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...
Let f:R → Z defined by f(x) = 23 – 2. Prove that f is a one-to-one correspondence (i.e., a bijection).
5. (a) (5 points) Let R F[x] for a field F. Let f, g E R be nonzero. Prove that (f(x)) = (g(x)) if and only if g(x) = af(x) for some constant a E F. (b) (5 points) Let R be any ring. Prove that the nilradical Vo is contained in the intersection of all prime ideals.
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
(1,3), с %3D (2,1), d (3,4) (1,2), b (4,2), f (5,3) and (5,5). Let 5. Let a = е 3 - {a, b, c, d, e, f, g} be the set of these 7 points. We define the following partial order on S: We have (r, y)(', y) iff x< x and y < / Draw the Hasse diagram of S S 6. We consider the same partial order as in Problem 5, but it is now defined on R2....
QUESTION 2 20 points Save Answer (a) Let A- 101 112 and let T: R 225) T: P = R o via maria menina dentar, TV6 – AR.20 - ( +R be the matrix mapping defined by T(x) = ist wens meer under T is the vector b. and determine whether X is unique (b) Let : R2 + R be the linear transformation that maps the vector - Cinto (6and maps v = ()ino (9) Use the fact that...
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such (9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t € [2 – 1, 2+1]}. Prove that g(x) is also continous. Hint: To prove g(x) is continous at x = xo. You can consider the continuity of f(x) at the two boundary point xo - 1 and xo +1. When x get close to xo, the points in (7 - 1, + 1) is close to xo - 1, xo + 1, or inside...
(18) Let f and g be functions from R to R that have derivatives of al orders. Let h(k) denote the kth derivative of any function. Prove using the product rule for derivatives, the fact that and induction that k +1 k=0 (19) The Fibonacci numbers are defined recursively by Fn+2 = Fn+1 Prove that the number of subsets of { 1, 2, 3, . . . , n} containing no two successive integers is E, (20) Prove that 7n...
How can this be done using a direct proof? bijection, Let f.122 defined by f(x)=x²-2 Prove that t is one-to-one correspondence (aka