### 数学代写|数理逻辑代写Mathematical logic代考|MATH4810

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数理逻辑代写Mathematical logic代考|An Example from Group Theory

In this and the next section we present two simple mathematical proofs. They illustrate some of the methods of proof used by mathematicians. Guided by these examples, we raise some questions which lead us to the main topics of the book.
We begin with the proof of a theorem from group theory. We therefore require the axioms of group theory, which we now state. We use o to denote the group multiplication and $e$ to denote the identity element. The axioms may then be formulated as follows:
(G1) For all $x, y, z: \quad(x \circ y) \circ z=x \circ(y \circ z)$.
(G2) For all $x: \quad x \circ e=x$.
(G3) For every $x$ there is a $y$ such that $x \circ y=e$.
A group is a triple $\left(G, \circ^G, e^G\right)$ which satisfies (G1)-(G3). Here $G$ is a set, $e^G$ is an element of $G$, and $\circ^G$ is a binary function on $G$, i.e., a function defined on all ordered pairs of elements from $G$, the values of which are also elements of $G$. The variables $x, y, z$ range over elements of $G, \circ$ refers to $\circ^G$, and $e$ refers to $e^G$.

As an example of a group we mention the additive group of the reals $(\mathbb{R},+, 0)$, where $\mathbb{R}$ is the set of real numbers, $+$ is the usual addition, and 0 is the real number zero. On the other hand, $(\mathbb{R}, \cdot, 1)$ is not a group (where – is the usual multiplication). For example, the real number 0 violates axiom (G3): there is no real number $r$ such that $0 \cdot r=1$.

We call triples such as $(\mathbb{R},+, 0)$ or $(\mathbb{R}, \cdot, 1)$ structures. In Chapter III we shall give an exact definition of the notion of “structure.”
Now we prove the following simple theorem from group theory:
1.1 Theorem on the Existence of a Left Inverse. For every $x$ there is a $y$ such that $y \circ x=e$.

## 数学代写|数理逻辑代写Mathematical logic代考|An Example from the Theory of Equivalence Relations

The thenry of equivalence relations is hased on the following three axions ( $x k y$ is to be read as ” $x$ is equivalent to $y$ “);
(E1) For all $x: x R x$.
(E2) For all $x, y$ : If $x R y$, then $y R x$.
(E3) For all $x, y, z$ : If $x R y$ and $y R z$, then $x R z$.
Let $A$ be a nonempty set, and let $R^A$ be a binary relation on $A$, i.e., $R^A \subseteq A \times A$. For $(a, b) \in R^A$ we also write $a R^A b$. The pair $\left(A, R^A\right)$ is another example of a structure. We call $R^A$ an equivalence relation on $A$, and the structure $\left(A, R^A\right)$ an equivalence structure, if (E1), (E2), and (E3) are satisfied. For example, $\left(\mathbb{Z}, R_5\right)$ is an equivalence structure, where $\mathbb{Z}$ is the set of integers and
$$R_5={(a, b) \mid a, b \in \mathbb{Z} \text { and } b-a \text { is divisible by } 5} .$$
We now prove a simple theorem about equivalence relations.

2.1 Theorem. If $x$ and $y$ are both equivalent to a third element, they are equivalent to the same elements. More formally: For all $x$ and $y$, if there is a $u$ such that $x R u$ and $y R u$, then for all $z, x R z$ if and only if $y R z$.
Proof. Let $x$ and $y$ be given arbitrarily; suppose that for some $u$ $x R u$ and $y R u$.
From (E2) we then obtain $u R x$ and $u R y$.
From $x R u$ and $u R y$ we get, using (E3),
$$x R y,$$
and from $y R u$ and $u R x$ we likewise get (using (E3))
$$y R x .$$
Now let $z$ be chosen arbitrarily. If
$$x R z \text {, }$$
then, using (E3), we obtain from (4) and (5)
$$y R z .$$
On the other hand, if
$$y R z \text {, }$$
then, using (E3), we get from (3) and (6)
$$x R z \text {. }$$
Thus the claim is proved for all $z$.
As in the previous example, this proof shows that every structure (of the form $\left(A, R^A\right)$ ) which satisfies the axioms (E1), (E2), and (E3), also satisfies Theorem 2.1, i.e., that Theorem $2.1$ follows from (E1), (E2), and (E3).

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|An Example from Group Theory

(G1) 对于所有 $x, y, z:(x \circ y) \circ z=x \circ(y \circ z)$.
(G) 对所有人 $x: \quad x \circ e=x$.
(G3) 对于每个 $x$ 有一个 $y$ 这样 $x \circ y=e$.

$1.1$ 左逆的存在性定理。对于每一个 $x$ 有一个 $y$ 这样 $y \circ x=e$.

## 数学代写|数理逻辑代写Mathematical logic代考|An Example from the Theory of Equivalence Relations

(E1) 对于所有人 $x: x R x$.
(E2) 对于所有人 $x, y$ : 如果 $x R y$ ，然后 $y R x$.
(E3) 对于所有人 $x, y, z$ : 如果 $x R y$ 和 $y R z$ ， 然后 $x R z$.

. 这对 $\left(A, R^A\right)$ 是结构的另一个例子。我们称之为 $R^A$ 上的等价关系 $A$, 和结构 $\left(A, R^A\right)$ 如果满足 (E1)、
(E2) 和 (E3)，则为等价结构。例如， $\left(\mathbb{Z}, R_5\right)$ 是一个等价结构，其中 $\mathbb{Z}$ 是整数集，并且
$R_5=(a, b) \mid a, b \in \mathbb{Z}$ and $b-a$ is divisible by 5.

$2.1$ 定理。如果 $x$ 和 $y$ 都等价于第三个元素，它们等价于相同的元素。更正式地说：对于所有人 $x$ 和 $y$ ，如果 有 $u$ 这样 $x R u$ 和 $y R u$ ，那么对于所有 $z, x R z$ 当且仅当 $y R z$.

$x R y$,

$y R x$

$$x R z$$

$$y R z$$

$$y R z$$

$$x R z$$

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## MATLAB代写

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