数学代考| Boole’s Symbolic Logic 离散数学代写代写 - 代考代写：100%准时可靠您的作业代写专家

George Boole (Fig. 14.2) was born in Lincoln, England, in 1815. His father (a cobbler who was interested in mathematics and optical instruments) taught him mathematics and showed him how to make optical instruments. Boole inherited his father’s interest in knowledge, and he was self-taught in mathematics and Greek. He taught at various schools near Lincoln, and he developed his mathematical knowledge by working his way through Newton’s Principia, as well as applying himself to the work of mathematicians such as Laplace and Lagrange.

He published regular papers from his early twenties, and these included contributions to probability theory, differential equations and finite differences. He developed his symbolic algebra, which is the foundation for modern computing, and he is considered (along with Babbage) to be one of the grandfathers of computing. His work was theoretical, and he never actually built a computer or calculating machine. However, Boole’s symbolic logic was the perfect mathematical model for switching theory, and for the design of digital circuits.

Boole became interested in formulating a calculus of reasoning, and he published a pamphlet titled “Mathematical Analysis of Logic” in 1847 [2]. This short book developed novel ideas on a logical method, and he argued that logic should be considered as a separate branch of mathematics, rather than a part of philosophy. He argued that there are mathematical laws to express the operation of reasoning in the human mind, and he showed how Aristotle’s syllogistic logic could be reduced to a set of algebraic equations. He corresponded regularly on logic with Augustus De Morgan. $^{2}$

He introduced two quantities ‘ 0 ‘ and ‘ 1 ‘ with the quantity 1 used to represent the universe of thinkable objects (i.e. the universal set), and the quantity 0 represents the absence of any objects (i.e. the empty set). He then employed symbols such as $x$, $y, z$, etc. to represent collections or classes of objects given by the meaning attached to adjectives and nouns. Next, he introduced three operators $(+,-$ and $\times)$ that combined classes of objects.

The expression $x, y$ (i.e. $x$ multiplied by $y$ or $x \times y$ ) combines the two classes $x$, $y$ to form the new class $x, y$ (i.e. the class whose objects satisfy the two meanings represented by the classes $x$ and $y$ ). Similarly, the expression $x+y$ combines the two classes $x, y$ to form the new class $x+y$ (that satisfies either the meaning represented by class $x$ or class $y$ ). The expression $x-y$ combines the two classes $x$, $y$ to form the new class $x-y$. This represents the class (that satisfies the meaning represented by class $x$ but not class $y$. The expression $(1-x)$ represents objects that do not have the attribute that represents class $x$.

Thus, if $x=$ black and $y=$ sheep, then $x y$ represents the class of black sheep. Similarly, $(1-x)$ would represent the class obtained by the operation of selecting all things in the world except black things; $x(1-y)$ represents the class of
${ }^{2}$ De Morgan was a 19 th British mathematician based at University College London. De Morgan’s laws in Set Theory and Logic state that: $(\mathrm{A} \cup \mathrm{B})^{c}=\mathrm{A}^{c} \cap \mathrm{B}^{c}$ and $\neg(\mathrm{A} \vee \mathrm{B}) \equiv \neg \mathrm{A} \wedge \neg \mathrm{B}$.

图论代考

George Boole（图 14.2）于 1815 年出生于英国林肯市。他的父亲（一个对数学和光学仪器感兴趣的鞋匠）教他数学并教他如何制造光学仪器。布尔继承了父亲对知识的兴趣，自学了数学和希腊语。他在林肯附近的多所学校任教，并通过学习牛顿原理以及将自己应用于拉普拉斯和拉格朗日等数学家的工作来发展他的数学知识。

他从 20 岁出头开始定期发表论文，其中包括对概率论、微分方程和有限差分的贡献。他开发了符号代数，这是现代计算的基础，他（与巴贝奇一起）被认为是计算的祖父之一。他的工作是理论性的，他从未真正建造过计算机或计算机。然而，布尔的符号逻辑是开关理论和数字电路设计的完美数学模型。

布尔对制定推理演算产生了兴趣，并于 1847 年出版了一本名为“逻辑数学分析”的小册子 [2]。这本简短的书在逻辑方法上提出了新的想法，他认为逻辑应该被视为数学的一个独立分支，而不是哲学的一部分。他认为存在数学定律来表达人类思维中的推理操作，并且他展示了亚里士多德的三段论逻辑如何可以简化为一组代数方程。他定期与奥古斯都·德·摩根就逻辑问题进行通信。 $^{2}$

他引入了两个量“0”和“1”，其中量1用于表示可思考对象的宇宙（即全集），而量0表示不存在任何对象（即空集）。然后，他使用诸如 $x$、$y、z$ 等符号来表示由形容词和名词所赋予的含义所赋予的对象的集合或类别。接下来，他介绍了三个运算符 $(+,-$ 和 $\times)$ 组合对象类。

表达式 $x, y$ （即 $x$ 乘以 $y$ 或 $x \times y$ ）将两个类 $x$, $y$ 组合成新的类 $x, y$ （即类其对象满足 $x$ 和 $y$ 类表示的两种含义）。类似地，表达式 $x+y$ 组合了两个类 $x, y$ 以形成新的类 $x+y$（满足类 $x$ 或类 $y$ 表示的含义）。表达式 $x-y$ 组合了两个类 $x$、$y$ 以形成新的类 $x-y$。这表示类（满足类 $x$ 表示的含义但不满足类 $y$ 的含义。表达式 $(1-x)$ 表示不具有表示类 $x$ 的属性的对象。

因此，如果 $x=$ black 和 $y=$sheep，那么 $x y$ 代表害群之马的类别。类似地，$(1-x)$ 将表示通过选择世界上除黑色事物之外的所有事物的操作获得的类； $x(1-y)$ 表示类
${ }^{2}$ De Morgan 是伦敦大学学院的第 19 位英国数学家。 De Morgan 在集合论和逻辑中的定律指出： $(\mathrm{A} \cup \mathrm{B})^{c}=\mathrm{A}^{c} \cap \mathrm{B}^{c }$ 和 $\neg(\mathrm{A} \vee \mathrm{B}) \equiv \neg \mathrm{A} \wedge \neg \mathrm{B}$。

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Cryptosystem
A system that describes how to encrypt or decrypt messages
Plaintext
Message in its original form
Ciphertext
Message in its encrypted form
Cryptographer
Invents encryption algorithms
Cryptanalyst
Breaks encryption algorithms or implementations

编码理论代写

编码理论（英语：Coding theory）是研究编码的性质以及它们在具体应用中的性能的理论。编码用于数据压缩、加密、纠错，最近也用于网络编码中。不同学科（如信息论、电机工程学、数学、语言学以及计算机科学）都研究编码是为了设计出高效、可靠的数据传输方法。这通常需要去除冗余并校正（或检测）数据传输中的错误。

编码共分四类：^[1]