Algebra Know-It-ALL

Maximal domain and co-domain

The large rectangles in Fig. 13-1 contain more points than those inside the ovals. The top
rectangle is called the maximal domain of the mapping, and the bottom rectangle is called the
co-domain. Some, but not all, of the points inside the maximal domain and the co-domain are
actively involved in the mapping. In our example, the maximal domain of the mapping is the
set of all people in the upper van, whether they’re sending messages or not. The co-domain is
the set of all people in the lower van, whether they’re receiving messages or not.
The domain is a subset of the maximal domain. The range is a subset of the co-domain.
Sometimes, the domain and the maximal domain of a mapping are identical. If that were
the case in Fig. 13-1, then the hatched region would fill the top rectangle. Similarly, the co-
domain and the range of a mapping might be identical. If that were true in Fig. 13-1, then the
hatched region would fill the bottom rectangle.

Ordered pairs

When you have a mapping from the elements of one set to the elements of another set, you can
define the mapping in terms of ordered pairs. An ordered pair is an expression in parentheses

Mapping “Territories” 209

Maximal domain

Co-domain

Range

a

b

c

d

e

v

w

x

y z

f

Domain

Figure 13-1 A mapping between two sets.