Video není dostupné.
Omlouváme se.
Relations and Functions: The Modern Definition of a Mathematical Function.
Vložit
- čas přidán 14. 08. 2024
- In this video, we discuss how the definition of a function has changed over time, largely due to the development of set theory.
Chapters
0:00 - Introduction and Motivation
01:50 - Products of Sets
02:53 - Relations
03:59 - Functions
The product links below are Amazon affiliate links. If you buy certain products on Amazon soon after clicking them, I may receive a commission. The price is the same for you, but it does help to support the channel :-)
In-video references
1. A history of Mathematics (Merzbach and Boyer) - amzn.to/33505x1
2. Concepts of Modern Mathematics (Stewart) - amzn.to/3zIJyut
3. Foundations of Mathematics (Stewart and Tall) - amzn.to/3JVY0UJ
FAQ : How do you make these animations?
Animations are made in Apple Keynote which has lots of functionality for animating shapes, lines, curves and text (as well as really good LaTeX). Editing and voiceover work in DaVinci Resolve.
Supporting the Channel.
If you would like to support me in making free mathematics tutorials then you can make a small donation over at
www.buymeacoff...
Thank you so much, I hope you find the content useful.
I didn't go into too much detail in the video but the reason this definition came about is because there was a phase in the 19th/early 20th century when mathematicians (notably, David Hilbert) wanted to establish every object/thing in mathematics in it's most basic or foundational form. Hence, here we have the definition of a function made entirely from sets and set theory!
I feel like this video deserves a follow-up concerning injectivity, surjectivity, and bijectivity. :)
Good point! These also nicely leading to the inverse of a function. Very tempted to do this vid!
That would be very useful
Interesting, thanks! For others wondering, the square wave is function then, even though it appears to have vertical sections with more than y per x, based on Wikipedia which defines its domain to exclude the integers at the vertical sections.
Hidden gem of a channel
Thank you so much. The first video I watch that explains what is a function with its pure meaning inside set theory.
I misread it as what is fuction of mathematics.
And now I know what function are in mathematics.
❤
A friend of mine came up with some new notation based on a programming language I forgot the name of.
He introduced 4 symbols, but this notation can be expanded.
Let A and B be sets. We call f a general mapping if it is just any subset of AxB. We will denote this by f: A * -> * B.
A fuction is a subset of AxB were for all x in A, there exist a y in B such that (x,y) is in the subset, and for any x in A, if (x,y) and (x,z) is in the subset, then y=z. This is to say that if we look at all the elements in the subset. We will find every element of A once and only once, while any element of B may appear anywhere between 0 to infinity many times. We will denote this by f: A !->* B
Sometimes we do a little bit of notational abuse and are not able to map every element from the domain to the output space. (Example, we write f: R -> R, for f(x) = 1/x, while we should write f: R/{0} -> R). If we look at the subset of AxB that discribes such a "function", we'll notice that we are missing some x in A such that (x,y) is in the subset for some y. We either see any elementen once, or not at all. We denote this by f: A ?->* B and called it a partial function.
Last example before I start explaining. We call f: A !->+ B a surjection.
So what do all these symbols mean? well, they indicate how many times an element of A (or B respectively) show up at part of an element in the subset of AxB that discribes the mapping.
!: 1
?: 0 or 1
*: 0 to ∞
+: 1 to ∞
With this notation, we exactly know how the mappings behave, and this can be expanded to include more symboly. Also, the behaviour of inverses of mappings are quite easy to determine. Example, we call f a injection if it is f: A !->? B. The inverse of this is simplily f^-1: B ?->! A. We see that it sort of behaves like a partial function, exept that any valid input has only one output. We called any mapping of this form an inverse injection.
I could go on with all the names for the 16 different types of maps you get with this notation, but this is already a long text. Try to see what mappings/functions have which behaviour for yourself.
Edit: third paragraph, youtube makes stars dissaprear when I type them too close to eachother, so I put extra spaces in the notation of a general mapping.
this looks like a mix of math and regexp
Great video as usual! Keep up the good work!
Thanks a lot!
This is amazing and a great refresher before my winter terms!
But what about injection, surjection and bijection?
As a tangent you could've mentioned some issues with this definition compared to the naive rules-based (constructive) approach: Take A to be the set of even natural numbers greater than 2 and B to be the set of primes. Define the relation to be that y of B is the smallest prime of all prime pairs that make up a given x of A. There is currently no quaranteed way to produce such y and it's not even known if a requisite prime pair exists to begin with (Goldbach's conjecture). Assuming the conjecture this relation still defines a function. Contrast this with a rules-based approach that always produces an answer in a straightforward way.
