Equivalence relations always create a partition of the set they are defined on, via a construction called equivalence classes. For the relation in the previous problem, the equivalence classes are the pre-images. Prove directly that the collection of pre-images partition $U$ by showing that (a) every $\vect{x}\in U$ is contained in some pre-image, and that (b) any two different pre-images do not have any elements in common.