Why do we need a property called surjectivity?

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 \mathbb{R} to \mathbb{R} 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 x \mapsto x^6-3x^3 +x^2 defined on \mathbb{R}. 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 \mathbb{R} to \mathbb{R} that isn’t surjective.

I suppose though that we could just say that it is a function from \mathbb{R} 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?

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One response to “Why do we need a property called surjectivity?

  1. The question is less “why do we need a property called surjectivity?” and more “why do we need an object called the codomain?” Once you notice that not all maps have equal image and codomain, it is surely at least convenient to give a name to those functions for which they are equal; and then you realize that actually this is extremely convenient. So we want a good justification for the notion of codomain.

    With the notion of codomain available, we can make statements like the following: given any continuous function $f\colon[0,1]\to\mathbb{R},$ the integral $\int_{0}^{1}f(t)\,\mathrm{d}t$ (which is just a number, intuitively the area under the graph of $f$) exists. This statement is saying that if we want to, then we can think of the act of integration as a function, whose domain is the set of continuous functions $[0,1]\to\mathbb{R}.$

    This is all well and good, but what if we didn’t have the notion of codomain available? Then, how would we describe the domain of this “integration” function? We might say “the set of continuous functions with domain $\mathbb{R}$ and with image contained in $\mathbb{R}$”, but this sounds clumsy, and after repeating it in enough situations I think you would eventually decide “okay, let’s agree to say that a function whose image is contained in $\mathbb{R}$ `has codomain $\mathbb{R}$’; that ought to save us some words.”

    You could instead try to argue that when you want to compute the image of a function like your polynomial function above, it is sometimes helpful to sketch the graph of the function; but as soon as you do that, you are admitting that you already have an idea of where the image should live (for the polynomial, you know the graph lives in the plane, so you are admitting that the image of the function should be a subset of $\mathbb{R},$ i.e., that you can treat your polynomial as a function $\mathbb{R}\to\mathbb{R}$).

    Ultimately – and I’ve just thought this now – I think a big problem might be that when we come up with names to describe functions, like “injective” and “surjective” and “continuous”, that’s because we are thinking of the functions themselves as objects of interest. Students who ask this question might be thinking only of functions as things that we might use to study objects that they believe should be of interest, like numbers, and not realizing that actually, most of the time, mathematicians are more directly concerned with the slightly more abstract notions of sets and functions (and it only gets “worse” from there!).

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