Discrete Mathematical Structures by Pearson

8-14 Discrete Mathematical Structures

Define f: A → B by

f (1) = f, f (3) = {1}, f (4) = {2}, f (12) = {1, 2}

Then, f is an isomorphism from (A, ≤ ) to ( B, ≤ ′).

Figure 8.17 Hasse diagrams of sets A and B

4

{1}

{2}

{1,2}

12

3

1

(A,≤)

(B,≤′)

f

(A, ≤ ) (B, ≤′ )

Note 8.4 Let a poset A be isomorphic to a poset B and f: A → B be an isomorphism from A onto B.

Then

1. An element a ∈ A is minimal of A iff f (a) is the minimal of B.

2. An element a ∈ A is the greatest element of A iff f (a) is the greatest element of B.

8.9 WELLORDERED SET

A set P with an ordering relation is said to be well-ordered, if every non-empty subset of P has

a least element.

8.10 PROPERTIES ...