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12:00 AM
@TedShifrin Thank you for notifying me. I was confused by Xanders posts in CURED. I did not vote to close the "good post", I just deleted the comment you condemn me for, automated when I voted to close as a dupe, when I had the two posts mixed up. I upvoted the "good post", and downvoted the "bad post". What next will you excoriate me for, @Ted???? Adios!
I did not intend to disturb your kingdom. But you were willing to sit on your resentment about the one post, vs. the other post, and two and more of us tried to do something about it. The way the posts were presented to me confused me. And I corrected the mistake you scathed me for. Let me now, having learned I am not welcome in your Kingdom depart.
@AkivaWeinberger Okay, so I think my question is more about this. We, in this context, know that $\alpha$ is $\sup a_n$. However, let's say we met $\alpha$ on the street and it was dressed like $\beta$ that is the output of a program $P$.
We interrogate $\beta$'s $n$th digit, using $P$
That gives us an algorithm to decide if $n\in\{c_i:i\in\Bbb N\}$
Okay, so $c_n$ is an uncomputable sequence?
but the hypothesis was that, though $\{c_i:i\in\Bbb N\}$ was computably enumerable, it was not computable. (Two explicit examples are Gödel numberings of halting Turing machines and Gödel numbers of provable statements in arithmetic.)
$c_n$ is a computable sequence, but if we rearrange it to be increasing, it is not computable anymore, essentially.
Think about how you would make a computable sequence that outputs a list of halting Turing machines
You would run machine 1 for one timestep, then machines 1 and 2 for one timestep, then machines 1, 2, and 3 for one timestep, etc
and whenever a machine halts, you write its ID down on a list
This list will not be given in order, because some short programs take a long time to halt and some long programs halt very quickly.
12:16 AM
It's been quite a while since I've done any computability theory.
This just shows I'm getting lame.
My CS legs aren't what they used to be.
If you want a better description then what I've given here, I recommend the book Reverse Mathematics: Proofs from the Inside Out by John Stillwell
He is a much much better teacher than I am
I'll add it to the list.
I recently cleaned out my book wishlist so it's not that long anymore.
1:13 AM
Anyone would be willing to help me fix my kernel panic problem.. i'm desperate
1 message moved from CURED
@UlysseCorbeil what's the question?
1:54 AM
Let $B$ and $A$ be nonempty sets such that $B \subset A$. Prove that $|A - B| \leq |A|$.

So, I know that I need to show an injection between the sets in order to prove this. My thinking is that $A - B = A$. So we can define a function $f: A - B \rightarrow A$ such that $f(a) = a$. And from here, an injection is simple to show.
that rule will work even without the assumption (not needed, and not implied by the given information) that A - B = A
Oh, hmm
e.g. B = {1,2} is a subset of A = {1,2,3,4} and A - B = {3,4} is not A
but the rule f(x) = x does define an injection from A - B to A
That always trips me up
Alright, that's good at least.
note that this is equivalent to showing that $|S| \leq |A|$ for any subset $S$ of $A$. the fact that $S$ has the form $A - B$ is kind of beside the point (although any subset $S$ of $A$ does have this form, for some $B$)
1:58 AM
Oooh, that point becomes much more clear when you present it that way.
2 hours later…
3:56 AM
Is there a term for functions that feel cheaty, in that they're defined to be the solution to something we don't otherwise have a solution for, and also don't have another way to evaluate? E.g., the error function and the lambert W function
injections hurt sometimes.
you mean that they are implicitly defined?
i can't think of a term. it isn't really a property of a function so much as a property of a way of defining a function
e.g. ln(x) can be defined in a cheaty way as the solution to e^y = x, or as int 1...x 1/t dt which maybe doesn't feel as cheaty
i think copper's notion of implicit definition captures a lot of what you might not like about W and erf
at least as defined in the usual ways
I don't think it's implicitly defined though. Doesn't that mean when f(x) doesn't appear on one side of the definition by itself? But for erf, it does, it's just that the other side is an integral we can't otherwise solve
some people seem more OK with some inverse definitions than others, e.g. i think many people who don't think arcsin or ln defined as inverses are cheaty, but would agree that W is.
being 'implicitly defined' isn't really a property of a function. it's a way of defining a function
it's like 'piecewise function' - that's not really a property of a function either, more of a property of a method of definition
you might be asking about which functions can or cannot be expressed in terms of 'eelmentary functions' which are a somewhat arbitrary but well known collection of functions. liouville proved some theorems on this
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is e − x 2 , {\displaystyle e^{-x^{2}},} whose antiderivative is (with...
