I love the dog listening and hopefully learning. Great job with these videos. Min. 12:00, isn't there a problem with your defining a surjective maps? Surjective means that every element of Y is hit. The definition of a map automatically means that every element of X is mapped to some element of Y.
Thank you! I have tried to make things as concise and logically consistent as possible, based on my own route through to understanding this subject. Appreciate the support!
What do you plan to do with all this knowledge? Please sent your reply to my website, it would be great to learn more about your plans to conquer the world.
@@HughJSloanIII Send me a link to your website? I plan to make more videos! Going to cover more differential geometry up to differential forms and Lie group theory, and then use all of the differential geometry covered to do a comprehensive treatment of special and general relativity!
Excellent videos, the dog makes me feel at home instead of in a classroom , which in a strange way enhances my ability to learn. Keep it up, you’re doing great ! 🙏🏻
Injective and surjective maps can occur in any combination. Eg the map you labeled injective is, in fact injective but not surjective because not all elements in the co-domain has an 'antecedent' in the domain. The map you have listed as being surjective is indeed surjective but is not injective because each point in Y does not have a unique 'antecedent' in X. In a surjective map the range of the map is equal to the co-domain. An injective map has only the trivial (right) nullspace (kernal). A surjective map has only the trivial left nullspace.
Excellent video! One little thing to point out: A given surjection may only require a subset of the domain for the map to be classified as such. Again, magnificent video! Liked and subbed, friend.
Just thinking of some quick examples. Perhaps f(x)=cos(x),f(x)=x^3-x, and f(x)=x as examples of injective, surjective, and bijective maps, respectively.
I'm so glad I could inspire you to keep at it! It really does get easier and more intuitive the more you do you just need to find the right route and method that works for you, glad I could help!:)
Isn't the way you defined surjective maps equivalent to your definition of maps in general? I think on the surjection you meant to say that every element of the codomain is an image of at least an element of the domain.
And if we work on a set theory, based on axioms without classes, according to the Wikipedia page I found (I don't really know anything about category theory, I just trust Wikipedia) then all maps are functions so an important thing you didn't make clear is that every element of the domain gets mapped to exactly one element of the codomain.
@@aggelosgkekas3113 Yes I got the definitions messed up, only just realised lol, thankyou for pointing them out though if I ever get round to making another set theory video I'll correct it!
Maybe it was discussed before but you mentioned surjective máps can have an inverse but from what I remember an inverse can only be defined when there is a bijection. Excellent explanations tho, I got into topics I never dreamed of thanks to your videos
How do you define the set membership relation ? My understanding of a relation between two sets A and B is that a relation is a ⊆ of the Cartesian product A × B = {(a,b) | a ∈ A ∧ b ∈ B}, thereby requiring that set membership is already defined?
The teaching style and presentation is very good. Would only like to suggest this small correction though: the injective definition is that every element in co domain is mapped from at most one element in domain. By the definition you gave, there could be two elements in domain mapping to one element in co domain which is wrong.
Thankyou! Someone else also pointed out I got the surjective definition confused with the definition of maps itself, I'll admit I improvised much of this video on the spot from memory so that's why the mistakes, glad people are spotting them though! Can only pin one comment at a time but hopefully this one will also be visible. Hope you enjoy my other videos!
@@rorywhybrow8979 i am learning manifold at the moment but i couldnt find good resources as the math textbooks are too hard (since the physics textbook didnt teach much thing about manifold...)
I'm glad you liked it! Yes I plan to cover many more topics in topology + differential geometry. I'm currently very busy with my studies but I'm hoping to get a few more videos out in the new year after January exams! Thanks for watching!
@@garytzehaylau9432 I would hope so as this is what I intended! Certainly not every detail but the basics and helpful intuition are covered here. You might want to check out the lectures by Alex Flournoy as he covered a similar level of depth but over a longer course (see my featured channels page for link) Textbook wise I would recommend Carrol - Spacetime and Geometry, I find it to be a good level of maths but not bogged down in formalism, and it's meant for a physics audience - the Flournoy lectures are based around this book and were what I used when first learning GR! Enjoy!
Maestro: A small question at 6:02 you say "elements of the x set always appear in the lefthand slot and elements of the y set always appear in the righthand slot"...But this is not always true, vis: in the second list of ordered pairs, they reverse slots?
