Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
(REV)00 Edition Edition
ISBN: 9780395977224
Author: Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher: McDougal Littell
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[画像:Task 1. Sets and Counterexamples 1. For each of the following statements about sets A, B and C, give a counterexample to show that the statement is false. (a) If A E B and B C C, then ACC. (b) If A E B and B E C, then A E C. Task 2. Set Operations 1. Consider the symmetric difference of the sets A and B, which is defined as: AAB := (AUB) - (ANB) to answer the following questions. (a) Using a Venn Diagram, represent AAB. (b) Is the operation A commutative? Why or why not? (c) Give a sufficient and necessary condition for AAB = AUB. 2. For every iЄ N, A1 = (-1, 1⁄2), where A; denotes the open interval from - to. =: (a) Determine U1 A1 = A1 UA2 U A3 and 1 A1 = A1 NA2 NA3. Provide an explanation for your answer. (b) Determine U111 and Ai. Provide an explanation for your answer. Note: ∞ represents infinity. Task 3. Proofs 1. Prove or disprove (AB) = AUB. 2. Prove the following statements directly from the definitions of the terms. Do not use any other facts previously proved in class or in the textbooks or in the exercises. (a) Vn Є Z, n2 + n is even. (b) Va, b, c Z, if a|b and a/c, then a|(56+ 3c).]
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Transcribed Image Text:Task 1. Sets and Counterexamples 1. For each of the following statements about sets A, B and C, give a counterexample to show that the statement is false. (a) If A E B and B C C, then ACC. (b) If A E B and B E C, then A E C. Task 2. Set Operations 1. Consider the symmetric difference of the sets A and B, which is defined as: AAB := (AUB) - (ANB) to answer the following questions. (a) Using a Venn Diagram, represent AAB. (b) Is the operation A commutative? Why or why not? (c) Give a sufficient and necessary condition for AAB = AUB. 2. For every iЄ N, A1 = (-1, 1⁄2), where A; denotes the open interval from - to. =: (a) Determine U1 A1 = A1 UA2 U A3 and 1 A1 = A1 NA2 NA3. Provide an explanation for your answer. (b) Determine U111 and Ai. Provide an explanation for your answer. Note: ∞ represents infinity. Task 3. Proofs 1. Prove or disprove (AB) = AUB. 2. Prove the following statements directly from the definitions of the terms. Do not use any other facts previously proved in class or in the textbooks or in the exercises. (a) Vn Є Z, n2 + n is even. (b) Va, b, c Z, if a|b and a/c, then a|(56+ 3c).
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