Equivalence Relation Examples

Set Theory · Axiom Academy

EXAMPLE Working with Equivalence Relations Prove that congruence modulo 3 on integers is an equivalence relation and find its equivalence classes Excellent work! You've successfully proven an equivalence relation. Here's what we learned: Equivalence Relations: A relation must satisfy three properties: reflexive, symmetric, and transitive. Reflexive: Every element is related to itself. For congruence mod 3, n - n = 0 is always divisible by 3. Symmetric: If a ~ b, then b ~ a. Divisibility works both ways: if 3|(a-b), then 3|(b-a). Transitive: If a ~ b and b ~ c, then a ~ c. Adding divisibility conditions preserves the property. Equivalence Classes: Every equivalence relation partitions a set into disjoint classes. For mod 3, we get exactly 3 classes. This proof structure applies to all modular arithmetic equivalence relations. Try proving mod 5 for practice!

This is the written version of the interactive lesson above. See the full Set Theory course.