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Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁W2, then (W₁,<) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ → g: W2 → W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.) W2 and

Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁W2, then (W₁,<) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ → g: W2 → W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.) W2 and

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 12EQ
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Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁W2, then (W₁,<) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ → g: W2 → W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.) W2 and
Transcribed Image Text:Exercise 2. Suppose that (W₁, <) and (W2, <) are well-orderings. (a) Prove that if there exists an order-preserving map f: W₁W2, then (W₁,<) is isomorphic to an initial segment of (W2, <). (b) Prove that if there exist order-preserving maps f: W₁ → g: W2 → W₁, then (W₁, <) and (W2, <) are isomorphic. (Hint: Apply the Comparability Theorem.) W2 and
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