Definition·D076
Disjoint union
Combine two sets into one where every element remembers which side it came from, even if the original sets overlapped.
For sets
:
Write
for
and
for
.
In words
The disjoint union of two sets tags every element of the first set with 0 and every element of the second set with 1, then takes the union, so elements from A and B never collide, even if A and B shared elements to begin with.
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 the products
and
(each a set as usual, via Power Set and Separation) and union. The maps
are injective with disjoint images (L61):
contains a faithful, non-overlapping copy of each of
and
, unlike the ordinary union
, which silently merges any elements the two sets happen to share. The standard way to combine several separately-described sets that must not be confused with one another, e.g. building a formal alphabet out of variables, logical connectives, and a language's own symbols.