Definition 5.2.1
Let and be two sets and be a mapping from to . The mapping
is injective (or is one-to-one, or is an injection) if every element of the range has at most one pre-image in by .

This means that the following holds:

The contrapositive statement is as follows:

The diagram in Figure 3(a) determines an injection and the diagram in Figure 3(b) determines a non injective mapping.

Figure 3:
Injection or not.

Example 5.2.2
Let be a mapping from
to
such that for any real we have . Take
and
; then we have:

Therefore is an injection.

Example 5.2.3
The mapping
such that is not injective, as there exists pairs of distinct reals which have the same square. For instance,
.

Proposition 5.2.4
Let ,, be three sets and let
and
be two mappings. If and are injective, then is an injection.

Proof.
Let and be two elements such that
, i.e.
. As is injective, we have:
, and as is injective it follows that . This proves that is injective.

Proposition 5.2.5
Let ,, be three sets and let
and
be two mappings. If is an injection, then is an injection.

Proof.
We use a contrapositive argument.
Assume that is not an injection, i.e. that there exists two different elements and in such that
. Then
, i.e.
. This proves that is not an injection.