Mathematics — Linear Algebra · equivalence-classes-pre-images
Problem
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 x∈U is contained in some pre-image, and that (b) any two different pre-images do not have any elements in common.