Image and preïmage

The preïmage of the image and image of the preïmage are not necessarily the identity

Given an arbitrary function , we understand the image and the preïmage . The result of composing these functions together is not necessarily the identity, but rather has the following properties: general

where and .

Proof

Let . Then , and thus . Therefore .

Similarly, let . It follows that there exists such that , whence .

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