Some Number Theory Problem

[fair cobalt]

Theorem 1.2 says there that there is a smallest positive integer belonging to 𝑛ℤ ∩ 𝑚ℤ and definition 1.3 says that this is the least common multiple of m and n.

But isn't the smallest positive integer EITHER m or n depending on which is smaller than the other? For example, if m < n, then m is the smallest positive integer of the set 𝑛ℤ ∩ 𝑚ℤ and if n < m then n is the smallest positive integer of the set 𝑛ℤ ∩ 𝑚ℤ.

Is that right?

[craggy comet]

if n = 2 and m = 3, then 2 is not in mZ.

[fair cobalt]

AH!

I totally missed the intersection for some reason, I see now

I was thinking union

[craggy comet]

happens to the best of us, dw

anything else?

[fair cobalt]

No, that's all for now, thanks though!

I'm going thru this book:

[craggy comet]

all the best! that's a new one for AA though

[fair cobalt]

Did you go thru it? Is it good?

[craggy comet]

nope! I use completely different books for AA.

[fair cobalt]

Please share!

[craggy comet]

Abstract Algebra; David S. Dummit, Richard M. Foote
Basic Algebra I; Nathan Jacobson
Basic Algebra II; Nathan Jacobson

and Algebra; Serge Lang as well if you don't mind a text that's a tad bit terse.

[proud estuary]

this would probably only work for nk and mi, where k, i in Z such that nk = m, where m < i

oh i didn't scroll below it

[craggy comet]

😅

[fair cobalt]

m < j?

[proud estuary]

oh sorry it's supposed to be i

i had to jumble between which letters i should use

m, n, k

or i, j, k

[fair cobalt]

Should/can this be proved? Or is it to be 'accepted' cuz it's very fundamental it seems like?

[craggy comet]

this is a generalized version of the well-ordering principle on the naturals

you can prove this using well-ordering and a clever construction

however, in ENT and basic AA, well-ordering itself is kinda a given basic axiom about N

[fair cobalt]

generalized version of the well-ordering principle

It looks like the well-ordering principle?

[craggy comet]

nope

well-ordering operates on N, not Z

the integers are not well-ordered under the usual definition of <

[fair cobalt]

What's ENT?

[craggy comet]

elementary number theory

[fair cobalt]

Oh, interesting.. that's very concise

[fair cobalt]

Because it extends to negative infinity? If I choose -9999999 as the least element then someone will say "-9999999 - 1"?

[proud estuary]

can S be countably infinite

[craggy comet]

intuitively speaking, yes. precisely worded, the integers are not bounded below.

[proud estuary]

this doesn't seem very intuitive to me because taking the even subset of the integers doesn't bound it below

[fair cobalt]

Is this theorem talking about finite sets though right? Or ?

[craggy comet]

the condition that there exists an integer m that is strictly less than all elements in S is already bounding it below!

[fair cobalt]

@craggy comet @proud estuary What does "bound below" mean?

[proud estuary]

oh it's just taking the sup{}

[craggy comet]

a set has a lower bound if there exists an element m such that $m \leq s$ for all s in S.

[proud estuary]

it has a sup{}

[mighty flume]

differential Towametry (Towa)

[proud estuary]

or min{}

[craggy comet]

inf*

[proud estuary]

depending on context they may be interchangable

yeah i have a habit of mistaking those two

[craggy comet]

sup is an upper bound

[fair cobalt]

Oh so m would be the least element if it's the lower bound?

[proud estuary]

i forgot to look at the quantifier

[craggy comet]

but no, bounded below does not imply the existence of an infimum.

but it does imply the existence of lower bounds

[craggy comet]

mm, inaccurate phrasing

[proud estuary]

no

[craggy comet]

a lower bound is not necessarily a member of the set itself

[proud estuary]

a lower bound could be anything

[craggy comet]

in fact in most circumstances it isn't

[proud estuary]

i. e. we can say m - 1 is a lower bound to m even if we suppose m is the minimum

[craggy comet]

so long as it is less than or equal to all elements in the set, it is a lower bound.

[fair cobalt]

Ah

[craggy comet]

that being said, I'm starting to see that you probably don't need two helpers at once and it's messing with my tracking, so I'll step back for now

[fair cobalt]

No, don't leave!

I appreciate you both 🙂

[proud estuary]

yeah i think the minimum exists because it's countable

although i don't think that'll suffice in the entire proof

[craggy comet]

it's still messing with my tracking and the other helper is doing fine, so

[proud estuary]

@fair cobalt ok just ping me whenever you need anything

[fair cobalt]

I didn't know what sup and inf means.. it looks like infimum and supremum?

[proud estuary]

yes sup is supremum

inf is infimum

the supremum is an alternative to the maximum

and the infimum is an alternative to the minimum

they're different because for both of these

only one may be defined for open intervals

i. e. take (0, 1)

this has neither a maximum nor a minimum

but it has a supremum and infimum

so we say sup(0, 1) = 1 and inf(0, 1) = 0

but here it's interchangable because certainly a countable subset of integers that has an element that's less than everything else in it

will be the "least element" or the minimum

although i doubt outside of finite sets that you can find a sup (or max) here

[tacit dragon]

woah what's that?

[fair cobalt]

No proofs are given to the theorems, you gotta prove everything yourself

That's the theme of the book

[tacit dragon]

woahhhhh

[fair cobalt]

I just started, I'm on the first chapter only, if anyone wants to continue along and finish this book as a co-op, hit me up

It looks like Supremum's and Infimum's are part of Real Analysis: https://www.youtube.com/watch?v=QRGIhqz9vh4

[fair cobalt]

So I need to know Supremum's and Infimum's to prove this ^ right?

[proud estuary]

nah you just need to work with minimums

i don't think you need to learn it yet for aa

because this is what it gives you

"for every n in S, there exists an m in Z such that m < n"

this is what it means to be bounded below

you can find an integer such that for every element in your set

it's less than that integer

[proud estuary]

i think it would be productive to think of this number line-wise

but remove everything except the integers

it'll just act like steps

for example suppose you were placed onto a random spot on that "integer line"

if you step forward well you're already at a spot greater than before

step backward you're a spot less than before

and you could also say this for the actual number line but

but what's different is that in the number line say you were to find the greater spots between your spot and the spot a step away

to do that you'd need a lot of very small steps forward

infinitely many ones in fact

[fair cobalt]

Shouldn't that be m < n?

[proud estuary]

should be n < m

i was jumbling between whether to use m or s

[fair cobalt]

If m is the lower bound, then m < n no?

[proud estuary]

oh i misplaced the inequality sign

yeah