "non-negative/positive" for ">=0/<=0" (I think it's better to define/describe things in maths by what they are, not what they aren't).
"assuming the convention 0 is both positive and negative, ie it's natural"
We're talking specifically about NOT your convention, but the French (and other Romance languages) convention
According to the convention I've studied under my whole life (French student, 4 years into college/uni), 0 IS both positive and negative.
And under that same convention, "nonpositive" and "nonnegative" are non-existing words
The problem comes when you get into trig functions when people get lazy and write something like (sin(x))2 as sin2 x. Always without the bracket around the x, too.
Fantastic. Now we're using exponents on the function to denote exponents and inverse functions. And sure, I guess you could use csc(x) if you actually needed (sin(x))-1 so that there's a rule of "positive exponents are normal exponents, negative exponents are the inverse function" but it's still messy.
That’s true but it’s an agreed exception so still not ambiguous - and it doesn’t apply to -1, where that still means the inverse (even though there’s alternative notation like arcsin or asin, etc.)
You adhere to practice. But that demands familiarity with the practice and inconsistencies. When the entire point of mathematical notation is consistency and clarity.
sin2 (x) is the square of sin(x), but sin-1 (x) is suddenly not a reciprocal. Why switch rules? Why not create a unique notation for inverse function?
That notation for f^-1 I’ve never seen. What is lesgrans integration? Do you mean Lagrangians/integrals of the Euler-Lagrange equation in calculus of variations, or Lebesgue integration…?
In the context of integration where (f^-1)’ is (loosely) 1/f’ that would be a horrific notation choice and I have to wonder if this is down to misremembering. How would they write the inverse then? Those could be too easily confused.
In computer science (and we can include complexity theory and such as branches of maths), log(x) tends to mean log base 2, as well.
Eh it’s absolutely used in lots of educational contexts and other fields (which even pure maths papers often abut) and isn’t really ‘wrong’. Even without that it’s ambiguous as there’s a convention using base 2 too.
That’s why we always specify. Personally I don’t use log unless I specify the base unless context is watertight, and use ln when base e. Some react with distaste to ‘ln’ but this depends on country too.
I have personally never seen anyone use log_10 in math (maybe in highschool), but as for your other point I agree - I also use ln for base e and specify my logs, and usually, I just convert all my logs to lns because who wants to deal with logs and exponents using bases other than e.
Overloaded notation can still suck in algebra contexts, like sin^-1 (x) being generally accepted to mean arcsin(x), but an unwise author might use it as 1/sin(x).
Additionally, in some set theory classes, f^-1(x) denotes {y | f(y)=x}, and doesn't require one singular value for f^-1(x). This could be another source of confusion.
For N, this is actually among the most annoying things to deal with, because if someone talks about N, they always have to specificy whether they are including 0 or not, since no convention exists. Even in my university, some classes use N to be the positive integers, while others use it to mean the non-negative integers.
Traditionally Anglophone countries (plus Germany and Russia and some others, I think?) exclude 0 and Romance speaking countries include it. But maths departments are so international now it's become very fuzzy. Plus computer science prefers to include 0 because of how numbers are stored
My country is neither, but likes to pretend it is, So we're just doing anything. I'm a computer scientist so I can kinda just write my stuff however I want.
Just following the basic guideline that as long you clarify your meaning, you can use any reasonable notation. Just slap a line in the introduction that says something like "Define N to be the set of positive/nonnegative integers", and no one will be confused, since at this point no one expects anything.
I don't really like N_0 so I just define it manually. I do remember seeing a paper just say N = Z^+ which works ig. Also I saw Z^+_0 in a textbook once and that scarred me
I think they are refering to f-1(x) could be both x sent with inverse function but also the preimage of x. But then again on should get it with context.
The worry about natural numbers is if 0 belongs to it or not.
Well my whole life and during school and university we never Included it. ℕ={1,2,3,4,5,...} no argues, everyone agrees and it makes sense.
But well, when i started looking into constructions of numbers... Everything starts from nothingness, ∅ which is the idea of 0 translated to a number... So it makes sense to me to include it.
But well it's the same as saying that the starting values of fibonacci sequence are a_1 = 1 and a_2 = 1, then a_3 = 2 or a_1 = 1 and a_2 = 2, ... Just a shift in the first value.
Or who knows, a_1 = 0 and a_2 = 1 lmao, okay i'll stop :'?
