We have seen that sets are building blocks of mathematics and have said a little about functions between sets. We shall now look more closely at functions. For functions f : X → Y we define injective, surjective and bijective functions. These definitions allow us to compare sets and in the case of bijective functions allow us to say whether one set is just a relabelling of the elements of the other.

Furthermore, using the notion of a bijection we can define two different types of infinite sets, those for which we can count the elements, such as ℕ, and those for which we can't count, such as ℝ. Thus we have two types of infinity!

Injective functions

Definition 30.1

A function f : X → Y is called injective or one-to-one if, for all x1 ∈ X, x2 ∈ X, x1 ≠ x2 implies that f(x1) ≠ f(x2).

The definition says that if I take two elements of X, then their values under f are the same if and only if the elements are the same. What we do not want is, for example, f(3) = f(5).