Definition·D023
Partition
A division of a set into non-empty, non-overlapping pieces that cover it.
A set
of subsets of
is a partition of
when
In words
A collection of subsets of A is a partition when the empty set is not one of the pieces, the union of the pieces is all of A, and for any C and D, if C and D are both pieces and C and D are different then C and D have empty intersection: every element lies in exactly one piece.
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 clauses use the empty set, the union of a family and intersection. An equivalent reading: every element of
belongs to exactly one member of
. Partitions and equivalence relations are two views of one concept: the classes of an equivalence relation partition the set (T04), and conversely declaring two elements related when they share a piece recovers an equivalence relation. Counting a finite set piece by piece (L27) is the combinatorial use of partitions, and the engine inside Lagrange's theorem.
Used by
Propose an edit1 published revision