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,...
Please provide detailed, line-by-line proofs. Thanks! 1. What is øer) is an injection Л X x(g(f(x)) = x)) (Hint: try to construct one such g) 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 x N : (n-ОЛУ-1) V (n 0A3y, E n(n-1, y,) є f Лу-у, + y'*)}, .) and show that this is a function bv induction 1. What is øer) is an injection Л X x(g(f(x)) =...
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 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...
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote xi for the ith element a the sequence x E S. Take for any x EL (i) Show that lic 12 (where recall 1-(x є s i Izel < oo)) (ii) Is l? Prove this or find a counterexample to show that these two sets do not coinside (iii) ls e c loc where recall looー(x є sl...
Solve the Dirichlet problem in an infinite strip uxx + uyy=0 for x ϵ R and 0 <y <b , u(x,0)=f(x) , u(x,b)=g(x). (Hint: first do the case f=0. The case g=0 reduces to this one by the substitution y→ b-y , and the case general is obtained by superposition) 4. Solve the Dirichlet problem in an infinite strip: uxx + Uyy 0 <у<b, u(x, 0) — S(x), и(х, b) — g(x). (Hint: First do the case The case g...
Use the definition of the fact that f(x) is O(g(x)) to show that: 1. 7x^2 is O(x^2) 2. x^4 + 9x^3 + 4x + 7 is O(x^4) 3. (x^2+1) / (x+1) is O(x) Hint: Try to simplify algebraic expression.
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...
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...