I tend to have major gripes with this particular subtopic of set theory, because in my experience, there is no agreement among mathematcians in their usage of terminology or in how they define these concepts, often using the words "function" and "map" interchangeably, while other times using both words to have different meanings. This inconsistency is what makes it difficult for teachers to teach the concept correctly, and therefore, what makes the concept difficult for students to learn. I consider the simplest formalization of the concept, and also ultimately the most useful and least ambiguous, to be with respect to relations. Using the terminology in the video, you would call a set R a "binary relation from A to B" iff it is a subset of Cartesian(A, B) (I am using this notation because I lack a mathematical keyboard on my phone). A binary relation f from A to B is called a partial function from A to B iff for all w, x, y, z, if (w, x) is an element of R and (y, z) is an element of R, then w = y implies x = z. A partial function f from A to B is called a total function from A to B iff, for all x in A, there exists some y in B such that (x, y) is an element of f. Done. This is simple, intuitive, practical, and equally rigorous, and it avoids the ambiguities of the different usages of the word "function" to refer to partial functions or total functions depending on the context, and it avoids the confusing usage of the word "map," which should just be rendered obsolete. If you want to use the word "map," then you can, but only if your usage of the word is not interchangeable with the phrase "total function" or "partial function," and it refers to something else or something more specific, or more general. An alternative nomenclature I would equally allow is if the word "function" referred to only "partial functions," and "map" only referred to "total functions." With this nomenclature, "partial function" and "total function" would become obsolete phrases in the terminology instead, and should absolutely never be used again. Having the idea of a binary relation as the primary notion in the subtopic is essential, since ideas such as "equivalence relations" and "ordering on a set," for example, can be formalized, generalized, and unified as just special cases of binary relations as defined above. Either nomenclature is fine, but the idea I am trying to present is that mathematicians should only use one nomenclature, or the other, and not both, as they are mutually incompatible, and especially because mathematicians have a lazy habit of never really specifying their nomenclature, expecting you to already know which one they are using, which is unacceptable under most circumstances. If you want to not specify which nomenclature you are using, then just use the one everyone uses! Why exactly is this so difficult for mathematicians to do? I have no idea. What confuses me is that using only one of the two across the whole community and having an agreement is objectively not hard, and there are no inconveniences with choosing consistency and strictness in the terminology, and there are no advantages in staying with the inconsistent and confusing usage of terminology as done today, so I fail to understand why mathematicians choose to make these terminology decisions that confuse people. What irritates me is that the entire intention behind the existence of mathematical rigor is that mathematics be expressed in precise and clear language, yet the formal framework of rigor is routinely used by authors to confuse people and achieve the complete opposite instead, with language being used inconsistently and ambiguously, ultimately leading to common abuse of notation and common abuse of language.
6:45 Shouldn't you introduce the symbol for (there exists a unique), because with the usual symbol it can still mean that one element in A being mapped to more than one element in B is allowed?
@@brightsideofmathses, the original definition is fine. It's only with the reformulation with the "there exists" quantifier that I see a possible problem. However, you do say that it must be unique, but that's not captured in writing.
@3:00 Wondering if we might run into problems in case x=y because (x,x):={{x},{x,x}} = {{x}}. Wouldn't (x,y):={ {x}, {{x}, y} } therefore be a better definition?
@@brightsideofmaths so could I write (x,y) := {{x},{y}}? My apologies, but I am very new to this broad range of mathematics, I've only done functions and a bit of calculus from school
@@brightsideofmaths Oh wait I thought about it a bit, so a cartesian product can either be a set that is (x) or (x,y)? (x is from the first set and y is from the second set) And there can't be just a set that is (y) because y is from the second set and you need something from the first set? If my understanding is correct, why is a set with only elements from the first set (x) still a cartesian product? Thank you!
From the ordered pair you want to have two positions: a first one and a second one. However, a set has no order. The set {x,y} is the same as {y,x}. So how can you bring order into this. You could define (x,y) = {{x,1}, {y,2}} to fix this problem. And a similar one is my definition from above.@@Lockout_31019
@@brightsideofmaths Ah I think I get it now, since (x,y) := {{x},{x,y}} is a definition, you are telling us that you have to choose an element from the first set (x) and (x,y) is the ordered pair you get from choosing something for (x) right? I understood everything else in the video, but I was just stuck on this part because I thought {{x},{x,y}} was a set of elements instead of a definition Thank you so much for the replies btw
@@brightsideofmaths well for example, f(x)=1/x there does not exist an ordered pair (0,y) which is an element of G_f. But the definition states that for all x there must exist an ordered pair which is an element of G_f
can we define ordered pair as = {y,{x,y}}. how did you introduce order in it. i don't understand means y can take 1 position and x can take second position.
