Problem 1. (2 credits) Let f: X +Y. Prove that f is injective if and only...
1. Prove that the function f: X → Y is injective if and only if it satisfies the following condition: For any set T and functions g: T → X and h : T → X, o g = f o h implies g = h.
8. Prove the following: a. A function, f: X Y, is injective if and only if If-2013 1 for each y EY b. A function, f:X + Y, is surjective if and only if \f-1(y) 2 1 for each y E Y c. A function, f:X → Y, is bijective if and only if \f-(y)= 1 for each y E Y
Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y . Prove that for every subset A ⊂ X: (a) (10 points) A ⊂ F^(−1) (F(A)). (b) (10 points) F ^(−1) (F(A)) ⊂ A
4. Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y. Prove that for every subset A CX: (a) (10 points) AC F-(F(A)). (b) (10 points) F-1(F(A)) C A.
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...
Problem 6.8. Let X = {1, 2, 3}, Y = {a, b, c, d, e}. (a) Let f : X → Y be a function, given by f(1) = a, f(2) = b, f(3) = c. Prove there exists a function g : Y → X such that g ◦f = id X . Is g the inverse function to f? (Hint: define g on f(X) to make g ◦ f = id X . Then define g on Y...
Problem 4. (15 points each) Let F(x, y, z) = (0, x, y) G(x, y, z) = (2x, z, y) + (x, y, z) = (3y, 2x, z). (a) For each field, either find a scalar potential function or prove that none exists. (b) For each field, either find a vector potential function or prove that none exists. (c) Let F(t) = (2, 2t, t2). For which of these vector fields is ñ a flow line? Justify your answer.
Let f : R2-R2 be a function defin ed by f(x,y) (3+ z +y,) (a) Determine if f is injective. Explain why. (b) Determine if f is surjective. Explain why Let f : R2-R2 be a function defin ed by f(x,y) (3+ z +y,) (a) Determine if f is injective. Explain why. (b) Determine if f is surjective. Explain why
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...
Prove If the functions are injective, surjective, or bijective. You must prove your answer. For example, if you decide a function is only injective, you must prove that it is injective and prove that it is not surjective and that it is not bijective. Similarly, if you claim a function is only surjective, you must prove it is surjective and then prove it is not injective and not bijective. - Define the function g: N>0 → N>0 U {0} such that g(x) = floor(x/2). You may use the fact that...