I think it's still right, if X = {a,b} , Y = {c,d} then X * Y = {(a, c), (a, d), (b, c), (b, d)}, which I wrote across two lines which might have been confusing. Regardless it should definitely always be the case that you respect the ordering of the sets in the pair!
English is not my mother tongue, but when you say "each element map to a single element", that doesn't seem to exclude the situation where two elements are mapped to one same element. Which is not injective anymore...
Also, a bijection is not "isomorphic". Without adding structures on the sets, it wouldn't really make sense to use the word "isomorphic". An isomorphism is a bijection that preserves the structure from both directions.
Just finished a masters in mathematical physics at the University of Nottingham UK! Was planning to do a Phd but decided against it as academia in general is abit rubbish and very competitive these days, but I still might end up doing one in the end!
I think the definition of "surjective" you give is slightly wrong. Surjection is not about every element of the domain mapping to at least one element of the image. Rather, surjection is about every element of the image being mapped onto from at least one element of the domain. This also makes your definition of bijection incorrect.
I love the dog listening and hopefully learning. Great job with these videos. Min. 12:00, isn't there a problem with your defining a surjective maps? Surjective means that every element of Y is hit. The definition of a map automatically means that every element of X is mapped to some element of Y.
Yes you are right thankyou, I just got the definitions mixed up! And yes she does love listening
He is waiting for the magic word ....... .... ..... ..... *WALKIES!*
I have to say I'm so grateful for the time lapses when you write a word or title you just said...
Thank you. Out of the many people that attempt to COGENTLY lecture on tensors, you have a very LUCID style, much appreciated. H. Silicon Valley
Thank you! I have tried to make things as concise and logically consistent as possible, based on my own route through to understanding this subject. Appreciate the support!
What do you plan to do with all this knowledge? Please sent your reply to my website, it would be great to learn more about your plans to conquer the world.
@@HughJSloanIII Send me a link to your website? I plan to make more videos! Going to cover more differential geometry up to differential forms and Lie group theory, and then use all of the differential geometry covered to do a comprehensive treatment of special and general relativity!
@@WHYBmaths well, this conversation went left real quick 😂
the dog is listening and seems understanding at some point which is amazing.
Elements Y i, is definitely for us geordies up in Newcastle.
Great videos! 👍🏻
Great class …. Basics covered
Also , the dog there also has a relaxing effect when brain may start to overload 😜…… Makes study more fun 😂😂
Cool! I am looking for this kind of tutorial video to learn Topology too, here I am! Helps a lot, thanks.
Excellent videos, the dog makes me feel at home instead of in a classroom , which in a strange way enhances my ability to learn. Keep it up, you’re doing great ! 🙏🏻
I swear that dog is so attentive I think that he actually understands the math in one way or another.
Injective and surjective maps can occur in any combination. Eg the map you labeled injective is, in fact injective but not surjective because not all elements in the co-domain has an 'antecedent' in the domain. The map you have listed as being surjective is indeed surjective but is not injective because each point in Y does not have a unique 'antecedent' in X. In a surjective map the range of the map is equal to the co-domain. An injective map has only the trivial (right) nullspace (kernal). A surjective map has only the trivial left nullspace.
So cool! Undergrad here just learning about this stuff!
Excellent video! One little thing to point out: A given surjection may only require a subset of the domain for the map to be classified as such. Again, magnificent video! Liked and subbed, friend.
Finally, I've been looking for something like this for a very long time! Great job!
Just thinking of some quick examples. Perhaps f(x)=cos(x),f(x)=x^3-x, and f(x)=x as examples of injective, surjective, and bijective maps, respectively.
Bravo! Excellent intro.
I'm quite happy because your explanation made me realise it's not very tough rather it is quite interesting 😀
I'm so glad I could inspire you to keep at it! It really does get easier and more intuitive the more you do you just need to find the right route and method that works for you, glad I could help!:)
Isn't the way you defined surjective maps equivalent to your definition of maps in general? I think on the surjection you meant to say that every element of the codomain is an image of at least an element of the domain.
And if we work on a set theory, based on axioms without classes, according to the Wikipedia page I found (I don't really know anything about category theory, I just trust Wikipedia) then all maps are functions so an important thing you didn't make clear is that every element of the domain gets mapped to exactly one element of the codomain.
