(related to Problem: To Construct a Partition of a Given Set)

Creating Partitions Using Fibers of Surjective Functions

Creating Partitions Using Equivalence Relations

Differences between the two possibilities

A partition built by a given equivalence relation can itself be reversely used to define this equivalence relation, i.e. if we are given partition, then its elements can be re-interpreted as equivalence classes of an equivalence relation. In other words, partitions and equivalence relations are mathematical concepts, which are very closely related to each other.

This one-to-one correspondence between partitions and equivalence relations cannot be observed for partitions built by means of fibers of a given surjective function. For a given partition, it is usually not possible to find an appropriate surjective function. The reason for this is simple. A given partition of a set $X$ is independent of another set $Y$, which we need to define a function between $X$ and $Y.$

Thank you to the contributors under CC BY-SA 4.0!




  1. Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition