Chapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since

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1 Chapter 2 1. Prove or disprove: A (B C) = (A B) (A C)., since A ( B C) = A B C = A ( B C) = ( A B) ( A C) = ( A B) ( A C). 2. Prove that A B= A B by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side). Ans: A B A B : Let x A B. x A B, x A or x B, x A or x B, x A B. Reversing the steps shows that A B A B. 3. Prove that A B= A B by giving an element table proof. Ans: The columns for A B and A B match: each entry is 0 if and only if A and B have the value Prove that A B= A B by giving a proof using logical equivalence. A B={ x x A B} Ans: ={ xx A B} ={ x ( x A B) } ={ x ( x A x B) } ={ x ( x A) ( x B) } ={ xx A x B} ={ xx A x B} ={ x x A B}= A B 5. Prove that A B = A B by giving a Venn diagram proof. Ans:. Page 22

2 6. Prove that A (B C) = (A B) (A C) by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side). Ans: A (B C) (A B) (A C): Let x A (B C). x A and x B C, x A and x B, or x A and x C, x (A B) (A C). Reversing the steps gives the opposite containment. 7. Prove that A (B C) = (A B) (A C) by giving an element table proof. Ans: Each set has the same values in the element table: the value is 1 if and only if A has the value 1 and either B or C has the value Prove that A (B C) = (A B) (A C) by giving a proof using logical equivalence. A ( B C) ={ x x A ( B C) } Ans: ={ xx A x ( B C) } ={ xx A ( x B x C) } ={ x ( x A x B) ( x A x C) } ={ xx A B x A C} ={ xx ( A B) ( A C) } = ( A B) ( A C) 9. Prove that A (B C) = (A B) (A C) by giving a Venn diagram proof. Ans: Page 23

3 10. Prove or disprove: if A, B, and C are sets, then A (B C) = (A B) (A C).. For example, let A = {1,2}, B = {1}, C = {2}. 11. Prove or disprove A (B C) = (A B) C., using either a membership table or a containment proof, for example. Use the following to answer questions 12-15: Use a Venn diagram to determine which relationship,, =,, is true for the pair of sets. 12. A B, A (B A). Ans: =. 13. A (B C), (A B) C. Ans:. 14. (A B) (A C), A (B C). Ans: =. 15. (A C) (B C), A B. Ans:. Use the following to answer questions 16-20: In the questions below determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. 16. {,{ },{a},{{a}},{{{a}}},{,a},{,{a}},{,{{a}}},{a,{a}},{a,{{a}}},{{a},{{a}}}, {,a,{a}},{,a,{{a}}},{,{a},{{a}}},{a,{a},{{a}}},{,a,{a},{{a}}}}. Ans: Yes {,a,{a},{{a}}}. 17. {,{a}}. Ans: Yes, {a}. 18. {,{a},{,a}}. Ans: No, it lacks { }. 19. {,{a},{ },{a, }}. Ans: Yes, {{a, }}. 20. {,{a, }}. Ans: No, it lacks {a} and { }. Page 24

4 21. Prove that S T = S T for all sets S and T. Ans: Since S T = S T (De Morgan's law), the complements are equal. Use the following to answer questions 22-33: In the questions below mark each statement TRUE or FALSE. Assume that the statement applies to all sets. 22. A (B C) = (A B) C. 23. (A C) (B C) = A B. 24. A (B C) = (A B) (A C). 25. A (B C) = (A B) (A C). 26. A B A= A. 27. If A C = B C, then A = B. 28. If A C = B C, then A = B. 29. If A B = A B, then A = B. 30. If A B = A, then B = A. 31. There is a set A such that P(A) = A A = A. 33. Find three subsets of {1,2,3,4,5,6,7,8,9} such that the intersection of any two has size 2 and the intersection of all three has size 1. Ans: For example, {1,2,3}, {2,3,4}, {1,3,4}. Page 25

5 + 34. Find [ 1/ i,1/ i]. i = 1 Ans: [-1,1] Find ( 1, 1). i i = 1 Ans: Find [ 1, 1]. i i = 1 Ans: {1} Find ( i, ). i = 1 Ans:. 38. Suppose U = {1, 2,..., 9}, A = all multiples of 2, B = all multiples of 3, and C = {3, 4, 5, 6, 7}. Find C - (B - A). Ans: {4,5,6,7}. 39. Suppose S = {1, 2, 3, 4, 5}. Find PS ( ). Ans: 32. Use the following to answer questions 40-43: In the questions below suppose A = {x,y} and B = {x,{x}}. Mark the statement TRUE or FALSE 40. x B. 41. P(B). 42. {x} A B. 43. P(A) = 4. Page 26

