Algebra Know-It-ALL

212 Mappings, Relations, and Functions

in set X corresponds to one, but only one, element y in set Y. A mapping of this type is called

aninjection. You may occasionally hear an injection called a one-to-one mapping, or simply

one-to-one. But that’s a little misleading, because an injection doesn’t necessarily involve all the

elements of either set X or set Y.

Surjection

The mapping in Fig. 13-4 is a little different. Here, elements of X are mapped to all the ele-

ments of Y. The domain is a subset of X,but the range is identical to Y. This type of mapping

is called a surjection. Because it maps elements of set X completely onto set Y, a surjection

is sometimes called an onto mapping, or simply onto. A surjection can be one-to-one, but it

doesn’t have to be, and in this example, it clearly isn’t!

Bijection

Figure 13-5 shows an example of a third type of mapping, called a bijection, between two sets

X and Y. This is an injection that is also a surjection. Another expression you might hear is, “A

bijection is both one-to-one and onto.” The old-fashioned term for a bijection is one-to-one cor-

respondence. That term is rarely used nowadays, because it sounds too much like “one-to-one,”

which some people call an injection.

Domain

Range = set Y= all possible

values of variable y

SetX= all possible

values of variable x

Figure 13-4 An example of a surjection. Every possible

elementy is accounted for, and is the result

of a mapping from at least one element x.