So in the quizzes, A = {1, 2} B = {5, 6, 7}, and a subset of A x B is a function f = {(1, 2), (5, 6), (6, 7)}. But none of the ordered pairs is in A x B, and the first ordered pair has only elements from A and the two other pairs have only elements from B. I assume these ordered pairs mean that f(1) = 2, f(5) = 6, and f(6) = 7. But this makes no sense for the given A and B. I wonder what I'm getting wrong. Is there actually a typo in the quiz? Or am I not understanding this?
What is tilde? I don't remember you introducing it, nor can I find anything about it in my book and online. Edit: i found another comment about it 👍. Still wanna leave this here because your channel is so amazing ❤🙏
You define a map as a function, but I have seen a definition of a map which doesn't include the limitations needed for a function, as a more general concept. Defined just as a subset of the Cartesian product. You won't need this forward so you just left it out? Or is there another reason?
In the definition of ordered pairs, why does proving (x,y) = (x~,y~) tell us that we have an actual first and second position? I only could grasp that by proving this we know that if two pairs are the same, their respective elements must be equal to each other i.e. x = x~ and y = y~ but I don't get how this is related to the position :(
@@brightsideofmaths hmm, perhaps you could explain to me what you mean by position? To me, mean position means x comes first then y in the pair (x,y). Not sure if I'm on the same page.
The math is really cool and all but can we just appreciate this guy’s amazing handwriting.
Finally understood what a codomain is. Thanks ! (Was initially confused between range, image and codomain)
Nice! I am glad that this was helpful!
I tend to have major gripes with this particular subtopic of set theory, because in my experience, there is no agreement among mathematcians in their usage of terminology or in how they define these concepts, often using the words "function" and "map" interchangeably, while other times using both words to have different meanings. This inconsistency is what makes it difficult for teachers to teach the concept correctly, and therefore, what makes the concept difficult for students to learn.
I consider the simplest formalization of the concept, and also ultimately the most useful and least ambiguous, to be with respect to relations. Using the terminology in the video, you would call a set R a "binary relation from A to B" iff it is a subset of Cartesian(A, B) (I am using this notation because I lack a mathematical keyboard on my phone). A binary relation f from A to B is called a partial function from A to B iff for all w, x, y, z, if (w, x) is an element of R and (y, z) is an element of R, then w = y implies x = z. A partial function f from A to B is called a total function from A to B iff, for all x in A, there exists some y in B such that (x, y) is an element of f. Done. This is simple, intuitive, practical, and equally rigorous, and it avoids the ambiguities of the different usages of the word "function" to refer to partial functions or total functions depending on the context, and it avoids the confusing usage of the word "map," which should just be rendered obsolete. If you want to use the word "map," then you can, but only if your usage of the word is not interchangeable with the phrase "total function" or "partial function," and it refers to something else or something more specific, or more general.
An alternative nomenclature I would equally allow is if the word "function" referred to only "partial functions," and "map" only referred to "total functions." With this nomenclature, "partial function" and "total function" would become obsolete phrases in the terminology instead, and should absolutely never be used again.
Having the idea of a binary relation as the primary notion in the subtopic is essential, since ideas such as "equivalence relations" and "ordering on a set," for example, can be formalized, generalized, and unified as just special cases of binary relations as defined above.
Either nomenclature is fine, but the idea I am trying to present is that mathematicians should only use one nomenclature, or the other, and not both, as they are mutually incompatible, and especially because mathematicians have a lazy habit of never really specifying their nomenclature, expecting you to already know which one they are using, which is unacceptable under most circumstances. If you want to not specify which nomenclature you are using, then just use the one everyone uses! Why exactly is this so difficult for mathematicians to do? I have no idea. What confuses me is that using only one of the two across the whole community and having an agreement is objectively not hard, and there are no inconveniences with choosing consistency and strictness in the terminology, and there are no advantages in staying with the inconsistent and confusing usage of terminology as done today, so I fail to understand why mathematicians choose to make these terminology decisions that confuse people. What irritates me is that the entire intention behind the existence of mathematical rigor is that mathematics be expressed in precise and clear language, yet the formal framework of rigor is routinely used by authors to confuse people and achieve the complete opposite instead, with language being used inconsistently and ambiguously, ultimately leading to common abuse of notation and common abuse of language.
6:45 Shouldn't you introduce the symbol for (there exists a unique), because with the usual symbol it can still mean that one element in A being mapped to more than one element in B is allowed?
But this is already in the definition of a "function" above, isn't it?
@@brightsideofmathses, the original definition is fine. It's only with the reformulation with the "there exists" quantifier that I see a possible problem. However, you do say that it must be unique, but that's not captured in writing.
The second one is in addition to the first one :)
3:53 Thank you for your videos! I think this logical equivalence is not obvious.
No, not immediately obvious but you can try to write it down in more details!
