Start Learning Sets 4 | Cartesian Product and Maps

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This is my video series about Start Learning Sets. We discuss basic definition, like union and intersection, and also go into the details of maps and functions. I hope that it will help everyone who wants to learn about it.

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#StartLearningMathematics
#Mathematics
#LearnMath
#calculus
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I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[Crazycrock]

The math is really cool and all but can we just appreciate this guy’s amazing handwriting.

[VictorHugo-xnjz]

Finally understood what a codomain is. Thanks ! (Was initially confused between range, image and codomain)

[angelmendez-rivera]

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.

[anupamadewpura]

I'm an absolute beginner. Can someone tell me what the tilde sign mean?

[nadeemhameedi]

Thank you SOOO much! This is absolutely perfect

[doyunnam]

3:53 Thank you for your videos! I think this logical equivalence is not obvious.

[me_hanics]

Hi! This might be a random question, but what program/app you use to draw like that, on pictures/presentations?

[Irenecometa]

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 ❤🙏

[gauravnainwal]

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?

[b.kpiano]

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.

[noros-troll]

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?

[Lockout_]

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?

[johnsu]

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 :(

[kodie]

@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?

[someperson]

6:51 wouldn't that discount discontinuous functions?

[rafaelschipiura]

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?

[CharlieYoutubing]

Thank you, some day could you talk about non standard analysis?, please <3