these lectures seem to be from early 2000 .... how do i know?... see professor's belt .. there is mobile carrier that was very famous in that time ... ... how amazing .. these all students must be doing really great in their life today .. some of them might be teaching in some IITs ...
Not true. That belt thing was famous even in early 2012 and all. And Hell No! It can't be from early 2000, coz just see the video quality of other NPTEL lectures that are from that decade, it's not that good. So this series has to be recorded after 2010 or so.
Following the lecture series from 1st of the July. Completed the third, today. This lecture was a bliss to eyes. Enjoying pure mathematics at its peak. Teaching was tranquil and continuous like the flow of water in a fall. Thank you professor!
@@shashvatshukla I am on lecture 4. Hopefully I finish it. I have never finished Gilbert Strang> I kept watching individual part as need arise. I may go watch it all over.
I think maximal elements are those elements in a set which are unable to establish the relation with other elements. A set can have one or more maximal elements and they dont have to be at the end of a set they can be anywhere. Lets have an example Let Set A={1,2,3,4,5,7,8,9,16,18,32,64} be a POSET under relation R which we will chose x~y such that y/x i.e x divides y (x and y belongs to A where x is any first element into consideration and y is any next another element into consideration where y can be same element too). But why set A is POSET under relation y/x Since for any x,y,z in set A relation y/x is 1.reflexive I.e if y=x then x is divisible by x (i.e x/x) ex.1/1 , 2/2 , 4/4 etc 2.transitive I.e if y/x and z/y then z/x ex. If 4/2 , 8/4 then 8/2 3.Anti-Symmetric I.e. if y/x then also x/y which is only possible when y=x ex. 2/1 but 1 is not divisible by 2 But if x,y=4 then y/x = 4/4 and x/y = 4/4. Now lets find out maximal elements in set A Look at elements 5,7,18,64 these are maximal elements of set A because they do not divide even single element other than themselves. While 1,2,3,4,8,9,16,32 atleast divide one element in set A thats why they are not maximal elements of set A. Now these maximal elements are maximal elements under relation y/x if we change the relation maximal elements also changes. Now lets talk about upper bound for same set A Here 64 is upper bound of set A because every element of set A is ≤64 and upper bound is independent of relation i.e even if we change the relation still 64 remains the upper bound of set A.
Not just 8 and 9, 6 and 7 are maximal elements as well. Why? Because an element is maximal if it is bigger than any number it can be compared with. 10 can be compared with, you guessed it right, all it's gcd(greatest common divisors) which are 1,2,5 and since 10 is the largest, it is the maximal element. Similar logic can be applied to find that 6,7,8, and 9 are maximal elements as well.
Watched a couple of videos. Pretty informative and well-paced. Only that the professor sometimes picks up a less-useful example and, realising it, drops it midway only to never touch it again. For a student taking notes, these parts are mildly irritating as even the littlest of point made in these little examples, when missed, disturb the conceptual clarity needed for the upcoming main concepts.
these lectures seem to be from early 2000 .... how do i know?... see professor's belt .. there is mobile carrier that was very famous in that time ... ... how amazing .. these all students must be doing really great in their life today .. some of them might be teaching in some IITs ...
Not true. That belt thing was famous even in early 2012 and all. And Hell No! It can't be from early 2000, coz just see the video quality of other NPTEL lectures that are from that decade, it's not that good. So this series has to be recorded after 2010 or so.
Following the lecture series from 1st of the July. Completed the third, today. This lecture was a bliss to eyes. Enjoying pure mathematics at its peak. Teaching was tranquil and continuous like the flow of water in a fall. Thank you professor!
Same bro i also started on 1 july.....
Bro, It's my first time studying analysis. Are you following rudin's book, with it?
@@ashutoshsharma2394 have u completed ?
Got a clearer understanding of Scröder-Bernstein Theorem from this lecture, when Sir mentions a mapping form A to C U (A\B) .
I've watched 3 lectures so far, lets see how much of this course I can finish before I procrastinate it into oblivion.
Did you finish it?
@@jbm5195 I got pretty far! but not all the way :P Though I did watch all of Gil Strang's Linear Algebra
@@shashvatshukla I am on lecture 4. Hopefully I finish it. I have never finished Gilbert Strang> I kept watching individual part as need arise. I may go watch it all over.