6 Use the following to answer questions 44-51: In the questions below suppose A = {a,b,c}. Mark the statement TRUE or FALSE. 44. {b,c} P(A). 45. {{a}} P(A). 46. A. 47. { } P(A). 48. A A. 49. {a,c} A. 50. {a,b} A A. 51. (c,c) A A. Use the following to answer questions 52-59: In the questions below suppose A = {1,2,3,4,5}. Mark the statement TRUE or FALSE. 52. {1} P(A). 53. {{3}} P(A). 54. A. 55. { } P(A). Page 27

7 56. P(A). 57. {2,4} A A. 58. { } P(A). 59. (1,1) A A. Use the following to answer questions 60-62: In the questions below suppose the following are fuzzy sets: F = {0.7Ann,0.1Bill,0.8Fran,0.3Olive,0.5Tom}, R = {0.4Ann,0.9Bill,0.9Fran,0.6Olive,0.7Tom}. 60. Find F and R. Ans: F = { 03. Ann, 09. Bill, 02. Fran, 07. Olive, 05. Tom}, R ={ 06. Ann, 01. Bill, 01. Fran, 04. Olive, 03. Tom} 61. Find F R. Ans: {0.7Ann,0.9Bill,0.9Fran,0.6Olive,0.7Tom}. 62. Find F R. Ans: {0.4Ann,0.1Bill,0.8Fran,0.3Olive,0.5Tom}. Use the following to answer questions 63-72: In the questions below, suppose A = {a,b,c} and B = {b,{c}}. Mark the statement TRUE or FALSE 63. c A B. 64. P(A B) = P(B). 66. B A. Page 28

8 67. {c} B. 68. {a,b} A A. 69. {b,c} P(A). 70. {b,{c}} P(B). 71. A A. 72. {{{c}}} P(B). Use the following to answer questions 73-85: In the questions below determine whether the set is finite or infinite. If the set is finite, find its size. 73. {x x Z and x 2 < 10}. Ans: P({a,b,c,d}), where P denotes the power set. Ans: {1,3,5,7, }. Ans: Infinite. 76. A B, where A = {1,2,3,4,5} and B = {1,2,3}. Ans: {x x N and 9x 2 1 = 0}. Ans: P(A), where A is the power set of {a,b,c}. Ans: A B, where A = {a,b,c} and B =. Ans: 0. Page 29

9 80. {x x N and 4x 2 8 = 0}. Ans: {x x Z and x 2 = 2}. Ans: P(A), where A = P({1,2}). Ans: {1,10,100,1000, }. Ans: Infinite. 84. S T, where S = {a,b,c} and T = {1,2,3,4,5}. Ans: {x x Z and x 2 < 8}. Ans: Prove that between every two rational numbers a/b and c/d (a) there is a rational number. (b) there are an infinite number of rational numbers. Ans: (a) Assume a a c c a b + d ad + bc c. Then =. b d b 2 2bd d (b) Assume a c a c <. Let m 1 be the midpoint of b d, b d a mi midpoint of, 1.. b 87. Prove that there is no smallest positive rational number. a a a a Ans: If 0 <, then 0 < < < < < a b 4 b 3 b 2 b b. Use the following to answer questions 88-96:. For i > 1 let m i be the In the questions below determine whether the rule describes a function with the given domain and codomain. 88. f : N N where f ( n) = n. Ans: Not a function; f(2) is not an integer. 89. h : R R where hx ( ) = x. Ans: Function. Page 30

10 90. g : N N where g(n) = any integer > n. Ans: Not a function; g(1) has more than one value F : R R where F( x) =. x 5 Ans: Not a function; F(5) not defined F : Z R where F( x) = 2 x 5 Ans: Function F : Z Z where F( x) =. 2 x 5 Ans: Not a function; F(1) not an integer. x+ 2 if x G : R R where G( x) = x 1 if x 4 Ans: Not a function; the cases overlap. For example, G(1) is equal to both 3 and 0. 2 x if x f : R R where f( x) = x 1 if x 4 Ans: Not a function; f(3) not defined. 96. G : Q Q where G(p/q) = q. Ans: Not a function; f(1/2) = 2 and f(2/4) = Give an example of a function f : Z Z that is 1-1 and not onto Z. Ans: f(n) = 2n. 98. Give an example of a function f : Z Z that is onto Z but not 1-1. n Ans: f( n ) = Give an example of a function f : Z N that is both 1-1 and onto N. 2 n if n 0 Ans: f( n ) = 2n 1 if x> 0 Page 31

