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Current time:0:00Total duration:22:41

Proof: Invertibility implies a unique solution to f(x)=y

Video transcript

I've got a function f and it's a mapping from the set X to the set Y and I also let's just say for the sake of argument let's say that f is invertible f is invertible what i want to know is what does this imply about this equation right here the equation f of X is equal to Y I want to know that for every Y that's a member of our of our codomain so for every y so let me write this down for every y that's a member of my co domain is there a unique I'll write in caps unique solution X that's a member of our domain such that and I could write such that well I'll just write it out I was going to write the Matthew way but I think it's nicer to write it in the actual word sometimes such that f of X is equal to Y so if we just let me just draw everything out a little bit we have our set X right here this is X we have our codomain y here we know that F if you if you take some point here let's call that a it's a member of X and you apply the function f to it it'll map you to some element and set Y so that's F of a right there this is what we this is so far what just this tells us now I want to look at this equation here and I want to know that if I can pick any Y in this set in or any lower case Y in this set Y so let's say I pick let's say I pick something here let's say that's B I want to know is there a unique solution to the equation f of X is equal to B is there a unique solution so one I guess you have to think is there a solution so is there a solution is saying look is there some X here that if I apply the transformation F to it that I get there and I also want to know is it unique for exhibit if this is the only one that it's unique but it's not unique if there's some other guy if there's more than one solution if there's some other guy in X that if I apply the transformation I also go to B this would create it make it non unique not unique yeah not unique so what I want to concern ourselves with in this video is somehow is invertibility related to the idea of a unique solution to this for any Y in our codomain so let's just work through our definitions of invertibility and see if we can get anywhere constructive so by definition f is invertible implies that implies that there exists this little backward-looking a3 looking thing this means there exists I think it's it's nice to be exposed sometimes to the Mathieu notation so let me just write there there exists that means that there exists some function that means that there exists some function let's call it F inverse F inverse so that's a mapping from Y to X from Y to X such that such that let me and actually the colons are also the shorthand for such that but I'll write it out such that such that the composition the composition of F inverse with F is equal to the identity on X so essentially it's saying look if I apply F to something in X and then I apply F inverse to that I'm going to get back to that point which is essentially equivalent or it isn't just essentially equivalent it is equivalent to just applying the identity the identity function so that's IX so you just get what you put into it such that the this inverse function the composition of the inverse with the function is equal to the identity function and but the composition of the function with the inverse function is equal to the identity function on Y so if you start in Y if you started Y and you apply the inverse and then you apply the function to that you're going to end up back in Y at that same point that's equivalent to just applying the identity function so this is what invertibility tells me this is how I defined invertibility in the last video now we're concerned our cell we are concerned with this equation up here we're concerned with the equation I'll write it in pink f of X is equal to Y 4 and when we want to know for any Y in our or any lowercase cursive Y and our big set why is there a unique x solution to this so what we can do is we know that f is invertible I told you that from the get-go so given that F is invertible we know that there is this F inverse function we know that there's this F inverse function and I can apply that F inverse function it's a mapping from Y to X the mapping from Y to X so I can apply it I can apply it to any element in Y so this for any Y let's say that this is my Y right there so I can apply my F inverse to that Y and I'm going to go over here and of course Y is equal to f of X y is equal to f of X these are the exact same points so let's apply our F inverse function to this so if I apply the F inverse function to both sides of the equation both sides this is a this right here is an element in Y and this is the same element in Y right they're the same element now if I apply the the mapping the inverse mapping to both of that that's going to take me to some element in X so let's do that so if I take the inverse function out of this if I take the inverse function on both sides of this equation both sides of the equation where some element over here and Y and I'm taking the inverse function to get to some element in X and what is this going to be equal to well on the right hand side we could just write the F inverse of Y that's going to be some element over here but what is the left-hand side of this equation translate to the definition of this inverse function is that when you take the composition with F you're going to end up with the identity function this is going to be equivalent to let me write it this way this is equal to the composition of F inverse with f of X which is equivalent to the identity function being applied to X and then the identity function being applied to X is what that's just X this thing right here just reduces to X this reduces to X