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An Introduction to Mathematical Reasoning by Peter J. Eccles

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9

Injections, surjections and bijections

In defining a function f: XY we insist that a unique element of Y is assigned to each element of X. However, we do not require that each element of Y is assigned to some element of X nor do we prevent the possibility of the same element of Y being assigned to several (or even all the) elements of X. By imposing additional conditions concerning the number of elements of X to which elements of Y are assigned we get functions with particular properties. In this chapter we consider functions with particularly good properties and in particular functions which are bijections for which we can define an inverse function.

9.1 Properties of functions

Definition 9.1.1 Suppose that f: XY is a function.

(i) ...

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