Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.)

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 4CM: Use a software program or a graphing utility to write v as a linear combination of u1, u2, u3, u4,...
See similar textbooks
icon
Related questions
Question

Set theory 

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.)
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS