Mathematics

they're all 3 logically equivalent, so it doesn't matter which you assume as an axiom

no mathematician would interpret "field" as a binary operation

math is a positive thing

It's nice to see the chat active, even if I totally ignore whatever is happening

just don't use math frak

if this post also gets downvotes, then what will happen? I wonder.

There is no reason for this example to be so complicated

imaging someone saying ooh to do 123454321 by 36 divide 123 by 36 first hahahaha like bro

@LuckyChouhan are you annoying whole day

@Shaun If you run into John Fraser, say "Hi" for me.

ignore this chat.stackexchange.com/transcript/message/65642334#65642334

(Telling another person that they have misunderstood is an invitation to further conversation.)

surely someone around here has an academic credential that can grab 4.10

I wish all of category theory to be a typo

the SE platform has an automated way of dealing with that

someone is serial upvoting my answers

I don't know, the star system is kinda useless anyway so the stars just indicate something that's relevant here and should be read

as for the proof you might have noticed already that $B((x, y), r) = B(x, r)\times B(y, r)$. Moreoever if $(t, z)\in B(x, r)\times B(y, s)$ then there exists $r_0$ and $s_0$ with $B(t, r_0)\subseteq B(x, r)$ and $B(z, s_0)\subseteq B(y, s)$. By replacing $r_0$ and $s_0$ with $\min(r_0, s_0)$ we can assume $r_0 = s_0$. Thus $B((t, z), r_0)\subseteq B(x, r)\times B(y, s)$

the "d(x_2, y_2)" issue is maybe the outstanding issue. is that a typo in your source, or is it your typo? please note that rigorous type checking of the problem statement would reveal that we do not have a given way of computing "d" between an element of X and an element of Y

i now click into the history of this question and note that "max" became "min" in an edit. if it was originally "max" it should have stayed "max," unless, again, this is something in a source you are drawing from, and not your own creation (in which case i would mention that)

[the putative typo in the formula defining your putative metric on the product space is that the second thing in the 'max' should probably be d(y_1, y_2). in the current context, there is no reason why "d(x_2, y_2)" would have any meaning because it purports to compute "d" between elements of different spaces]

@ThomasFinley have you checked if that min metric satisfies triangular inequality or not?

horrible notational practice to use the same letter, "d", for three different metrics, as you seem to acknowledge by changing the name of the metric on X x Y. if that is the source's fault and not yours i would mention it. a lot of people will skip right by a question like this if they can't tell whether it is a question about an unreliable source (like a textbook that has a typo in it), or a question from an unreliable source (i.e. an OP who makes typos a lot)

maybe try "max" instead of "min." try stating the source of the problem so people can check which errors are yours and which errors are in the source.

there are typos all over that question and it is too long, as usual. the question is whether that formula, or that formula redefined so that it involves y_1 in an obvious way, defines a metric. it doesnt.

Coming up with "hard problems" is not particularly hard. The goal should be to come up with interesting, well-motivated problems.

@shintuku We all suck at something until we give a try

We are all pseudo-anonymous semi-equals here.

you are making a lot of work out of something straightforward.

iff A, tthen B.

@XanderHenderson I think copper.hat would consider $-\frac1{12}$ to be a prime number.

Definitions can be more or less useful, but I don't like the idea of a definition being correct.

Amann Escher's first album, with some of the psychedelic guitar stuff and stockhausen inspired tape experiments, was their best work

@XanderHenderson 'cause there's no ale.

@uhoh I believe this, but others may, or do not. This is the age old question of "inventing" and "discovering" that many people, even experts, disagree with each other

If you have a proof of a theorem, you have to learn to read it, word by word and line by line.

@jasmine If $a'\in \mathfrak{m}$, that doesn't necessarily mean $a'\in \mathfrak{m}^3$. It's $\mathfrak{m}^3\subseteq \mathfrak{m}$ that's true, not necessarily the other way around

@Shaun no need to take a look. We found basic errors... -__- anyhow back on my ArrowGlue code for now, and I'll put twin primes on pause. It takes a lot of mental energy to work on it and doing it all day just isn't productive ^_^

ah, that's a cute proof

What is going on with all the stars?!

hey chat. wanted to share here: i finally got my BSc degree in math, and now I'm enrolled in the MSc math program!

@RyderRude from the post: "The difference between a trick and a method is understanding."

@leslietownes that shows how radical left ideals can be annihilating

@Shaun what's up man?

(Thank goodness that she is very supportive. I can't tell y'all how happy I am to have good administrators.)

@Jakobian when earth's crust had partially solidified just enough, and thermal and electric conditions were such that random chance created amino acids that became encased in cell-like containers, these somehow congealed together into the first edition of "multivariable mathematics."

is the set of points in $\Bbb R^3$ with exactly one rational coordinate connected?

@Peter it is not nonsensical. it comes out of Einstein's equations