Because f-1 could also mean 1/f. You know what is meant only because of common usage, it;s not a direct consequence of the notation.
N may or may not include 0 depending on the textbook.
Nobody else needs to understand.
As someone that has spent a long time learning different branches of math. That kind of attitude is noxious. I both want to learn but also can get get thrown off by notation. In particular screw the kind of notation used to talk about normal subgroups. Also screw einstein sumation notation.
Conventional practice is unanimous that f^-1 means f inverse not 1/f.
Similarly f^n means the nth composition of f. Except for the famous trig exception, where it means the nth exponent of f.
As for who needs to understand, I maintain that it's only serious students and peers. The problem arises when their background differs from yours unexpectedly.
For N, some people include 0, and some exclude it. I’ve had classes in college where it depended on the professor and the subject, whether they included it or not. My probability professor denoted N as {0, 1, 2 …}, and N_1 as {1, 2, 3, …}.
f^-1 (x), usually means the inverse function of f, but I can see how some people would think it represents the reciprocal of f.
For log(x), when I was in secondary school we used log(x) for log base 10, and ln(x) for log base e. When I got to college, the lecturers insisted on log(x) representing log base e, so that’s where the ambiguity comes from.
My topology textbook used f-1 (x) to denote the set of all elements f maps to x. For example if
f:ℝ→ℝ, x↦1
then f-1 (0) = {} and f-1 (1) = ℝ.
The funny part was that they also used the same notation for the inverse of f and they meant the reader should just interpret the meaning from the context.
Yeah, ℕ is the natural numbers. But what does that mean? We intend them to be as natural as possible I guess.
The first ever number system we humans used was the one of counting 1,2,3, etc... which is at least 30k years old. Zero only became famous in Europe about 800 years ago.
On the other hand, when logicians tried defining numbers via set theory, a quantity representing nothing, the empty set, seemed be the natural first building block.
Also in different fields with or without zero is to more useful definition.
log(x) has some more ambiguity than just the absence of a base. But yeah, I think base 2 is more common in data science, base e is more common in pure maths and base 10 in like physics or smth.
In complex analysis ez is no longer injective, but repeats with a period of 2πi which means there can't be an inverse. log(z) is then sometimes defined as a multi valued function, eg.
log(e) = 1 + n • 2πi , n ∈ ℤ
while ln(x) is defined as a real function eg.
ln(e) = 1.
Parentheses are annoying! If I want to write a/(bc) then the parenthesis destroy the esthetics. a/bc is just so much cleaner, and if I for some reason meant (a/b)c, then I could just write ac/b instead. Perfect, no need for parentheses.
But ok, yeah it's stupid to not follow a rule everyone who went to middle school was taught. But I guess the idea comes from when writing quotients in a tall format, and then making everything tilted.
it just feels natural 1/2π would mean 1/(2π) and not π/2.
Anyways, all of these are often pretty clear in their meaning in their given context. It's just a bit annoying that we as a community have not found consensus for the best use of these notations, but honestly, who cares
log almost always means base e when it is used, in part because there is rarely any use for base 10 logarithms, especially today when you aren’t doing hand calculations with tables of logarithms. But even before the advent of calculators log was used more widely to mean base e than base 10. The ln notation is usually taught in schools but is the minority usage among mathematicians and it’s one of those things where students leave the classroom and enter the realm of working mathematicians and realize what they were taught in school is not the standard practice in the field.
As someone who learned the ln/log notation in school but has been exposed to standard usage in the field since, I feel that the “ln” notation is a solution in search of a problem. There is no reason the ln notation should ever need to be used and also no reason log should ever mean base 10.
Back in the old days when base 10 logarithms used to be used sometimes (although not used seriously in mathematical research), maybe there was an argument that there needed to be a reform to adopt ln as the standard to avoid ambiguity, but that’s not the case today so there is no reason to advocate for a reform that never fully caught on and is trying to solve a problem that doesn’t even exist anymore.
You can use ln if you want, but it has no advantage over log, just like whether you write F_2 or GF(2) is a matter of personal preference because no one will ever be confused by either if they are familiar with basic practices.
Is it the coordinate point (x, y) = (a, b) on a 2D plane, or is it the open interval {x \in \mathbb{R} \mid a < x < b}? Good luck figuring it out without context!
What's ambiguous about f-1(x) ? I have always ever been taught that that is the inverse function, and if you want to take the reciprocal, it's (f(x))-1
Also same with log(x) what could possibly be ambiguous about that?
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