Let S= (1,2) union (2,3). Find intS and bdS. Define open set and closed set and give examples of each.
Let S= (1,2) union (2,3). Find intS and bdS. Define open set and closed set and...
1.16 Let 2 = {1,2,3,4,5,6). Define three events: A = {1,2}, B = {2,3), and C = {4,5,6). The probability measure is unknown, but it satisfies the three axioms. (a) What is the probability of An C? (b) What is the probability of AU BUC? (C) State a condition on the probability of either B or C that would allow them to be independent events.
3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a set G (a countable intersection of open sets), and a set F (a countable union of closed sets) such that F CE C G and m* (F) the Lebesgue measure of a set Hint: The Lebesgue measure can be calculated in terms of open and closed sets m* (E) m* (G), where m* denotes 3. Let E E Lm* (Lebesgue measurable set). Prove that there exist a...
Let the universal set S be S = {1,2,...,10}, and A = {1,2,3}, B = {3,4,5,6,7} and C = {7,8,9,10} 1) Find (A∪C)−B 2) Find A^c ∩(B^c ∪C)
. Let C be a collection of open subsets of R. Thus, C is a set whose elements are open subsets of R. Note that C need not be finite, or even countable. (a) Prove that the union U S is also an open subset of R. SEC (b) Assuming C is finite, prove that the intersection n S is an open subset of R. SEC (c) Give an example where C is infinite and n S is not open....
(5) For each set, figure out whether it is open, closed or neither, and find its interior, boundary and limit points (a) S [3, 4) (b) T 2-n e N} (c) the Cantor set
11. (a) Let A be the open interval (1,5), and let B be the interval (0,8). Define a bijection from A to B (b) Let A = (0,00) and let B = [0,00). Define a bijection from A to B. 12. Is it possible to find two infinite sets A and B such that If your answer is yes, then construct an example 13. Is it possible to find a finite set A such that [AAI = 27? 11. (a)...
Let S be the set of all subsets of Z. Define a relation,∼, on S by “two subsets A and B of Z are equivalent,A∼B, if A⊆B.” Prove or disprove each of the following statements: (a)∼is reflexive(b)∼is symmetric(c)∼is transitive
So the set {0} is not open in the Euclidean line, is it closed? Please explain why? Is this the same for any single element sets in the Euclidean line with the Euclidean metric? How does this change in other metric spaces? Please give some examples.
Question 1. In this question, for brevity we define an open sector in the complex plane to be a set β), where 0 < β-α < 2π. Consider : α < arg(z) < β} and a closed sector to be a set {z : α < arg(z) the following transcendental elementary functions: e* cos(z) f(z) = e:cos(z): g(z)= In which sectors, if any, do each of these functions decay to zero as 0o? Explain your answers and distinguish clearly between...
5- Recall that a set KCR is said to be compact if every open cover for K has a finite subcover 5-1) Use the above definition to prove that if A and B are two compact subsets of R then AUB is compact induction to show that a finite union of compact subsets of R is compact. 5-2) Now use 5-3) Let A be a nonempty finite subset of R. Prove that A is compact 5-4) Give an example of...