Definition·D022
Quotient set
The set of all equivalence classes.
the set of all equivalence classes of the equivalence relation
on
.
In words
The quotient of A by R is the set of those members C of the power set of A for which there is some a with a in A and C equal to the equivalence class of a: its elements are exactly the equivalence classes.
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
Each class
is a subset of
, hence a member of the power set
(Power set), and Separation collects exactly the classes. The quotient is the set-theoretic rendering of "
up to
-equivalence": passing from
to
merges equivalent elements into a single point. T04 shows
is a partition of
; the map
is the canonical surjection onto the quotient, used for instance in Lagrange's theorem where the classes are the cosets of a subgroup.
Used by
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