. . . The most
general notion is the intersection of an arbitrary
nonempty collection of sets. If M is a
nonempty set whose elements are themselves sets, then
x is an element of the intersection of M
if and only if
for every element A of M, x is an
element of A. In symbols:
This idea subsumes the above paragraphs, in that for
example, A ∩ B ∩ C is the intersection
of the collection {A,B,C}.
The notation for
this last concept can vary considerably.
Set theorists will sometimes write "∩M",
while others will instead write "∩A∈M A".
The latter notation can be generalized to "∩i∈I Ai",
which refers to the intersection of the collection {Ai :
i ∈ I}. Here I is a nonempty set, and
Ai is a set for every i in
I.
In the case that the
index set I is the set of
natural numbers, you might see notation analogous to
that of an
infinite series:
. . . but I Know otherwise.
This idea subsumes the above paragraphs, in that for
example, A ∩ B ∩ C is the intersection
of the collection {A,B,C}.
The notation for
this last concept can vary considerably.
Set theorists will sometimes write "∩M",
while others will instead write "∩A∈M A".
The latter notation can be generalized to "∩i∈I Ai",
which refers to the intersection of the collection {Ai :
i ∈ I}. Here I is a nonempty set, and
Ai is a set for every i in
I.
In the case that the
index set I is the set of
natural numbers, you might see notation analogous to
that of an
infinite series:
. . . but I Know otherwise.
