Relating invertibility to being onto and one-to-one

a couple of videos ago we learned that a function that is a mapping from the set X to the set Y is invertible invertible if and only if and i'll write that is it if with two F's if if and only if for every y so let me write this down for every I'll do this in yellow for every Y that is a member of our codomain there exists there exists a unique there exists a unique and I'll make that a little bit bold a unique X that is a member of our domain such that such that f of X F of this X is equal to this Y so that's just saying that if I take my domain right here that's X and then I take a co domain here that is why we say that the function f is invertible we say that the function f is invertible and we know what invertibility means it means that there's this other function called the inverse that can essentially if you apply if you take that in composition with f it's like taking the identity on X or if you take F in composition with it it's like taking the identity on Y it we've done that multiple times so I won't repeat that there we know what invertibility means but we we learned that if it's invertible if and only if for every y here so you take any y here any y that's a member of your co-domain there exists a unique X such that f of X is equal to that Y f of X is equal to let me write it this way if this is an X let's say that's an X naught f of X naught would be equal to Y so this Y would be equal to f of X naught you apply the function here it's going to map it to this point here it wouldn't be invertible if you had this if you had two members of X mapping year that would break invertibility if you had the situation because then you wouldn't have the the unique condition you have to have a unique X that maps to this thing and what I just drew here with this other pink mapping we don't have a unique X that maps to Y we have two x's that map to Y now based on what I just told you on that last video what does this mean if we have a unique exit map to each Y that means that we have to have a one-to-one mapping that F has to be one-to-one so let me write that so another way of saying this is that F F is one-to-one or injective injective so if we have two guys mapping to the same Y that would break down this condition we wouldn't be one-to-one and we couldn't say that there exists a unique X such a unique x solution to this equation right here now the other part of this is this for every Y you could pick any Y here and you're going to there exists a unique X that maps to that so I don't care there cannot be some Y here let's say that there's some Y here and no one maps to that no one no one maps if that's the case then we don't have then we don't have our conditions for invertibility so that would be not invertible so everything in Y every element of Y has to be mapped to all of these guys have to be mapped to and they can only be mapped to by one of the elements of X everything here has to be mapped to by a unique guy now in the last video what did we call it when a function maps to every element of your codomain so this this every here what is another way of saying that that a function maps to every element of your codomain on the last video I explained that that notion is called a surjective surjective or on to function so the whole reason why I'm doing this video is because I really just want to restate the condition for invertibility in in using the vocabulary that I that I introduced to you in the last video so given that we can given that this statement for every why that's a member of our codomain there exists a X that maps to it we could just say that f is surjective f is surjective and then the fact is if we just said that f is surjective that means that everything here is mapped to but it could be mapped to maybe you know this person right here could be mapped to by more than one surjective doesn't doesn't by itself make the condition that there's a unique mapping from a member of X to that element of Y so in order to get that in order to satisfy the unique condition of this condition for invertibility we have to say that f is also on is also injective is injective and obviously may be the less formal terms for either of these you call this on to and you could call this one-to-one so using the terminology that we learned in the last video we can restate this condition for invertibility we can say we can say that a function that is a mapping from the domain X to the codomain y is invertible is invertible if and only if I'll write it out and only if F is both surjective surjective and injective and inject it or we could have said that f is invertible if and only if f is on - I could write that here on - and one-two-one one-two-one and these are really just fancy ways of saying for every Y in our codomain there is a unique X that F maps to it there isn't more than 1 and every Y does get mapped to