Amazing professor. Greetings from Perú
Very beautiful explanation in a
very systematic way
Even Prof.Balki taught in this same class where Prof.Kulkarni is teaching !! ^-^
That was my thought too the very first time I saw that
very nice lectures sir... Thank u sir..
sir @13:00 , will the set (X,
yes
1 is the minimal element as well as g.l.b.
does this course correlates to real analysis which is there in upsc cse maths optional?
yes it is. tho, to me, the first 4 lectures are the most difficult and they are all just elementary set theory. Real Analysis really starts from lec6
@@oscarobioha595 !!ط
Made it to lecture 3!!!
i still dont understand diffrence between maximal number and upper bound...can u please suggest me
maximal element always belong to that set while an upper bound may not be an element of that set
eg [0,1) has no maximal element (because for any number you give me, i can find a bigger number in the set) but it has an upper bound of 1.
Shashvat Shukla yeah u r right
I think maximal elements are those elements in a set which are unable to establish the relation with other elements.
A set can have one or more maximal elements and they dont have to be at the end of a set they can be anywhere.
Lets have an example
Let Set A={1,2,3,4,5,7,8,9,16,18,32,64} be a POSET under relation R which we will chose x~y such that y/x i.e x divides y (x and y belongs to A where x is any first element into consideration and y is any next another element into consideration where y can be same element too).
But why set A is POSET under relation y/x
Since for any x,y,z in set A relation y/x is
1.reflexive
I.e if y=x then x is divisible by x (i.e x/x) ex.1/1 , 2/2 , 4/4 etc
2.transitive
I.e if y/x and z/y then z/x
ex. If 4/2 , 8/4 then 8/2
3.Anti-Symmetric
I.e. if y/x then also x/y which is only possible when y=x
ex. 2/1 but 1 is not divisible by 2
But if x,y=4 then y/x = 4/4 and x/y = 4/4.
Now lets find out maximal elements in set A
Look at elements 5,7,18,64 these are maximal elements of set A because they do not divide even single element other than themselves.
While 1,2,3,4,8,9,16,32 atleast divide one element in set A thats why they are not maximal elements of set A.
Now these maximal elements are maximal elements under relation y/x if we change the relation maximal elements also changes.
Now lets talk about upper bound for same set A
Here 64 is upper bound of set A because every element of set A is ≤64 and upper bound is independent of relation i.e even if we change the relation still 64 remains the upper bound of set A.
Mathematics is the best.
Hello Sir, I want to know that is there a mistake at 6:19 where u said p< or equal to m in lub. isnt it m
no ... p is least upper bound then it must be least in all the upper bounds
Thanks sir lecture are very interesting.
Sir at 13:26 why 9 and 8 are maxi. Elements i didn't get it :'(
Not just 8 and 9, 6 and 7 are maximal elements as well. Why? Because an element is maximal if it is bigger than any number it can be compared with. 10 can be compared with, you guessed it right, all it's gcd(greatest common divisors) which are 1,2,5 and since 10 is the largest, it is the maximal element. Similar logic can be applied to find that 6,7,8, and 9 are maximal elements as well.
The key is that, in the particular example, the a
it took me a while to understand tbh
@@dhirajupadhyay01 thanks dhiraj
@@thisbobful pls elaborate more specifically
19:15. "When does a POSET have a maximal element?" Zorn's Lemma. Say hi to your classmates 13:15
Best class
Superb
Very nice lecture.
...
Is this course for MSc or BSc?? Please reply.
amazing lecture
:) good lectures!
Watched a couple of videos. Pretty informative and well-paced. Only that the professor sometimes picks up a less-useful example and, realising it, drops it midway only to never touch it again. For a student taking notes, these parts are mildly irritating as even the littlest of point made in these little examples, when missed, disturb the conceptual clarity needed for the upcoming main concepts.
Really nice sir
Good
nm - nC1(n-1)m + nC2(n-2)m - nC3(n-3)m+….- nCn-1 (1)m. Please give me proof sir
good lecture :)
hg
Hood
not so good
IIT walo ko padha rhe the, aapke liye nhi banaya tha🥺
Explanation is not that good