Infinite union & infinite intersection

[kingwinner]

I don't quite understand the meaning of "infinite union" and "infinite intersection".

Is an infinite union
∞
U A_k
k=1

being defined as a limit

lim (A₁ U A₂ U ... U A_n) ?
n->∞

How about an infinite intersection?

Thanks!

 

[Hurkyl]

There are no limits involved. The definition of union is unchanged no matter how many different things you are taking the union of. Similarly for intersection.

Union and intersection are different from an operation such as addition.

Addition is only defined on a pair of arguments. You can add a finite number of arguments by repeated addition (because of associativity and commutativity), but you have to do something fundamentally different to generalize addition to infinitely many objects.

Unions and intersections, on the other hand, are inherently defined for any class of sets. The binary versions of union and intersection are just special cases where the class has two elements.


nitpick 1: set theory and category theory define notions of limit and colimit, but they have nothing to do with calculus
nitpick 2: the intersection of an empty class of sets is defined, but it's a proper class, not a set. Similarly, the union of a proper class of sets is sometimes a proper class[/size]

 

[kingwinner]

So an element x is in an infinite union of A_k if and only if the element x is in at least one of the A_k's, am I interpretating it correctly?

(But to have ∞, don't you have to relate it in some sense to the "limit"?)

Thanks!

 

[honestrosewater]

So an element x is in an infinite union of A_k if and only if the element x is in at least one of the A_k's, am I interpretating it correctly?

Right.

(But to have ∞, don't you have to relate it in some sense to the "limit"?)

The limit of what? You are not dealing with a distance function. You are dealing with a membership relation on a class of sets. The infinity symbol is used to denote the size or cardinality of your class of sets. Whether or not x is in A for any x and A is always well-defined. There is no question of whether or not an element is in a set. The question is just how many sets you are dealing with, and there is no problem with dealing with countably many sets. Consider the class of singletons whose members are all n in N, i.e., { {1}, {2}, {3}, ... }. How many sets are in this class? What is its union?

 

[Hurkyl]

In this context, the symbols

$$\bigcup_{n=0}^{\infty}$$​

and

$$\bigcup_{n \in \mathbb{N}}$$​

are synonymous.

 

[Poopsilon]

I apologize for necromancing this thread but one further clarification:

A countably infinite intersection of sets contains an element x if and only if every set in the intersection contains x.

Also in answer to honestrosewater's post about the set {{1}, {2}, {3},... }, just to make sure I understand this, there are countably many sets in this class, the union is N and the intersection is ø.

 

[natasha d]

it really does look look like a limit in the case of ∞ intersections, as in the sets are tending towards their intersection but not actually attaining it . Consider the intersection of the sets
∞
π (1-1/n, 2+ 1/n)
n=1
would the smallest set be an infinitesimally small ε on either side of the closed set [1,2], which would hence be their infinite intersection?