Definition·D021
Equivalence class
All the elements equivalent to a given one.
for an equivalence relation
on
and
; any
is called a representative of the class.
In words
The equivalence class of a is the set of those x in A such that a is related to x.
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
Exists by Separation. By reflexivity
, so classes are never empty. The fundamental fact, proved in T04, is that two classes are either identical or disjoint:
exactly when
. Classes let one treat "equivalent" as "equal": the class forgets everything about its representative except what the relation sees.