Generating equivalence relationsIf two equivalence relations over the set X are given, then their intersection (viewed as subsets of X×X) is also an equivalence relation. This allows for a convenient way of defining equivalence relations: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. |
Equivalence relations are often used to group together objects that are similiar in some sense.
Every equivalence relation on X defines a partition of X into subsets called equivalence classes: all elements equivalent to each other are put into one class. Conversely, if a set can be partitioned into subsets, then we can define an equivalence relation R by the rule "a R b if and only if a and b lie in the same subset".
For example, if G is a group and H is a subgroup of G, then we can define an equivalence relation ~ on G by writing a ~ b if and only if ab-1 lies in H. The equivalence classes of this relation are the right coset?s of H in G.
If two equivalence relations over the set X are given, then their intersection (viewed as subsets of X×X) is also an equivalence relation. This allows for a convenient way of defining equivalence relations: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R.