Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 6 - Set Theory - Exercise Set 6.1 - Page 350: 3

Answer

a. No, R ⊈ T because there are elements in R that are not in T. If R = {2, 4, 6, 8, 10, …} and T = {3, 6, 9, 12, 15, …}, then 2 is one example of an element in R that is not in T. b. Yes, T⊆R because all elements in T are divisible by 6, all elements in R are divisible by 2, and all elements divisible by 6 are divisible by 2. The set of elements in T can be described as z = 6a = 2(3a) = 2 * (some integer) The set of elements in R can be described as y = 2b = 2 * (some integer). The set of elements in T and R can both be described as 2 * (some integer). Therefore, T ⊆ R is true. c. Yes, T⊆S because all elements in T are divisible by 6, all elements in S are divisible by 3, and all elements divisible by 6 are divisible by 3. The set of elements in T can be described as z = 6a = 3(2a) = 3 * (some integer) The set of elements in S can be described as x = 3b = 3 * (some integer). The set of elements in T and R can both be described as 3 * (some integer). Therefore, T ⊆ S is true.

Work Step by Step

a. No, R ⊈ T because there are elements in R that are not in T. If R = {2, 4, 6, 8, 10, …} and T = {3, 6, 9, 12, 15, …}, then 2 is one example of an element in R that is not in T. b. Yes, T⊆R because all elements in T are divisible by 6, all elements in R are divisible by 2, and all elements divisible by 6 are divisible by 2. The set of elements in T can be described as z = 6a = 2(3a) = 2 * (some integer) The set of elements in R can be described as y = 2b = 2 * (some integer). The set of elements in T and R can both be described as 2 * (some integer). Therefore, T ⊆ R is true. c. Yes, T⊆S because all elements in T are divisible by 6, all elements in S are divisible by 3, and all elements divisible by 6 are divisible by 3. The set of elements in T can be described as z = 6a = 3(2a) = 3 * (some integer) The set of elements in S can be described as x = 3b = 3 * (some integer). The set of elements in T and R can both be described as 3 * (some integer). Therefore, T ⊆ S is true.
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