4.2 INFINITE UNIONS AND INTERSECTIONS 125The collection d is also sometimes denoted {Ai),. or (Ai 1 i =
1,2,3,.. .).
EXAMPLE 1 (a) Let A, = {i) for each i = 1,2,3,.... Then d =
(Ail i E N} = {(I), (21, (31,.. .} is a collection of singleton sets. Note
that if positive integers i and j are two distinct indices, then A, n Aj =
@. For this reason we say that this family of sets is pairwise disjoint
(or mutually disjoint).
(b) Let B, = (1,2,3,... , ij for each i = 1, 2, 3,... , so that B =
((11, (1,2), {1,2,3),.. .). In this example, for any two indices i and j,
i < j implies Bi r Bj. For this reason B is called an increasing family
of sets.
(c) Let Ci= [i, GO) for each i= 1, 2, 3,.. ., so that %'= {c,li~N)
is a family of closed, unbounded intervals satisfying the condition i < j
implies C, 2 Cj. Any family of sets indexed by N possessing this prop-
erty is called a decreasing family of sets. In particular, a collection of
intervals satisfying this property is called a family of nested intervals.
(d) Let Di = [0, 1 - (lli)] for each i = 1,2, 3,.... Then each set Di
in the collection 9 = {Di},,, is a closed and bounded interval. Which
of the properties defined in (a), (b), and (c) does 9 possess?
The reader should easily grasp the meaning of expressions such as
A, u A, u.. u A,. [Calculate this expression in (a) of Example I] and
B, n B, n... n B,. [What does this set equal in (b) of Example I?] We
use the notation uy=, A, and r)y==, Bj, respectively, as shorthand repre-
sentation of the preceding two sets. This notation is also suggestive of the
next definition, in which we generalize union and intersection to certain
types of infinite collections of sets.
DEFINITION 2
Let &' = (A,[i = 1, 2, 3,.. .) be a collection of sets indexed by N. We define:
(a) The union of the collection d, denoted Us, Ai (also denoted Ui, A, and
U {Ail A, E d)) to be the set {XI x E A, for some i E N) = {XI 3i E N such that
x E Ai)
(b) The intersection of the collection d, denoted T)sl Ai (also denoted
nip A, a-nd n {A,I A, E d)) to be the set {XI x E A, for every i E N) =
{xlx~ Ai V~E N).If d is a collection of sets A,, A,, A,,... , with U as universal set, then
both U,p"= Ai and n.p"=, Ai are sets and both have U as universal set. The
former consists of all the elements in any of the sets A,, grouped together
into one set, whereas the latter consists only of those objects common to all.
the sets A,. Note also that the letter i, in this context, is a dummy variable
(recall the discussion following Definition 1, Article 3.2).