Definition·D020
Equivalence relation
A reflexive, symmetric, transitive relation: a notion of sameness.
A relation
on
is an equivalence relation when it is
In words
R is an equivalence relation when it is reflexive, symmetric, and transitive. Reflexive: for every a, if a lies in A then a is related to itself. Symmetric: for any a and b, if a is related to b then b is related to a. Transitive: for any a, b and c, if a is related to b and b to c then a is related to c.
Rests onno axioms yet
Never needed: F02 · F03 · F04 · F05 · F06 · F08 · F09 · F10 · F11 · F12 · F13 · F14 · A01 · A02 · A03 · A04 · A05 · A06 · A07 · A08 (computed from the citation graph, not asserted).
Remarks
The three laws are exactly what a notion of "sameness in some respect" must satisfy, and they are the laws of equality itself (F11, F12, F13) restated for a relation inside a set. The payoff of the definition is T04: an equivalence relation slices its set into disjoint classes of mutually equivalent elements. Examples in this library: parallelism of lines (T14), congruence modulo a subgroup (L29), and equinumerosity.