126 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4
EXAMPLE 2 Consider the collection of intervals % = (J, 1 n E N], where
each Jn = [l/n, 23. Calculate n:=, Jn and U;=, J,.
Solution First, note that the collection J? is an increasing family of inter-
vals, since if i < j, then l/j < l/i so that [lli, 21 G [llj, 21, and thus
Ji G Jj.
Now, to calculate J,, we must ask which real numbers are
common to all of the intervals J,. Surely, each such number is contained
in J, = [I, 21 and J,, in turn, is a subset of every other set in the col-
lection (since the collection is increasing). Hence we reason that
n:= J, = J, [note Exercise 5(a)].
On the other hand, to find U:=, J,, we must ask which real numbers
are in at least one of the sets J,. Note the pattern J, = [I, 21, J, u J, =
[i, 21, J, u J, u J, = [$, 21, and so on. Note especially that the right-
hand end point of each such interval is fixed at 2, so that we need worry
only about the left-hand end point. It is intuitively clear (although hard
to prove rigorously) that every positive real number less than or equal
to 2 will eventually fall into one of the intervals J,. Also, clearly no
negative number is in any of the J,'s. Thus the desired union equals
either (0,2] or [O, 21. Will zero eventually fall into an interval J, for
some sufficiently large n? The answer is "no!" Since 0 < l/n for any
positive integer n, no matter how large, then 0 4 [lln, 21 for any positive
integer n. We conclude that U:', Jn = (0,2].
The theoretical property needed to justify the "hard to prove" statement
in Example 2 is the so-called Archimedean property of the set N of all posi-
tive integers. This property (in one of its many forms) states that, to any
positive real number p, no matter how small, there corresponds a positive
integer n such that l/n < p. This property is, among other things, the basis
of the important fact that the infinite sequence (l/n} converges to zero in
the real number system. We will encounter the Archimedean property
again in Chapters 9 and 10.
As Example 2 suggests, infinite unions and intersections behave differ-
ently from finite unions and intersections in a variety of ways. Intuitive
expectations should be adjusted carefully to correspond to these differences.
Some cases in point are provided in Exercise 6.
One general property of infinite collections of sets, whose proof is as-
signed as Exercise 5(a), was hinted at in the solution to Example 2. Proofs
of other general properties of infinite collections of sets constitute Exercises
4, 5, and 7. The general approach to these proofs is outlined in the pre-
ceding article. You should keep in mind especially the "choose" approach
to proving inclusion and the "mutual inclusion" approach to proving equal-
ity of sets.