An equivalence relation on a set 
is a subset of 
, i.e., a collection 
of ordered pairs of elements of 
, satisfying certain properties. Write "
" to mean 
is an element of 
, and we say "
is related to 
," then the properties are
1. Reflexive: 
for all 
,
2. Symmetric: 
implies 
for all 
3. Transitive: 
and 
imply 
for all 
,
where these three properties are completely independent. Other notations are often used to indicate a relation, e.g., 
or 
.
