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数学代写|现代代数代考Modern Algebra代写|Postulates for the Integers (Optional)

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数学代写|现代代数代考Modern Algebra代写|Postulates for the Integers (Optional)

The material in this chapter is concerned exclusively with integers. For this reason, we make a notational agreement that all variables represent integers. As our starting point, we shall take the system of integers as given and assume that the system of integers satisfies a certain list of basic axioms, or postulates. More precisely, we assume that there is a set $\mathbf{Z}$ of elements, called the integers, that satisfies the following conditions.
Postulates for the Set Z of Integers

Addition postulates. There is a binary operation defined in $\mathbf{Z}$ that is called addition, is denoted by + , and has the following properties:
a. $\mathbf{Z}$ is closed under addition.
c. $\mathbf{Z}$ contains an element 0 that is an identity element for addition.

d. For each $x \in \mathbf{Z}$, there is an additive inverse of $x$ in $\mathbf{Z}$, denoted by $-x$, such that $x+(-x)=0=(-x)+x$.

Multiplication postulates. There is a binary operation defined in $\mathbf{Z}$ that is called multiplication, is denoted by $\cdot$, and has the following properties:
a. $\mathbf{Z}$ is closed under multiplication.
b. Multiplication is associative.
c. $\mathbf{Z}$ contains an element 1 that is different from 0 and that is an identity element for multiplication.
d. Multiplication is commutative.

The distributive law,
$$x \cdot(y+z)=x \cdot y+x \cdot z$$
holds for all elements $x, y, z \in \mathbf{Z}$.

$\mathbf{Z}$ contains a subset $\mathbf{Z}^{+}$, called the positive integers, that has the following properties:
a. $\mathbf{Z}^{+}$is closed under addition.
b. $\mathbf{Z}^{+}$is closed under multiplication.
c. For each $x$ in $\mathbf{Z}$, one and only one of the following statements is true.
i. $x \in \mathbf{Z}^{+}$
ii. $x=0$
iii. $-x \in \mathbf{Z}^{+}$

Induction postulate. If $S$ is a subset of $\mathbf{Z}^{+}$such that
a. $1 \in S$, and
b. $x \in S$ always implies $x+1 \in S$,
then $S=\mathbf{Z}^{+}$.

数学代写|现代代数代考Modern Algebra代写|Mathematical Induction

From this point on, full knowledge of the properties of addition, subtraction, and multiplication of integers is assumed. A study of divisibility begins in Section 2.3.

As mentioned in the last section, the induction postulate forms a basis for the method of proof known as mathematical induction. Some students may have encountered this method of proof in calculus or in other previous courses. In this case, it is possible to skip this section and continue with Section 2.3.

Proof by Mathematical Induction In a typical proof by induction, there is a statement $P_n$ to be proved true for every positive integer $n$. The proof consists of three steps:

1. Basis Step The statement is verified for $n=1$.
2. Induction Hypothesis The statement is assumed true for $n=k$.
3. Inductive Step With this assumption made, the statement is then proved to be true for $n=k+1$.

The assumption that is made in step 2 is called the inductive assumption or the induction hypothesis.
Principle of Mathematical Induction
The logic of the method is that
a. if $P_n$ is true for $n=1$, and
b. if the truth of $P_k$ always implies that $P_{k+1}$ is true,
then the statement $P_n$ is true for all positive integers $n$. This logic fits the induction postulate perfectly if we let $S$ be the set of all positive integers $n$ for which $P_n$ is true. When the induction postulate is used in this form, it is frequently called the Principle of Mathematical Induction.

现代代数代写

数学代写|现代代数代考Modern Algebra代写|Postulates for the Integers (Optional)

A. $\mathbf{Z}$在加法下封闭。
b.加法是结合律。
C. $\mathbf{Z}$包含一个用于加法的单位元素0。

d.对于每个$x \in \mathbf{Z}$, $\mathbf{Z}$中都有一个$x$的加性逆，用$-x$表示，使得$x+(-x)=0=(-x)+x$。
e.加法是可交换的。

A. $\mathbf{Z}$在乘法下封闭。
b.乘法是结合律。
C. $\mathbf{Z}$包含一个不同于0的元素1，它是乘法的单位元素。
d.乘法是可交换的。

$$x \cdot(y+z)=x \cdot y+x \cdot z$$

$\mathbf{Z}$ 包含一个子集$\mathbf{Z}^{+}$，称为正整数，具有以下属性:
A. $\mathbf{Z}^{+}$在加法下封闭。
B. $\mathbf{Z}^{+}$在乘法下封闭。
c.对于$\mathbf{Z}$中的每个$x$，下列表述中有且只有一个是正确的。

2$x=0$
3 $-x \in \mathbf{Z}^{+}$

A. $1 \in S$，和
B. $x \in S$总是暗示$x+1 \in S$;

数学代写|现代代数代考Modern Algebra代写|Mathematical Induction

A.如果$P_n$对于$n=1$是正确的，并且
B.如果$P_k$的真理总是暗示$P_{k+1}$是真的，

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