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Exercise 1. Let (W₁, <1) and (W2, <2) be linear orders such that W₁ W20; and let be the linear ordering on W₁ × W2 defined by (a,b) and only if either: a <1 c; or ac and b <2 d. 0 and (c,d) if Prove that (W₁x W2, ) is a well-ordering if and only if (W₁, <1) and (W2, <2) are both well-orderings. Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁→ W2, then (W₁, <) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ - → W₂ and g: W2 W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.)

Exercise 1. Let (W₁, <1) and (W2, <2) be linear orders such that W₁ W20; and let be the linear ordering on W₁ × W2 defined by (a,b) and only if either: a <1 c; or ac and b <2 d. 0 and (c,d) if Prove that (W₁x W2, ) is a well-ordering if and only if (W₁, <1) and (W2, <2) are both well-orderings. Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁→ W2, then (W₁, <) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ - → W₂ and g: W2 W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 74EQ
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Exercise 1. Let (W₁, <1) and (W2, <2) be linear orders such that W₁ W20; and let be the linear ordering on W₁ × W2 defined by (a,b) and only if either: a <1 c; or ac and b <2 d. 0 and (c,d) if Prove that (W₁x W2, ) is a well-ordering if and only if (W₁, <1) and (W2, <2) are both well-orderings. Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁→ W2, then (W₁, <) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ - → W₂ and g: W2 W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.)
Transcribed Image Text:Exercise 1. Let (W₁, <1) and (W2, <2) be linear orders such that W₁ W20; and let be the linear ordering on W₁ × W2 defined by (a,b) and only if either: a <1 c; or ac and b <2 d. 0 and (c,d) if Prove that (W₁x W2, ) is a well-ordering if and only if (W₁, <1) and (W2, <2) are both well-orderings. Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁→ W2, then (W₁, <) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ - → W₂ and g: W2 W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.)
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