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An equivalence relation is a binary relation on a set that satisfies three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every element in the set is related to itself. For example, if 'a' is an element of the set, then 'a' is related to 'a'. Symmetry means that if an element 'a' is related to an element 'b', then 'b' is also related to 'a'. So, if 'a R b', then 'b R a'. Transitivity means that if an element 'a' is related to an element 'b', and 'b' is related to an element 'c', then 'a' must also be related to 'c'. This can be written as: if 'a R b' and 'b R c', then 'a R c'.

Equivalence relations partition a set into disjoint subsets called equivalence classes. All elements within an equivalence class are related to each other, and no element in one equivalence class is related to any element in a different equivalence class. A common example of an equivalence relation is the equality relation on any set, where every element is equal to itself, if a is equal to b, then b is equal to a, and if a is equal to b and b is equal to c, then a is equal to c. Another example is the relation "has the same remainder as when divided by n" for integers.