Discrete Mathematical Structures by Pearson

Ordered Sets and Lattices 8-9

Example: Let A = {a, b, c, …, z} be English alphabet, which is linearly ordered, that is, a ≤ b,

b ≤ c, …, y ≤ z. Then, the set A

n

= A × A × … × A (n factors) can be regarded as a set of all

words of length n. The lexicographic ordering on A

n

has the property that if v

1

< v

2

, then v

1

would

precede v

2

in the dictionary listing. Thus, grape < graph, man < map, can < cap and so on.

8.6 UPPER AND LOWER BOUNDS

Let ( P, ≤ ) be a partially order set and let A ⊆ P.

Definition 8.1 Upper Bound

An element x ∈ P is called an upper bound of A, if a ≤ x for every a ∈ A.

Example 1 Let B = {1, 2, 3}, then (P( B), ⊆) with set inclusion ...