Algebra Know-It-ALL

214 Mappings, Relations, and Functions

This mapping is an injection. The domain is the whole set X,and the range happens to be a proper subset

ofY, as shown in Fig. 13-6. This mapping is one-to-one, because for any x in the domain, there’s a single

y in the range, and vice-versa. But it’s not onto the entire set Y.

Now let’s consider a different mapping. If we take the reciprocal of any number x in set X, we get a

numbery in set Y that’s larger than 1. We can write

y= 1/x

This mapping is a bijection. No matter what x between 0 and 1 we choose, the reciprocal is always a

unique (one and only one, or exactly one) real number y larger than 1. Conversely, no matter what real

numbery larger than 1 we choose, we can always find a unique real number x between 0 and 1 that has

y as its reciprocal. This mapping, shown in Fig. 13-7, is a bijection, because every element in set Y is

accounted for.

1

2

3

0

4

5

1

2

3

0

4

5

SetX

SetY

y=x+ 1

Figure 13-6 An injection from

the set X of all reals

between, but not

including, 0 and 1 to

the set Y of all reals

strictly larger than 1.

Open circles indicate

points not in the

domain and range,

which are shown as

heavy gray lines.

1

2

3

0

4

5

1

2

3

0

4

5

SetX

SetY

y= 1/x

Figure 13-7 A bijection between

the set X of all reals

between 0 and 1 and

the set Y of all reals

strictly larger than 1.

Open circles indicate

points not included

in the domain and

range, which are

shown as heavy gray

lines.