A theorem on equivalence relation states that for any set S, the set of equivalence classes of S under an equivalence relation R constitutes a partition of a set. Moreover, given any partition of a set, one can define an equivalence relation on the set.(adsbygoogle = window.adsbygoogle || []).push({});

What allows you to "create" a partition of a set, say the set of its equivalence classes? If the set is finite, then it is intuitively easy to see that a partition can be created since the elements of the set must eventually be exhausted. But if the set is infinite, say Z, then what guarantees we can create a partition, of say, cosets of some subgroup of Z.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Partitioning an infinite set

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**