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Solution: (related to "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 $Y.$

| | | | created: 2019-09-08 11:36:46 | modified: 2019-09-08 11:42:54 | by: bookofproofs | references: [8297]

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Bibliography (further reading)

[8297] Flachsmeyer, Jürgen: “Kombinatorik”, VEB Deutscher Verlag der Wissenschaften, 1972, Dritte Auflage