Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is numerical equality. Any number \(a\) is equal to itself (reflexive). If \(a = b\), then \(b = a\) (symmetric). If \(a = b\) and \(b = c\), then \(a = c\) (transitive).

Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.

Various notations are used in the literature to denote that two elements \(a\) and \(b\) of a set are equivalent with respect to an equivalence relation \(R;\) the most common are "\(a \sim b\)" and "\(a \equiv b\)", which are used when \(R\) is implicit, and variations of "\(a \sim_R b\)", "\(a \equiv_R b\)", or "\({a\mathop{R}b}\)" to specify \(R\) explicitly. Non-equivalence may be written "\(a \not\sim b\)" or "\(a \not\equiv b\)".

A binary relation \(\,\sim\,\) on a set \(X\) is said to be an equivalence relation if it is reflexive, symmetric and transitive. That is, for all \(a, b,\) and \(c\) in \(X:\)

• \(a \sim a\) (reflexivity).
• \(a \sim b\) if and only if \(b \sim a\) (symmetry).
• If \(a \sim b\) and \(b \sim c\) then \(a \sim c\) (transitivity).

\(X\) together with the relation \(\,\sim\,\) is called a setoid. The equivalence class of \(a\) under \(\,\sim,\) denoted \([a],\) is defined as \([a] = \{x \in X : x \sim a\}.\)

In relational algebra, if \(R\subseteq X\times Y\) and \(S\subseteq Y\times Z\) are relations, then the composite relation \(SR\subseteq X\times Z\) is defined so that \(x \, SR \, z\) if and only if there is a \(y\in Y\) such that \(x \, R \, y\) and \(y \, S \, z\). This definition is a generalisation of the definition of functional composition. The defining properties of an equivalence relation \(R\) on a set \(X\) can then be reformulated as follows:

• \(\operatorname{id} \subseteq R\). (reflexivity). (Here, \(\operatorname{id}\) denotes the identity function on \(X\).)
• \(R=R^{-1}\) (symmetry).
• \(RR\subseteq R\) (transitivity).

On the set \(X = \{a, b, c\}\), the relation \(R = \{(a, a), (b, b), (c, c), (b, c), (c, b)\}\) is an equivalence relation. The following sets are equivalence classes of this relation: \[[a] = \{a\}, ~~~~ [b] = [c] = \{b, c\}.\]

The set of all equivalence classes for \(R\) is \(\{\{a\}, \{b, c\}\}.\) This set is a partition of the set \(X\). It is also called the quotient set of \(X\) by \(R\).

The following relations are all equivalence relations:

• "Is equal to" on the set of numbers. For example, \(\tfrac{1}{2}\) is equal to \(\tfrac{4}{8}.\)
• "Is similar to" on the set of all triangles.
• "Is congruent to" on the set of all triangles.
• Given a function \(f:X \to Y\), "has the same image under \(f\) as" on the elements of \(f\)'s domain \(X\). For example, \(0\) and \(\pi\) have the same image under \(\sin\), viz. \(0\). In particular:
  • "Has the same absolute value as" on the set of real numbers
  • "Has the same cosine as" on the set of all angles.
  • Given a natural number \(n\), "is congruent to, modulo \(n\)" on the integers.
  • "Have the same length and direction" (equipollence) on the set of directed line segments.
  • "Has the same birthday as" on the set of all people.

• The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7.
• The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1.
• The empty relation R (defined so that aRb is never true) on a set X is vacuously symmetric and transitive; however, it is not reflexive (unless X itself is empty).
• The relation "is approximately equal to" between real numbers, even if more precisely defined, is not an equivalence relation, because although reflexive and symmetric, it is not transitive, since multiple small changes can accumulate to become a big change. However, if the approximation is defined asymptotically, for example by saying that two functions f and g are approximately equal near some point if the limit of f − g is 0 at that point, then this defines an equivalence relation.

