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In: the set {1,...,n} consisting of the positive integers 1 up to n (n included). P(S):...
Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of...
Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of S belongs to the set 10, 11,...,99) Use the Pigeonhole Principle to prove that this set S contains two distinct elements r and y, such that the sum of the two digits of r is equal to the sum of the two digits of y. Question 10: Let S be a set consisting of 9 people. Every person r in S has an age...
Let P(n) be the proposition that a set with n elements has 2" subsets. What would...
Let P(n) be the proposition that a set with n elements has 2" subsets. What would the basis step to prove this proposition PO) is true, because a set with zero elements, the empty set, has exactly 2° = 1 subset, namely, itself. 01 Ploi 2. This is not possible to prove this proposition. 3. po 3p(1) is true, we need to show first what happens a set with 1 element. Because, we can't do P(O), that is not allowed....
Question 5. Let r, n be positive integers. 1. (6 pts) Consider the random binary r...
Question 5. Let r, n be positive integers. 1. (6 pts) Consider the random binary r n matrix M , where each entry is equal to 0 or 1 with probability 1/2 (so each entry follows the Bernoulli random variable with parameter 1/2), and these entries are (jointly) independent random variables. What is the probability that each column in M has at most one entry with 1? 1 2. (*, 4 pts) Let S1, . . . , Srbe identically...
(1 point) Determine whether the given set S is a subspace of the vector space V....
(1 point) Determine whether the given set S is a subspace of the vector space V. A. V = R", and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed mxn matrix. B. V is the vector space of all real-valued functions defined on the interval (-oo, oo), and S is the subset of V consisting of those functions satisfying f(0) 0 C. V Mn (R), and S is the...
(A and C) Exercise 1.14. If n and k are integers, define the binomial coeffi- cient...
(A and C)
Exercise 1.14. If n and k are integers, define the binomial coeffi- cient (m), read n choose k, by n! if 0 <k <n, = 0 otherwise. k!(n - k)! (a) Prove that ("#") = (m) + (-2) for all integers n and k. (b) By definition, () = 1 if k = 0 and 0 otherwise. The recursion relation in (a) gives a computational procedure, Pascal's triangle, for calculating binomial coefficients for small n. Start with...
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, w...
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si
3. (20 pts) Let ụ be a finite set, and let...
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3}...
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
Question 29 1 pts In an application of the RSA cryptosystem, Bob selects positive integers p,...
Question 29 1 pts In an application of the RSA cryptosystem, Bob selects positive integers p, q, e, and d, where p and a are prime. He publishes public key (e, N), where N =p'q. the number d is the decryption key. 0 = (p-1)(q-1). Select all the statements that are correct. Ifm is not equal to por q, then (m) mod N=m It must be the case that d'e mod 0 - 1 If mis not equal to por...
1. Let U be the universal set with disjoint subsets A and B, where n(U-46, n(A-15,...
1. Let U be the universal set with disjoint subsets A and B, where n(U-46, n(A-15, and n(B-14. Find nAn B 2. A merchant surveyed 300 people to determine the way they leaned about an upcoming sale. The survey showed that 180 learned about the sale from the radio, 170 from television, 130 from the newspaper, 120 from radio and television, 70 from radio and newspapers, 80 from television and newspapers, and 60 from all three sources. How many people...
11. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g}...
11. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: A ∪ B 12. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: b. A ∩ B 13. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: AC...
Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of S belongs to the set 10, 11,...,99) Use the Pigeonhole Principle to prove that this set S contains two distinct elements r and y, such that the sum of the two digits of r is equal to the sum of the two digits of y. Question 10: Let S be a set consisting of 9 people. Every person r in S has an age...
Let P(n) be the proposition that a set with n elements has 2" subsets. What would the basis step to prove this proposition PO) is true, because a set with zero elements, the empty set, has exactly 2° = 1 subset, namely, itself. 01 Ploi 2. This is not possible to prove this proposition. 3. po 3p(1) is true, we need to show first what happens a set with 1 element. Because, we can't do P(O), that is not allowed....
(1 point) Determine whether the given set S is a subspace of the vector space V. A. V = R", and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed mxn matrix. B. V is the vector space of all real-valued functions defined on the interval (-oo, oo), and S is the subset of V consisting of those functions satisfying f(0) 0 C. V Mn (R), and S is the...
(A and C)
Exercise 1.14. If n and k are integers, define the binomial coeffi- cient (m), read n choose k, by n! if 0 <k <n, = 0 otherwise. k!(n - k)! (a) Prove that ("#") = (m) + (-2) for all integers n and k. (b) By definition, () = 1 if k = 0 and 0 otherwise. The recursion relation in (a) gives a computational procedure, Pascal's triangle, for calculating binomial coefficients for small n. Start with...
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si
3. (20 pts) Let ụ be a finite set, and let...
Let n > 1, and let S = {1, 2, 3}" (the cartesian product of {1,2,3} n times). (a) What is Sl? Give a brief explanation. (b) For 0 <k <n, let T be the set of all elements of S with exactly k occurrences of 3's. Determine |Tx I, and prove it using a bijection. In your solution, you need to define a set Ax that involves subsets and/or cartesian products with known cardinalities. Then clearly define your bijection...
Question 29 1 pts In an application of the RSA cryptosystem, Bob selects positive integers p, q, e, and d, where p and a are prime. He publishes public key (e, N), where N =p'q. the number d is the decryption key. 0 = (p-1)(q-1). Select all the statements that are correct. Ifm is not equal to por q, then (m) mod N=m It must be the case that d'e mod 0 - 1 If mis not equal to por...
1. Let U be the universal set with disjoint subsets A and B, where n(U-46, n(A-15, and n(B-14. Find nAn B 2. A merchant surveyed 300 people to determine the way they leaned about an upcoming sale. The survey showed that 180 learned about the sale from the radio, 170 from television, 130 from the newspaper, 120 from radio and television, 70 from radio and newspapers, 80 from television and newspapers, and 60 from all three sources. How many people...
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