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...

• 46

This preview shows page 1 - 3 out of 46 pages.

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 } .
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.

#### You've reached the end of your free preview.

Want to read all 46 pages?

• Summer '10
• Halpern

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes