ch1ans.pdf - 1.2 Exercises 1 Let X be the set of all students at a university Let A be the set of students who are first-year students B the set of

ch1ans.pdf - 1.2 Exercises 1 Let X be the set of all...

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1.2 Exercises 1. Let X be the set of all students at a university. Let A be the set of students who are first-year students, B the set of students who are second-year students, C the set of students who are in a discrete mathematics course, D the set of students who are international relations majors, E the set of students who went to a concert on Monday night, and F the set of students who studied until 2 A.M. on Tuesday. Express in set theoretic notation the following sets of students: (a) All second-year students in the discrete mathematics course. Sample Solution. { x X : x B and x C } . (b) All first-year students who studied until 2 AM on Tuesday. (c) All students who are international relations majors and went to the concert on Monday night. (d) All students who studied until 2 AM on Tuesday, are second-year students, and are not international relations majors. (e) All first- and second-year students who did not go to the concert on Monday night but are international relations majors. (f) All students who are first-year international relations majors or who studied until 2 AM on Tuesday. (g) All students who are first- or second-year students who went to a concert on Monday night. (h) All first-year students who are international relations majors or went to a concert on Monday night. 1. (a) { x : x X and x B and x C } 1. (b) { x : x X and x A and x F } 1. (c) { x : x X and x D and x E } 1. (d) { x : x X and x F and x B and x 6∈ D } 1. (e) { x : x X and ( x A or x B ) and x 6∈ E and x D } 1. (f) { x X : ( x A and x D ) or x F } 1. (g) { x X : ( x A or x B ) and x E } 1. (h) { x X : x A and ( x D or x E ) } 3. Write three descriptions of the elements of the set { 2, 5, 8, 11, 14 } . 3. { 2 , 5 , 8 , 11 , 14 } ; { n N : n = 3 k - 1 for some k N such that 1 k 5 } ; or { n Z ; n = 3 k + 2 for some k Z such that - 1 < k < 5 } .
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5. Which of the following pairs of sets are equal? For each pair that is unequal, find an element that is in one but is not in the other. (a) { 0, 1, 2 } and { 0, 0, 1, 2, 2, 1 } (b) { 0, 1, 3, { 1, 2 }} and { 0, 1, 2, { 2, 3 }} (c) {{ 1, 3, 5 } , { 2, 4, 6 } , { 5, 5, 1, 3 }} and {{ 3, 5, 1 } , { 6, 4, 4, 4, 2 } , { 2, 4, 4, 2, 6 }} (d) {{ 5, 3, 5, 1, 5 } , { 2, 4, 6 } , { 5, 1, 3, 3 }} and {{ 1, 3, 5, 1 } , { 6, 4, 2 } , { 6, 6, 4, 4, 6 }} (e) and { x N : x > 1 and x 2 = x } (f) and {∅} 5. (a) equal 5. (b) not equal; 3 is an element of the first but not the second; 2 is an element of the second but not the first. 5. (c) equal 5. (d) not equal; { 6,4 } is an element of the second but not the first. 5. (e) equal 5. (f) not equal; is an element of the second but not the first. 7. Let A = { n : n N and n = 2 k + 1 for some k N } , B = { n : n N and n = 4 k +1 for some k N } , and C = { m N : m = 2 k - 1 and k N and k 1 } . Prove the following: (a) 35 A (b) 35 C (c) 35 6∈ B (d) A = C (e) B A (f) B C (g) B A (h) B C 7. (a) Solve 35 = 2 k + 1 and verify k 0. 7. (b) Solve 35 = 2 k - 1 and verify k 1.
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