Proving that Functions are Injective and Surjective (One-to-One and Onto)

0:00 Introduction
0:20 Functions
3:30 Injective/one-to-one functions
6:33 Proving that a function is injective
8:42 Proving that a function is not injective
9:28 Why is injectivity confusing?
12:02 Surjective/onto functions
14:41 Proving that a function is surjective
18:27 Proving that a function is not surjective
19:07 Why is surjectivity confusing?
20:41 Bijective functions
21:52 Proving that a function is bijective

[@mnka-be2vm]

video deserves more views

[@dodecahedra]

Excellent video. I might only add one or two small things. If students find the other definition of injective more intuitive, then I would just let them use that one. x₁≠x₂→f(x₁)≠f(x₂) is just the contrapositive of the "real definition", so they're totally equivalent of course. The proofs from this definition are slightly less elegant, but perhaps that's a sacrifice worth making. The other thing that makes surjections hard to understand is that you have to have already accepted the reality and importance of the codomain. Without the codomain concept, surjections don't make any sense. In more casual contexts (like high school) the codomain gets downplayed or totally ignored and functions are represented as sets of ordered pairs. The great thing about a bijection is that its inverse is also a (bijective) function. So we want our functions to be bijections. So we need to talk about surjections. So we need codomains in our function definition. This might have been worth mentioning to convince a sceptic that surjectivity is a useful concept.

[@DavidRicheson]

Thanks! I actually had the thought, after getting too far into the video making process, that I should have said more about the importance of the codomain. I talk to my students about that. The function f(x)=x^2 is not surjective if the codomain is R, but it is surjective if the codomain is [0,infty), as I know you know. I order to talk about a function f we need to know the domain, the codomain, and what f does to the elements of the domain. All three are essential.