so that's one way of making the notion rigorous, it might not quite match up with what feels 'cheaty'
Ah, that gets close to what I'm going for. I think "functions that can't be expressed in terms of elementary functions" would be a necessary but not sufficient condition for what I'm calling "cheaty"
4:05 AM
arxiv.org/abs/math/9805045 is a kind of fun article that speaks about numbers instead of functions
W is not an 'elementary function,' nor is erf
The redesigned Activity page has major MathJax issues.
people have been mentioning this
It was clearly designed without MathJax in mind.
4:32 AM
Are there any theorems that state that an uncountable set unioned with a countable set is uncountable?
omg ;-;
Does it have a specific name? I'm not finding anything about it in my textbook.
you proved something like it earlier. if S is a subset of A then $|S| \leq |A|$
i don't think it has a name
[uncountable set] is a subset of [uncountable set] union [countable set]
I keep telling myself I'm never going to complete a problem set the night before it's due ever again. This takes all of the fun out of math.
that's pretty common. everybody does it.
i don't know if it's that bad. if the instructor is not assigning insane stuff on a weird timetable, it ought to be workable.
i don't recommend a similar approach for studying and exams.
once i graded homework for a prof who assigned homework due every monday, wednesday, and friday. they were short assignments, so it wasn't an unreasonable as an amount of work to assign in a week. but the sheer amount of paper that everyone had to deal with was unreasonable. and you have to allow people a little more flexibility in their scheduling than that.
4:43 AM
I try not to let this habit bleed over into exams, but sometimes it does.
And that many assignments in a week for one class is crazy.
the pedagogical purpose, i think, was to keep people engaged with the class on a near daily basis. but it also discouraged deeper thinking and led to a lot of hastily done work.
it also sort of increased the feeling among students that they were constantly being evaluated, which i didn't think was helpful. people need space to practice. the same amount of work due once a week would have been just fine.
I agree.
I found it. It's labeled as "Theorem 11.10" in the text.

"If $A \subseteq B$ and $A$ is uncountable, then $B$ is uncountable."
it's a practice that might only make sense in a setting where all students have the same life. i think frequent homework in high school can sometimes work on this basis. with college students people can have much more widely varying courseloads, jobs, etc. it's not reasonable to expect work every MWF.
Oh definitely. Especially these days where pretty much everyone works.
i'm against takehome exams in college for this reason. they really exacerbate any differences in the amount of time that people have available outside of class.
4:48 AM
I think there are a lot of facets of higher education that could stand to be adjusted for the times.
Wow, I was just telling someone I wish I had take-home finals today. I hadn't even considered this.
i mean, if i have the time, i definitely want a take-home final, as a student.
but in a group of 25 people, some of them won't. so it's harder as a prof.
I would prefer just because of the more relaxed environment -- assuming you start within a reasonable time frame.
if everyone has the time, they eliminate the time element. which i think is helpful.
but it's tough if it's like "you have a week to do this!" and some of your students have three finals in three other classes during that week, and some of your students don't.
if ted were here he would say that if you study properly along the way, it should not make that much of a difference, which is true.
how boring
yes. nobody should be studying, they should be at the bar with copper.hat, mortgaging their futures.
4:56 AM
I tend to get stuck on the whole "studying properly" part
I never had to study before college, so the adjustment has been really rough
well, you recognize the need for adjustment. that is good fortune. for some, that comes too late, or never.
Yeah, I guess that's true.
the pub...
math is deceptive because if you 'get it' early on, you definitely can cultivate habits that aren't sustainable, and it works for a time
the low page numbers have a lot to do with it. i don't think anyone would try to cram 1500 pages of reading into a few nights' reading, but if an exam is just on 25 pages of a math textbook, it seems like less of a bad idea to wait. it's only 25 pages!
That's so true!
5:01 AM
and maybe actually that's not true, people actually did try to cram hundreds of pages into last minute reading in law school.
studying just sucks and people don't like to do it.
Even today for my calc quiz. I waited until 30 min before to study one of the sections because my notes were only 3 pages. But it was very dense and I ended up needing to ask my professor for a clarification before the quiz.
it's only three pages!
