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 $f:X\to Y$ be a function, and $A\subseteq X$ a subset.

Let $x\in A$. Then

$$x\in A⟹f(x)\in f(A)$$

that is, $f(x)$ is in its image.

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

$$x\in f^{-1}(f(A))$$

Counterexample of equality

Note that $f(x)\in f(A)$ need not imply that $x\in A$: there can be some point $c\in X\setminus A$ such that $f(c)\in f(A)$. 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