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Exercise 1. Suppose that

Exercise 1. Suppose that

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 1E
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Set theory

Exercise 1. Suppose that <A and <B 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 (BA) 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). (ii) Prove that is not a well-ordering of P(N). Exercise 3. Prove that if m, n Ew satisfy m < n, then there exists pЄ w such that nm+p+. (Hint: Argue by induction on n.)
Transcribed Image Text:Exercise 1. Suppose that <A and <B 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 (BA) 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). (ii) Prove that is not a well-ordering of P(N). Exercise 3. Prove that if m, n Ew satisfy m < n, then there exists pЄ w such that nm+p+. (Hint: Argue by induction on n.)
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