Supremum of a set

Supremum of a set
In this video, which is the most important video of the chapter, I define the supremum of a set of real numbers. It is like a maximum, except that it always exists, and will be super useful in the rest of our analysis adventure.
Check out my Real Numbers Playlist: • Real Numbers

[@bertrandspuzzle]

Where we're going, we won't need bounds.

[@CliffStamp]

That is an elegant way to define something like a general maximum for a set.

[@LucaIlarioCarbonini]

Definetively something I was looking for! Thanks!

[@Kdd160]

1:32 "SOUP"?? IN THE SUBTITLES LOLLLL

[@jupitersolarsystem1014]

Thank you for the clear explanation. I am a self-learner so this lectures help a lot when I do not understand a concept from textbooks.

[@valelopez5931]

please don't change. love this content

[@Na-eo1gx]

this was extremely helpful, thanks!

[@omaymaouhadi9315]

Thaaanks a lot this is such as a great explication!!!

[@imansaid2321]

The wtf part killed me

[@dgrandlapinblanc]

Ok. Thank you very much.

[@something2doTV]

Thanks, really helped :)

[@justpassingby6090]

I have a question sir : In this video you stated if M1 < 4 and S1 > M1 then 4 has to be a sup. Doesn't that mean 5 is also a sup(least upper bound) in that definition? Thanks in advance.

[@drpeyam]

5 is an upper bound but not the least upper bound, since 4 is smaller

[@justpassingby6090]

@@drpeyam , thanks for the clarification sir.I really love your videos!

[@muyangyan]

This was made on my birthday lol. Thanks for the tutorial

[@drpeyam]

Happy birthday!!! 🎂

[@zjc7353]

I love this titleXD

[@tomkerruish2982]

How do you intend to prove this as a theorem? Are you going to construct the reals as Dedekind cuts or Cauchy sequences of the rationals, or perhaps in some other fashion?

[@drpeyam]

Both

[@moshadj]

This works for any ordered set as well. But not all ordered sets have the Least Upper Bound property (the property that a bounded, non-empty set has a supremum in the containing ordered set). Consider the following subset of the rationals, the rationals who's square is lesser equal to 2. Clearly from high school math we know the sup of this set would be sqrt(2), but also from that same math class we know sqrt(2) is irrational and in particular there is no sup in the rationals even though the set is clearly non-empty and bounded.

[@moshadj]

In fact one "definition" for the real field is the "smallest" ordered field containing the rationals that has the LUB property.

[@IoT_]

I was wondering : does it work for field of the rational numbers? We could divide at half the interval infinitely many times as well but I have a feeling that it can messed up because Rationals are countable.

[@drpeyam]

Well you could define it the same way as for the reals, but it wouldn’t exist

[@IoT_]

@@drpeyam so for the interval (-inf;4) € Q there is no supremum?

[@drpeyam]

Well in this case the sup is 4. But if you take the set of rational numbers r such that r^2 < 2, then the sup doesn’t exist (in Q)

[@IoT_]

@@drpeyam this is very obvious for sure, sqrt(2) is not rational) I was talking about the interval from the video

[@foreachepsilon]

Eg. M the supremum is 4. 4 is in Q.

[@sanderneckebroeck843]

Best joke I heard for supremum... Will do it on the exam! :)

[@sonusaini-nm9xc]

Nice

[@jacobgoldman5780]

Is there a sup for lower bounds?

[@leo8879]

Yes, for a set bounded below there is the infinum, which the largest lower bound. (If you think of the supremum as the smallest upper bound)

[@drpeyam]

Yep, the inf (next video)

[@kartikraturi9888]

Wass sup my dawg

[@xavierplatiau4635]

1:00 You were right to say « non empty bounded subset of R » For exemple R which is not bounded has no sup. So I wonder, will you be talking about the extended real line in which every subset has an inf and a sup one day? That would be amazing !

[@pichass9337]

Are you talking about the projective line?

[@xavierplatiau4635]

I’m talking about that : en.m.wikipedia.org/wiki/Extended_real_number_line I’m French, we call it « R barre », no idea how it’s commonly called in English !

[@drpeyam]

Yeah, I will actually!

[@SartajKhan-jg3nz]

Say we have a set S, and X is less or equal to the Sup(S). If we can show that, if Y is an element of S greater than X, X=Y then does that imply X=Y=sup(S)?

[@IoT_]

If you say that Y is strictly greater than X ,it means that X cannot be sup(S) and it means that X is NOT less or equal but just less than sup(S).

