If sets are determined by their elements, then if a set such that $z\in x$ iff $\varphi(z)$ exists, then we can write $x = \{z : \varphi(z)\}$ because it will be unique
@EE18 we mean $\{z : \varphi(z)\}$ and all other variations
Well, you use more axioms than just this one to show a set exists
as for the latter, axiom of extensionality tells you that for each formula $\varphi$ of set theory there can exist at most one set $x$ with $z\in x\iff \varphi(z)$
the set doesn't have to exist, but if it does, there is only one
True. I guess I should amend the above to say that given some set $X$ exists, we justify set-builder notation to build a set $Y$ from $X$ by using specification/comprehension to guarantee the existence of the set and extensionality to ensure its uniqueness
EE18: "set-builder notation" is a term that a lot of books would not define and does not generally have an established place i know of in set theory
but at least some books would use it roughy to refer to a use of {x in A: f(x)} where A is a known set and f is a nice enough formula (perhaps also depending on things besides x)
and not just any use of curly braces, some of which are much more fundamental
or "{x: f(x)}" where the role of an ambient set is understood
its maybe helpful to separate stuff that exists within your chosen formalization of set theory because of how you're formalizing set theory, and stuff that can be carried from place to place outside of that
as jakobian notes it might be related to some use of a specific axiom but it is a correspondence in the sense of how humans talk about things and not a theorem in the sense of something that is in a textbook
jakobian: you don't get to, i'm sorry
jakobian: here's a pedagogical point you might consider. if someone is reading set theory out of a book that gives very explicit definitions for some things, and does not define others, and "set-builder notation" is not one of those other things, it is pedagogically unwise to tell such a person that "set-builder notation" is morse code for one of the things they're reading about
it is also pedagogically unwise to do this if they might be reading out of such a text and you just don't know either way
regardless of your own feelings on the subject
this is just an enormous blind spot you have about how you talk about math
i think you'll find that many set theory and logic books at the levels of formality that you find respectable do not use the term "set builder notation" anywhere
it's just not common. i honestly don't know if you're trolling or if this is actually an issue of english use vs. whatever "Set builder notation" is in a language you also read
It's also well taken (I think) that it's at least interesting and useful to think about how one might formalize set-builder notation if they had to, if for no other reason than that's useful for flexing the relevant brain muscles for a rookie like me :)
EE18: to get back to the point, i've definitely read or taught out of books where "almost anything with curly braces and commas in it" would be described as set builder notation, including e.g. {1,2,3}, although [again all of this depends on axiomatization] the axioms you need to make sense of something like {1,2,3} would not necessarily draw on comprehension/abstraction/whatever you call it
But ya I think a lot of heartache I had in the first chapter of Amann Escher was that they are so damn precise about everything...except for everything underneath where they start...so it can be hard to delineate the two
@TedShifrin ;) it's a bad habit i need to avoid, at least for now. it's just so damn tempting
I think you can do and learn plenty of rigorous mathematics without spending two years studying foundational ZFC and axiomatic set theory. At least, I sure did.
The amount of foundational set theory in Munkres's Topology is sufficient for at least 99% of mathematicians.
i'll give an example, i opened the one set theory book i have, and herbert enderton's "elements of set theory" refers to what you call comprehension as "abstraction" and informally refers to the use of "B = {x in c: ___}" (where c is a known set and ___ is a symbolic formula not containing B) as "abstraction notation"
Let’s say I have two continuous functions $f(x)$ and $g(x)$ , and both have the same derivative $h(x)$.
How could I formally show that $f(x)=g(x)+c$ where $c$ is a constant.
I know I have to show that $f(x)-g(x)$ is a constant function but not sure how?
Thanks
@TedShifrin ok just real quick question. When I've used the dot product in cosine similarity, in that formula, the $\theta = \text{arccos} \left(\frac{\mathbf{x}\cdot \mathbf{y}}{\lVert \mathbf{x}\rVert \lVert \mathbf{y}\rVert }\right)$
that itself is a measure of closeness of two vectors, right?
soumik: so, it's not a well-formed question because of its silence about the (preusmably common) domains of f and g, and i'd hope that any answer would point that out, and expressly state what they were assuming about the domains of f and g in order to give a "proof"
@TedShifrin its you that assumed that I'm talking about everyone. And its also you that seems to be assuming that just because you didn't have to deal with foundational issues, everyone wouldn't have to
lots of people don't have to, but there are some that do
i wouldn't think it unreasonable to post an answer that expressly provides that assumption and gives that proof. i.e. the kind of condition that you'd need to assume for the question to make sense is not an arbitrary choice, coming out of nowhere
@leslietownes If someone asks to prove a false statement, what would be the correct approach at answering it? providing a counterexample or proving after having further assumptions?
@leslietownes Yes, all the other answers except the accepted one are wrong
soumik: well in the example you've given, the missing hypotheses (needed to make the statement true) are pretty clear, and so commonly made in a lot of "calculus class" type pedagogical environments that the OP might not even be aware that they are hypotheses. so maybe this isn't a very representative example of the world of "somebody asking to prove a false statement"
soumik: and literally none of them have upvotes? is there something that upsets you about this question specifically? it looks like MSE voting actually works in this case
i'm not sure if you are asking something specific to this question (from 2019??) or are you asking about some more general phenomenon
@TedShifrin i'm glad you pointed this out. i've not remembered that's a requirement and just accepted passively when people told me its a measure of similarity
Certainly I could make something less symmetric than the usual tetrahedron. But your requirement that every edge is in a cycle of length 3 destroys my examples. You certainly did not say that to start with.
