Instructions: Words: 3700 Words. This assignment should have detailed explanations and has to be done to such a high standard that you get full marks. Done by yourself fully in your own words and to get full marks and also to have a lot of examples. You must show that you understand the assignment well and also don't use Turnitin but can use grammerly and some others if you tell me which one. The first document attached is the assignment. The second document attached is to help you which is the lecture notes and the third document attached is to how to write a good math dissertation.

Here are some tips on writing a good essay once you have finished the planning stage and are ready to start writing: Give your dissertation a structure of sections and possibly subsections, but don’t go too far with dividing things into subsubsections. It’s often tempting to provide lots of titles, headings and subheadings to indicate the structure of an essay, but you should observe that professionally written mathematics doesn’t do this. Instead you should try to make the text flow by indicating to the reader where you are going next. It’s much more pleasant to read what will come next and see where the essay is going than to be jolted by a sudden change of direction signalled by a heading.

You should imagine that the reader of your essay is an undergraduate who understood the key material in the relevant lecture, but might need reminding of it. So you don’t need to describe standard material from the first years of your degree. You should include plenty of examples, wherever possible examples that you have invented yourself. It helps the reader to confirm that they have understood what you are saying. It also means that if there is a slip somewhere in a definition, such as a missing minus sign, its correct use in an example will show the examiners that the slip was just a slip, and not an error of understanding on your part.

Use simple, clear language, forming proper sentences grouped carefully into paragraphs. Each sentence should generally express one idea. You may need to think very carefully about the best way to explain a complex idea: it is much better to explain it precisely, briefly and accurately just once, than to describe it in multiple different but imprecise ways. Include a bibliography, listing all the texts that you have cited. Label and describe the items clearly and consistently, by putting them in alphabetical order by the first author. When using a reference, always cite it properly, and include guidance as to where in the reference to look. If you state a theorem and then say that the proof is in [3], when [3] is a 600 page book, it’s not very helpful! It’s much better to say that the theorem is proved as Proposition 4.6 of [3].