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# Injections

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.

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.

Next: Surjections Up: Mappings Previous: Mappings   Contents
root 2002-06-10