so we started with the idea that f is invertible we use the definition of invertibility that there exists this inverse function right there and then we essentially apply the inverse function to both sides of this equation say look you give me any Y any lowercase cursive Y in this set Y and I will find you a unique X this is the only X that satisfies this equation this is the only X remember if and how do I notice the only X because this is the only possible inverse function this is the only possible inverse function only one inverse function or which is true I prove that to you in the last video that if f is invertible it only has one unique inverse function we tried before to have maybe two inverse functions but we saw that they have to be the same thing so since we only have one inverse function and it applies to anything in this big upper case set Y we know we have a solution and because it's only one inverse function and functions only map to one value in this case then we know this is a unique solution so let's write this down so we have established that F if f is invertible I'll do this in orange if f is invertible vertical then the so then the equation f of X is equal to Y for all for all that little V with that looks like it's filled up with something for all y the member of our set Y has a unique solution as a unique solution unique solution and that unique solution that unique solution if you know you really care about is going to be the inverse function applied to why it might not it might seem like a bit of a no-brainer but you can see you have to be a little bit precise about it in order to get to the point you want let's see if the opposite is true let's see if we assume let's see if we start from the assumption that for all Y that our that is a member of our set Y that the solution that the equation f of X is equal to Y has a unique unique solution let's assume this and see if it can get us the other way if we given this we can prove invertibility so let's think about it the first way so we're saying that for any Y so let me pick let me draw my sets again so this is my set X and this is my set Y right there now we're working for the assumption that you can pick any any element in Y right here and I and then the equation right here has a unique solution as a unique solution let's call that unique solution well we could call whatever with a unique solution X so you can pick any point here and I've given you we're assuming now that look you pick a point in Y I can find you I can find you some point in X such that f of X is equal to Y and not only can I find that for you that that is a unique solution so given that let me define a new a new function let me define the function S the function S is a mapping from Y to X it's a mapping from Y to X and s of let's say s of Y where of course Y is a member of our set capital y s of Y is equal to D unique the unique let me write the unique solution in X in X to f of X is equal to Y now you're saying hey Sal that looks a little convoluted but this is it think about this is a completely valid function definition right we're starting with the idea that you give me any Y here you pick me you give me any member of this set and I can always find you a unique solution to this equation well like okay so that means that any guy here can be associated with a unique solution in the set X where the unique solution is unique solution to this equation here so why don't I just define a function that says look I'm going to associate every member Y with its unique solution to f of X is equal to Y that's how I'm defining this function right here and of course this is a completely valid mapping from Y to X and we know that this is this only has one legitimate value because this any value Y any lower case value Y in this set has a unique solution to f of X is equal to Y so this can only equal one value so it's well-defined so let's apply let's let's take some element here let's take SM element right let me do a good color let's say this is B and B is a member of Y so let's find so let's just map it using our new function right here so let's take it and map it and this is s of B right here s of B which is a member of X now we know that s of B is a unique solution by definition I know it seems a little circular but it's not we know that s of B is a solution so we know that s of B is the unique solution is a unique solution to f of X is equal to B well if this is the case if this is good we just got this because that's this is what this function does it Maps every y2 it's the unique solution to this equation because we said that every Y has a unique solution so this is the K so what happens if I take F of s of B well I just said this unique solution to this so if I put this guy in here what am I going to get I'm going to get B or what another way of saying this is that the composition the composition of F with s applied to B is equal to B or another way to say it is that when you take the composition of F with s this is the same thing because if I take s I apply s to B and then I apply F back to that that's the composition I just get back to B that's what's happening here so this is the same thing as the identity function on Y being applied to B so it's equal to B so we can say that the composition we can say that there exists and we know that this function exists I or that we can always construct this so we already know that this exists this existed by me constructing it but I've hopefully shown you that this is well-defined that you know from our assumption that Y that this always has a unique solution in X for any Y here I can define this in a fairly reasonable way so it definitely exists and not only does it exist but we know that the composition of F with this function that I just constructed here is equal to is equal to the identity the identity