Math board is dead, hoping one of you techheads can help me with this.
I can't wrap my head around this. It seems like it would work if dedekind cuts were defined as (-inf, a), but that is supposed to just be a 'representation' of them. The actual definition is (-inf, a) U (a, inf), which means that the union of any two cuts where a!=b should equal the entirety of Q.
What am I missing?
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Q = R you goon
No, R is the completion of Q, that is, it also includes all of the Algebraic Reals (roots of polynomials with integer coefficients) and the Transcendentals (e, pi, etc). Q is countable, R is not. They are certainly not equivalent.
And even if you were correct, that still would not answer the question. Let us suppose that Q is indeed R, and thus Dedekind cuts correspond to unions of sets (-inf, r) U (r, inf) for r in R. In this case the union of any two cuts would equal the entire numberline, which by the definition of cuts would not equate to the suprenum of any set. My question stands.
No, Q = R, it was pqoven a few yeaqs ago.
Please stop this nonsense
Also why is Z cyclic if its generators and can only ever travel in one 'direction', e.g. should only give Z+ and should only give Z-.
It's an infinite cycle. If you look at the definitionon wikipedia, it means that every element is generated by a single element with an associative inverse (1 or -1).
Doesn't that mean a is just irrational?
'a' is a real number not in Q, but all numbers in Q are either greater or less than it...
Sorry if I didn't understand
I think I understand my confusion, I was thinking of a dedekind cut as a union of the lower and upper set, so for a rational number it would just be all Q for any number thus making it meaningless, whereas if you leave it as a lower and upper set it has a meaning.
Likewise for the irrationals its the same lower and upper set just without the number itself included in the upper set.
Now for the suprenum definition I understood it as the entire cut representing a real number, but the only logical explanation is that its ONLY the lower set that represents the number, that way it makes sense for the union of all numbers in the set A = lower set of Sup(A) because all dedekind lower sets in A are a subset of Sup(A).
Perhaps you can confirm my suspicion.
I get the first paragraph, but not the second one...
are you saying if there are two elements {a, b} with a < b, then the union of the dedekind cuts would be Q,
whereas if you take the lower half of the dedekind cuts it would correspond to the suprenum of the two numbers.
Is that what you're saying?
simple, (-inf, a) is not a subbase for the topology of R, while (-inf,a) U (a, inf) is a topology.
Also if you really need to understand this the Rudin as a nice construction of R with Dedekin cuts of about 9 pages.
The autism gene
Ok I'll try and make it clearer
Say there is a set A in R such that b = Sup(A)
My assumptions:
-By using the dedekind cut representation, any number in R can be represented as the lower set of the cut, for example the number 1 is represented as (-inf, 1).
-Now if you have a set of numbers say [0, 2) then its suprenum (in this case, 2) can be represented by the union of the lower parts of the cuts. Because:
>0 = (inf, 0)
>1 = (inf, 1)
> 1.5 = (inf, 1)
> therefore 0 subset of 1 subset of 1.5 ....
> therefore their union just equals the largest number which is 2
So based on this assumption it would appear correct, that the representation you are supposed to use is the lower set of the cut. But I am unsure.
I understand that (-inf, a) U (a, inf) is a topology of R, but at the same time its basically every q in Q, so I don't know how that is unique compared to (-inf, b) U (b, inf), assuming that a and b are both irregular. Rudin is still a little impenetrable for me at present.
welp, I meant 1.5 = (inf, 1.5)
seems like some pleb doesn't know that the diagonalization proof used by cantor to prove the uncountability of real numbers was the basis for the turing\church works on computation that built the modern computer science.
furthermore, if you're not even able the beauty of math, you really are a pleb. now get the fuck out, braindead faggot
Go play with your number lines faggot
I prefer shitting on your bullshits
WOW AMAZING
By giving precise names to things they can be used as elements to construct more complex arguments without having to write a whole paragraph of text for each element. It's not esoteric for the sake of it, you are just repelled by it because years of exposure to a poor education system has made you fear it. The next time you try to make a sound argument and find yourself tripping over your own words and thoughts I hope you are reminded of this and perhaps rethink your stance.
...
Is there some magical point on a number line that doesn't fit that definition?
Ok then feel free to come up with a consistent definitions for the gaps in the Q numberline so as to construct R without using Dedekind cuts. Remember it has to be consistent.
Do your own homework.
It's not my homework, I was indirectly telling you what Dedekind cuts are good for.
Oh, I see what you're saying.
I think you're right according to this page:
en.wikipedia.org
Thanks for that link, I actually got my answer:
Supremum. If a nonempty set S of real numbers has any upper bound in R then it has a least upper bound in R that is equal to Union S
OP, wherever said that the cut itself was defined to be (-inf, a) U (a, inf) was wrong, since clearly for irrational a, (-inf, a) U (a, inf) = Q, as you said. You were right all along you just have to believe in yourself.
Some people play fast and loose with language especially if they aren't mathematicians themselves. When they say the cut is defined to be (-inf, a) U (a, inf), what they really mean is the cut is defined to be two sets (-inf, a) and (a, inf).
Polite sage for non-tech thread.
Thanks but I don't see why math talk is prohibited on this board since math is a form of technology and the math board is dead nor is there a science board. Perhaps you can direct me to somewhere active where this type of discussion is permitted?
That's why.
New board owner of >>>/math/ here, I'm trying to revive the board. I found a way to get the TeX rendering to work. Can you please make the same thread there and continue discussion?
Just make it yourself and link to it.
It looks as though OP's question has already been answered but yes I plan to make some new threads on >>>/math/ about other math topics. When the fall term starts, I will advertise the board to other students at my school's campus. I would appreciate it if OP and any other mathematically inclined people here do the same.