Only three ;-;
Law school kind of sounds like a nightmare.
there are only like, what, not even 10 axioms for an order-complete ordered field? i've got this. real analysis, piece of cake.
i tend not to recommend it to others.
5:20 AM
@Fargle i think it is fairly easy to compute a solution to your problem, but it is not an explicit formua.
subdifferential calculus has got to be good for something.
it's good for domdifferential calculus. it has to be.
it involves tying up parts of one's differentials, preferably while wearing black thigh highs.
did the round trip to santa cruz today!
quite a bike ride.
or does that mean you made contact with copper jr
by car, unlike my 2 yrs younger brother, i am not up for such cycles :-). yes, he's home. for some reason he does not like the little badges i collected yesterday at the trade sho.
however, gordos went down well.
i wouldn't mind a trip to gordos.
5:31 AM
on another note, i overheard my wife talking to my daughter who had he ears pierced recently
i was in berkeley the week before the pandemic really took off and got a burrito at the cactus location in rockridge.
the line i hear was "the bigger the hoops the bigger the ho"
that cactus is better than the solano one
the solano one serves burritos for dieting moms
i remember housesitting in north berkeley and going to the solano one and it was like a different restaurant.
5:33 AM
just heads up when your daughter gets here ears pierced...
most of my female friends either have no piercings whatsoever, or quite a lot of them. there's no middle ground. i don't know what this means.
i was against ear piercing
labia is out of sight (mostly)
im joking.
we expect this. it's to be expected.
i'm fine with visible labia
my wife is very anti-piercing. i'll let her handle those conversations with my daughter.
5:36 AM
there used to be a clothing optional section behind the bleachers at the RSF
my friends would be bad influences.
i had a friend who got a long lasting infection as a result of a shady piercing
why is your wife opposed?
i honestly have no idea. but she's opposed.
i just don't like permanent body modifications
except for the chip injections
my wifi was starting to suck, so i did have to get the booster.
but yeah, other than that.
5:38 AM
whew, i read wife
of course you did.
old age, sight and all that
strangely, my son was interested in a cadence branded laptop bag
you just never know...
personally i think the "handshake approved" badge is pretty darn cool
huh. i don't know cadence except from ip litigation.
are they a real company? in my job you lose track of which companies are real and which ones just sue people a lot.
my life is cadence.
i don't work there, however, their versions of my products are still generating mullions yearly...
and they bought my company
there were two people there that inspired me, both long gone.
i have a marketing mug that was inspired by me. the words on the side of the mug are "size matters"
and i have my tempus fugit balls.
hmm, he wasn't so impressed with the siemen's hand sanitiser
it was tim cook that nixed our request to have the apple logo on our tempus fugit site (they were using our product). i'll never forgive him
we were allowed to say that apple was a customer, but not the nice logo.
nvidia let us put the logo up. i like nvidia.
and sun.
its wine time i think
i have some sun swag from something in the 90s. i forget why
and a silicongraphics frisbee
5:58 AM
the trade show (DAC) used to be a great supply of kiddie soft toys/animals.
flashlights, etc
@copper.hat when covid first hit, I remember gifs suggesting rolling back to UDP to reduce handshakes.
@Avra are you really?
@robjohn excuse the repeat post, badges from a trade show yesterday: imgur.com/Jt2PXK3
6:02 AM
Prove or disprove that $A = \left\{\frac{m}{2^n}: m, n \in \mathbb N\right\}$ is countable.

I'm pretty sure this is true if you think of this as the product of two countable sets.
@robjohn. Yes Professor. It's me :|
Passed exam with your efforts and patience
@UnderMathUate surely you can find a surjective function from $\mathbb{N}^2$ to $A$?
@copper.hat It's been a long time since I've been to an electronics trade show.
@robjohn the show yesterday (and tomorrow if I can generate the energy) was the deadest show I have ever been to
they used to be the highlight of the working year
Yeah, that was my first thought. I was just trying to find a possibly shorter work around.
6:04 AM
what could possibly be shorter?
I honestly don't know.
under: it's a subset of Q. do you know Q is countable?
it's hard to rank levels of shortness on this time scale
Ah, yeah. We proved that in class.
And on that note, I'm going to bed. I'll write it up in the morning.
so you are going to spent longer trying to find a marginally shorter proof?
copper's point of course is that if S is any set and f is any function from N^2 to S, {f(m,n): (m,n) in N^2} is countable
6:07 AM
I liked Q in TNG
speaking of countable
I see it now
i can die peacefully npw
oops pwned myself
@robjohn. What are you doing for Christmas? Hopefully not math?