function on Y now let's do another little experiment let's take a particular let me just draw our sets again let me take some this is our set X and let me take some member of set X call it a now let's do it let me take my set Y right there and so we can apply the function to a and we'll get a member of set Y let's call that right there let's call that F let's call that F of a right there now if I apply my magic function here that always I can get you any member of set Y and I'll give you the unique solution in X to this equation so let me apply that to this let me apply s to this so if I apply s to this we'll give me the unique solution so let me write this down so if I apply s to this I'm going to apply s to this I'm going to apply s to this and maybe I shouldn't point it back at that I don't want to imply that necessarily points back at that so let me apply s to that s to this so what is this going to point to what is that point going to be right there so that's going to be s of this point which is f of a which we know is the unique solution so this is equal to the unique solution to the equation to the equation f of X is equal to this Y right here or this Y right here is just called F of a right remember the the mapping s just maps you from any member of a to the unique solution to the equation f of X is equal to that so this is the mapping from F of a to the unique so this s of f of a is going to be a mapping to the or this right here it's going to be the unique solution to the equation f of X is equal to this member of wine what's this member of Y called it's called F of a it's called F of a well what is you know we you could go say this in a very convoluted way but if I were to just you know before you learned any out linear algebra if I said look if I have the equation f of X is equal to f of a what is the unique solution to this equation what does X equal well X would have to be equal to a X would have to be equal to a so the unique solution to the equation f of X is equal to f of a is equal to a and we know that there's only one solution to that we only we only we know that there's only one solution to that because that was one of our starting assumptions so this thing is equal to a or we could write s of s of f of a is equal to a or that the composition of s with F is equal or applied to a is equal to a or that the composition of s with F is just the identity the identity function on the set X right this is a mapping right here from X to X so we could write that s the composition of s with F is the identity on X so what have we done so far we started with the idea that you pick any y in our set capital y here and we're going to have a unique solution X such that this is true such that f of X is equal to Y that's what the assumption we started off with we constructed this function S that immediately Maps any member here with its unique solution to this equation fair enough now from that we say ok this definitely exists not only does exist but we figured out the composition of F with our constructed function is equal to the identity in on the set y and then we also learned that s the composition of s with F is the is the identity function on X let me write this so we learned this and we also learned that the composition of F with s is equal to the identity on Y and s clearly exists because I constructed it and we know it's well-defined because every Y for every y here there is a solution to this so given given that I was able to find for so for my function f I was able to find a function that these two things are true this is by definition what it means to be invertible to me invertible remember so this means that f is invertible remember f being invertible in order for f to be invertible that means that it must be there must be there must exist some function from so if f is a mapping from x to y invertibility means that there must be some function f inverse that is a mapping from Y to X such that so that I can write there exists a function such that the inverse function composed with composed with our function should be equal to should be equal to the identity on X and the inverse and the function in the composition of the function with the inverse function should be the identity on Y well we just we found a function it exists and that function is s where both of these things are true we can say that s is equal to F inverse so f is definitely invertible so hopefully you found this satisfying this proof is very subtle and very nuanced because we kind of you know keep bouncing between our six sets x and y but what we've shown is is that if f is in the beginning part of this video we show that if f is invertible then there's for any y there is a unique solution to the equation f of x equals y and in the second part of the video we showed it that the other way occur the other way is true that if if let me put it this way that if for all y a member of capital y there is is a unique solution to f of X is equal to X then F is invertible invertible so the fact that both of these both of these assumptions imply each other we can write our final conclusion of the video that f being invertible if f which is a mapping from x to y is invertible this is true if and only if and we could write that either as a two-way arrow or we could write if for if and only if so both of these statements imply each other if and only if for all y for every y that is a member of our set Y there exists a unique there exists a unique I can actually write that like that that means there exists a unique X there exists a unique X for the or let me write it this way there exist a unique solution to the equation f of X f of X is equal to Y so that was our big takeaway in this video that invertibility of a function implies there's a unique solution to this equation for any Y that's in the co-domain of our function