Difference between revisions of "Equivalence relation"
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3. Transitive: if a~b and b~c, then a~c. | 3. Transitive: if a~b and b~c, then a~c. | ||
| − | [[Equality]] and set [[ | + | [[Equality]] and set [[member]]ship are the most basic equivalence relations, which is why they are often called trivial equivalence relations. Every equivalence relation is constructed from [[Irreducibility|irreducible]] equivalences. |
| − | [[Congruence|Congruence relations]] are examples of equivalence relations, as is the j-invariant of [[ | + | [[Congruence|Congruence relations]] are examples of equivalence relations, as is the j-invariant of [[elliptic curve]]s. |
| − | An equivalence relation on a set S [[ | + | An equivalence relation on a set S [[partition]]s S into [[disjoint]] subsets, known as '''equivalence classes'''. Two objects in the same equivalence class are said to be '''equivalent'''. |
| − | Equivalence relations are the mathematical foundation for forming [[ | + | Equivalence relations are the mathematical foundation for forming [[quotient]]s. |
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Latest revision as of 12:04, July 13, 2016
An equivalence relation is a mathematical relation (denoted by ~) which is reflexive, symmetric, and transitive.
1. Reflexive: a~a.
2. Symmetric: if a~b then b~a.
3. Transitive: if a~b and b~c, then a~c.
Equality and set membership are the most basic equivalence relations, which is why they are often called trivial equivalence relations. Every equivalence relation is constructed from irreducible equivalences.
Congruence relations are examples of equivalence relations, as is the j-invariant of elliptic curves.
An equivalence relation on a set S partitions S into disjoint subsets, known as equivalence classes. Two objects in the same equivalence class are said to be equivalent.
Equivalence relations are the mathematical foundation for forming quotients.