代写Assignment 1 (M365)代写数据结构语言程序

日期：2025-02-11 05:09

Assignment 1 (M365)

Problem 1. Suppose Ω is an infinite set and Ω ∩ N = ∅, where N = {1, 2, . . .}. Prove that Ω ∪ N has the same cardinality as Ω.

Problem 2. Having the following definition for a fixed non-empty set X ⊂ R

” inf{x : x ∈ X} = a if

[∀x ∈ X x ≥ a] and [∀y > a ∃x ∈ X : x < y] ”,

formulate the formal statement that a ≠ inf{x : x ∈ X} using only the words ”and”, ”or”, and mathematical symbols like ∀, ∃, ∈, . . ..

Problem 3. (i) Give an example of two fields F1 and F2 over the same universal set Ω such that F1 ∪ F2 is not a field.

(ii) Let Ω = {1, 2, 3} and construct the minimal field containing the subset {1, 2}. Is this field also a σ-field? Justify your answer.

Problem 4. (i) Prove that if F is a σ-field (as defined on the lectures) and A1, A2, . . . ∈ F, then An ∈ F.

(ii) Prove that any σ-field is a monotone class.

By submitting solutions to this assessment you affirm that you have read and understood the Academic Integrity Policy detailed in Appendix L of the Code of Practice on Assessment and have successfully passed the Academic Integrity Tutorial and Quiz in the course of your studies.

You also affirm that

• the work you are submitting is your own and you have not commissioned production of the work from a third party or used artificial intelligence software in an unacceptable manner to generate the work*,

• you have not copied material from another person or source, nor com-mitted plagiarism, nor fabricated, falsified or embellished data when com-pleting the attached piece of work.

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Marks achieved on this assessment remain provisional until they are ratified by the Board of Examiners in June 2024. Please note that your submission will be analysed by Turnitin for plagiarism and an Artificial Intelligence detection tool.

*software applications include, but are not limited to, ChatGPT, Bing Chat, DALL.E, Bard.