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: #m/thm/general

where and .

Proof

Let . Then , and thus . Therefore .

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

#state/tidy | #lang/en | #SemBr