The image of the preimage of a set is contained in the set

up:: Image of Function

For a given set , not all of its points are reached by . Thus, premapping from then mapping back via may yield a smaller set than .

Let be a function, and a subset.

Let . Then, by the definition of Image of Function, we have that

By the definition of Preimage of Function, we have that this implies that .

Counterexample of equality

Note that need not imply that : for all points we have . That is why The surjective image of the preimage of a set is equal to the set.

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 01>)