Reading about indexed sets in my analysis textbook was making no sense at all so I came here and with your demonstrations the concept clicked immediately; thank you!
Thanks so much for clearly stating what it is that your indexing - “finite set”; countable infinite set; uncountable infinite set etc.! The lack of explanation of this particular detail in the book is what made something this simple confusing.
This is a thousand times better than whatever was going on in my book It’s like this… Definition: long, drawn out, bland pages with no points of emphasis Example: The most basic or complex thing you’ve seen regarding the essay you just read above in the definition. Practice: Equivalent of trying to hit a home run against an MLB-level pitcher throwing you a fastball
First, let me tell you that I enjoy your work quite a lot. You're doing a great job. Second, please allow me to point out that if your indexing set is empty the corresponding intersection cannot be a set as it would actually be the (proper) class of all sets. Keep up the great work!
@@angelogandolfo4174 Everything is true for every member of the empty set. In particular, for every set A, we have that A is an element of B for every B an element of Ø. Why? No possibility of a counterexample! Thus, every set A is an element of the intersection over the elements of Ø, and so said intersection must actually be V, the (proper) class of all sets.
@@angelogandolfo4174 No, it can't. For reference you can look that up here: en.wikipedia.org/wiki/Intersection_(set_theory)#Nullary_intersection There the set of all sets is being referred to as the "universal set".
you can do it yourself. the first containment (the union inside integers) is because every A_i is contained in Z. the second containment is because: given z an integer then either z or -z is positive (the case for zero is simple) so it is in A_|z| and so
Apologies for simple question. In the beginning of the video, when talking about the union/intersection of the index sets, why are we switching the subscript from I to j inside of the "such that" statement? Wouldn't i = j if we're looking at the entirety of the index set?
As i understood it, A1,A2,A3 can be all unrelated sets, meaning that they are not necessarily subsets of a bigger A, that their union makes that bigger A. is this correct? I mean instead of writing A1,A2,A3 I could write also sets A,B,C?
@@parnashri_wankhede The union is easier to explain. The union of a family of sets consists of those elements which belong to at least one of the sets in the family. If the family of sets is empty, then there are no sets in it, and thus no element can be an element of a set in the family. Consequently, the union is the empty set. The intersection is trickier. The intersection of a family of sets consists of those elements which are in every set in the family. If the family is empty, then (vacuously) every element is in every set in the family, and thus the intersection is the universal class. Another way to look at it is that an element _isn't_ in the intersection iff we can find a set in the family to which the element _doesn't_ belong; since we can't find such a set in an empty family (there being no sets, period), _everything_ is an element of the intersection. Note: I'm considering "pure" set theory, in which every object is itself a set; that's why I'm equating "the class of all elements" with "the class of all sets".
Reading about indexed sets in my analysis textbook was making no sense at all so I came here and with your demonstrations the concept clicked immediately; thank you!
Short + Sharp = Perfect Explanation => Great Teaching Techniques
Thanks so much for clearly stating what it is that your indexing - “finite set”; countable infinite set; uncountable infinite set etc.! The lack of explanation of this particular detail in the book is what made something this simple confusing.
Very clear examples. Thank-you.
Man you're helping me so much with abstract mathematics
Great stuff. This information is essential for so many proofs! 😃
This is a thousand times better than whatever was going on in my book
It’s like this…
Definition: long, drawn out, bland pages with no points of emphasis
Example: The most basic or complex thing you’ve seen regarding the essay you just read above in the definition.
Practice: Equivalent of trying to hit a home run against an MLB-level pitcher throwing you a fastball
Flashing back to my Intro to Topology class.
Thank you
First, let me tell you that I enjoy your work quite a lot. You're doing a great job.
Second, please allow me to point out that if your indexing set is empty the corresponding intersection cannot be a set as it would actually be the (proper) class of all sets.
Keep up the great work!
Beat me to it.
Are you sure about that? If the indexing set is empty, the corresponding intersection could be set arbitrarily, couldn’t it?
@@angelogandolfo4174 Everything is true for every member of the empty set. In particular, for every set A, we have that A is an element of B for every B an element of Ø. Why? No possibility of a counterexample! Thus, every set A is an element of the intersection over the elements of Ø, and so said intersection must actually be V, the (proper) class of all sets.
@@angelogandolfo4174 No, it can't. For reference you can look that up here: en.wikipedia.org/wiki/Intersection_(set_theory)#Nullary_intersection
There the set of all sets is being referred to as the "universal set".
@@tomkerruish2982
Yes, my bad, I get it now doh!
Thank you so much for everything. I have learned a lot.
4:59 I’m curious to see the proof of that... I can’t wait for your video then!
you can do it yourself.
the first containment (the union inside integers) is because every A_i is contained in Z. the second containment is because: given z an integer then either z or -z is positive (the case for zero is simple) so it is in A_|z| and so
Thanks teacher
Thank you so much 💌
Thank youu! I was quite confused about this notion but you explained it really well
This was it? I was going crazy trying to comprehend this, now I just GET IT.
You are doing an excellent job!!!
Best explanation
My textbook was making too hard going of the subject. Thank you for a concise intro to the subject matter!
Thanks man, I always get confused with indexing sets and families
it was really useful video cleared all my doubts regarding index set
Keep them going!
Great video! Thanks!
Example 1: there is i={1,2,3}. Shouldn't it be I instead of i?
Amazing video!
good video master Penn keep it up
Apologies for simple question.
In the beginning of the video, when talking about the union/intersection of the index sets, why are we switching the subscript from I to j inside of the "such that" statement?
Wouldn't i = j if we're looking at the entirety of the index set?
As i understood it, A1,A2,A3 can be all unrelated sets, meaning that they are not necessarily subsets of a bigger A, that their union makes that bigger A. is this correct? I mean instead of writing A1,A2,A3 I could write also sets A,B,C?
Pretty awesome!
Thanke you for ur Nice Course!
9:36
Ahhh amazing!!
great job very clear sir, 😘😘
Thanks :)
Imma claiming at 3:24 it should be {0, 1, 2}, no?
Really great! But why always stop when things are starting to fly?
Please explain the case
when
index set is empty ?
@@iang0th i don't think so , google the result
@@iang0th wrong！！！！！，it is a proper class of all sets.
The union is the empty set. The intersection is the proper class of all sets. Can't explain now, it's been a long day.
@@tomkerruish2982 Can you please explain?
@@parnashri_wankhede The union is easier to explain. The union of a family of sets consists of those elements which belong to at least one of the sets in the family. If the family of sets is empty, then there are no sets in it, and thus no element can be an element of a set in the family. Consequently, the union is the empty set.
The intersection is trickier. The intersection of a family of sets consists of those elements which are in every set in the family. If the family is empty, then (vacuously) every element is in every set in the family, and thus the intersection is the universal class. Another way to look at it is that an element _isn't_ in the intersection iff we can find a set in the family to which the element _doesn't_ belong; since we can't find such a set in an empty family (there being no sets, period), _everything_ is an element of the intersection.
Note: I'm considering "pure" set theory, in which every object is itself a set; that's why I'm equating "the class of all elements" with "the class of all sets".
7:54 waiting for A Scorpion
buff Neil Patrick Harris teaches Discrete Mathematics
I guess you will talk about topology, compact set, etc so you need some set theory.
❤❤
Hi,
For fun:
2 "so let's go ahead and",
1 "and so on and so forth".
Can u help me?
Gosh this awfully sounds like programming...
auto x = std::indexed_set{ - 4, -2, 0, 2, 4 };
UAI? Como assim!
wwoowww!!
Thank you