These videos are awesome, thanks so much! I got a book on real analysis, but it was waaaayyy too terse to be used as a learning tool on its own. These are really helpful!
Given the definition, it's very natural. If you search for hyperreal numbers, then the maximal and minimum of an open subset of the hyperreal numbers is just the supremum of the real - infinitesimal for the maximum and infimum of the real + infinitesimal for the minimum, if you like to think in that way, the supremum is the standard part of the maximum of an "open" interval.
Why is the infimum of an empty set is equal to positive infinity? And supremum of an empty set is equal to negative infinity? Is there anything hidden here?
Good question. Of course, it is just a definition. However, it is a useful one because it guarantees that the supremum does not get smaller when you add points to a set.
For the definition of supremum of M, is it important to say that the real S is greater or equal to any X of M ? Instead, could it be "S is strictly greater than any X of M" ? Or it is a kind of convention, so a supremum can also be a maximum if the the real S is part of M ? Also, for the 2nd point definiting the supremum of M, could we say instead "For any E greater than 0, S - E belongs to M" ? I feel like it could work the same way, without using x tild as an intermediate. What do you think?
I'm back here to answer my own 2nd question It is better to use an x~ because then we dont have to care that the precise number S-E is inside M. Maybe M is not a continous line but a collection of points. Then it is possible that S-E is not in the set M. But S-E would still be interesting if you compare it to another point of M that is greater than it. That's why x~ is interesting, as it allows E to be any value greater than zero, and still be used to define the supremum
@@brightsideofmaths , since sup of an empty set is minus infinity, and inf of an empty set is plus infinity, then (at least heuristically) we can say that for the empty set inf is greater than sup. This would be avoided if inf and sup of the empty set were left undefined, I guess.
The way i learned in my book (which is quite old) is the following definition: x is called a supremum of a set M if it it is true that: 1) x is an upper bound; 2) for all upper bounds y, x is smaller or equal to y. The infimum is similarly defined, and upper/lower bounds has the same definitions as in your video. Is this definition of supremum and infimum still used today, or is it "outdated"? And also, great video! Thanks so much!
In the definition of supremum, you said that all "x" in "M" are such that "x" is less than or equal to "s" that means "x" can be equal to "s" and "s" may be the part of "M" but "s" isn't part of "M" . I thought definition of supremum should be all "x" in "M" are such that "x" should be less than "s".
@@brightsideofmaths In which case supremum could be maximum?Can you give any example set. For example- In set [1,3] 3 is the maximum and in set (1,3) 3 is the supremum like these.
I would use ]1, 2] to denote your (1, 2]. That means using [,] as a closed subset, ], [ as an open subset, and choosing that for each borders It is most of the time similar, but it is easier when you want to distinct (1, 2) and ]1, 2[
Of course, use another notation if it fits your style and helps you! However, my notation here is also very common and therefore one needs to understand it as well.
I didn't know (or didn't remember) that definition for the supremum/infimum of the empty set, interesting!
These videos are awesome, thanks so much! I got a book on real analysis, but it was waaaayyy too terse to be used as a learning tool on its own. These are really helpful!
Thank you :)
Thank you so much for making these videos!
Glad you like them!
thanks a lot!
Thank you it was very useful 😊
Thank you!
Thank you so much. Great explanation!!!
Thank you very much :)
Sir ...what are the supremum and infimum of (0,1) intersection {m+n√2 : m,n are integers}?
Very nice and clear explanation!
Glad you liked it
Supremum? More like "This is free, man!" Thanks for making and sharing these amazing videos.
Bonjour !😎
It's me again. Another little difference! In France, we write ]a,b] instead of (a,b].
Yes, this is common notation and I also like it.
another reason to hate france
Unhinged notation😮 but it makes sense
It always blows my mind that open intervals don't have maximal / minimum elements. weird stuff
Given the definition, it's very natural. If you search for hyperreal numbers, then the maximal and minimum of an open subset of the hyperreal numbers is just the supremum of the real - infinitesimal for the maximum and infimum of the real + infinitesimal for the minimum, if you like to think in that way, the supremum is the standard part of the maximum of an "open" interval.
beautifully presented
Thank you! Cheers!
Thank you so much that was extremely helpful !
thankyou!!
You're welcome!
you are my sunshine❤TY for your videos good❤
In the final example, is s - epsilon not an upper bound because it’s the maximal element?
Be careful: A maximal element is always an upper bound by definition.
