Section 10.5 Composition of functions
Our last step before defining inverse functions is to explain a generic way of combining functions. Now since we are dealing with general sets and not just sets of numbers, we can’t add or subtract or divide or differentiate or any of the things we usually do with functions in calculus courses. Really the only thing we can with our abstract functions defined on abstract sets is composition — that is, take an element, apply the first function and then apply the second function.
This standard notation for compositions looks a bit backwards — we read things from left to right, but when we actually evaluate the composition we do the rightmost function first.
Theorem 10.5.2.
The proof of associativity is not difficult but is quite long; we leave it to the interested reader to work through it. Instead we’ll look at the following useful (and nice) theorem. This tells us that composition of functions interacts nicely with injections, surjections and bijections.
Theorem 10.5.3.
Scratch work:
-
Injective: Assume both
and are injective. So if then Similarly if then We can just put these together. If then and so Thus The diagram below should help. -
Surjective: Assume both
and are surjective. Then for each there is some such that Similarly for each there is some such that Thus Again, we refer the reader to the diagram below. - Bijectiveness follows from surjectiveness and injectiveness. That is, if
are both bijective, then they are both injective and surjective. Hence their composition is injective and surjective, and so bijective.
We are ready to write things up nicely for the reader.
Proof of Theorem 10.5.3.
It suffices to prove the first two points, since the final point follows immediately from them.
- Let
be injective functions. And let such that Since is injective, it follows that And since is injective it followst that Thus Thus is injective. - Let
be surjective functions and let Since is surjective, for some Since is surjective, there is some such that It follows that and so is surjective.
Here is another nice result about compositions, injections and surjections. In particular, if we know the composition is injective, then what can we say about Similarly, if is surjective, then what can we say about Similarly
Theorem 10.5.4.
Proof.
We prove each in turn.
- Assume that
is injective, and let so that To show that is injective it suffices to show that Since we know that and since is injective we have that - Now assume that
is surjective and let To prove that is surjective it suffices to find so that Since is surjective, we know that there is so that Now set Then as required.
Example 10.5.5.
The above theorem proves that when is injective, we know that is injective, the following example show that need not be injective. Further, it also shows that just because and are surjective, we need not have that is surjective.