May 2024
Intermediate to advanced
687 pages
18h 35m
English
Content preview from Discrete Mathematical Structures by Pearson
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8-12 Discrete Mathematical Structures
8.6.3 Greatest Lower Bound or Infimum
Let ( P, ≤ ) be a poset and A ⊆ P. An element x ∈ P is called the greatest lower bound (GLB) of
A or infimum of A if x is the lower bound and y ≤ x for all lower bounds y of A.
A GLB is unique if it exists.
Example 1 Let (N, ≤ ) be the poset of natural numbers. Let A = {3, 5, 7, 9, 11}, then 1, 2 and 3
are lower bounds of A and GLB or inf (A) = 3.
Example 2 Consider the partially ordered set P = {1, 2, 3, 4, 6, 9, 12, 18, 36} with divisibility
relation ≤. Find GLB and LUB of the sets {9, 18} and {4, 6, 9}.
18
9
3
6
1
2
4
12
36
Figure 8.16
Solution: For the set {9, 18}, the upper ...
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