Why is the infimum of an empty set is equal to positive infinity? And supremum of an empty set is equal to negative infinity? Is there anything hidden here?
Good question. Of course, it is just a definition. However, it is a useful one because it guarantees that the supremum does not get smaller when you add points to a set.
For the definition of supremum of M, is it important to say that the real S is greater or equal to any X of M ? Instead, could it be "S is strictly greater than any X of M" ? Or it is a kind of convention, so a supremum can also be a maximum if the the real S is part of M ?
Also, for the 2nd point definiting the supremum of M, could we say instead "For any E greater than 0, S - E belongs to M" ? I feel like it could work the same way, without using x tild as an intermediate. What do you think?
I'm back here to answer my own 2nd question
It is better to use an x~ because then we dont have to care that the precise number S-E is inside M.
Maybe M is not a continous line but a collection of points. Then it is possible that S-E is not in the set M. But S-E would still be interesting if you compare it to another point of M that is greater than it. That's why x~ is interesting, as it allows E to be any value greater than zero, and still be used to define the supremum
danke:]]
Do you have an explanation on your channel about the topic of quantum numbers q-binolmial?
q-derivative etc ?
Not yet, sorry!
But then it means phi is such a set for whom sup
Yes, for the empty set, this inequality is not fulfilled :)
Thank you very much sir.
How can the inf of a set be greater than the set's sup?
No this can't happen. Why do you think that?
@@brightsideofmaths , since sup of an empty set is minus infinity, and inf of an empty set is plus infinity, then (at least heuristically) we can say that for the empty set inf is greater than sup. This would be avoided if inf and sup of the empty set were left undefined, I guess.
@@MrScattterbrain Yeah, you are right. The empty set is an exception here. Nevertheless, the definition is very helpful in a lot of contexts.
If a number is excluded from a set such as (2,5] then how can 2 be the infimum of a set?
By definition of the infinum: inf and sup are always well-defined real numbers. That's the difference to the minimum and maximum :)
The infimum doesn't need to be in the set
5:53 two properties of sup
7:20 sup of empty set
Can I use your bookmarks as the official video bookmarks? :)
@@brightsideofmaths Yes!sure! I am glad to help!
@@qiaohuizhou6960 Of course, I am always if one helps with bookmarks, subtitles and so on. I am not always able to do everything myself :)
The way i learned in my book (which is quite old) is the following definition: x is called a supremum of a set M if it it is true that: 1) x is an upper bound; 2) for all upper bounds y, x is smaller or equal to y. The infimum is similarly defined, and upper/lower bounds has the same definitions as in your video.
Is this definition of supremum and infimum still used today, or is it "outdated"?
And also, great video! Thanks so much!
Yes, it's the same thing :)
Sir what is x (tilde)??
Just a name for a number :)
love supwemum and weal numbers! so gweat
:D Don't make fun of people with speech disorder!
In the definition of supremum, you said that all "x" in "M" are such that "x" is less than or equal to "s" that means "x" can be equal to "s" and "s" may be the part of "M" but "s" isn't part of "M" . I thought definition of supremum should be all "x" in "M" are such that "x" should be less than "s".
s could be an element of M.
@@brightsideofmaths if s is the element of M then it should be called as maximum and not the supremum.
@@dhirajghadage4342 The supremum could also be a maximum. This doesn't invalidate the properties of the supremum.
@@brightsideofmaths In which case supremum could be maximum?Can you give any example set. For example- In set [1,3] 3 is the maximum and in set (1,3) 3 is the supremum like these.
@@dhirajghadage4342 For the set [1,3], the point 3 is the supremum (and also the maximum).
I would use ]1, 2] to denote your (1, 2].
That means using [,] as a closed subset, ], [ as an open subset, and choosing that for each borders
It is most of the time similar, but it is easier when you want to distinct (1, 2) and ]1, 2[
Of course, use another notation if it fits your style and helps you! However, my notation here is also very common and therefore one needs to understand it as well.
Is supremum the same thing as least upper bound?
Yes!
Sir did u learn all these things from Book
100% teaching level❤
Thanks! I studied mathematics for many years :)
@@brightsideofmaths that's why so
Woaw just woaw..
Kannst bitte mehr deutsche Videos machen :( das ist genau mein Thema und ich versteh kein Wort
Deutsche Videos veröffentliche ich dann auf dem deutschen Channel.
s - ε < x
Spell out something familiar
This is like that in the video :)