128 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4(e) (U;P= A,) n B = U,"=, (A, n B) [Generalized distributivitr, recaII
Exercise 1 l(c), Article 3.3.)
(f) (0; , A,) u B = OF=, (A, u B) [Recall Exercise 110, Article 3.3.1- (a) Prove that if {A,\ k = 1,2,3,.. .) is an increasing family of sets, then
(n?= A,) = A,. dl Example 2)
*(b) Prove that if {A,I k = 1,2,3,.. .) is a decreashg family of sets, then
(U?= A,) = A,.
(c) Prove that if {A, 1 k = 1,2,3,.. .) is a mutually disjoint family of sets, then
A, = 0.
(d) Prove or disprove: {A,lk = 1,2,3,.. .) is a mutually disjoint family of sets
if and only if A, = 0.
(e) Prove or disprov: If {A,l k = 1,2,3,.. .) is a family of sets, then for any pair
of positive integers m and n: - Let (A,lk = 1,2,3,.. .) be a collection of subsets of U = R:
*(a) Suppose each A, is an open and bounded interval, that is, of the form (a, 6).
Is it possible for A, to be an interval of another form? (Recall Definition
3, Article 1.1.)
(b) Suppose each A, is a closed and bounded interval, that is, of the form [a, b].
Is it possible for U,"=, A, to be an interval of another form?
(c) Suppose A, = 0. Must there exist some positive integer n such that
fl;= 1 A, = -a?
(d) Suppose U?=, A, = U. Need it be true that V = U;, , A, for some positive
integer n? - Let (A, (k = 1,2,3,... ) be a family of subsets of a universal set U. According to
Exercise 4(a), the set UP= A, contains each set in this family as a subset, whereas
by Exercise 4(b), the set ng, A, is contained in each set in the family.
(a) Prove that A, is the smallest subset of U having the property described
above. Specifically, prove that if B is any subset of U having the property that
A, c B for each k = 1,2,3,... , then U,"!, A, G B. (For this reason it is often
said that U,"=, A, is the least upper bound of the given family of sets)
*(b) Prove that ng, A, is the lurgast subset of U having the property described
.-- -. above. Specifically, prove that if C is any subset of U having the property that
C c A, for each k = 1,2,3,... , then C E OF=, A,. (For this reason it is often
said that ng =, A, is the greatest lower bound of the family (A, 1 k = 1,2,3,.. -1.)
The Limit Concept (Optional)
In the introduction to Chapter 2 we noted that the limit concept, especially
the epsilon-delta definition of limits, is one of the most difficult ideas for
most students of elementary calculus. At the same time, however, we sug-
gested that a thorough grounding in logic, especially quantification, could