Mathematical logic and foundations Definition
Equivalence relation
Builds on Reflexive relationSymmetric relationTransitive relation
Definition
Let be a set. An equivalence relation on is a binary relation on that is reflexive, symmetric and transitive. For all :
- ;
- implies ;
- and imply .
Remarks
For a fixed integer , congruence modulo on is an equivalence relation: when divides . An equivalence relation cuts the set into equivalence classes, and these form a partition. Each function gives an equivalence relation. See equivalence relation induced by a function.
Reading path
To understand this page from the ground up, read these 14 pages in order. The foundations come first.
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- no evidence level
- Element definition
- Proposition concept
- Set definition
- Implication definition
- Predicate definition
- Ordered pair definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- Binary relation definition
- Reflexive relation definition
- Symmetric relation definition
- Transitive relation definition
- Equivalence relation definition
Where this leads
66 pages build on this one.
- A finite group is nilpotent iff it is the product of its Sylow subgroups
- Equivalence classes form a partition
- Torus
- Möbius strip
- Groups of prime order are cyclic
- Correspondence between equivalence relations and partitions
- Equivalence class
- Order of an element divides the order of the group
- Fixed points of a p-group action
- Second Sylow theorem
- Groups of order p squared are abelian
- The number of Sylow subgroups is the index of the normaliser
- Universal property of the quotient topology
- The alternating group A5 is simple
- Abelian simple groups have prime order
- Cauchy's theorem
- The number of Sylow subgroups divides the index
- Converse of Lagrange's theorem fails for the alternating group A4
- Third Sylow theorem
- Equivalence relation induced by a function
- Frattini argument
- Size of an orbit times the size of the stabiliser
- Path component
- Quotient topology
- First Sylow theorem
- Circle as a quotient of the real line
- Groups of order pq
- Burnside's lemma
- g to the order of the group is the identity
- Quotient set
- Relative CW complex
- Translation surface
- Weak ordering
- Discrete series representation
- Classification of surfaces
- Lagrange's theorem
- Oscillator representation
- Kolmogorov–Arnold–Moser theorem
- Galois group of a prime degree polynomial with two nonreal roots
- Lp space
- Nichols–Zoeller theorem
- Standard map
- Hall's theorem on solvable groups
- Proof that the order of an element divides the order of the group
- Arnold's cat map
- Integral operator
- Hall subgroup
- Sylow theorems
- Frattini subgroup
- Proof that groups of prime order are cyclic
- Action-angle coordinates
- Sinai billiard
- Milnor's isospectral flat tori
- Orbifold
- Torus knot
- Orientation of a manifold
- Real projective plane
- Liouville–Arnold theorem
- Proof of Lagrange's theorem
- Pseudo-Anosov map
- Geometric realization
- Torus as a group
- Bunimovich stadium
- Maximal torus
- Symmetric and self-reciprocal kernels
- Stone–von Neumann theorem
Further reading
- Web https://en.wikipedia.org/wiki/Equivalence_relation, lead section
About this page
| Tag | 0SGTMBEWRF |
|---|---|
| URI | m2://0SGTMBEWRF |
| Kind | definition |
| Writing | W1 stated |
| MSC | 03E20 |
| Defines | Equivalence relation |
| Uses | Reflexive relation, Symmetric relation, Transitive relation |
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What this page needs for W2
- edge instance_of in: one example (0 of 1)
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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