Question

In: Advanced Math

Problem 17.5. Consider the function χ(0,1) : R → R (this is the characteristic of Definition...

Problem 17.5. Consider the function χ(0,1) : R → R (this is the characteristic of Definition 14.4). Find: (a) χ(0,1)((0,1)); (b) χ(0,1)((−1,3)); (c) and (in general) χ(0,1)((a,b)), where a,b ∈ R and a < b; prove that the set you found is correct; (d) χ−1 (0,1) ((−2,−1)); (e) χ−1 (0,1) ((0,2)); (f) and (in general) χ−1 (0,1) ((a,b)), where a,b ∈ R and a < b; prove that the set you found is correct,

Definition 14.4 (for Problems 14.6 through 14.9). Let X be a nonempty set and let
A be a subset of X. The characteristic function or indicator function of the set A
in X is

χA : X → {0,1} defined by χA(x) = 1 if x ∈ A
0 if x ∈ X\A

Expert Solution

From definition, is given by .

(a). . This follows trivially from the above stated definition.

(b). as for any x in , so, and , combining them we get .

(c). Case 1.:- . Then, for any we have . So, .

Case 2.:- . Here, 0 is in (a,b) and and , so . Hence, .

Case 3.:- . Then, similarly as case 2, .

Case 4:- . Here, (a,b) is a subset of (0,1) so, .

Case 5:- . Then, (a,1) is a subset of (0,1), so and since , . Therefore, .

Case 6:- . Here , so .

Now observe that, and .

(d). As 0 and 1 are both not in (-2,-1), we get where denotes the empty set.

(e). As we get .

(f). Case 1.- . Here, so we get .

Case 2.- . Here, so we get .

Case 3.- . Here, , so we get .

Case 4.- . Here, , so we get .

Case 5.- . Here, so we get .

Case 6.- . Here, so we get .

Colby Messinger answered 2 years ago

Consider R with the cofinite topology. Is [0,1] compact? Can you describe the compact sets? Consider...

Consider R with the cofinite topology. Is [0,1] compact? Can you describe the compact sets? Consider R with the cocountable topology. Is [0, 1] compact? Can you describe the compact sets? Consider R with the lower limit topology. Is [0,1] compact? Can you describe the compact sets?

Consider the probability distribution of the discrete random vector [Χ,Y ] where Χ represents the number...

Consider the probability distribution of the discrete random vector [Χ,Y ] where Χ represents the number of orders for chickens in August at neighbouring supermarket and Y represents the number of orders in September. The joint distribution is shown in the following table: 6 Χ Y 100 200 300 400 500 100 0.06 0.05 0.05 0.01 0.01 200 0.07 0.05 0.01 0.01 0.01 300 0.05 0.10 0.10 0.05 0.05 400 0.05 0.02 0.01 0.01 0.03 500 0.05 0.06 0.05 0.01...

Find the critical value Χ 2/r corresponding to a sample size of 13 and a confidence...

Find the critical value Χ 2/r corresponding to a sample size of 13 and a confidence interval of 98%. (Draw the graph)

Problem(3) Consider a r.v. representing coin throws (Bernoulli Variable with Σ = {0,1} ). Let the...

Problem(3) Consider a r.v. representing coin throws (Bernoulli Variable with Σ = {0,1} ). Let the true probability distribution be p(0) = r, p(1) = 1-r. Someone guesses a different distribution q(0) = s, q(1) = 1-s. (a) Find expressions for the Kullback–Leibler distances D(p||q) and D(q||p) between the two distributions in terms of r and s. (b) Show that in general, D(p||q) ≠ D(q||p) and that equality occurs iff r = s. (c) Compute D(p||q) and D(q||p) for the...

The normalized wave function of an electron in a linear accelerator is ψ = (cos χ)eikx...

The normalized wave function of an electron in a linear accelerator is ψ = (cos χ)eikx + (sin χ)e–ikx, where χ (chi) is a parameter. (a) What is the probability that the electron will be found with a linear momentum (a) +kħ, (b) −kħ? (c) What form would the wave function have if it were 90% certain that the electron had linear momentum +kħ? (d) Evaluate the kinetic energy of the electron.

1. Consider the function f: R→R, where R represents the set of all real numbers and...

1. Consider the function f: R→R, where R represents the set of all real numbers and for every x ϵ R, f(x) = x3. Which of the following statements is true? a. f is onto but not one-to-one. b. f is one-to-one but not onto. c. f is neither one-to-one nor onto. d. f is one-to-one and onto. 2. Consider the function g: Z→ {0, 1, 2, 3, 4, 5}, where Z represents the set of all integers and for...

Please Consider the function f : R -> R given by f(x, y) = (2 -...

Please Consider the function f : R -> R given by f(x, y) = (2 - y, 2 - x). (a) Prove that f is an isometry. (b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C = (3, 2), and the triangle with vertices f(A), f(B), f(C). (c) Is f a rotation, a translation, or a glide reflection? Explain your answer.

The Cantor set, C, is the set of real numbers r for which Tn(r) ϵ [0,1]...

The Cantor set, C, is the set of real numbers r for which Tn(r) ϵ [0,1] for all n, where T is the tent transformation. If we set C0= [0,1], then we can recursively define a sequence of sets Ci, each of which is a union of 2i intervals of length 3-i as follows: Ci+1 is obtained from Ci by removing the (open) middle third from each interval in Ci. We then can define the Cantor set by C= i=0...

Show that {xx^R | x,y ∈ {0,1}*} is a context-free language. Note that x^R is the...

Show that {xx^R | x,y ∈ {0,1}*} is a context-free language. Note that x^R is the reversal of x. Show all work. Question is for Discrete Math Structures

Write a function softMax_M(Q) to return a 2darray of softmax function values ??ℎ=???ℎ−?∑??=1???ℎ−? for ?=0,1,⋯,?−1,ℎ=0,1,⋯,?−1 ,...

Write a function softMax_M(Q) to return a 2darray of softmax function values ??ℎ=???ℎ−?∑??=1???ℎ−? for ?=0,1,⋯,?−1,ℎ=0,1,⋯,?−1 , where ? is a 2darray of floats with shape ( ? , ? ), ??ℎ is the element at the (?+1) -th row and the (ℎ+1) -th column of array ? , and ? is the largest element in array ? . (Hint: operations should be performed down the rows; a returned 2darray and ? are of the same shape.) Sample: if D =...