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Exercise 4. Prove that if f : A → B and g : B → C are bijective, then (gof) : A→C is bijective. then r 3 for all intogomo Exercise 5, Prove that if T. = 1 and r

Exercise 4. Prove that if f : A → B and g : B → C are bijective, then (gof) : A→C is bijective. then r 3 for all intogomo Exercise 5, Prove that if T. = 1 and r

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 29E
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Exercise 4. Prove that if f : A → B and g :B → C are bijective, then (go f) : A→C is bijective. Exercise 5. Prove that if x1 1 and xn+1 V6 + xn, then In <3 for all integers n > 1. Exercise 6. Let ECR such that E is nonempty and bounded. Prove that the set -E = {-x : xE E} satisfies inf(E) = – sup(-E). Exercise 7. If x is a rational number and y is irrational, prove that x +y and - y are irrational. Exercise 8. If x, y ER with x < y, prove that x < tx + (1 – t)y < y for all t E (0, 1). Exercise 9. Prove that if A and B are countable sets, then A x B and AU B are countable. Exercise 10. Is the set of all finite sequences of 0s and 1s countable? Justify your answer. Exercise 11. Is the set of all sequences of Os and ls countable? Justify your answer.
Transcribed Image Text:Exercise 4. Prove that if f : A → B and g :B → C are bijective, then (go f) : A→C is bijective. Exercise 5. Prove that if x1 1 and xn+1 V6 + xn, then In <3 for all integers n > 1. Exercise 6. Let ECR such that E is nonempty and bounded. Prove that the set -E = {-x : xE E} satisfies inf(E) = – sup(-E). Exercise 7. If x is a rational number and y is irrational, prove that x +y and - y are irrational. Exercise 8. If x, y ER with x < y, prove that x < tx + (1 – t)y < y for all t E (0, 1). Exercise 9. Prove that if A and B are countable sets, then A x B and AU B are countable. Exercise 10. Is the set of all finite sequences of 0s and 1s countable? Justify your answer. Exercise 11. Is the set of all sequences of Os and ls countable? Justify your answer.
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