The preimage of the image of a set contains the set

up:: 020 MOC Mathematics

The intuition is that the preimage of a set can be bigger than the set itself (due to different points mapping to the same output).

Let be a function, and a subset.

Let . Then

that is, is in its image.

By the definition of the Preimage of Function , this is equivalent to

Counterexample of equality

Note that need not imply that : there can be some point such that . That is why The injective preimage of the image 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