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:
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, notation analogous to that of an infinite series may be seen:
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 (⋂).
Read more about this topic: Intersection (set Theory)
Famous quotes containing the word arbitrary:
“We do the same thing to parents that we do to children. We insist that they are some kind of categorical abstraction because they produced a child. They were people before that, and they’re still people in all other areas of their lives. But when it comes to the state of parenthood they are abruptly heir to a whole collection of virtues and feelings that are assigned to them with a fine arbitrary disregard for individuality.”
—Leontine Young (20th century)