6:08 AM
@UnderMathUate $(2m+1)2^n$ uniquely determines an element of $A$
I thought everyone did math for Christmas
i will be summing a turkey
@Avra math is a Christmas present, why would I not want to get another present?
sorry, consuming
6:09 AM
@robjohn You're referring to what I said about the product of two countable sets? Sorry if I'm being slow.
you are addiction guys
maths is a dangerous siren
nothing but pain
its a universal property
nothing but pain = infinite gain
infinite gain is a recipe for instability, need to keep eastern Europeans out of the starboard side of the aircraft.
@robjohn. Can you use your profile photo :|?
6:15 AM
@copper.hat I'm so tired, I literally cannot parse this sentence.
in linear control, one needs to ensure that there are no poles in the right half plane.
old, really bad joke.
that's old even to me.
but i qualify on all counts.
6:16 AM
da da da dumf
you should have your own big bang show
Alright, good night guys. Thanks for the help.
i'm too old for stage shows.
6:17 AM
been a while since i was in amsterdam...
but it used to be walking down the street you would get dragged into stage shows
even if you were with a girl...
Amesterdam is a unique city
i loved the Netherlands
i have family there now
Netherlanders are very similar to Germans
paving the driveways
6:22 AM
except that Germans can't pronounce Scheveningen properly
the paving comment was a rude remark about being Irish
Irish people are similar to what? I would never say French.
it was very non pc. it was really aimed at Lesie
i love it
i don't feel comfortable explaining it. discussing labia piercings is fine, but travelers, no....
6:24 AM
lesile you are Irish?
in spirit
my most irish friend is actually african.
leslie is definitely from United Kindgom
i have 3 close family members living there.
i think the irish are going through a major image change. i mean, the old image of leprechauns, shamrocks, guinness. horses running through council estates. toothless simpletons. people with eyebrows on their cheeks. badly tarmacked drives, in this country. men in platform shoes being arrested for bombings. lots of rocks, and beamish. i think people are saying "yes, there’s more to ireland than this."
and a brother's brother-in-law whom i say next to in high school
because nowadays, everybody knows where everybody lives
6:27 AM
@copper.hat $\sum\unicode{x1F983}$
@Avra for what?
@robjohn i am trying to interest my son in the idea of cooking a turkey for Christmas dinner
@UnderMathUate yes
@robjohn. Can I use your photo as my profile photo :|
I think he wants Indo food, but that is so passe.
it was the drink driving laws that ruined Ireland.
sorry, i was reciting something from alan partridge.
6:29 AM
@copper.hat My wife cooked our turkey breast side down, and the white meat was much moister.
@robjohn i hate to admit it, having tried many techniques, but cooking on one of those oven bags seems to be the trick.
@leslietownes don't get rid of the Guinness!
@copper.hat it was in an oven bag as well
My lifetime integrated consumption of Guinness is less than a pint.
@robjohn I want to stuff the turkey, but my wife is afraid of some microbes.
@copper.hat cook it to the required temp (165°F, I believe) and it should be okay
just realised i forgot to pick up my frozen tamales on my run to tjs an hour ago. blast.
@robjohn i agree. but i have no say when it comes to such matters.
her family regularly cooks meat to the point of dessication.
6:32 AM
@copper.hat You need to have the stuffing from inside the turkey to mix with the stuff made outside the turkey to add the flavor.
signs of pink in a steak are clear warning signs.
@copper.hat ACK! I like my steak to moo!
@robjohn i think the turkey fat adds flavour
My mom used to overcook stuff as well. I think it is cultural.
I loved my Mom's cooking, but I like it a lot less cooked too.
It might be a generational thing. My mom and her friends liked shoe leather for dinner
I only discovered spinach when I was 40+
before that i thought it was dark green slop.
god forbid, raw spinach
6:35 AM
I like raw spinach okay, but cooked spinach makes me gag, actually.
me too
My mom stopped giving it to me when I threw it up
i 'discovered' cabbage the same way
it does not have to be a sodden mass.
Now, cooked cabbage is fine with me
cooked spinach is amazing with lemon and chickpeas
6:36 AM
i like it cooked, but not to mush
sure, no mush
i only like cooked spinach in palak paneer
i love garbanzos
raw is better
they are so chic
6:37 AM
too chic for me
just hummusing along...
Have you ever went to gym guys or you prefer running, jogging?
three bean salad is fine, but they keep giving me more than three of the beans.
@Avra i'm 61. i have done all the above.
That's great. Wish you live till 1000
6:39 AM
i prefer to exercise outside.
They now are studying methods to increase life up to 200 years :|
i appreciate the wishes, but it this is what it is like at 61 i don't want 100
even young guys once they stop walking, they get heavy. That's normal :|
heavy is not the issue. wear & tear is :-)
nothing stops math
6:43 AM
@UnderMathUate did you understand the idea?
$\frac{m}{2^n}\to(2m+1)2^n$ is an injection from $A\to\mathbb{N}$
You don't even need to worry about reducing fractions.
How does it feel when God always roll unlucky dice on you?
Cormac McCarthy had something to say about that
7:23 AM
Is there a name for the space of matrices that are the sum of skew symmetric matrices and symmetric matrices of signature $(p,q)$?
7:42 AM
@Astyx I think i might be starting to understand this better. this Ismorphism you presented. Is it now between R^2 and C as fields or as vector spaces? Since we can identify them "depending on the operations acting" as such.
This function you have defined, we used " i checked my old linear algebra papers" to define R^2 as a FIELD with the vector addition and vector multiplication, then with identifying (0,1) to be the element i , we recieve the complex field.
However if we speak about them as vector spaces, then we are not using this multiplication anymore right? because we have to identify then scalar multiplication and vector addition for them.. or am i wrong?
8:03 AM
@robjohn I remember there were MathJax issues the last time they redesigned the Activity page, but it wasn't this bad.
@RandomVariable The layout is just horrible
I was just saying elsewhere that even the mobile interface has too much white space. All I can see is my reputation and badge totals. everything else requires scrolling down
@Alex hey there
@RandomVariable It is like in both interfaces, someone has decided what I want to see without scrolling, but gotten it wrong.
In the full interface (or whatever it is called currently), things also have too much space around them or frames. It eats up the available screen real estate.
I'm new to this section, it's a nice place to talk about math. I'm on my cell phone, but I can't use LaTeX :-(
Things may be "neater", but you have to scroll a lot to see anything
8:10 AM
in this room
I think it’s bit complicated to render latex in android.
@Alex is it that you cannot install ChatJax, or that you don't know LaTeX?
@Alex I know in the mobile interface, one does not see the sidebar which has the installer link for ChatJax.
@robjohn Thanks, I know LaTeX but here I cannot use it. For example I don't see the LaTeX on this line $(0,1)$.
@Alex Have you installed ChatJax?
@robjohn No :-( Just right now I'm reading about this
8:14 AM
See if that helps.
@RandomVariable Each entry has a frame around it in each panel: each answer, question, reputation, badge total, bookmark, etc has a frame around it, which adds white space as well as the frame.
This is a misuse of screen real estate, IMO
If I were to use my 4K TV as a monitor, maybe the layout might look neater, but to get enough stuff on my screen to be useful, the resolution would require me to stand a few feet from the screen to read anything.
also, the sidebar with "Summary", "Answers", "Questions", etc. links takes up a small vertical space, but it takes up horizontal space along the whole page Rather than have those set out like that, maybe a pop-up menu would be better or just make the titles to the same sections in the main part of the page should be live links.
In case anyone is wondering, I think the new layout for the Activity and Profile pages is horrid
@robjohn Even if they fix the MathJax issues, it's still going to look awful.
There is a huge white space along the left side for a small amount of data at the top of the left side.
@RandomVariable You mean the multiple lines for a single line of MathJax?
That is bad, too, but even when there is no MathJax, the layout is atrocious
Quite wasteful of screen real estate
8:29 AM
@robjohn Yes. I've seen titles spread out over like 10 lines.
@RandomVariable more than that for some that I have on my screen
As I say, it is horrible
I couldn't install LaTeX since Chrome on my phone, I'm sad about that. By the other hand, in the last part in this solution math.stackexchange.com/a/4326466/941425 is it correct? I think should be the denominator above in this fraction.
@robjohn Empty space seems to be the new design trend.
they butchered the activity page
@Alex That should not prevent installing chatjax
@BalarkaSen they butchered that and the profile page
8:45 AM
at least the top questions/answers section doesnt have massive fonts with broken TeX
but you have to scroll so far to see those
@robjohn I think I did not understand the instructions, in the link.
