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# Bijections

Definition 5.4.1   The function is bijective if it is both injective and surjective, i.e. if every element of the range has a unique pre-image by in .

The mapping shown in Example 3.2 is a bijection from to : when we looked for a pre-image of an element in , we actually proved that every real number has a unique pre-image by . The following proposition is a direct consequence of Prop. 2.4 and Prop. 3.4.

Proposition 5.4.2   Let ,, be three sets and let and be two mappings. If and are bijective, then is a bijection.

The following proposition is a direct consequence of Prop. 2.5 and Prop. 3.5.

Proposition 5.4.3   Let ,, be three sets and let and be two mappings. If is a bijection, then is injective and is surjective.

Definition 5.4.4   Let be a set. The identity mapping of is the mapping such that for any , .

Definition 5.4.5   Let and be two sets and let be a mapping from to . The mapping is invertible if there exists a mapping such that and (see Figure 6). If these conditions hold, we say that is the inverse mapping of and we denote it by .

Example 5.4.6   Let such that and such that . We have:

Thus . We have two comments here:
1. The mapping is not an inverse mapping for .
2. This does not mean that is not invertible! Perhaps there exists another mapping for which the conditions hold. Actually, there is none, but we still did not prove it!

Example 5.4.7   Let such that . We look for a mapping such that the conditions of Definition 4.5 hold. First we have:

Now we must check whether the mapping verifies the second condition:

Thus . we proved that is invertible and that .

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root 2002-06-10