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# 数学代写|抽象代数代写Abstract Algebra代考|MATH413 Writing proofs

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## 数学代写|抽象代数代写Abstract Algebra代考|Writing proofs

Reading proofs more effectively will help you to understand them more fully and become better at constructing proofs of your own. But there is no doubt that proof construction is difficult. Just as mathematical reading is not like ordinary reading, mathematical writing is not like ordinary writing. A proof is not like a message to tell a housemate that you have gone to the supermarket. Nor is it like a standard algorithm that you can learn and apply. There are standard proof strategies, and you should look out for those. But they are not that standard-to some extent, each proof is different, which can leave students feeling overwhelmed.

However, specific approaches can help. Some people find it useful to think of proof construction as involving two main processes: a formal part and a problem solving part. The formal part involves using the structure of a theorem to write a ‘frame’ for its proof: writing the premises, leaving a gap, then writing the conclusion. With that done, it is often possible to work forward from the premises and backward from the conclusion by formulating relevant things in terms of definitions or by making standard or obvious deductions. If you let it, a formal approach will shoulder quite a bit of the burden of proving. Indeed, for simple proofs, it might on its own be enough.

For more complex proofs, you also need problem solving to fill in the gap. This requires insight, which can come from reasoning about familiar examples, or from writing down possibly relevant theorems and thinking about whether they usefully apply. A proof will not flow from your pen in a single stream of mathematically correct argument-probably it will involve some false starts and periods of being stuck, and some cleaning up so that the writing makes sense. But writing one requires no magic. To demonstrate what I mean, I will reason through a formal part and a problem-solving part for this theorem.

## 数学代写|抽象代数代写Abstract Algebra代考|Who are you as a student?

Whappy to learn, and willing to put in the hours. The problem you will face is that it is easy to be like that for the first two weeks of a course like Abstract Algebra, but hard to sustain for more than about four. By week eight, you might want to lie on the floor, moan quietly and wish for someone to make it all easy. I can’t make it easy-undergraduate mathematics just isn’t easy. But if you find Abstract Algebra difficult, that is not because you are stupid or incapable. It’s because it is difficult. If this is your first full-on theorems-and-proofs course, it is likely to seem both difficult and alarmingly different from earlier mathematics. This can make students wonder whether they have topped out-whether they cannot cope with mathematics at this level or, more prosaically, whether they just don’t like it.

I would encourage you, though, to avoid making either judgement too soon. Many students transitioning to advanced mathematics have to adjust their expectations in two ways, accepting that they will not understand everything and learning to tolerate longer periods of intellectual discomfort. But most do manage that, and reach a point where they are satisfied with what they have learned. Of course, some then decide that pure mathematics is not their thing and that where possible in future they will avoid it. But better to decide from a position of strength, I think; better to know that you could do more but choose not to. Others experience not only new understanding but real joy in trading the more routine aspects of earlier work for logical reasoning and theory building.

To reach a positive position with minimal pain, I think it helps to reflect on your study trajectory, on the decisions you have made. For instance, you probably chose to study at the most prestigious accessible institution. A natural consequence of this is that the material you are taught will be only just within your intellectual reach. If you wished, you could switch to an easier degree or major, switch to a lower-prestige institution, or drop out of higher education and take a different route into professional life. Some students choose to do those things, and more power to themeveryone should think about how to use their time. But most students don’t. Most, when they reflect, decide that although it might be difficult, they do want to stick with their degree. Reflection and recommitment help, though. If you recognize that you’re doing what you’re doing by choice, it becomes easier to put up with its downsides and keep your eye on the prize.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。