Equivalence relation
From AoPSWiki
Let 
be a set. A binary relation 
on is said to be an equivalence relation if satisfies the following three properties:
1. For every element 
, 
. (Reflexive property)
2. If 
such that 
, then we also have 
. (Symmetric property)
3. If 
such that and 
, then we also have 
. (Transitive property)
Some common examples of equivalence relations:
- The relation
(equality), on the set of real numbers.
- The relation
(congruence), on the set of geometric figures in the plane.
- The relation (similarity), on the set of geometric figures in the plane.
- For a given positive integer
, the relation 
, on the set of integers. (Congruence mod n)
