6. (Extra Credit) Let I be the interval (0,1). Define F(I) = {f:I+I:f is a function},...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
5. Recall that if the domain of a function f:B-C is the same as the codomain of a function g: A-B, we can define the composition of these functions fog:A-C given by fºg(a) = f(g(a)). (a) Prove that if f,g: A - A are bijections, then fog: A - A is a bijection. (b) If A is finite with n elements, how many bijections A - A are there? That is, how many elements are in the set Bij(A) :=...
hello sir, solve both questions Problem 5: Let f : A → B be a function, and let X-A and Y-B. Show that X S(x)) Problem 6: Recall that BA denot es the set of all functions A the function f : P(A) → {0,1}A by B. Fix a set A and defi ne f (X)Xx (the charact erist ic function), VX EP(A) Prove that f is a bijection
Let f : R → R , f ( x ) = x^2 ( x − 3). (a) Given a real number b , find the number of elements in f ^(-1) [ { b } ]. (The answer will depend on b . It will be helpful to draw a rough graph of f , and you probably will need ideas from calculus to complete this exercise.) (b) Find three intervals whose union is R , such that f...
B2. (a) Let I denote the interval 0,1 and let C denote the space of continuous functions I-R. Define dsup(f,g)-sup |f(t)-g(t) and di(f.g)f (t)- g(t)ldt (f,g E C) tEI (i) Prove that dsup is a metric on C (ii) Prove that di is a metric on C. (You may use any standard properties of continuous functions and integrals, provided you make your reasoning clear.) 6 i) Let 1 denote the constant function on I with value 1. Give an explicit...
5. Let V-Pi(R), and, for p(x) E V, define f, f2 E V by 2 fi (p(x))p(t) dt and f2(p(xp(t) dt 0 0 Prove that (fi, f2) is a basis for V", and find a basis for V for which it is the dual basis
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,...
A function f is said to be invertible with respect to integration over the interval (a, b) if and only if f is one-to-one and continuous on the interval (a, b), and in addition [r"() de = ["s(e) dr. In the list below, some functions are described either by their rules or by their graphs. Select all the functions which are invertible with respect to integration over the interval (0,1). (A) f(x) = x2 + cos(-x) (D) 2 f(x) =...
Consider the sequence of functions fn : [0,1| R where each fn is defined to be the unique piecewise linear function with domain [0, 1] whose graph passes through the points (0,0) (, n), (j,0), and (1,0) (a) Sketch the graphs of fi, f2, and f3. (b) Computefn(x) dx. (Hint: Compute the area under the graph of any fn) (c) Find a function f : [0, 1] -> R such that fn -* f pointwise, i.e. the pointwise limit of...
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q. II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...