hello sir, solve both questions Problem 5: Let f : A → B be a function,...
6. (Extra Credit) Let I be the interval (0,1). Define F(I) = {f:I+I:f is a function}, the set of all functions from the interval (0,1) to itself. (a) Thinking about the graph of a function, define a one-to-one function F(1) ► PIXI). Prove your function is one-to-one (remember that functions fi and f2 are equal when they have the same domain and codomain, and fi(x) = f2(x) for every x in the domain). (b) Given a set A CI, define...
solve #5 only please 5 Prove that the function f in problem 4 is integrable and sf = 0. Suggestion: Use the suggestion for problem 4(a) to show that given €>0, there is a partition Pof [0, 1] with Uff, P) < 2€ , while Laf, P) =0. Do this by enclosing the points of the finite set where f(x) 2e in a finite set of disjoint closed intervals, each contained in (0,1), with the sum of the lengths <€....
1. (a) (6 points) Let f : A + B and g:B + C be two functions. Suppose that the composition of functions go f is a bijection. Prove that the function f : A + B must be one-to-one and that the function g:B + C must be onto. (b) (4 points) Give an example of a pair of functions, f and g, such that the composition gof is a bijection, but f is not onto and g is...
P2.9.7 Let A be a set and f a bijection from A to itself. We say that f fixes an element r of A if f(x) = r. (a) Write a quantified statement, with variables ranging over A, that says "there is exactly one element of A that f does not fix." (b) Prove that if A has more than one element, the statement of part (a) leads to a contradiction. That is, if f does not fix 2, and...
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) :=...
5. Let f R2 ->R2 be the function given by f(x, y) (х + у, х — у). (i) Prove that f is linear as a function from R2 to R2. (ii) Calculatee the matrix of f. (iii) Prove that f is a one-to-one function whose range is R2. Deduce that f has an inverse function and calculate it. (iv) If C is the square in R2 given by C = [0,1] x [0, 1], find the set f(C), illustrating...
Problem 5 Let f : [0,1] → R be continuous and assume f(zje (0, 1) for all x E (0,1). Let n E N with n 22. Show that there is eractly one solution in (0,1) for the equation 7L IC nx+f" (t) dt-n-f(t) dt.
Answer each question in the space below. 1. Let A = {0,1} U... U{0,1}5 and let be the order on A defined by (s, t) €< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element & is minimal if there does not erist Y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give...
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT A1. Define g: C -> D by f(x)-g(x). Briefly, if g is not injective, then explain why f is not injective either. Let j : B → { 1, 2, 3, . . . , BI}...
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers) and define function f: ℝ → {0,1}, where f(x) = XA(x) That is, f is the characteristic function of set A; it maps elements of the domain that are in set A (ie, those that are even integers) to 1 and all other elements of the domain to 0. By demonstrating a counter-example, show that the function f is not injective (not one-to-one). b)...