Yeah, I think he messed that up.
@@aggelosgkekas3113 Yes I got the definitions messed up, only just realised lol, thankyou for pointing them out though if I ever get round to making another set theory video I'll correct it!
very informative and intensive thank you
Your dog is enjoying the lesson :-D
or maybe they dont know to bunk classes
Cool fractal gears in the background
Maybe it was discussed before but you mentioned surjective máps can have an inverse but from what I remember an inverse can only be defined when there is a bijection. Excellent explanations tho, I got into topics I never dreamed of thanks to your videos
How do you define the set membership relation ? My understanding of a relation between two sets A and B is that a relation is a ⊆ of the Cartesian product A × B = {(a,b) | a ∈ A ∧ b ∈ B}, thereby requiring that set membership is already defined?
oh boy. this seems scary. i hope i get trough the whole playlist.
Great videos! Succinct and clear. Could you make some videos on bundles and gauge theory?
The dog has to ve a genius by now
She's a smart puppy
Saludos desde Medellin Colombia.
thanks so much
Great lecture sir.
Great video!
Hello! Do you have the notes of this course? Thank you for the lesson.
The teaching style and presentation is very good. Would only like to suggest this small correction though: the injective definition is that every element in co domain is mapped from at most one element in domain. By the definition you gave, there could be two elements in domain mapping to one element in co domain which is wrong.
Thankyou! Someone else also pointed out I got the surjective definition confused with the definition of maps itself, I'll admit I improvised much of this video on the spot from memory so that's why the mistakes, glad people are spotting them though! Can only pin one comment at a time but hopefully this one will also be visible. Hope you enjoy my other videos!
thank for your video ! Very good explanation
could you make more video in Topology
thank you
@@rorywhybrow8979 i am learning manifold at the moment but i couldnt find good resources as the math textbooks are too hard
(since the physics textbook didnt teach much thing about manifold...)
@@rorywhybrow8979 are your videos enough for me to understand spacetime itself as 4-D manifold?THANNNKK for your quick reply
I'm glad you liked it! Yes I plan to cover many more topics in topology + differential geometry. I'm currently very busy with my studies but I'm hoping to get a few more videos out in the new year after January exams! Thanks for watching!
@@garytzehaylau9432 I would hope so as this is what I intended! Certainly not every detail but the basics and helpful intuition are covered here. You might want to check out the lectures by Alex Flournoy as he covered a similar level of depth but over a longer course (see my featured channels page for link)
Textbook wise I would recommend Carrol - Spacetime and Geometry, I find it to be a good level of maths but not bogged down in formalism, and it's meant for a physics audience - the Flournoy lectures are based around this book and were what I used when first learning GR! Enjoy!
thanks
Future Roger Penrose.
Wow that's a real compliment! He's one of my faves, saw him briefly in person when he visited my uni, amazing mathematician!
thanks a lot man
Maestro: A small question at 6:02 you say "elements of the x set always appear in the lefthand slot and elements of the y set always appear in the righthand slot"...But this is not always true, vis: in the second list of ordered pairs, they reverse slots?
I think it's still right, if X = {a,b} , Y = {c,d} then X * Y = {(a, c), (a, d), (b, c), (b, d)}, which I wrote across two lines which might have been confusing. Regardless it should definitely always be the case that you respect the ordering of the sets in the pair!
Doesn't injective map mean "different elements in the domain are mapped to different elements in the codomain"?
English is not my mother tongue, but when you say "each element map to a single element", that doesn't seem to exclude the situation where two elements are mapped to one same element. Which is not injective anymore...
Also, a bijection is not "isomorphic". Without adding structures on the sets, it wouldn't really make sense to use the word "isomorphic". An isomorphism is a bijection that preserves the structure from both directions.
lovely dog
I love your cute student, on the right❤
When it came to bijective functions, the dog lost interest.
Where are u studying ?
Just finished a masters in mathematical physics at the University of Nottingham UK! Was planning to do a Phd but decided against it as academia in general is abit rubbish and very competitive these days, but I still might end up doing one in the end!
I think the definition of "surjective" you give is slightly wrong. Surjection is not about every element of the domain mapping to at least one element of the image. Rather, surjection is about every element of the image being mapped onto from at least one element of the domain. This also makes your definition of bijection incorrect.