8. Prove or disprove the following:(a) If X and Y are path-connected, then X x Y is path-connected.(b) If A C X is path-connected, then A is path-connected.(c) If X is locally path-connected, and AC X, then A is locally path-connected.(d) If X is path-connected, and f: X Y is continuous, then f(X) is path-connected.(e) If X is locally path-connected, and f: X→ Y is continuous, then f(X) islocally path-connected.

Let X and Y be topological spaces. We need to prove or disprove the following: If X and Y are…

Answered: Prove or disprove the following: (a) If…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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For each n E N, let fn → R be a (F,B(R))- measurable function. (i) Then, each of the functions supnen fn, infnen fn, lim supn→∞o fn, and lim infn oo fn is (F, B(R))-measurable. (ii) The set A = {w : limn→∞o fn(w) exists and is finite} lies in F and the function h = (limn→∞ fn) · IA is (F, B(R))-measurable.
2
Exercise 1. A bucket contains 10 ping pong balls, of which 3 are orange and 7 are blue. You draw two ping pong balls out of the bag (randomly and without replacement). Let X be the number of blue balls that you draw. (a) Explain why X meets the criteria to be a discrete r.v., and give its space. (b) Find the pmf of X.
A set of real numbers D is called dense if and only if every open interval contains at least one point of D. Assume a set E and its complement E = {x € R : x ¢ E} are both dense. Let f : [0, 1] → R be given by f(x) = 1 if x E E and f(x) = 0 if x ¢ E. (Thus f(x) = XE (x) for x E [0, 1].) Prove that f is not integrable on [0, 1].
Exercise 1. Suppose that X - f(x) =÷1(x = 1,2,3,4,5). Compute... (a) E(X), (b) E(X²), and (c) E(X+2)² Hint: (X + 2)2 = X² + 4X + 4
Recall the definition of Linear Independence: The set {V1,..., Vn} is linearly independent if, whenever a₁ V₁ +...+ an Vn = 0, it must be that a₁ = = an = 0. Suppose that {v2, V3} is linearly independent. Briefly describe what is wrong with the following "proof" that {V1, V2, V3} is linearly independent (where V₁ is some nonzero vector): Since (v2, V3} is linearly independent, a2V2 + α3 V3 = 0. Then, if a₁V₁ + a2 V2 + a3 V3 = 0, we have that a₁ V₁ +0=0 so that a₁ = 0. Therefore, {V1, V2, V3} is linearly independent.
8. The null space of A is a. NS (A) = {X € R|AX = 1} d. NS (A) = {X € R²\AX # 0} b. NS (A) = (X € R"|AX = 0) C.NS (A) = {X E R³|AX = 0}

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