4. Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, X = {2, 3, 4}, Y = {1, 4, 5}, Z = {2, 5, 7}. Find a) (X Y) b) X Y c) X (Y Z) d) (X Y) Z
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1 Exercises 1. Write formal descriptions of the following sets. a) The set containing the numbers 1, 10, and 100 b) The set containing all integers that are greater than 5 c) The set containing all natural numbers that are less than 5 d) The set containing the string aba e) The set containing the empty string f) The set containing nothing at all 2. Give a description of the following statements in your own words: = {x x A or x B} b) A B = {x x A and x B} c) A B = {x x A and x B} d) A B = {x (x A or x B) and x A B} e) Ā = { x x A } Or U A f) A g) P(A) or 2 A or 2 n 3. Let A be the set {x, y, z} and B be the set {x, y}. a) Is A a subset of B? (True or False) b) Is B a subset of A? (True or False) c) What is A B? d) What is A B? e) What is A B? f) What is the power set of B? 4. Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, X = {2, 3, 4}, Y = {1, 4, 5}, Z = {2, 5, 7}. Find a) (X Y) b) X Y c) X (Y Z) d) (X Y) Z 5. Let U = {1, 2, 3, 4}, A = {2, 3}, and B = {3, 4}. Draw Venn diagrams showing the sets: a) A b) (A B) c) A (B ) d) A B e) A B 1 P a g e
2 6. Let X = {1, 2, 3} and Y = {0, 2, 4, 6}. Give a) X Y (the union of X and Y) b) X Y (the intersection of X and Y) c) X Y (the complement of Y relative to X) d) Y X (the complement of X relative to Y) 7. Let X = {a, b} and Y = {1, 2, 3}. a) List all the subsets of X. b) List all the members of X Y. 8. Enumerate (write out all elements) each of the following sets: a) {a, b, {a, b}, {{a, b}}} {a, b} b) {{a}, {a, b}, {{c, d}, {d, e, f}}} c) 2 {1, 2, 3} {1, 3} 2 d) 2 {a,b} x 2 {c,d} e) 2 {} f) 2 {a,b} {a, b} g) {1} {1, 2} {1, 2, 3} = 9. How many elements are there in the set {, { }, {, }, {, { }}, {, {{, { }, {, { }}}}}}? 10. If A has a elements and B has b elements, how many elements are in the power set of A B? Explain your answer. 11. If X is a set with x elements, how many elements are in the power set of X? Explain your answer. 12. Examine the following formal descriptions of sets so that you understand which members they contain. Write a short informal English description for each set. a) {1, 3, 5, 7,...} b) {..., -4, -2, 0, 2, 4,...} c) {n n = 2m for m in N} d) {n n = 2m for m in N, and n = 3k for some k in N} e) {w w is a string of 0s and 1s and w is equals the reverse of w} f) {n n is an integer and n = n + 1} 13. Given U = {18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, A = {21, 23, 25, 27, 29}, and B = {26, 27, 28, 29}. Find A B. Hint: Do not skip to the answer. List A and B, then find their Union. 2 P a g e
3 14. Given U = {l, m, n, o, p, q, r, s, t, u, v, w}, A = {l, m, n, o, p, q}, B = {n, o, r, s, v, w}, and C = {l, m, p, q, r, t}, find (A C ) B. Hint: Do not skip to the answer. List A and C, then find their Union. Then find B. Lastly, find the intersection of (A C ) and B. 15. Given A = {1, 2, 3}, B = {3, 4, 5, 6}, and C = {3, 5, 7}. Evaluate each set b) A C c) A C d) B C e) (A B) C f) A (B C) g) (A B) C h) (A B) C 16. Given U = {All the lowercase letters of the alphabet}, A = {c, d, e, f}, and B = {e, f, g, h, k}. List the elements of set b) A B c) A B d) A B e) A B f) (A B ) B g) (A B) (A B ) 17. Given U = {All the lowercase letters of the alphabet}, A = {b, c, d} and B = {c, e, f, g}. List the elements of set b) A B c) A B d) A B e) A B f) (A B ) B g) (A B) (A B ) 18. Set K contains 30 elements, set J contains 46 elements, and 11 elements are common to both sets. Find the number of elements in the set (K J). 3 P a g e
4 19. For sets A and B, determine whether A = B, A is a subset of B, or B is a subset of A? A = {x x N and 13 < x < 18} B = {12, 13, 14, 15, 16, 17, 18} 20. If Set A has 9 letters and 3 numbers. Set B has 12 letters and 1 number. 9 letters and 1 number is common to both sets. What is the number of elements in set A or set B? 21. In a survey of a college with 50 members, 21 were taking mathematics, 36 were taking English, and 9 were taking both. How many were not taking either of these subjects? 22. Set D = {Lower cost, Educational, Less time to see the sights, Can visit relatives along the way} Set F = {Higher cost, Educational, More time to see the sights, Cannot visit relatives along the way} Set U = {Higher cost, Educational, More time to see the sights, Cannot visit relatives along the way, Lower cost, Less time to see the sights, Can visit relatives along the way} Find the set of elements common to both sets: F' and D' 23. In a group of 80 high school student, 30 study Spanish, 40 study French, and 25 study neither Spanish nor French. How many of 80 students study both Spanish and French? 24. The Wildlife Federation surveyed 140 San Diego residents. The results are as follows: 22 had a membership to the Birch Aquarium. 26 had a membership to the San Diego Zoo. 15 had both. Construct a Venn diagram in order to answer the following questions: i. How many had only a zoo membership? ii. How many had only an aquarium membership? iii. How many belonged to either one or the other or both? iv. How many belonged to neither? 25. In a survey of 100 college students, 35 were registered in college algebra, 52 were registered in history 1, and 18 were registered in both courses. How many students were registered in college algebra or history 1. And how many were not registered in either course. 26. A survey of a group of people produced the following results: there were 25 people with brown eyes and 15 people with blonde hair. If 10 people had both brown eyes and blonde hair and 23 people had neither, how many people were interviewed? 4 P a g e
5 27. In a survey of 75 resorts, it was reported that: 34 provided refrigerators in the guest rooms 30 provided laundry services 37 provided business centers 15 provided refrigerators in the guest rooms and laundry services 17 provided refrigerators in the guest rooms and business centers 19 provided laundry services and business centers 7 provided all three features. Construct a Venn Diagram and use it to answer the following questions: a) How many of the resorts provided only refrigerators in the guest rooms? b) How many of the resorts provided exactly one of the features? c) How many of the resorts provided at least one of the features? d) How many of the resorts provided exactly two of the features? e) How many of the resorts provided none of the features? 5 P a g e
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