# 数学代写|数理逻辑代写Mathematical logic代考|Terms and Formulas

#### Doug I. Jones

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## 数学代写|数理逻辑代写Mathematical logic代考|Terms and Formulas

Suppose that $\mathcal{L}$ is the language ${0,+,<}$, and we are going to use $\mathcal{L}$ to discuss portions of arithmetic. If I were to write down the string of symbols from $\mathcal{L}$,
$$\left(v_1+0\right)<v_1$$
and the string
$$\left.v_{17}\right)(\forall++(((0,$$
you would probably agree that the first string conveyed some meaning, even if that meaning were incorrect, while the second string was meaningless. It is our goal in this section to carefully define which strings of symbols of $\mathcal{L}$ we will use. In other words, we will select the strings that will have meaning.

Now, the point of having a language is to be able to make statements about certain kinds of mathematical systems. Thus, we will want the statements in our language to have the ability to refer to objects in the mathematical structures under consideration. So we will need some of the strings in our language to refer to those objects. Those strings are called the terms of $\mathcal{L}$.

Definition 1.3.1. If $\mathcal{L}$ is a language, a term of $\mathcal{L}$ is a nonempty finite string $t$ of symbols from $\mathcal{L}$ such that either:

1. $t$ is a variable, or
2. $t$ is a constant symbol, or
3. $t$ is $f t_1 t_2 \ldots t_n$, where $f$ is an $n$-ary function symbol of $\mathcal{L}$ and each of the $t_i$ is a term of $\mathcal{L}$.

This is a definition by recursion, as you notice that in the third clause of the definition, $t$ is a term if it contains substrings that are terms. Since the substrings of $t$ are shorter (contain fewer symbols) than $t$, and as none of the symbols of $\mathcal{L}$ are made up of other symbols of $\mathcal{L}$, this causes no problems.

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

You are familiar, no doubt, with proofs by induction. They are the bane of most mathematics students from their first introduction in high school through the college years. It is my goal in this section to discuss the proofs by induction that you know so well, put them in a different light, and then generalize that notion of induction to a setting that will allow us to use induction to prove things about terms and formulas rather than just the natural numbers.

Just to remind you of the general form of a proof by induction on the natural numbers, let me state and prove a familiar theorem, assuming for the moment that the set of natural numbers is ${1,2,3, \ldots}$.
Theorem 1.4.1. For every natural number n,
$$1+2+\cdots+n=\frac{n(n+1)}{2} \text {. }$$
Proof. If $n=1$, simple computation shows that the equality holds. For the inductive case, fix $k \geq 1$ and assume that
$$1+2+\cdots+k=\frac{k(k+1)}{2}$$
If we add $k+1$ to both sides of this equation, we get
$$1+2+\cdots+k+(k+1)=\frac{k(k+1)}{2}+(k+1),$$
and simplifying the right-hand side of this equation shows that
$$1+2+\cdots+(k+1)=\frac{(k+1)[(k+1)((k+1)+1)]}{2},$$
finishing the inductive step, and the proof.

# 数理逻辑代写

## 数学代写|数理逻辑代写Mathematical logic代考|Terms and Formulas

$$\left(v_1+0\right)<v_1$$

$$\left.v_{17}\right)(\forall++(((0,$$

1.3.1.定义如果$\mathcal{L}$是一种语言，则$\mathcal{L}$的项是一个由来自$\mathcal{L}$的符号组成的非空有限字符串$t$，使得:

$t$ 是变量还是

$t$ 是一个不变的符号，还是

$t$ 是$f t_1 t_2 \ldots t_n$，其中$f$是$\mathcal{L}$的一个$n$ -任意函数符号，每个$t_i$是$\mathcal{L}$的一个项。

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

$$1+2+\cdots+n=\frac{n(n+1)}{2} \text {. }$$

$$1+2+\cdots+k=\frac{k(k+1)}{2}$$

$$1+2+\cdots+k+(k+1)=\frac{k(k+1)}{2}+(k+1),$$

$$1+2+\cdots+(k+1)=\frac{(k+1)[(k+1)((k+1)+1)]}{2},$$

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## 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。

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