Page 212 - HOW TO PROVE IT: A Structured Approach, Second Edition

P1: PIG/ P2: OYK/
0521861241c04 CB996/Velleman October 20, 2005 2:54 0 521 86124 1 Char Count= 0

198 Relations
2. Let A be the set of all English words, and let R be the partial order on A
described after Example 4.4.3. Let B ={house, boat}. Does B have any
upper or lower bounds? Does it have a least upper bound or a greatest lower
bound?

Solutions
1. Clearly the largest element of B is 1. It is also an upper bound for B,asis
any number larger than 1. By definition, an upper bound for B must be at
least as large as every element of B, so in particular it must be at least as
large as 1. Thus, no number smaller than 1 is an upper bound for B, so the
set of upper bounds for B is {x ∈ R | x ≥ 1}. Clearly the smallest element
of this set is 1, so 1 is the l.u.b. of B.
Clearly 0 is a lower bound for B, as is any negative number. On the other
hand, suppose a is a positive number. Then for a large enough integer n
we will have 1/n < a. (You should convince yourself that any integer n
larger than 1/a would do.) Thus, it is not the case that ∀x ∈ B(a ≤ x), and
therefore a is not a lower bound for B. So the set of all lower bounds for B
is {x ∈ R | x ≤ 0}, and the g.l.b. of B is 0.
2. Clearly houseboat and boathouse are upper bounds for B. In fact, no shorter
word could be an upper bound, so they are both minimal elements of the
set of all upper bounds. According to part 2 of Theorem 4.4.6, a set that has
more than one minimal element can have no smallest element, so the set of
all upper bounds for B does not have a smallest element, and therefore B
doesn’t have a l.u.b.
The only letter that the words house and boat have in common is o, which
is not a word of English. Thus, B has no lower bounds.

Notice that in part 1 of Example 4.4.10, the largest element of B also turned
out to be its least upper bound. You might wonder whether largest elements are
always least upper bounds and whether smallest elements are always greatest
lower bounds. You are asked to prove that they are in exercise 20. Another
interesting fact about this example is that, although B did not have a smallest
element, it did have a greatest lower bound. This was not a coincidence. It
is an important fact about the real numbers that every nonempty set of real
numbers that has a lower bound has a greatest lower bound and, similarly,
every nonempty set of real numbers that has an upper bound has a least upper
bound. The proof of this fact is beyond the scope of this book, but it is important
to realize that it is a special fact about the real numbers; it does not apply to all
partial orders or even to all total orders. For example, the set B in the second
part of Example 4.4.10 had upper bounds but no least upper bound.