Injections, Surjections, and Bijections Functions are frequently used in mathematics to define and describe certain rela- tionships between sets and other mathematical objects. In addition, functions can be used to impose certain mathematical structures on sets. In this section, we will study special types of functions that are used to describe these relationships that are called injections and surjections. Before defining these types of functions, we will revisit what the definition of a function tells us and explore certain functions with finite domains. Preview Activity 1 (Functions with Finite Domains) Let A and B be sets.

Given a function f :A B, we know the following: For every x E A. f(x e B. That is, every element of A is an input for the function f. This could also be stated as follows: For each x EA, there exists a y E B such that y f(x). For a given x E A. there is exactly one y E B such that y f(A). The definition of a function does not require that different inputs produce different outputs. That is, it is possible to have xi, x2 E A with xi x2 and for fCr2).

The amow diagram for the function f in Figure 6.5 illustrates such a function. Also, the definition of a function does not require that the range of the function must equal the codomain. The range is always a subset of the codomain, but these two sets are not required to be equal.

That is, if g A B, then it is possible to have a y B such that g() y for all x E A. The amow diagram for the function g in Figure 6.5 illustrates such a function.