• A partial order is a relation that is reflexive, antisymmetric, and transitive.
• Equality is both an equivalence relation and a partial order. Equality is also the only relation on a set that is reflexive, symmetric and antisymmetric. In algebraic expressions, equal variables may be substituted for one another, a facility that is not available for equivalence related variables. The equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class.
• A strict partial order is irreflexive, transitive, and asymmetric.
• A partial equivalence relation is transitive and symmetric. Such a relation is reflexive if and only if it is total, that is, if for all \(a,\) there exists some \(b \text{ such that } a \sim b.\) Therefore, an equivalence relation may be alternatively defined as a symmetric, transitive, and total relation.
• A ternary equivalence relation is a ternary analogue to the usual (binary) equivalence relation.
• A reflexive and symmetric relation is a dependency relation (if finite), and a tolerance relation if infinite.
• A preorder is reflexive and transitive.
• A congruence relation is an equivalence relation whose domain \(X\) is also the underlying set for an algebraic structure, and which respects the additional structure. In general, congruence relations play the role of kernels of homomorphisms, and the quotient of a structure by a congruence relation can be formed. In many important cases, congruence relations have an alternative representation as substructures of the structure on which they are defined (e.g., the congruence relations on groups correspond to the normal subgroups).
• Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle.
• Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.

If \(\,\sim\,\) is an equivalence relation on \(X,\) and \(P(x)\) is a property of elements of \(X,\) such that whenever \(x \sim y,\) \(P(x)\) is true if \(P(y)\) is true, then the property \(P\) is said to be well-defined or a class invariant under the relation \(\,\sim.\)

A frequent particular case occurs when \(f\) is a function from \(X\) to another set \(Y;\) if \(x_1 \sim x_2\) implies \(f\left(x_1\right) = f\left(x_2\right)\) then \(f\) is said to be a morphism for \(\,\sim,\) a class invariant under \(\,\sim,\) or simply invariant under \(\,\sim.\) This occurs, e.g. in the character theory of finite groups. The latter case with the function \(f\) can be expressed by a commutative triangle. See also invariant. Some authors use "compatible with \(\,\sim\)" or just "respects \(\,\sim\)" instead of "invariant under \(\,\sim\)".

More generally, a function may map equivalent arguments (under an equivalence relation \(\,\sim_A\)) to equivalent values (under an equivalence relation \(\,\sim_B\)). Such a function is known as a morphism from \(\,\sim_A\) to \(\,\sim_B.\)

A subset \(Y\) of \(X\) such that \(a \sim b\) holds for all \(a\) and \(b\) in \(Y\), and never for \(a\) in \(Y\) and \(b\) outside \(Y\), is called an equivalence class of \(X\) by \(\sim\). Let \([a] := \{x \in X : a \sim x\}\) denote the equivalence class to which \(a\) belongs. All elements of \(X\) equivalent to each other are also elements of the same equivalence class.

The set of all equivalence classes of \(X\) by \(\sim,\) denoted \(X / \mathord{\sim} := \{[x] : x \in X\},\) is the quotient set of \(X\) by \(\sim.\) If \(X\) is a topological space, there is a natural way of transforming \(X / \sim\) into a topological space; see Quotient space for the details.

The projection of \(\,\sim\,\) is the function \(\pi : X \to X/\mathord{\sim}\) defined by \(\pi(x) = [x]\) which maps elements of \(X\) into their respective equivalence classes by \(\,\sim.\)

Theorem on projections: Let the function \(f : X \to B\) be such that if \(a \sim b\) then \(f(a) = f(b).\) Then there is a unique function \(g : X / \sim \to B\) such that \(f = g \pi.\) If \(f\) is a surjection and \(a \sim b \text{ if and only if } f(a) = f(b),\) then \(g\) is a bijection.

The equivalence kernel of a function \(f\) is the equivalence relation ~ defined by \(x \sim y \text{ if and only if } f(x) = f(y).\) The equivalence kernel of an injection is the identity relation.

A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single element of P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their union is X.