Notice that the following claim is among one of the multiple steps of proving an important result...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is...
need help with proving discrete math HW, please try write clearly and i will give a thumb up thanks!! Let A and be B be sets and let f:A B be a function. Define C Ax A by r~y if and only if f(x)f(y). Prove thatis an equivalence relation on A. Let X be the set of~-equivalence classes of A. L.e. Define g : X->B by g(x) Prove that g is a function. Prove that g is injective. Since g...
Exercise 3 (Cantor-Bernstein-Schröder). Let f: A → B and g: B → A be injective maps. We define recursively the sets C = UCn Co = A \ g(B), Cn+1 = g(f(Cn)), nƐN and a new map h: A → B by if x E C, f (x) h(x) = if x 4 C, g='(x) where the preimage g¬1(x) is well-defined since g is injective and x E g(B) in that case (check that!). Show that h is bijective. Conclude...
Answer the questions in the space provided below. 1. The definition of a function f: X + Y is as a certain subset of the product X x Y. Let f: N + N be the function defined by the equation f(n) = n2. For each pair (x, y) listed below, determine whether or not (x,y) ef. a) (2,4) b) (5, 23) c) (1,1) d) (-3,9) 2. For each function defined below, state whether it is injective (one-to-one) and whether...
A function f : A - B is said to be injective (or one-to-one) provided Va, a2 € A, f(a) = f(az) ► a1 = . A function g: A + B is said to be surjective (or onto) provided W6 € B, 3 some a € A such that g(a) = b. A function h: A → B is said to be bijective (or a bijection or a one-to-one correspondence) if it is both injective and surjective. The following...
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)...
You're the grader. To each "Proof", assign one of the following grades: A (correct), if the claim and proof are correct, even if the proof is not the simplest, or the proof you would have given. C (partially correct), if the claim is correct and the proof is largely a correct claim, but contains one or two incorrect statements or justifications. . F (failure), if the claim is incorrect, the main idea of the proof is incorrect, or most of...
Exercise 2: Möbius Transformations I (a) [10 points] Denote A := {z € C: |z| < 1}. Prove the following statement. Every Möbius transformation g: A → A who maps A onto A can be written as 9(2) = e® (2- 20 Zoz – 1 with 0 eR and |zo| < 1. Conversely, each such function maps A onto A. (b) [6 points] Find a Möbius transformation f with f(i) = i, f (0) = 0 and f(-i) = 0....
Let h : X −→ Y be defined by h(x) := f(x) if x ∈ F g −1 (x) if x ∈ X − F Now we must prove that h is injective and bijective. Starting with injectivity, let x1, x2 ∈ X such that h(x1) = h(x2). Assume x1 ∈ F and x2 ∈ X −F. Then h(x1) = f(x1) ∈ f(F) and h(x2) = g −1 (x2) ∈ g −1 (X − F) = Y...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...