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Ordering Relations 197
minimal element, because the set A ={y ∈ R | 6 ≤ y} is smaller – in other

words, A ⊆ A and A = A. But A is also not the smallest element, since

A ={y ∈ R | 7 ≤ y} is still smaller. In fact, this family has no smallest,
or even minimal, element. You’re asked to verify this in exercise 12. This
example shows that we must be careful when talking about the smallest set
with some property. There may be no such smallest set!
You have probably already guessed how to define maximal and largest ele-
ments in partially ordered sets. Suppose R is a partial order on A, B ⊆ A, and
b ∈ B. We say that b is the largest element of B if ∀x ∈ B(xRb), and it is a
maximal element of B if ¬∃x ∈ B(bRx ∧ b = x). Of course, these definitions
are quite similar to the ones in Definition 4.4.4. You are asked in exercise 14 to
work out some of the connections among these ideas. Another useful related
idea is the concept of an upper or lower bound for a set.

Definition 4.4.8. Suppose R is a partial order on A, B ⊆ A, and a ∈ A. Then
a is called a lower bound for B if ∀x ∈ B(aRx). Similarly, it is an upper bound
for B if ∀x ∈ B(xRa).

Note that a lower bound for B need not be an element of B. This is the only
difference between lower bounds and smallest elements. A smallest element
of B is just a lower bound that is also an element of B. For example, in part 1
of Example 4.4.5, we concluded that 7 was not a smallest element of the set
C ={x ∈ R | x > 7} because 7 /∈ C. But 7 is a lower bound for C. In fact, so
is every real number smaller than 7, but not any number larger than 7. Thus,
the set of all lower bounds of C is the set {x ∈ R | x ≤ 7}, and 7 is its largest
element. We say that 7 is the greatest lower bound of the set C.

Definition 4.4.9. Suppose R is a partial order on A and B ⊆ A. Let U be the
set of all upper bounds for B, and let L be the set of all lower bounds. If U has
a smallest element, then this smallest element is called the least upper bound
of B.If L has a largest element, then this largest element is called the greatest
lower bound of B. The phrases least upper bound and greatest lower bound
are sometimes abbreviated l.u.b. and g.l.b.

Example 4.4.10.

1. Let L ={(x, y) ∈ R × R | x ≤ y}, a total order on R. Let B ={1/n | n ∈
+
Z }={1, 1/2, 1/3, 1/4, 1/5,...}⊆ R. Does B have any upper or lower
bounds? Does it have a least upper bound or greatest lower bound?