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)