Equivalence relation

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

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

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

${\displaystyle X}$ together with the relation ${\displaystyle \,\sim \,}$ is called a setoid. The equivalence class of ${\displaystyle a}$ under ${\displaystyle \,\sim ,}$ denoted ${\displaystyle [a],}$ is defined as ${\displaystyle [a]=\{x\in X:x\sim a\}.}$[1][2]

• 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 ${\displaystyle a,}$ there exists some ${\displaystyle b{\text{ such that }}a\sim b.}$[proof 1] 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 ${\displaystyle X}$ 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 ${\displaystyle \,\sim \,}$ is an equivalence relation on ${\displaystyle X,}$ and ${\displaystyle P(x)}$ is a property of elements of ${\displaystyle X,}$ such that whenever ${\displaystyle x\sim y,}$ ${\displaystyle P(x)}$ is true if ${\displaystyle P(y)}$ is true, then the property ${\displaystyle P}$ is said to be well-defined or a class invariant under the relation ${\displaystyle \,\sim .}$

A frequent particular case occurs when ${\displaystyle f}$ is a function from ${\displaystyle X}$ to another set ${\displaystyle Y;}$ if ${\displaystyle x_{1}\sim x_{2}}$ implies ${\displaystyle f\left(x_{1}\right)=f\left(x_{2}\right)}$ then ${\displaystyle f}$ is said to be a morphism for ${\displaystyle \,\sim ,}$ a class invariant under ${\displaystyle \,\sim ,}$ or simply invariant under ${\displaystyle \,\sim .}$ This occurs, e.g. in the character theory of finite groups. The latter case with the function ${\displaystyle f}$ can be expressed by a commutative triangle. See also invariant. Some authors use "compatible with ${\displaystyle \,\sim }$" or just "respects ${\displaystyle \,\sim }$" instead of "invariant under ${\displaystyle \,\sim }$".

More generally, a function may map equivalent arguments (under an equivalence relation ${\displaystyle \,\sim _{A}}$) to equivalent values (under an equivalence relation ${\displaystyle \,\sim _{B}}$). Such a function is known as a morphism from ${\displaystyle \,\sim _{A}}$ to ${\displaystyle \,\sim _{B}.}$

Let ${\displaystyle a,b\in X}$, and ${\displaystyle \sim }$ be an equivalence relation. Some key definitions and terminology follow:

A key result links equivalence relations and partitions:[5][6][7]

• An equivalence relation ~ on a set X partitions X.
• Conversely, corresponding to any partition of X, there exists an equivalence relation ~ on X.

In both cases, the cells of the partition of X are the equivalence classes of X by ~. Since each element of X belongs to a unique cell of any partition of X, and since each cell of the partition is identical to an equivalence class of X by ~, each element of X belongs to a unique equivalence class of X by ~. Thus there is a natural bijection between the set of all equivalence relations on X and the set of all partitions of X.

If ${\displaystyle \sim }$ and ${\displaystyle \approx }$ are two equivalence relations on the same set ${\displaystyle S}$, and ${\displaystyle a\sim b}$ implies ${\displaystyle a\approx b}$ for all ${\displaystyle a,b\in S,}$ then ${\displaystyle \approx }$ is said to be a coarser relation than ${\displaystyle \sim }$, and ${\displaystyle \sim }$ is a finer relation than ${\displaystyle \approx }$. Equivalently,

• ${\displaystyle \sim }$ is finer than ${\displaystyle \approx }$ if every equivalence class of ${\displaystyle \sim }$ is a subset of an equivalence class of ${\displaystyle \approx }$, and thus every equivalence class of ${\displaystyle \approx }$ is a union of equivalence classes of ${\displaystyle \sim }$.
• ${\displaystyle \sim }$ is finer than ${\displaystyle \approx }$ if the partition created by ${\displaystyle \sim }$ is a refinement of the partition created by ${\displaystyle \approx }$.

The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest.

