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1. What is øer) is an injection Л X x(g(f(x)) = x)) (Hint: try to construct one such g) ing the d...
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summatio...
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol, we see that f(n)- f(n-1)+f(n-1) hence we can define f as f-{(my) є N N : (n-ОЛ у-1) v (n > 0A3y, E N(n-1, y') є f Лу = y, +y®)), and show that this is a function by induction.)
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol,...
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summatio...
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol, we see that f(n)- f(n-1)+f(n-1) hence we can define f as f-{(my) є N N : (n-ОЛ у-1) v (n > 0A3y, E N(n-1, y') є f Лу = y, +y®)), and show that this is a function by induction.)
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol,...
1. What is ∅^(∅^(∅^∅))? ∅ is empty se...
1. What is ∅^(∅^(∅^∅))? ∅ is empty set and ^ means raised to 2. ∀X∀Y∀f ∈ Y X((f is an injection ∧ X ̸= ∅) =⇒ ∃g ∈ XY∀x ∈X (g(f(x)) = x)) (Hint: try to construct one such g) 3. ∃f ∈N^N(f(0) = 1∧∀n ∈N(n > 0 ⇒ f(n) =∑︀i=0to n-1 f(i)^2)) (Hint: Using the definition of summation symbol, we see that f(n) = f(n−1)+f(n−1)2, hence we can define f as f = {(n,y) ∈N×N : (n = 0∧y...
1. What is ∅^(∅^(∅^∅))? ∅ is empty se...
1. What is ∅^(∅^(∅^∅))? ∅ is empty set and ^ means raised to 2. ∀X∀Y∀f ∈ Y X((f is an injection ∧ X ̸= ∅) =⇒ ∃g ∈ XY∀x ∈X (g(f(x)) = x)) (Hint: try to construct one such g) 3. ∃f ∈N^N(f(0) = 1∧∀n ∈N(n > 0 ⇒ f(n) =∑︀i=0to n-1 f(i)^2)) (Hint: Using the definition of summation symbol, we see that f(n) = f(n−1)+f(n−1)2, hence we can define f as f = {(n,y) ∈N×N : (n = 0∧y...
1. Assume a consumer has as preference relation represented by u(c1, 2) for g E (0,...
1. Assume a consumer has as preference relation represented by u(c1, 2) for g E (0, 1) and oo > n > 2, with x E C = Ri. Answer thefollow (x1+x2)" ing: a. Show the preference relation that this utility function induces "upper b. Show the preference relation these preferences represent are strictly C. Give another utility function that generates exactly the same behavior as level sets that are convexif U(x) is Convex for any xeX monotonic. this one....
Please do exercise 129: Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N...
Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that t...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below we describe a ? test for the null and alternative hypotheses We divide the sample space into 5 disjoint subsets refered to as bins A1(-00,-2), A2 -(-2,-0.5), As -(-0.5,0.5), A4 (0.5,2) As -(2, oo). as functions of X, by Now, define discrete random variables For example, if Xi --0.1, then Xi є Аз and so Y;-3. In other words, Y, is the label of...
Solve f, g, l, m x{n - 1] = 1.30. Determine if each of the following...
Solve f, g, l, m
x{n - 1] = 1.30. Determine if each of the following systems is invertible. If it is, construct the inverse system. If it is not, find two input signals to the system that have the same output. (a) y(t) = x(t - 4) (b) y(t) = cos(x(t)] (c) y[n] = nx[n] (d) y(t) = x(t)dt ( x[n - 1], n>1 (e) y[n] = {0, n = 0 (f) y[n] = x[n]x[n – 1] ( x[n],...
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol, we see that f(n)- f(n-1)+f(n-1) hence we can define f as f-{(my) є N N : (n-ОЛ у-1) v (n > 0A3y, E N(n-1, y') є f Лу = y, +y®)), and show that this is a function by induction.)
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol,...
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol, we see that f(n)- f(n-1)+f(n-1) hence we can define f as f-{(my) є N N : (n-ОЛ у-1) v (n > 0A3y, E N(n-1, y') є f Лу = y, +y®)), and show that this is a function by induction.)
1. What is x(g(f(x)) = x)) (Hint: try to construct one such g) n-1 ing the definition of summation symbol,...
1. Assume a consumer has as preference relation represented by u(c1, 2) for g E (0, 1) and oo > n > 2, with x E C = Ri. Answer thefollow (x1+x2)" ing: a. Show the preference relation that this utility function induces "upper b. Show the preference relation these preferences represent are strictly C. Give another utility function that generates exactly the same behavior as level sets that are convexif U(x) is Convex for any xeX monotonic. this one....
Please do exercise 129:
Exercise 128: Define r:N + N by r(n) = next(next(n)). Let f:N → N be the unique function that satisfies f(0) = 2 and f(next(n)) =r(f(n)) for all n E N. 102 1. Prove that f(3) = 8. 2. Prove that 2 <f(n) for all n E N. Exercise 129: Define r and f as in Exercise 128. Assume that x + y. Define r' = {(x,y),(y,x)}. Let g:N + {x,y} be the unique function that...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
Please all thank you
Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
iid Let X1,, X, ^ X~P for some unknown distribution P with continuous cdf F. Below we describe a ? test for the null and alternative hypotheses We divide the sample space into 5 disjoint subsets refered to as bins A1(-00,-2), A2 -(-2,-0.5), As -(-0.5,0.5), A4 (0.5,2) As -(2, oo). as functions of X, by Now, define discrete random variables For example, if Xi --0.1, then Xi є Аз and so Y;-3. In other words, Y, is the label of...
Solve f, g, l, m
x{n - 1] = 1.30. Determine if each of the following systems is invertible. If it is, construct the inverse system. If it is not, find two input signals to the system that have the same output. (a) y(t) = x(t - 4) (b) y(t) = cos(x(t)] (c) y[n] = nx[n] (d) y(t) = x(t)dt ( x[n - 1], n>1 (e) y[n] = {0, n = 0 (f) y[n] = x[n]x[n – 1] ( x[n],...
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