Abstract: 
This entry provides formulae for counting the number of equivalence
relations and partial equivalence relations over a finite carrier set
with given cardinality. To count the number of equivalence relations,
we provide bijections between equivalence relations and set
partitions, and then transfer the main results of the two AFP entries,
Cardinality of Set Partitions and Spivey's Generalized Recurrence for
Bell Numbers, to theorems on equivalence relations. To count the
number of partial equivalence relations, we observe that counting
partial equivalence relations over a set A is equivalent to counting
all equivalence relations over all subsets of the set A. From this
observation and the results on equivalence relations, we show that the
cardinality of partial equivalence relations over a finite set of
cardinality n is equal to the n+1th Bell number. 
BibTeX: 
@article{Card_Equiv_RelationsAFP,
author = {Lukas Bulwahn},
title = {Cardinality of Equivalence Relations},
journal = {Archive of Formal Proofs},
month = may,
year = 2016,
note = {\url{http://isaafp.org/entries/Card_Equiv_Relations.shtml},
Formal proof development},
ISSN = {2150914x},
}