The relation "${\displaystyle \sim }$ is finer than ${\displaystyle \approx }$" on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice.[8]

• Given any set ${\displaystyle X,}$ an equivalence relation over the set ${\displaystyle [X\to X]}$ of all functions ${\displaystyle X\to X}$ can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.
• An equivalence relation ${\displaystyle \,\sim \,}$ on ${\displaystyle X}$ is the equivalence kernel of its surjective projection ${\displaystyle \pi :X\to X/\sim .}$[9] Conversely, any surjection between sets determines a partition on its domain, the set of preimages of singletons in the codomain. Thus an equivalence relation over ${\displaystyle X,}$ a partition of ${\displaystyle X,}$ and a projection whose domain is ${\displaystyle X,}$ are three equivalent ways of specifying the same thing.
• The intersection of any collection of equivalence relations over X (binary relations viewed as a subset of ${\displaystyle X\times X}$) is also an equivalence relation. This yields a convenient way of generating an equivalence relation: given any binary relation R on X, the equivalence relation generated by R is the intersection of all equivalence relations containing R (also known as the smallest equivalence relation containing R). Concretely, R generates the equivalence relation

${\displaystyle a\sim b}$ if there exists a natural number ${\displaystyle n}$ and elements ${\displaystyle x_{0},\ldots ,x_{n}\in X}$ such that ${\displaystyle a=x_{0}}$, ${\displaystyle b=x_{n}}$, and ${\displaystyle x_{i-1}\mathrel {R} x_{i}}$ or ${\displaystyle x_{i}\mathrel {R} x_{i-1}}$, for ${\displaystyle i=1,\ldots ,n.}$

The equivalence relation generated in this manner can be trivial. For instance, the equivalence relation generated by any total order on X has exactly one equivalence class, X itself.

• Equivalence relations can construct new spaces by "gluing things together." Let X be the unit Cartesian square ${\displaystyle [0,1]\times [0,1],}$ and let ~ be the equivalence relation on X defined by ${\displaystyle (a,0)\sim (a,1)}$ for all ${\displaystyle a\in [0,1]}$ and ${\displaystyle (0,b)\sim (1,b)}$ for all ${\displaystyle b\in [0,1],}$ Then the quotient space ${\displaystyle X/\sim }$ can be naturally identified (homeomorphism) with a torus: take a square piece of paper, bend and glue together the upper and lower edge to form a cylinder, then bend the resulting cylinder so as to glue together its two open ends, resulting in a torus.

Much of mathematics is grounded in the study of equivalences, and order relations. Lattice theory captures the mathematical structure of order relations. Even though equivalence relations are as ubiquitous in mathematics as order relations, the algebraic structure of equivalences is not as well known as that of orders. The former structure draws primarily on group theory and, to a lesser extent, on the theory of lattices, categories, and groupoids.

Equivalence relations are a ready source of examples or counterexamples. For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.

An implication of model theory is that the properties defining a relation can be proved independent of each other (and hence necessary parts of the definition) if and only if, for each property, examples can be found of relations not satisfying the given property while satisfying all the other properties. Hence the three defining properties of equivalence relations can be proved mutually independent by the following three examples:

• Reflexive and transitive: The relation ≤ on N. Or any preorder;
• Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation;
• Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3." Or any dependency relation.

Properties definable in first-order logic that an equivalence relation may or may not possess include:

• The number of equivalence classes is finite or infinite;
• The number of equivalence classes equals the (finite) natural number n;
• All equivalence classes have infinite cardinality;
• The number of elements in each equivalence class is the natural number n.

• Borel equivalence relation
• Cluster graph – Graph made from disjoint union of complete graphs
• Conjugacy class – In group theory, equivalence class under the relation of conjugation
• Equipollence (geometry) – Property of parallel segments that have the same length and the same direction
• Hyperfinite equivalence relation
• Quotient by an equivalence relation
• Topological conjugacy
• Up to – Mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)

• Brown, Ronald, 2006. Topology and Groupoids. Booksurge LLC. ISBN 1-4196-2722-8.
• Castellani, E., 2003, "Symmetry and equivalence" in Brading, Katherine, and E. Castellani, eds., Symmetries in Physics: Philosophical Reflections. Cambridge Univ. Press: 422–433.
• Robert Dilworth and Crawley, Peter, 1973. Algebraic Theory of Lattices. Prentice Hall. Chpt. 12 discusses how equivalence relations arise in lattice theory.
• Higgins, P.J., 1971. Categories and groupoids. Van Nostrand. Downloadable since 2005 as a TAC Reprint.
• John Randolph Lucas, 1973. A Treatise on Time and Space. London: Methuen. Section 31.
• Rosen, Joseph (2008) Symmetry Rules: How Science and Nature are Founded on Symmetry. Springer-Verlag. Mostly chapters. 9,10.
• Raymond Wilder (1965) Introduction to the Foundations of Mathematics 2nd edition, Chapter 2-8: Axioms defining equivalence, pp 48–50, John Wiley & Sons.

• "Equivalence relation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Bogomolny, A., "Equivalence Relationship" cut-the-knot. Accessed 1 September 2009
• Equivalence relation at PlanetMath
• OEIS sequence A231428 (Binary matrices representing equivalence relations)