I'm an absolute beginner. Can someone tell me what the tilde sign mean?
It's just there to denote a different variable :)
@@brightsideofmaths ahh okay. Thank you so much for replying. I thought that was another symbol 😅❤️
Thank you SOOO much! This is absolutely perfect
Glad it helped!
Hi! This might be a random question, but what program/app you use to draw like that, on pictures/presentations?
Xournal :)
@@brightsideofmaths Thank You! Danke Schön!
@3:00 Wondering if we might run into problems in case x=y because (x,x):={{x},{x,x}} = {{x}}. Wouldn't (x,y):={ {x}, {{x}, y} } therefore be a better definition?
Very good question. There are a lot of different possibilities that reach the same goal we want for the ordered pair.
Why did you write (x,y) := {{x},{x,y}}? I just don't understand the point of that, could you explain it to me please?
One wants a definition for the ordered pair while using sets. This is one possibility.
@@brightsideofmaths so could I write (x,y) := {{x},{y}}? My apologies, but I am very new to this broad range of mathematics, I've only done functions and a bit of calculus from school
@@brightsideofmaths Oh wait I thought about it a bit, so a cartesian product can either be a set that is (x) or (x,y)? (x is from the first set and y is from the second set)
And there can't be just a set that is (y) because y is from the second set and you need something from the first set?
If my understanding is correct, why is a set with only elements from the first set (x) still a cartesian product?
Thank you!
From the ordered pair you want to have two positions: a first one and a second one. However, a set has no order. The set {x,y} is the same as {y,x}. So how can you bring order into this. You could define (x,y) = {{x,1}, {y,2}} to fix this problem. And a similar one is my definition from above.@@Lockout_31019
@@brightsideofmaths Ah I think I get it now, since (x,y) := {{x},{x,y}} is a definition, you are telling us that you have to choose an element from the first set (x) and (x,y) is the ordered pair you get from choosing something for (x) right? I understood everything else in the video, but I was just stuck on this part because I thought {{x},{x,y}} was a set of elements instead of a definition
Thank you so much for the replies btw
great lesson, thanks x )
6:51 wouldn't that discount discontinuous functions?
Why do you think that?
@@brightsideofmaths well for example, f(x)=1/x there does not exist an ordered pair (0,y) which is an element of G_f. But the definition states that for all x there must exist an ordered pair which is an element of G_f
@@someperson9052 What is the domain of definition for f(x) = 1/x?
@@brightsideofmaths oh of course, I see. Thank you very much, and these videos are great
can we define ordered pair as = {y,{x,y}}. how did you introduce order in it. i don't understand means y can take 1 position and x can take second position.
You see that the symmetry is broken here. That brings the order in.
So in the quizzes, A = {1, 2} B = {5, 6, 7}, and a subset of A x B is a function f = {(1, 2), (5, 6), (6, 7)}. But none of the ordered pairs is in A x B, and the first ordered pair has only elements from A and the two other pairs have only elements from B. I assume these ordered pairs mean that f(1) = 2, f(5) = 6, and f(6) = 7. But this makes no sense for the given A and B. I wonder what I'm getting wrong. Is there actually a typo in the quiz? Or am I not understanding this?
Oh, some typos could always happen. I check it immediately.
Oh my.. I really screwed this up with the answers. Thanks for telling me. I immediately corrected the typos. Sorry for the confusion.
@@brightsideofmaths That's a relief! Thank you for correcting it
What is tilde? I don't remember you introducing it, nor can I find anything about it in my book and online.
Edit: i found another comment about it 👍.
Still wanna leave this here because your channel is so amazing ❤🙏
Thanks. The tilde is just to denote a different element, so a different name.
Thanks
شكرا و جزاك الله خيرا
You define a map as a function, but I have seen a definition of a map which doesn't include the limitations needed for a function, as a more general concept. Defined just as a subset of the Cartesian product. You won't need this forward so you just left it out? Or is there another reason?
It's exactly the definition I gave here :)
G_f is a subset of the Cartesian product.
@@brightsideofmaths I saw. What mean is: do you have a name for such object?
@@rafaelschipiura9865 Map or functional relation.
In the definition of ordered pairs, why does proving (x,y) = (x~,y~) tell us that we have an actual first and second position? I only could grasp that by proving this we know that if two pairs are the same, their respective elements must be equal to each other i.e. x = x~ and y = y~ but I don't get how this is related to the position :(
Just a set has no position. However, you already explained that you could distinguish the positions in a pair.
@@brightsideofmaths hmm, perhaps you could explain to me what you mean by position? To me, mean position means x comes first then y in the pair (x,y). Not sure if I'm on the same page.
@@johnsu9949 Yes, it means exactly that :)
Thank you, some day could you talk about non standard analysis?, please