@leslietownes Maybe I'm not understanding the statement being made, but how is this an example? Isn't the claim essentially "let f and g be two continuous functions with same codomain and domain. Then f' = g' is sufficient but not necessary to guarantee f = g + c for some constant c"?
@TedShifrin I mean, I can only speak from it based on experience. in many language ML classes, the vector representations of words. Suppose I have the word "happy" as a vector and I ask it to use this cosine formula and comptue the angle between "happy" vector and other words. It will often return words with very highly similar meaning such as like "elated" etc.
So in this sense, it can be used to find "similar" words. there are other contexts where this works too. So I had typically taken it as an assumption that this dot product cosine angle formula was a measure of closeness.
no one pointed out that, if you think intuitively about it, obviously if you just took two vectors in 2D and computed their angle, whithout a restriction on magnitudes, the vectors may not be anywhere near each other.
@EE18 you need to read this as a calculus student will read it. keep in mind like 99% of people using MSE have never heard the word "codomain" and certainly aren't thinking about varying the domain.
a standard hypotheses in calculus books, sometimes buried in a paragraph of remarks and used only inconsistently later, is that when you make statements of whatever type, you are restricting to a domain that is an interval
Oof! So the statement is ""let f and g be two differentiable functions with same codomain and domain. Then f' = g' is sufficient but not necessary to guarantee f = g + c for some constant c"? I vaguely recall this from Spivak, but are you saying my comment about sufficiency is wrong even in this restatement Ted?
@EE18 oh sorry i now see what you're replying to. if f and g are different "slidings" of the halves (the idea is shift the two pieces by different constants) they will have the same derivative but not be the same.
soumik: isn't the top-voted answer (which points out the implicit hypotheses of connectedness for the theorem to hold, and what can happen if the domain has more than one 'piece') still at least somewhat informative in whatever example that is?
@leslietownes Hmm, still not sure I follow. If I look at the two pieces of 1/x then they have the same derivative if I "read" the left piece from right to left and the right piece from left to right? But not sure I follow. anyway, no stress, don't want to waste your time with my misunderstanding
I am saying $f' = g' \implies f = g+c$. Isn't that "$f' = g'$ is sufficient for $f = g+c$"? @TedShifrin
@EE18 an alternative way to think of this is the "calculus formula" integral -1/x^2 dx "=" 1/x + c is kinda wrong because the most general antiderivative to -1/x^2 [i.e. having the same largest domain as that thing] does not require "the c" for "the two 1/x pieces" to be "the same"
EE18 when i was using the word "pieces" i was speaking informally. consider any numbers A, B and define f for x negative by f(x) = 1/x + A and for x positive by f(x) = 1/x + B. then f'(x) = -1/x^2 for all x regardless of what A and B are
I guess that means I am totally forgetting some underlying hypotheses Spivak was making. That was a few years ago but dang that's bad still :( will hopefully get this better once I am deeper into Zorich and AE. Thanks for clarifying Ted
$\int f = g+C$ only making sense on intervals (i.e. the $+C$ depends on component you are in) is one of the first encounters of people with the notion of connectedness
@Simd So it seems your 3-cycle condition gives two different graphs up to isomorphism. It depends on whether there is any net boundary to the structure. For the tetrahedron with AB, BC, CA, AD, BD, CD, there is no boundary. But if you remove one of these edges, you now have boundary.
I think everyone does it the same, we first encounter a concept, and then have a sequence of, mostly guided, realizations about the subject so that we understand them more
@TedShifrin The counter example I saw also assumes the function to be continuous everywhere. The reference is S. Saks, Theory of the integral, page 205-206
@SoumikMukherjee I'd suggest starting with Bartle or Fonda for Kurzweil-Henstock integrals, both have advantages, Bartle is a great book to read from, and Bartle is side-tracked a lot, so you gain exposure to more in integration theory. On the other hand Fonda does things more straightforward, it only discusses the relevant theory. Fonda also discusses generalizations to multiple dimensions, and to differential topology
The last book I'd recommend is the one by Gordon, when he goes very extensive more akin to Saks, but more modern
the last book expositions you to various concepts of integrals
I'm not willing to recommend anything for point set topology without knowing whats it for
Dugundji added some of his own flavour into his book, so while its interesting you might encounter some things you might never find useful
if you want to read further into topology while already knowing a chunk of it, I recommend Engelking
its a great book, and I wish I had read it in full at some point
if you want to further your knowledge in some subfield of topology, there are options there too
if don't know any topology, not even metric spaces, you'd do better learning that first
also Dugundji goes very informal with algebraic topology part of his book, I didn't enjoy that part
I don't have a solid go to book for furthering your knowledge in point set topology after exposure to metric spaces, but Munkres or even Dugundji, its alright
im still not sure, do you want something that covers similar material to Munkres, but in a different style? or do you want to learn something new altogether?
Let’s say I have two continuous functions $f(x)$ and $g(x)$ , and both have the same derivative $h(x)$.
How could I formally show that $f(x)=g(x)+c$ where $c$ is a constant.
I know I have to show that $f(x)-g(x)$ is a constant function but not sure how?
Thanks
