如果你也在 怎样代写现代代数Modern Algebra 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。现代代数Modern Algebra有时被称为代数结构或抽象代数，或者仅仅在高等数学的背景下被称为代数。虽然这个名字可能只是暗示了一种新的方式来表示微积分之前的代数，但实际上它比微积分更广泛、更深入。
现代代数Modern Algebra这门学科的思想和方法几乎渗透到现代数学的每一个部分。此外，没有一门学科更适合培养处理抽象概念的能力，即理解和处理问题或学科的基本要素。这包括阅读数学的能力，提出正确的问题，解决问题，运用演绎推理，以及写出正确、切中要害、清晰的数学。
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数学代写|现代代数代写Modern Algebra代考|Sets
Abstract algebra had its beginnings in attempts to address mathematical problems such as the solution of polynomial equations by radicals and geometric constructions with straightedge and compass. From the solutions of specific problems, general techniques evolved that could be used to solve problems of the same type, and treatments were generalized to deal with whole classes of problems rather than individual ones.
In our study of abstract algebra, we shall make use of our knowledge of the various number systems. At the same time, in many cases we wish to examine how certain properties are consequences of other, known properties. This sort of examination deepens our understanding of the system. As we proceed, we shall be careful to distinguish between the properties we have assumed and made available for use and those that must be deduced from these properties. We must accept without definition some terms that are basic objects in our mathematical systems. Initial assumptions about each system are formulated using these undefined terms.
One such undefined term is set. We think of a set as a collection of objects about which it is possible to determine whether or not a particular object is a member of the set. Sets are usually denoted by capital letters and are sometimes described by a list of their elements, as illustrated in the following examples.
数学代写|现代代数代写Modern Algebra代考|Mappings
The concept of a function is fundamental to nearly all areas of mathematics. The term function is the one most widely used for the concept that we have in mind, but it has become traditional to use the terms mapping and transformation in algebra. It is likely that these words are used because they express an intuitive feel for the association between the elements involved. The basic idea is that correspondences of a certain type exist between the elements of two sets. There is to be a rule of association between the elements of a first set and those of a second set. The association is to be such that for each element in the first set, there is one and only one associated element in the second set. This rule of association leads to a natural pairing of the elements that are to correspond, and then to the formal statement in Definition 1.9.
By an ordered pair of elements we mean a pairing $(a, b)$, where there is to be a distinction between the pair $(a, b)$ and the pair $(b, a)$, if $a$ and $b$ are different. That is, there is to be a first position and a second position such that $(a, b)=(c, d)$ if and only if both $a=c$ and $b=d$. This ordering is altogether different from listing the elements of a set, for there the order of listing is of no consequence at all. The sets ${1,2}$ and ${2,1}$ have exactly the same elements, and ${1,2}={2,1}$. When we speak of ordered pairs, however, we do not consider $(1,2)$ and $(2,1)$ equal. With these ideas in mind, we make the following definition.
现代代数代考
数学代写|现代代数代写Modern Algebra代考|Sets
抽象代数的起源是试图解决数学问题，比如用根式解多项式方程，用直尺和指南针构造几何结构。从特定问题的解决方案，发展出可用于解决同一类型问题的通用技术，并且处理方法被一般化以处理整个类别的问题，而不是单个问题。
在学习抽象代数时，我们将利用各种数制的知识。同时，在许多情况下，我们希望研究某些属性是如何由其他已知属性导致的。这种考察加深了我们对制度的理解。在我们继续进行的过程中，我们将仔细区分我们已经假定并可供使用的性质和那些必须从这些性质中推导出来的性质。我们必须不加定义地接受一些术语，它们是我们数学系统中的基本对象。每个系统的初始假设是用这些未定义的术语来表述的。
其中一个未定义的术语就是set。我们认为集合是一组对象的集合，通过这些对象可以确定某个特定对象是否为集合的成员。集合通常用大写字母表示，有时用其元素的列表来描述，如下面的例子所示。
数学代写|现代代数代写Modern Algebra代考|Mappings
函数的概念是几乎所有数学领域的基础。函数这个术语是我们脑海中使用最广泛的概念，但是在代数中使用映射和变换已经成为传统。使用这些词很可能是因为它们表达了对相关元素之间联系的直观感觉。其基本思想是两个集合的元素之间存在某种类型的对应关系。在第一个集合的元素和第二个集合的元素之间必须有一个关联规则。这种关联是这样的:对于第一个集合中的每个元素，在第二个集合中有且只有一个关联元素。这个关联规则导致要对应的元素的自然配对，然后是定义1.9中的形式声明。
我们所说的有序元素对是指一对$(a, b)$，如果$a$和$b$不同，则对$(a, b)$和对$(b, a)$之间是有区别的。也就是说，存在第一位置和第二位置，使得$(a, b)=(c, d)$当且仅当$a=c$和$b=d$。这种排序与列出集合的元素完全不同，因为列出的顺序根本无关紧要。集合${1,2}$和${2,1}$具有完全相同的元素，并且${1,2}={2,1}$。然而，当我们谈到有序对时，我们不认为$(1,2)$和$(2,1)$相等。考虑到这些想法，我们做出以下定义。
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