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In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i.e., if the relation is written as ~ it holds for all a, b and c in X that
Equivalence relations are often used to group together objects that are similar 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 the set X can be partitioned into subsets, then we can define an equivalence relation ~ on X by the rule "a ~ 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 a'b^{1} lies in H. The equivalence classes of this relation are the right cosets of H in G.
If an equivalence relation ~ on X is given, then the set of all its equivalence classes is the quotient set of X by ~ and is denoted by X/~.
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.
Concretely, the equivalence relation ~ generated by R can be described as follows: a ~ b if and only if there exist elements x_{1}, x_{2},...,x_{n} in X such that x_{1} = a, x_{n} = b and such that (x_{i},x_{i+1}) or (x_{i+1},x_{i}) is in R for every i = 1,...,n1.
Note that the resulting equivalence relation can often be trivial! For instance, the equivalence relation ~ generated by the binary relation <= has exactly one equivalence class: x~y for all x and y. More generally, the equivalence relation will always be trivial when generated on a relation R having the "antisymmetric" property that, given any x and y, either x R y or y R x must be true.
In topology, if X is a topological space and ~ is an equivalence relation on X, then we can turn the quotient set X/~ into a topological space in a natural manner. See quotient space for the details.
One often generates equivalence relations to quickly construct new spaces by "glueing things together". Consider for instance the square X = [0,1]x[0,1] and the equivalence relation on X generated by the requirements (a,0) ~ (a,1) for all a in [0,1] and (0,b) ~ (1,b) for all b in [0,1]. Then the quotient space X/~ can be naturally identified with a torus: take a square piece of paper, bend it to glue together the upper and lower edge, then bend the resulting cylinder to glue together the two mouths.
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