Bridge to Abstract Mathematics: Mathematical Proof and Structures

4.2 INFINITE UNIONS AND INTERSECTIONS 127

EXAMPLE 3 Prove that if (A, 1 k = 1,2,3,... ) is a decreasing collection of

sets, then U,", , A, c A,.

Solution Let x E Ur= , A,, so that x E Aj for some positive integer j. We

must prove x E A,. Clearly either j = 1 or j > 1. If j = 1, then x E Aj =

A,, so x E A,, as desired. If j > 1, then Aj G A, (by the definition of

"decreasing family") so that x E Aj c A, and x E A,, again as desired. In

either case we have the desired conclusion x E A,, so that our theorem is

proved. 0

In Article 8.3 we will study cardinality of sets, a means of distinguishing

the "relative size" of infinite sets. At that point we will consider so-called

arbitrary collections of sets, that is, families of sets indexed by any set, not

necessarily N. In that context we will come to recognize the kinds of collec-

tions we have studied in this article as countably infinite collections of sets.

Exercises

1. Find U,"=, A, and n;", , A, for each of the collections of sets (A,[ k = 1,2,3,.. .)
   that follow:
   (4 Ak={-k) (b) Ak=(1,2,3 ,..., k)
   (c) Ak=(k,k+ 1,k+2, ...) (d) A, = (k, k + 1, k + 2,... ,2k)
   (4 A, = (0, ilk) (f) A, = [O, ilk]
   (9) Ak=(-l/k,l/k) (h) Ak = [O, 1 + llk)
   (i) A, = [lO/(k + I), 101 (I) A,=(-m,k]
   (k) A, = (- m, - k) (I) A, = (0, k - 1)
   (m) A, = [Mk + 21, (k + l)l(k + 211
   (n) Ak = [O, ll(k + 2)] u [(k + l)l(k + 2),^11


2. Label each of the collections in Exercise 1 as either increasing, decreasing,
   mutually disjoint, or none of the above.


3. Give examples other than those in the text or in Exercise 1 of collections of sets
   1' {~,(k = 1,2, 3,.. .) that are:

(a) Increasing

(c) Mutually disjpint

(b) Decreasing

(d) None of the above

4. Suppose that {A, I k = 1,2, 3,.. .) is an infinite collection of sets from a universal
   set U, B is a subset of U, and n is an arbitrary positive integer. Prove that:
   (a) A, c U,"=, A, [Recall Theorem 2(e), Article 3.3.1
   (b) n,", , A, c A, [Recall Theorem 2(d), Article 3.3.1
   (c) (nF= A,)' = U,"= , A; [Generalized De Morgan's law; see Theorem
   l(a), Article 3.3.1
   (d) (uG1 Ar)' = AA; [See Theorem l(b), Article 3.3.1