11 100. Give an example of a function f : N Z that is both 1-1 and onto Z. n, neven 2 Ans: f(n) = 2 n + 1, n odd Give an example of a function f : Z N that is 1-1 and not onto N. 2 n, n 0 Ans: f( n ) = 2n+ 1, n> Give an example of a function f : N Z that is onto Z and not 1-1. n, n odd Ans: f( n ) = 2 n-1, n even Suppose f : N N has the rule f(n) = 4n + 1. Determine whether f is 1-1. Ans: Yes Suppose f : N N has the rule f(n) = 4n + 1. Determine whether f is onto N. Ans: No Suppose f : Z Z has the rule f(n) = 3n 2 1. Determine whether f is 1-1. Ans: No Suppose f : Z Z has the rule f(n) = 3n 1. Determine whether f is onto Z. Ans: No Suppose f : N N has the rule f(n) = 3n 2 1. Determine whether f is 1-1. Ans: Yes Suppose f : N N has the rule f(n) = 4n Determine whether f is onto N. Ans: No. Page 32

12 109. Suppose f : R R where f(x) = x/2. (a) Draw the graph of f. (b) Is f 1-1? (c) Is f onto R? Ans: (a) (b) No. (c) No Suppose f : R R where f(x) = x/2. (a) If S = {x 1 x 6}, find f(s). (b) If T = {3,4,5}, find f 1 (T). Ans: (a) {0,1,2,3} (b) [6,12) Determine whether f is a function from the set of all bit strings to the set of integers if f(s) is the position of a 1 bit in the bit string S. Ans: No; there may be no 1 bits or more than one 1 bit Determine whether f is a function from the set of all bit strings to the set of integers if f(s) is the number of 0 bits in S. Ans: Yes Determine whether f is a function from the set of all bit strings to the set of integers if f(s) is the largest integer i such that the ith bit of S is 0 and f(s) = 1 when S is the empty string (the string with no bits). Ans: No; f not defined for the string of all 1's, for example S = Let f(x) = x 3 /3. Find f(s) if S is: (a) { 2, 1,0,1,2,3}. (b) {0,1,2,3,4,5}. (c) {1,5,7,11}. (d) {2,6,10,14}. Ans: (a) { 3, 1,0,2,9}. (b) {0,2,9,21,41}. (c) {0,41,114,443}. (d) {2,72,333,914}. Page 33

13 115. Suppose f : R Z where f(x) = 2x 1. (a) Draw the graph of f. (b) Is f 1-1? (Explain) (c) Is f onto Z? (Explain) Ans: (a) (b) No. (c) Yes Suppose f : R Z where f(x) = 2x 1. (a) If A = {x 1 x 4}, find f(a). (b) If B = {3,4,5,6,7}, find f(b). (c) If C = { 9, 8}, find f 1 (C). (d) If D = {0.4,0.5,0.6}, find f 1 (D). Ans: (a) {1,2,3,4,5,6,7}. (b) {5,7,9,11,13}. (c) ( 9/2, 7/2]. (d) Suppose g : R R where (a) Draw the graph of g. (b) Is g 1-1? (c) Is g onto R? x 1 gx ( ) = 2. Ans: (a) (b) No. (c) No. Page 34

14 x Suppose g : R R where gx ( ) = 2. (a) If S = {x 1 x 6}, find g(s). (b) If T = {2}, find g 1 (T). Ans: (a) {0,1,2}. (b) [5,7) Show that x = x. Ans: Let n = x, so that n 1 < x n. Multiplying by 1 yields n + 1 > x n, which means that n = x Prove or disprove: For all positive real numbers x and y, x y x y. : x = y = Prove or disprove: For all positive real numbers x and y, x y x y. : x x, y y ; therefore xy x y ; since x y least as great as xy, then xy x y. is an integer at 122. Suppose g : A B and f : B C where A = {1,2,3,4}, B = {a,b,c}, C = {2,7,10}, and f and g are defined by g = {(1,b),(2,a),(3,a),(4,b)} and f = {(a,10),(b,7),(c,2)}. Find f g. Ans: {(1,7),(2,10),(3,10),(4,7)} Suppose g : A B and f : B C where A = {1,2,3,4}, B = {a,b,c}, C = {2,7,10}, and f and g are defined by g = {(1,b),(2,a),(3,a),(4,b)} and f = {(a,10),(b,7),(c,2)}. Find f 1. Ans: {(2,c),(7,b),(10,a)}. Use the following to answer questions : In the questions below suppose g : A B and f : B C where A = B = C = {1,2,3,4}, g = {(1,4),(2,1),(3,1),(4,2)} and f = {(1,3),(2,2),(3,4),(4,2)} Find f g. Ans: {(1,2),(2,3),(3,3),(4,2)} Find g f. Ans: {(1,1),(2,1),(3,2),(4,1)} Find g g. Ans: {(1,2),(2,4),(3,4),(4,1)}. Page 35

