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We explicitly describe here set-theoritic bijection.
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Let S be a finite set with cardinality n>0. a. Prove, by constructing a bijection, that...
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...
all parts A-E please. Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some numb...
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
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...
where Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.)...
where
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
Additional 9-14 Prove that the language {a"b" n, k ε N and n S k} is...
Additional 9-14 Prove that the language {a"b" n, k ε N and n S k} is not regular Hint: I go over this proof in the lecture. You can watch it again to make sure you follow it before doing it. 2 Additional 9-15 Prove that the language (a"b n, k E N and n Hint: a little different... 2 k Is not regular 3 Additional 9-16 Prove that the language (w w (a, b and w has an equal...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random...
(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random variables {Xn} with pdf, fx, (x) = xht where 1<x<. Obtain Fx (2) and hence find the limiting distribution of X, as noo. c) Consider a random sample of size n from Fx (x) = where - <I<0. Find the limiting distribution of Yn as n + if (a)' = n max{X1, X2, X3,...,xn). and X(n) [17 marks]
Please help me solve 3,4,5 3- For all n € N, let an = 1. Let...
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
Let Σ = {0, 1). (a) Give a recursive definition of Σ., the set of strings from the alphabet Σ. (b) Prove that for every n E N there are 2" strings of length n in '. (c) Give a recursive d...
Let Σ = {0, 1). (a) Give a recursive definition of Σ., the set of strings from the alphabet Σ. (b) Prove that for every n E N there are 2" strings of length n in '. (c) Give a recursive definition of I(s), the length of a string s E Σ For a bitstring s, let O(s) and I(s) be number of zeroes and ones, respectively, that occur in s. So for example if s = 01001, then 0(s)...
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...
all parts A-E please.
Problem 8.43. For sake of a contradiction, assume the interval (0,1) is countable. Then there exists a bijection f : N-> (0,1). For each n є N, its image under f is some number in (0, 1). Let f(n) :-0.aina2na3n , where ain 1s the first digit in the decimal form for the image of n, a2 is the second digit, and so on. If f (n) terminates after k digits, then our convention will be...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
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...
where
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) < 00, then {x E X (z) > 0} is a countable set. (HINT: Show that for every k E N the set {x E X | f(x) > k-1} is finite.) f(x)-sup f(x) | F is any finite subset of X TEF
Problem 36. Assume f : X → [0, oo]. Prove that if Σ f(x) 0} is a countable set. (HINT: Show that...
Additional 9-14 Prove that the language {a"b" n, k ε N and n S k} is not regular Hint: I go over this proof in the lecture. You can watch it again to make sure you follow it before doing it. 2 Additional 9-15 Prove that the language (a"b n, k E N and n Hint: a little different... 2 k Is not regular 3 Additional 9-16 Prove that the language (w w (a, b and w has an equal...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
(x-2) 5. a) Let S Prove that s? Po? n-1 b) Consider a sequence of random variables {Xn} with pdf, fx, (x) = xht where 1<x<. Obtain Fx (2) and hence find the limiting distribution of X, as noo. c) Consider a random sample of size n from Fx (x) = where - <I<0. Find the limiting distribution of Yn as n + if (a)' = n max{X1, X2, X3,...,xn). and X(n) [17 marks]
Please help me solve 3,4,5
3- For all n € N, let an = 1. Let S = {an in€ N}. 3-1) Use the fact that lim - = 0 and the result of Exercise 1 to show that 0 ES'. Ron 3-2) Use the result of Exercise 2 to show that S = {0}. 4- Prove that 4-1) N' = 0. 4-2) Q =R. 5- Recall that a set KCR is said to be compact if every open cover...
Let Σ = {0, 1). (a) Give a recursive definition of Σ., the set of strings from the alphabet Σ. (b) Prove that for every n E N there are 2" strings of length n in '. (c) Give a recursive definition of I(s), the length of a string s E Σ For a bitstring s, let O(s) and I(s) be number of zeroes and ones, respectively, that occur in s. So for example if s = 01001, then 0(s)...
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