Naming the elements of a set

Let E_V be the set of inhabitants of a village V, and F the set of every conceivable name. Clearly, each person—and in particular, each inhabitant of V—can be associated with their name. This gives us a map f from E_V to F: f: E_V $\mapsto \backslash mapsto$ F. It sends each element of E_V, that is, each inhabitant of V, to an "image": that inhabitant's name.

There may be people with the same name in the village V. If two people share a name, the map f is not injective, since two elements of its domain have the same image.

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From injection to inclusion ----------------------------

If, on the other hand, all the inhabitants of V have different names, the map is said to be injective. Each inhabitant can then be called by name without any risk of confusion! It is only a small—and easily taken—step from there to regarding the inhabitants of V as nothing more than names. And if, instead of names, the inhabitants of V were assigned numbers, they would be entitled to exclaim, like Patrick Mac Goohan, Number 6 in the village where he was held prisoner in The Prisoner, the famous television series of the 1960s: "I am not a number, I am a man!"