290 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8
contain only a countable number of sets. Thus the collections of sets con-
sidered in Article 4.2 were countably infinite collections only. Consider,
however, the collection consisting of all intervals of the form [r, a), where
r is an arbitrary real number. Since R is uncountable, this collection clearly
contains an uncountable number of sets and so represents for us a mathe-
matical object different from (although conceptually similar to) those stud-
ied in Article 4.2. All circumstances involving an infinite collection of sets
indexed by a set of unknown cardinality (possibly KO, c, 2" or any other
infinite cardinal number) are encapsulated under the heading arbitrary
collections of sets, or collections of sets indexed by an arbitrary indexing set.
DEFINITION 1
Let I be an arbitrary set. The collection of sets d = {A,ll E I), containing a set A,
corresponding to each element I E I (where some universal set U contains each
set A, in the collection) is called a family (or collection) of sets indexed by I.An arbitrary collection of sets may also be viewed as a mapping f: I +
U, where f(i2) = A, for each i2 E I, and where U is a set containing each
A, as an element. Clearly, countably infinite collections of sets are a special
case of arbitrary collections. As with all generalizations, you must be care-
ful in dealing with arbitrary collections not to attribute to them character-
istics that are particular to the special (i.e., countable) case. For example,
we cannot in general talk about "increasing" or "decreasing" families of
sets [recall Example l(b, c), Article 4.21, since there may be no notion of
"less than" in the indexing set. On the other hand, the idea of "pairwise
disjoint" [Example l(a), Article 4.21 does carry over to arbitrary collections
(Exercise 3); as do the concepts of union and intersection.DEFINITION 2
= (A,II E I) is a collection of sets indexed by the arbitrary set I, we define:
The union of the collection d, denoted U,,, A, (also denoted U (A,I
1 E I)), to be the set, {xlx E A, for some 1 E I); that is, (~(31, E I such that x E
A,).
The intersection of the collection d, denoted n,, , A, (also denoted
n {A,II E I)), to be the set, (xlx E A, for every 1 E I); that is, (xlx E A,V1 E I).With reference to terminology introduced earlier in this article, note that
we dealt in Article 4.2 with countable unions and countable intersections, the
preceding definition introduces arbitrary unions and arbitrary intersections.
From the point of view of the undergraduate student of mathematics,
the primary importance of arbitrary collections, including arbitrary unions
and intersections, lies in the formulation of statements and execution of
proofs of theorems involving arbitrary collections. The main difficulty for
most students is adapting to the notational requirements of arbitrary col-
lections, and especially, avoiding the temptation to represent all collections
of sets as if they were countable collections.