Intersection set theory: Encyclopedia II - Intersection set theory - Arbitrary intersections

Intersection set theory - Arbitrary intersections

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 A_i", which refers to the intersection of the collection {A_i : i ∈ I}. Here I is a nonempty set, and A_i 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:

When formatting is difficult, this can also be written "A₁ ∩ A₂ ∩ A₃ ∩ ...", even though strictly speaking, A₁ ∩ (A₂ ∩ (A₃ ∩ ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for an example see the article on σ-algebras.)

Finally, let us note that whenever the symbol "∩" is placed before other symbols instead of between them, it should be of a larger size. (Eventually this will be available in HTML as the character entity &bigcap;, but until then, try <big>&cap;</big>.)

Adapted from the Wikipedia article "Arbitrary intersections", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki