An equivalence relation is a binary relation on a set that is reflexive, symmetric, and transitive. Reflexivity means that for all elements a, b in the...
An equivalence relation is a binary relation on a set that is reflexive, symmetric, and transitive.
Reflexivity means that for all elements a, b in the set, if a == b, then b == a.
Symmetry means that for all elements a, b in the set, if a == b, then b == a.
Transitivity means that for all elements a, b, c in the set, if a == b and b == c, then a == c.
Equivalence relations allow us to divide a set into disjoint equivalence classes. An equivalence class is a set of elements that are all equivalent to each other under the equivalence relation.
For example, in the set of real numbers, the equivalence relation "is equal to" is an equivalence relation. The equivalence classes are the sets of elements that are equal to each other under this equivalence relation.
The equivalence classes are often denoted by the symbol "=". For example, the equivalence class of the number 3 under the equivalence relation "is equal to" is the set {3}.
Equivalence relations can be used to partition sets into disjoint sets. A partition of a set is a set of disjoint sets that cover the entire set. An equivalence relation can be used to partition a set into its equivalence classes