In our example, to have an injection, we are not allowed to have an Australian city that has 2 (or more) roads coming into it from different English cities. There are a few types of road arrangement that have special names: InjectionĪn injection or ‘injective function’ is when each element in the co-domain is mapped to by at most one of the elements in the domain. The roads are one-way only in our case, so sadly no Aussies can return to the homeland. All the English cities must take us somewhere – so a set of roads is a valid function only if each city in England has a road going to some Australian city. In this scenario, a function is a set of roads that takes us from somewhere in England to somewhere in Australia. England and Australia both contain cities, which are represented by the nodes within each set. England is the domain – the place that we are mapping from, whilst Australia is the co-domain (also called range), and it is the place that we are mapping to. There are 2 spaces or sets, which in this case are the countries England and Australia. I find that it’s easiest to explain them using diagrams and by giving real names to things, so here goes… The ones that people are used to seeing are things like f(x) = x 2 but functions don’t have to involve algebra at all. In maths, a function is a ‘mapping’ from one ‘space’ to another. Before I can talk about it though, I need to introduce you to the world of mathematical functions. The Pigeonhole Principle is one of my favourites because of the fact that it is so powerful and quite tricky to prove, yet so intuitive and easy to understand.
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