Equivalence Relation

An equivalence relation in mathematics is a special relation that has three basic properties: reflexive, symmetric, and transitive. These together ensure that the relation expresses 'equivalence' among the elements of the set.

The above formula shows all three conditions of the equivalence relation in one line. But it is more understandable if we describe the three properties separately.

This is reflexivity: every element is related to itself.

This is symmetry: if a is related to b, then b is related to a.

This is transitivity: if a is related to b and b to c, then a is related to c.

Examples of Equivalence Relations

• Parity relation on integers: even or odd. Two numbers are equivalent if they have the same parity.
• Congruence modulo n on integers: a ≡ b mod n if n divides (a - b).
• Same birthday relation among people: two people are equivalent if they were born on the same day.

Counterexamples (Non-Equivalence Relations)

• Less than (<) relation on natural numbers: not reflexive (no number is less than itself).
• Parent of relation among people: not symmetric (if A is parent of B, B is not parent of A).
• Friend of relation among people: not transitive (if A is friend of B and B of C, A may not be friend of C).

Equivalence Classes

An equivalence relation partitions the set into equivalence classes. An equivalence class contains all elements that are equivalent to each other. These classes are mutually disjoint and together cover the entire set.

For example, the 'same remainder mod 3' relation partitions the integers into three classes: {…, -6, -3, 0, 3, 6, …}, {…, -5, -2, 1, 4, 7, …}, {…, -4, -1, 2, 5, 8, …}.

Summary

An equivalence relation is thus a relation that is reflexive, symmetric, and transitive. These properties ensure that the set's elements can be divided into 'equivalent groups', i.e., equivalence classes. This concept plays a central role in mathematics, as many structures and concepts are built on it.

Practice Exercise