Homework Answers
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) =...
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
Hi. I'm having trouble with this question in my Topology class. Can I get some help on this?? Tha...
Hi. I'm having trouble with this question in my Topology
class.
Can I get some help on this?? Thank you.
(3) Define a function d: R 2 x R2 → R by d(x, y) = max(ki-yil. 12.2-U21) for any two points x-(xi,T2), y-(yi,y2) є R2. Then d is a metric for R2. Prove that {(r,y) є R2lr+y > 0} is an open subset of the metric space (R2, d) and that {(x, y) є R2 1 x + y >...
(a) On R2, prove that di ((zı, y), (z2W2)) := Izı-zal + ly,-Val is a metric. (b) Assume that doc ( (zi, yī), (z2,p)) := maxlz-zal, lyi-yl} is a metric on R2 for each p 21. Prove that di and d induce...
(a) On R2, prove that di ((zı, y), (z2W2)) := Izı-zal + ly,-Val is a metric. (b) Assume that doc ( (zi, yī), (z2,p)) := maxlz-zal, lyi-yl} is a metric on R2 for each p 21. Prove that di and d induce the same topology on R2. You may use the following lemma (but do not need to prove it): Lemma: Let d and d' be two metrics on aset X; let T and T' be the topologies the induce...
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (...
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that:
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2 {(r,...
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
A function f : R^2 ↦R is called a metric if f(x, y) >= 0 for...
A function f : R^2 ↦R is called a metric if f(x, y) >= 0 for all x, y that are an element of R, f(x,y) = 0 if and only if x = y, and for any x, y, z we have f(x, z) <= f(x,y) + f(y, z). Is the function f(x, y) = the square root of |x - y| a metric or not?
For each of the following metric spaces (X, d) and subsets A S X decide whether...
For each of the following metric spaces (X, d) and subsets A S X decide whether A is open, closed, neither or both. You do not need to justify your answers. (a) X-Z, d is the discrete metric. ACX is any subset (b) X = R2, d = d2 is the Euclidean metric. A = {(x, yje R2 : x-y) (c) X = R2, d = d2 is the Euclidean metric. A (0, 1) × {0).
6. Show that the followings define metrics on R2: For r = (11, 12), y =...
6. Show that the followings define metrics on R2: For r = (11, 12), y = (y1, y2) ER, the company = 139-un +100 - 247 91.42.), y =(1,9) ER di(x,y) = |21 - y1| + |22 - y2), doo (I, y) = max{\21 – yı], \12 - y2|}.
1. (a) Let d be a metric on a non-empty set X. Prove that each of...
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
8) Prove that C([O, 1]) is a metric space with the metric .1 d(f, g) = / If(x)-g(x)| dx. 9) Let (X, di) and (Y, d2) be metric spaces. a) Prove that X × Y is a metric space with the metric b) Prove that X x Y is a metric space with the metric
Hi. I'm having trouble with this question in my Topology
class.
Can I get some help on this?? Thank you.
(3) Define a function d: R 2 x R2 → R by d(x, y) = max(ki-yil. 12.2-U21) for any two points x-(xi,T2), y-(yi,y2) є R2. Then d is a metric for R2. Prove that {(r,y) є R2lr+y > 0} is an open subset of the metric space (R2, d) and that {(x, y) є R2 1 x + y >...
(a) On R2, prove that di ((zı, y), (z2W2)) := Izı-zal + ly,-Val is a metric. (b) Assume that doc ( (zi, yī), (z2,p)) := maxlz-zal, lyi-yl} is a metric on R2 for each p 21. Prove that di and d induce the same topology on R2. You may use the following lemma (but do not need to prove it): Lemma: Let d and d' be two metrics on aset X; let T and T' be the topologies the induce...
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2 If(x,y)| 〈 M(x2 + y2)· for all (a·y) E R2 Prove that:
1(a) Let f : R2 → R b constant M > 0 such that livf(x,y)|| (0.0)-0. Assume that there exists a e continuously differentiable, with Mv/r2 + уг, for all (z. y) E R2...
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
{(r, y) E R2 y r} Let A = {A,:r e R} be a collection of sets given by A, = Prove that A is a partition of R2
For each of the following metric spaces (X, d) and subsets A S X decide whether A is open, closed, neither or both. You do not need to justify your answers. (a) X-Z, d is the discrete metric. ACX is any subset (b) X = R2, d = d2 is the Euclidean metric. A = {(x, yje R2 : x-y) (c) X = R2, d = d2 is the Euclidean metric. A (0, 1) × {0).
6. Show that the followings define metrics on R2: For r = (11, 12), y = (y1, y2) ER, the company = 139-un +100 - 247 91.42.), y =(1,9) ER di(x,y) = |21 - y1| + |22 - y2), doo (I, y) = max{\21 – yı], \12 - y2|}.
1. (a) Let d be a metric on a non-empty set X. Prove that each of the following are metrics on X: a a + i. d(1)(, y) = kd(x, y), where k >0; [3] ii. dr,y) d(2) (1, y) = [10] 1+ d(,y) The proof of the triangle inequality for d(2) boils down to showing b + > 1fc 1+a 1+b 1+c for all a, b, c > 0 with a +b > c. Proceed as follows to prove...
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