Equivalence Relations and Partitions

Abstract Algebra · Axiom Academy

EXAMPLE Equivalence Relations & Partitions Constructing equivalence classes for congruence modulo 5. Fantastic! You've successfully built the equivalence classes for mod 5 and verified the required properties. Equivalence Relation: A relation is an equivalence relation if it is Reflexive ( a ~ a ), Symmetric (if a ~ b then b ~ a ), and Transitive (if a ~ b and b ~ c then a ~ c ). Equivalence Classes: For an element a , its equivalence class [a] is the set of all elements related to it. Partition: The set of all equivalence classes for a relation on a set S forms a partition of S . This means every element of S is in exactly one equivalence class. The classes are disjoint and their union is S . Congruence Modulo n: For any integer n > 1 , congruence modulo n is an equivalence relation on ℤ with exactly n distinct classes: [0], [1], ..., [n-1] . This set of classes is denoted ℤₙ .

This is the written version of the interactive lesson above. See the full Abstract Algebra course.