Algebra Know-It-ALL

Types of Mappings 213

Are you confused?

In Fig. 13-3, the domain is shown as a proper subset of set X, and the range is shown as a proper subset
ofY. However, the domain could be the entire set X, or the range could be the entire set Y, or both. (If both
were true, we’d have a bijection, which is a special sort of injection!) In Fig. 13-4, the domain is shown as
a proper subset of X, but the range is shown as the entire set Y. Again, the domain could contain all of the
elements in set X.

Here’s a challenge!

LetX be the set of all real numbers x larger than 0 but smaller than 1. Let Y be the set of all real numbers
y strictly larger than 1. Give an example of an injection from X into Y. Give an example of a bijection
between X and Y.

Solution

If we add 1 to any number x in set X, we get a number y in set Y that’s larger than 1 but smaller than 2.
We can write

y=x+ 1

Range = set Y= all possible

values of variable y

For every x,there is

exactly one y

For every y,

there is

exactly one x

Domain = set X=

all possible values

of variable x

Figure 13-5 An example of a bijection. It is both an injection

and a surjection.