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Ordering Relations 199
We end this section by looking once again at how these new concepts apply
to the subset partial order on P (A), for any set A. It turns out that in this partial
order, least upper bounds and greatest lower bounds are our old friends unions
and intersections.

Theorem 4.4.11. Suppose A is a set, F ⊆ P (A), and F = ∅. Then the least
upper bound of F (in the subset partial order) is ∪F and the greatest lower
bound of F is ∩F.
Proof. See exercise 23.

Exercises

∗
1. In each case, say whether or not R is a partial order on A. If so, is it a total
order?
(a) A ={a, b, c}, R ={(a, a), (b, a), (b, b), (b, c), (c, c)}.
(b) A = R, R ={(x, y) ∈ R × R || x |≤| y |}.
(c) A = R, R ={(x, y) ∈ R × R || x | < | y | or x = y}.
2. In each case, say whether or not R is a partial order on A. If so, is it a total
order?
(a) A = the set of all words of English, R ={(x, y) ∈ A × A | the word
y occurs at least as late in alphabetical order as the word x}.
(b) A = the set of all words of English, R ={(x, y) ∈ A × A | the first
letter of the word y occurs at least as late in the alphabet as the first
letter of the word x}.
(c) A = the set of all countries in the world, R ={(x, y) ∈ A × A | the
population of the country y is at least as large as the population of the
country x}.
3. In each case find all minimal and maximal elements of B. Also find, if they
exist, the largest and smallest elements of B, and the least upper bound and
greatest lower bound of B.
(a) R = the relation shown in the following directed graph, B ={2, 3, 4}.