Here it's says that: Using with Chrome on Android

Evidently, Chrome on Android does not remember the current page when
opening a bookmark from the standard bookmark menu. The bookmark can
still be used; however, the procedure is a bit more involved than simply
opening from the bookmark menu.

Go to the Search bar and start typing the name you've given to the bookmark
(e.g. "start ChatJax"). The bookmarklet should appear in the list of items
found. Open that item and ChatJax should start running on the current page.
When you refresh the page, the MathJax issue goes away for like a split second.
Where can I find "search bar" here?
Have they increased the size of LaTeX? O_o
@Wolgwang thank you, the video in your link it helped me a lot!
Now, I suppose that $\frac{1}{2^{\frac{n-1}{n}}+\cdots+1}\cdot \frac{2^{\frac{1}{n}-1}}{2^{\frac{1}{n}-1}}=2^{\frac{1}{n}-1}$.
9:24 AM
@Alex do you not have a field to search for things (files, etc.)? (often it says search or has an icon of a magnifying glass, but not always)
you should be able to start typing the name you gave to the bookmark there, and it should show up.
@RandomVariable Yeah, like in interior design. Lots of empty space is what is desired these days. On screen, it is a waste.
@Alex Ah, so I assume you have ChatJax (or some other MathJax enabling script) installed and working?
ACK!! I just noticed that the dates have been removed from the posts on the activity page! This, probably to make room for the borders and frames.
9:41 AM
Is any of you aware of how to find a math pen pal?
Ugh! I just looked at the Answers page. The same horrible layout and space waste has been added there, too.
@robjohn Yes, now I can use ChatJax here. Thank you so much :-)
Is my observation about the answer in the link correct? I think the sequence of functions is UC 🤔
10:07 AM
@Alex see my comment to that question.
10:21 AM
$AXB$ is a semicircle and $AYB$ is some curve. Is the length of $AXB$ always greater than all curves like $AYB$ ?
@Wolgwang No. The inner curve can wiggle and have a much greater length. With constraints on the tangents to the inner curve, it might be possible.
maybe if you require it to be concave then it's ok
@robjohn If AYB is an arc of another semicircle/circle(with different diameter) then...?
@Wolgwang That would be enough
Ohk Thanks :-)
10:39 AM
Is there any proof ?
11:09 AM
@MadSpaces what do you mean by vector multiplication? $\mathbb R^2$ isn't a field per se (although it inherits a field structure from the isomorphism with $\mathbb C$)
@Wolgwang yes. note that the triangle in the wedge has $2r\sin(\theta/2)=AB$
For $AXB$, $\theta=\pi$, for $AYB$, $\theta\lt\pi$. The length of the arc is $r\theta=2r\sin(\theta/2)\frac{\theta/2}{\sin(\theta/2)}=AB\frac{\theta/2}{\sin(\theta/2)}$.
11:25 AM
Note that $\frac{\theta/2}{\sin(\theta/2)}$ decreases from $\frac\pi2$ to $1$ as $\theta$ decreases from $\pi$ to $0$
11:50 AM
@Wolgwang basically the arc length formula $\int_a^b |f'(x)|^2 \ \mathrm dx$
so if AYB is always flatter than AXB, then the f'(x) would be smaller, so the whole integral would be smaller
12:08 PM
@Astyx we can define a multiplication on R^2 like that of a complex vetor with another, then it is a field.
12:41 PM
Algebra question. Suppose we have a monoid.
Can an element exists , that behaves with all other elements just like the nutural element of that monoid but when it is paired with the nutural element it just gives itself back and not the natural element?
$ c \circ a = a = a \circ c $ but $ c \circ 1 = c = 1 \circ c \neq 1 $
I know that if it behaved like the nutural element for all elements then it is because the nutural element is clearly defined, the same as the nutural element, but what about that little twist i added? can we have such a meme?
12:53 PM
@MadSpaces consider {0,1} with multiplication
@robjohn Ah, yeah. I think I see it now.
@MadSpaces that's what I meant by the phrase between parentheses
So your isomorphism you defined, is defined on R^2 and C as fields..and the perspective multiplication on those fields.. not as vector spaces.. which have different multiplication.
Thus C and R^2 are isomorph to each other as fields as well
Btw i was able to answer my algebra question so ignore it :D
The answer "apprently" is "Yes"
The isomorphism I gave is a R-linear isomorphism, so the structure if vector space
Oh Okay.

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