Definition·D008
Intersection
The members common to two sets, or to every set of a non-empty family.
For a non-empty family
, the intersection
collects the objects belonging to every member:
.
In words
x belongs to the intersection of A and B exactly when x belongs to A and x belongs to B. For a non-empty family, x belongs to the intersection of the family exactly when for every set A, if A is in the family then x is in A.
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
Both forms exist by Separation:
, and for non-empty
pick any
and carve
out of it; the choice of
does not matter, by Extensionality. Emptiness matters:
(
the empty set) would have to contain every set whatsoever, and no such set exists (T02), so the family intersection is only defined for non-empty families. Two sets with
are called disjoint.