The function exists (if the conjecture is true), we just don't have a formula for it. Just like we don't have a formula for each real number
There are, however, such things are multi-valued functions. I encountered this when doing complex analysis in my undergraduate, though we didn't dwell on it long enough, it was just for defining the logarithm function on the entire complex plane, so I don't know much about them if it goes any further than that and if the idea of a function goes even further than that, would be interested in seeing that.
The way I understand it, multi-valued functions can be functions if we add a "branch index" to the input. For instance, ln(z) = ln|z| + arg(z)*i + 2π*n*i. The multi-valued function definition of this would have a domain of (z, n) where z is a complex number and n is an integer.
@@potaatobaked7013 I see, but is that true for all multivalued functions or just some? It makes sense in the case of the logarithm because of the geometry of the exponential on the complex plane, but I'm not too convinced it would work for general multivalued functions. Is it possible there are two classes of these functions, ones that can and can't be indexed in this manner, or is it inherently so that you can only defined a multivalued function if it has this underlying property somewhere of being indexed by something? These would be my follow up questions
Multivalued functions are not functions as it is defined in mathematics.
Multivalued "functions" usually arise when we want to find an inverse map to a forward map (say f(x)=x^2). When the function is not bijective this will fail to be a function for 2 reasons. 1) if the function is not surjective then the inverse "function" will not have a value for each valid input and 2) if the function is not injective then it will give multiple outputs for a single input (what we sometimes call being multivalued).
Notice that for x in *R* the inverse "function" to f(x) = x^2 will not be a function for both reasons. -1 will not have an output (in the reals) and x>0 will have 2 outputs.
In fact, you can have a multivalued function take as many values as you want. Let X be a topological space and f:X -> *R* be the function f(x) = 0. Then the "value" for the inverse "function" at 0 will be the entire space X. So the function is _very_ multivalued 😅
When we invert functions we can play 2 tricks to make the inverse "function" a true function. 1 is to restrict the domain and 2 is to restrict the co-domain. So if we say x^2 is a function from R+ -> R+ then it is a bijection and has an inverse function, and we call that the square root.
The function f: A->B being a subset of set AxB reminds me of Curry-Howard Isomorphism... like how exponents is related to multiplication.
Any possible combination of input (A) and output (B) may constitute a function.
This channel is the best
Love the video! Clear as Crystal! Furthermore, love the black background :)
I do have a question, given the definition of a function you give there, does the square wave you introduced at 0:37 is a function ?
Here my note well formed arguments :
· for all (x,y) | x \in R, y \in R and for all (v,w)| v in R, w \in R, v≠x , y=w, YES (existence and unicity of the ordered paire) which describes paires of the "horizontal lines"
· for all (x,y) | x in R, y in R and for all (v,w)| v in R, w in R, v=x , y≠w, NO (existence and ¬unicity of the ordered paire) which describes paires of the "vertical lines"
So the overall answer for me is no according to this argument
Thank you for what you do :)
For another video talk about the definition of the cartesian product as a set, such as Kuratowski's:
AxB={{{x},{x,y}} for x in A, y in B}
Its pretty amazing how things that seem to be "primitive ideas" in fact are not that foundational :)
And what happened after set theory
I would have mentioned
Riemann surface
This definition of function seems to differentiate from the usual 'process' conception of function that we have .
It does in the sense that a 'process' is an unclear concept in the abstract.
Its certainly a different definition from what I first got taught! The good thing is that the notion of a function being a rule or a process etc is still generally quite correct and appropriate for things like a calculus course! for example I wouldn't be going around like "well, actually that's not a function on its own" to everyone who said something like "consider the function 1/x"
@@DrWillWood yeah ,True.
I wonder what it was before
Meanwhile programmers:
Look at my function. It takes no parameters and returns no value.
A function from the empty set to the empty set is a completely valid function in mathematics. It has no inputs and no outputs.
Mr. Motormouth.
not a solution. just obfuscated the difficulties under set theory fog.
Today I will add "codomain of a function" to the list of maths concept that give me nightmares, though this one seems more like David Hilbert's fault rather than God's.
I reject rule 2, there are plenty of multi-valued functions, why are people so hung up on single-valued functions?
Multi-valued functions are not functions as we use the term in mathematics. It is very hard to use multivalued functions for things, and most definitions of calculus for example fall apart once you start introducing multivalued "functions"
Multivalued functions are essential to complex analysis and you can do traditional calculus on their branches as well as you can on any single valued function. Furthermore allowing multivalued functions simplifies many processes, e.g. by ensuring that every function has an inverse and avoiding the obsessive distinction between relations and functions(functions are relations). You'll eventually work with multi-valued inverses regardless, but usually you just disregard their functional nature and just consider them as "sets" or something. @@MK-13337
I am a math students and I realise later that function is not a formula but a set of co-ordinates(ordered pairs) but my question is why we read f(x) as f of x instead image of x under f. Because a function is defined as a set of ordered pairs. So saying f of x means nothing meaningful