代写Assignment 2 (M365)代写留学生数据结构程序

Assignment 2 (M365)

Problem 1. Let I be the generic notation for an interval I = (a, b) (a < b) and l(I) be the generic notation for its length l(I) = b − a.

(i) Using only the mathematical symbols ε, >, ∪ etc, and the words ‘EITHER’, ‘AND’ and ‘OR’, write down the formal definition of a ‘null’ set A ⊂ R.

(ii) In the similar way, write down the formal statement saying that a set A ⊂ R is not null.

Problem 2. (a) State the definition of a fucntion f : R → R to be continuous, in terms of preimages of open sets (p.14, lecture notes-1).

(b) Using this definition, prove that the function f(x) = sin(x) is con-tinuous.

Problem 3. The additivity property of a measure reads:

A ∩ B = ∅ =⇒ m(A ∪ B) = m(A) + m(B).

Using this property only, find, with explanations, formulae describing m(A∪B) and m(A ∪ B ∪ C) in terms of measures of the individual sets and their intersections. (Do not assume that the sets are pairwise disjoint).

Problem 4. Consider the probability space with

Ω = {(t, t),(t, h),(h, t),(h, h)},

describing tossing a coin twice: see the example presented on p.23 of lecture notes-1.

(i) Describe mathematically the events

A – meaning the first trial gives tails;

B – meaning the second trial gives tails;

C – meaning exactly one tail appears.

Write down the generated σ-fields FA, FB, FC, i.e., the minimal σ-fields containing A, B and C correspondingly.

(ii) Calculate P(A), P(B) and P(C). After that, calculate P(A∩B∩C). Are the σ-fields FA, FB, FC independent?