Here is a question I am often asked by my first-year students. Why do we need a name for the condition of surjectivity, when you can always change the codomain to be equal to the image, and make your function surjective that way? (I am avoiding the word ‘range’, because that turns out to be used differently by different authors.)
I don’t know currently what the best answer to this is, though I have some ideas.
I could refer to mathematical tradition, and talk about the flexibility of the standard approach. For example, I think it is quite useful that we have a large set of functions from to given by polynomial functions with real coefficients, and it would be inconvenient if we couldn’t say where these functions were mapping to without calculating the image. Off the top of my head, consider, for example, a polynomial function defined on . What is its image? Do we really want to have to calculate the image before we can say what the codomain is? It seems relatively easy just to treat it as a function from to that isn’t surjective.
I suppose though that we could just say that it is a function from to its image. Is there any problem with that? Well maybe it gets a bit complicated later when you start looking at homomorphisms in algebra, but you can probably work round that.
I expect that a category theorist would have something to say on this issue! But what is the best thing to say to a first-year undergraduate?