Discrete Mathematical Structures by Pearson

8-10 Discrete Mathematical Structures

(2) Let P = {1, 2, 3, 4, 5} be ordered as shown in Fig. 8.11( b) and let A = {3, 4} be its subset.

The upper bounds of A are 5, 6 and 7; the lower bounds are 1 and 2.

5

6

7

4

3

2

1

Figure 8.11( b)

8.6.1 Extremal Elements of Partially Ordered Sets

Let (A, ≤ ) be a poset with partial order ≤.

Maximal Elements: An element a ∈ A is called a maximal element of A if there is no element

c ∈ A such that a ≤ c.

Minimal Elements: An element b ∈ A is called a minimal element of A if there is no element

c ∈ A such that c ≤ b.

Note 8.3 There can be more than one maximal and minimal element in a poset.

Example 1 Consider the ...