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Exercise 1. Suppose that <₁ and < are linear orders on the sets A and B; and let f AB be order-preserving. (i) Prove that f is an injection. (ii) Prove that if f is a bijection, then f¹ BA is also order-preserving. Definition 1. If A and B are sets, then their symmetric difference is defined to be AAB = (AB) U (B \ A) i.e. AB is the set of elements which lie in exactly one of the sets A or B. Exercise 2. Let < be the relation on P(N) defined by A

Exercise 1. Suppose that <₁ and < are linear orders on the sets A and B; and let f AB be order-preserving. (i) Prove that f is an injection. (ii) Prove that if f is a bijection, then f¹ BA is also order-preserving. Definition 1. If A and B are sets, then their symmetric difference is defined to be AAB = (AB) U (B \ A) i.e. AB is the set of elements which lie in exactly one of the sets A or B. Exercise 2. Let < be the relation on P(N) defined by A

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 12EQ
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Exercise 1. Suppose that <₁ and < are linear orders on the sets A and B; and let f AB be order-preserving. (i) Prove that f is an injection. (ii) Prove that if f is a bijection, then f¹ BA is also order-preserving. Definition 1. If A and B are sets, then their symmetric difference is defined to be AAB = (AB) U (B \ A) i.e. AB is the set of elements which lie in exactly one of the sets A or B. Exercise 2. Let < be the relation on P(N) defined by A<B AB and min(AAB) € A. (i) Prove that is a linear ordering of P(N). is not a well-ordering of P(N). (ii) Prove that Exercise 3. Prove that if m, n Є w satisfy m < n, then there exists p E w such that n = m +p+. (Hint: Argue by induction on n.)
Transcribed Image Text:Exercise 1. Suppose that <₁ and < are linear orders on the sets A and B; and let f AB be order-preserving. (i) Prove that f is an injection. (ii) Prove that if f is a bijection, then f¹ BA is also order-preserving. Definition 1. If A and B are sets, then their symmetric difference is defined to be AAB = (AB) U (B \ A) i.e. AB is the set of elements which lie in exactly one of the sets A or B. Exercise 2. Let < be the relation on P(N) defined by A<B AB and min(AAB) € A. (i) Prove that is a linear ordering of P(N). is not a well-ordering of P(N). (ii) Prove that Exercise 3. Prove that if m, n Є w satisfy m < n, then there exists p E w such that n = m +p+. (Hint: Argue by induction on n.)
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