Equivalence relation

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

1. (Reflexivity) a ~ a
2. (Symmetry) if a ~ b then b ~ a
3. (Transitivity) if a ~ b and b ~ c then \a ~ c

Equivalence relations are often used to group together objects that are similar in some sense.

Table of contents

1 Examples of equivalence relations 2 Examples of relations that are not equivalences 3 Partitioning into equivalence classes 4 Generating equivalence relations 5 Common Notions in Euclid's Elements

Examples of equivalence relations

• The equality ("=") relation between real numbers or sets.
• The relation "is congruent to (modulo 5)" between integers.
• The relation "is similar to" on the set of all triangles.
• The relation "has the same birthday as" on the set of all human beings.
• The relation of logical equivalence on statements in first-order logic.
• The relation "is isomorphic to" on models of a set of sentences.

Examples of relations that are not equivalences

• The relation "≥" between real numbers is not an equivalence relation, because although it is reflexive and transitive, it is not symmetric. E.g. 7 ≥ 5 does not imply that 5 ≥ 7!
• The relation "has a common factor with" between natural numbers is not an equivalence relation, because although it is reflexive and symmetric, it is not transitive (2 and 6 have a common factor, and 6 and 3 have a common factor, but 2 and 3 do not have a common factor).
• The empty relation R on a non-empty set X (i.e. a R b is never true) is not an equivalence relation, because although it is vacuously symmetric and transitive, it is not reflexive (except when X is also empty).
• The relation "is approximately equal" between real numbers or other things, even if more precisely defined, is not an equivalence relation, because although it is reflexive and symmetric, it is not transitive (it may seem so at first sight, but many small changes can add up to a big change).

Partitioning into equivalence classes

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 ab^-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/~.

Generating equivalence relations

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 x₁, x₂,...,x_n in X such that x₁ = 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,...,n-1.

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 "gluing 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.

Common Notions in Euclid's Elements

The first person who introduced the idea of equivalence relations is Euclid in his book the Elements under Common Notions.

Common Notion 1. Things which equal the same thing also equal one another.

Nowadays, a binary relation is called Euclidean if it satisfies this property.

Unfortunately, he didn't mention symmetry or reflexitivity. But this suggests an alternative formulation: An equivalence relation is a relation which is Euclidean, symmetric and reflexive.