Give examples of a universal set U and sets A, B and C such that each...

Transcribed Image Text:

Give examples of a universal set U and sets A, B and C such that each of the following sets contains exactly one element: An BOC. (AB)-C. (ANC)-B. (BOC)-A.A-(BUC), B- (AUC), C-(AUB), AUBUC. Draw the accompanying Venn diagram. 1.36. For a real number r. define S, to be the interval [r-1.r+2). Let A (1,3,4). Determine Usea Sua and OA Sw 1.37. Let A= (1, 2, 5). B= (0, 2, 4). C=(2, 3, 4) and S= (A. B. C). Determine Uxes X and nxes X. 1.38. For a real number r, define A, (2), B, as the closed interval [r-1.r+1] and C, as the interval (r, 00). For S (1, 2, 4), determine (a) Uses Au and Nuss Aa (b) Uaes Ba and Nes Ba (e) Uaes Co and Nes Ca 1.39. Let A = (a. b.....) be the set consisting of the letters of the alphabet. For a € A, let A, consist of a and the two letters that follow it, where A,= (y.. a) and A, = (z.a. b). Find a set SCA of smallest cardinality such that Uues A = A. Explain why your set S has the required properties. 1.40. For i Z, let A; = (i-1,i+1). Determine the following: S (a) 4₂ (b) (4,0 A..) (e) (A₂-142+1) jal inl 1.41. For each of the following, find an indexed collection (AleeN of distinct sets (that is, no two sets are equal) satisfying the given conditions. (a) (b) il A = (0) and UA, = [0, 1] A=(-1,0, 1) and UA, Z. 1.42. For each of the following collections of sets, define a set A, for each # EN such that the indexed collection (AnlaeN is precisely the given collection of sets. Then find both the union and intersection of the indexed collection of sets. (a) ([1,2+1). [1.2+1/2), [1, 2+1/3)....) (b) [(-1,2), (-3/2, 4). (-5/3,6). (-7/4, 8)....). 1.43. For r e R¹, let A, = (x € R: [x]<r). Determine UeR+ A, and eR- Ar. 1.44. Each of the following sets is a subset of A= (1, 2,..., 10): A₁ (1.5, 7, 9, 10), A₂= (1, 2, 3, 8, 9), A3 (2, 4, 6, 8, 9), A₁ = (2, 4, 8). As =(3, 6, 7), A6 (3, 8, 10). A, (4, 5,7,9), A = (4.5. 10), A9 = (4, 6, 8). A10=(5, 6, 10), A1 = (5. 8, 9), A12= (6, 7, 10), A₁ = (6,8,9). Find a set / C (1. 2..... 13) such that for every two distinct elements j, k el. A, A₁ = 0 and User A is maximum. 1.45. For ne N. let A, = (-1.2-). Determine Unes An and nex An- Give examples of a universal set U and sets A, B and C such that each of the following sets contains exactly one element: An BOC. (AB)-C. (ANC)-B. (BOC)-A.A-(BUC), B- (AUC), C-(AUB), AUBUC. Draw the accompanying Venn diagram. 1.36. For a real number r. define S, to be the interval [r-1.r+2). Let A (1,3,4). Determine Usea Sua and OA Sw 1.37. Let A= (1, 2, 5). B= (0, 2, 4). C=(2, 3, 4) and S= (A. B. C). Determine Uxes X and nxes X. 1.38. For a real number r, define A, (2), B, as the closed interval [r-1.r+1] and C, as the interval (r, 00). For S (1, 2, 4), determine (a) Uses Au and Nuss Aa (b) Uaes Ba and Nes Ba (e) Uaes Co and Nes Ca 1.39. Let A = (a. b.....) be the set consisting of the letters of the alphabet. For a € A, let A, consist of a and the two letters that follow it, where A,= (y.. a) and A, = (z.a. b). Find a set SCA of smallest cardinality such that Uues A = A. Explain why your set S has the required properties. 1.40. For i Z, let A; = (i-1,i+1). Determine the following: S (a) 4₂ (b) (4,0 A..) (e) (A₂-142+1) jal inl 1.41. For each of the following, find an indexed collection (AleeN of distinct sets (that is, no two sets are equal) satisfying the given conditions. (a) (b) il A = (0) and UA, = [0, 1] A=(-1,0, 1) and UA, Z. 1.42. For each of the following collections of sets, define a set A, for each # EN such that the indexed collection (AnlaeN is precisely the given collection of sets. Then find both the union and intersection of the indexed collection of sets. (a) ([1,2+1). [1.2+1/2), [1, 2+1/3)....) (b) [(-1,2), (-3/2, 4). (-5/3,6). (-7/4, 8)....). 1.43. For r e R¹, let A, = (x € R: [x]<r). Determine UeR+ A, and eR- Ar. 1.44. Each of the following sets is a subset of A= (1, 2,..., 10): A₁ (1.5, 7, 9, 10), A₂= (1, 2, 3, 8, 9), A3 (2, 4, 6, 8, 9), A₁ = (2, 4, 8). As =(3, 6, 7), A6 (3, 8, 10). A, (4, 5,7,9), A = (4.5. 10), A9 = (4, 6, 8). A10=(5, 6, 10), A1 = (5. 8, 9), A12= (6, 7, 10), A₁ = (6,8,9). Find a set / C (1. 2..... 13) such that for every two distinct elements j, k el. A, A₁ = 0 and User A is maximum. 1.45. For ne N. let A, = (-1.2-). Determine Unes An and nex An-

Answer rating: 100% (QA)

136 For a real number r define A to be the interval r1 r2 Let A 1 3 4 Determine A and U To determine A we substitute the values from A into the interv... View the full answer