15 127. Find g (g g). Ans: {(1,1),(2,2),(3,2),(4,4)}. Use the following to answer questions : In the questions below suppose g : A B and f : B C where A = {1,2,3,4}, B = {a,b,c}, C = {2,8,10}, and g and f are defined by g = {(1,b),(2,a),(3,b),(4,a)} and f = {(a,8),(b,10),(c,2)} Find f g. Ans: {(1,10),(2,8),(3,10),(4,8)} Find f 1. Ans: {(2,c),(8,a),(10,b)} Find f f 1. Ans: {(2,2),(8,8),(10,10)} Explain why g 1 is not a function. Ans: g 1 (a) is equal to both 2 and 4. Use the following to answer questions : In the questions below suppose g : A B and f : B C where A = {a,b,c,d}, B = {1,2,3}, C = {2,3,6,8}, and g and f are defined by g = {(a,2),(b,1),(c,3),(d,2)} and f = {(1,8),(2,3),(3,2)} Find f g. Ans: {(a,3),(b,8),(c,2),(d,3)} Find f 1. Ans: {(2,3),(3,2),(8,1)} For any function f : A B, define a new function g : P(A) P(B) as follows: for every S A, g(s) = {f(x) x S}. Prove that f is onto if and only if g is onto. Ans: Suppose f is onto. Let T P(B) and let S = {x A f(x) T}. Then g(s) = T, and g is onto. If f is not onto B, let y B f(a). Then there is no subset S of A such that g(s) = {y}. Use the following to answer questions : In the questions below find the inverse of the function f or else explain why the function has no inverse. Page 36

16 135. f : Z Z where f(x) = x mod 10. Ans: f 1 (10) does not exist f : A B where A = {a,b,c}, B = {1,2,3} and f = {(a,2),(b,1),(c,3)}. Ans: {(1,b),(2,a),(3,c)} f : R R where f(x) = 3x x Ans: f ( x ) = f : R R where f(x) = 2x. 1 1 Ans: f () does not exist. 2 { x - 2 if x f : Z Z where f(x) = x + 1 if x 4. Ans: f 1 (5) is not a single value Suppose g : A B and f : B C, where f g is 1-1 and g is 1-1. Must f be 1-1? Ans: No Suppose g : A B and f : B C, where f g is 1-1 and f is 1-1. Must g be 1-1? Ans: Yes Suppose f : R R and g : R R where g(x) = 2x + 1 and g f(x) = 2x Find the rule for f. Ans: f(x) = x + 5. Use the following to answer questions : In the questions below for each partial function, determine its domain, codomain, domain of definition, set of values for which it is undefined or if it is a total function: 143. f : Z R where f(n) = 1/n. Ans: Z, R, Z {0}, {0} f : Z Z where f(n) = n/2. Ans: Z, Z, Z, total function f : Z Z Q where f(m,n) = m/n. Ans: Z Z, Q, Z (Z {0}), Z {0}. Page 37

17 146. f : Z Z Z where f(m,n) = mn. Ans: Z Z, Z, Z Z, total function f : Z Z Z where f(m,n) = m n if m > n. Ans: Z Z, Z, {(m,n) m > n}, {(m,n) m n} For the partial function f : Z Z R defined by f( m, n) =, determine its 2 2 n m domain, codomain, domain of definition, and set of values for which it is undefined or whether it is a total function. Ans: Z Z, R, {(m,n) m n or m n}, {(m,n) m = n or m = n} Let f : {1,2,3,4,5} {1,2,3,4,5,6} be a function. (a) How many total functions are there? (b) How many of these functions are one-to-one? Ans: (a) 6 5 = 7,776. (b) = 720. Use the following to answer questions : In the questions below find a formula that generates the following sequence a 1,a 2,a ,9,13,17,21,. Ans: a n = 4n ,3,3,3,3,. Ans: a n = ,20,25,30,35,. Ans: a n = 5(n + 2) ,0.9,0.8,0.7,0.6,. Ans: a n = 1 (n 1)/ ,1/3,1/5,1/7,1/9,. Ans: a n = 1/(2n 1) ,0,2,0,2,0,2,... Ans: a n = 1 + ( 1) n ,2,0,2,0,2,0,... Ans: a n = 1 + ( 1) n. Page 38

18 157. Find the sum 1/4 + 1/8 + 1/16 + 1/ Ans: 1/ Find the sum Ans: Find the sum Ans: + (2 ) Find the sum 1 1/2 + 1/4 1/8 + 1/16... Ans: 2/ Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/ Ans: 8/ Find the summ Ans: 220, Find 164. Find i= 1 Ans: i i (( 2) 2 ). j j= 1 i= 1 Ans: 25. ij Rewrite ( i + 1) so that the index of summation has lower limit 0 and upper limit 7. 7 i= 0 2 Ans: ( i 3) + 1). 4 i= 3 Page 39

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