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Surjections

Definition 5.3.1   The function is surjective (or onto) if each has at least one preimage .

The diagram in Figure 4(a) determines a surjection and the diagram in Figure 4(b) determines a non surjective mapping.

Example 5.3.2   Let such that . Take any real number (in the range), and look for a pre-image by . We have:

The real number is a pre-image of by .

Example 5.3.3   The mapping such that is not surjective, because a negative real number has no pre-image by .

Proposition 5.3.4   Let ,, be three sets and let and be two mappings. If and are surjective, then is an surjection.

Proof. Let be any element in . As is a surjection, there exists at least one element in such that . As is a surjection, this element has at least one pre-image , and this element is a pre-image of by . This shows that every element of has at least one pre-image in by , i.e. that is surjective.

Proposition 5.3.5   Let ,, be three sets and let and be two mappings. If is a surjection, then is a surjection.

Proof. Let be any element in . As is surjective, there exists at least one element such that . Thus is a pre-image of by . This shows that every element of has a pre-image by , i.e. is surjective.

Next: Bijections Up: Mappings Previous: Injections   Contents
root 2002-06-10