[@xavierplatiau4635]

Ok let’s start : Let S be a non empty bounded subset of R. Let x be in S such that for all y, x== sup(S) to prove x=sup(S) Let y be in S, either y =< x or x =< y and x=y and so y =< x So x is an upper bound of S and so sup(S) =< x as sup(S) is the min of the upper bound. So x = sup(S)

[@IoT_]

@@xavierplatiau4635 It's true but the comment was about that y just greater than x, not great or equal)

[@SartajKhan-jg3nz]

@@xavierplatiau4635 Thanksss got it

[@raydencreed1524]

If, for a given set, we define a new set of upper bounds, could we define the supremum as the minimum of that set instead?

[@drpeyam]

Yes because it’s always attained. That’s a really nice observation

[@GabrielRamirez-sd9fz]

Dr.Peyam could you explain some theory of Gauss Eliminitation please. From Perú 😃

[@drpeyam]

kzbin.info/aero/PLJb1qAQIrmmDBodVKfa0qmXmZwvzN4hx7

[@GabrielRamirez-sd9fz]

Thanks for the video. I am huge fan of your work 💪🏽

[@bjarnivalur6330]

Not much, how 'bout you? ... ok, now to watching the video.

[@foreachepsilon]

Why might it be reductive to say “supremum is a maximum with a built-in limit”?

[@drpeyam]

Well, you need supremum to define limits, so it’s a bit circular

[@sanelprtenjaca9776]

Let S = [3, 4). Pick 3.9: there is always bigger number then 3.9: 3.99 < 3.999 < ... Pick any number a in S, there is always b in S such that b > a (for example b = (a + 4)/2). And sup(S) = 4? If yes, then this definiton is obvious to me :)

[@Kdd160]

Actually in my opinion we should write it to be LIM as x->inf of 4-(1/x) (JUST KIDDING BRO, RELAX :)))))

[@drpeyam]

Yes, sup(S) = 4

[@deera_phina]

@@drpeyam Are we simply saying that the supremum is the largest number in the set?

[@joefuentes2977]

Why isnt supremeum defined like for all elements x in S then x

[@drpeyam]

Because then there are many M that satisfy this if x

[@drpeyam]

Sup is the least upper bound, the smallest one of all the M

[@joefuentes2977]

😄 thanks! That makes sense!

[@laviekolchinsky9441]

Could you still say the maximum is 4-epsilon?

[@laviekolchinsky9441]

(minus)

[@rabindranathghosh31]

@@laviekolchinsky9441 the maximum could be defined as the limit as epsilon goes to zero of 4-epsilon. But here epsilon is a variable but maximum is a constant. I think that's the issue here.

[@beatoriche7301]

Not really - I get the intuition of wanting to grasp this open interval as ending infinitesimally close to its endpoint, but the concept of an open interval really refers to all real numbers strictly between the two endpoints (without the endpoints themselves). Even if you introduce infinitesimals (which do not exist in the standard real numbers) via the hyperreals, subtracting an infinitesimal _dx_ from _4_ would make the number smaller than just a regular _4_ - so _4 - dx_ would, by definition, have to be in the interval. But you could do the same thing with _4 - dx/2,_ _4 - dx/3,_ and so on and so forth precisely because every hyperreal number has infinitely many hyperreal numbers infinitesimally close to it. In other words, the maximum of this set cannot exist - even if you allow the use of infinitesimals.

[@Kdd160]

We can write it to be LIM as x->0 of 4-x (RELAX, JUST KIDDING :} )

[@sugarfrosted2005]

ClUB results in puns though.

[@rogerkearns8094]

Wasn't Bugs Bunny always asking you about that, Doc?

[@propleyer2867]

ahaaa I am bigger than you hhhhhhh . thanks sir for this clear explanation

[@preetcharan]

He took year (Sup), but we aim to complete the understanding in 13 minutes (Inf). lol

[@drpeyam]

Haha

[@janouglaeser8049]

Dr. Peyam, I sent you an email on April 20 (the subject was: "[Proof!] All solutions for f'=ffffffffffff"). I would greatly appreciate if you could take a look. Sorry if I'm being annoying.

[@adfr1806]

Ez

[@TheMazyProduction]

Not much, how are you?

[@ajiwibowo8736]

Sup bro! How r u?

[@drpeyam]

Sup

[@DanieleCapellini]

not much, hbu?

[@ChienChiWang]

want to find as WTF 😂

[@jacobgoldman5780]

I don't think you know what WTF means Peyam

[@Kdd160]

Everyone knows; he used that in one of the videos in which he calculated the sum of 1/(x^2+1)

[@drpeyam]

Want to find, what else could it possibly mean? 😝

[@iabervon]

@@drpeyam Well, if someone gives you a weird expression and asks you for a limit, you say "WTF?" You need to know the domain, so your first thought should be "What's The Function?" (according to 3blue1brown)

[@shayanmoosavi9139]

@@drpeyam it could also mean "what's the force?" according to nick lucid :)) (I hope you've heard of him. He's awesome.)