计算机科学家的全面数学基础：集合、数论与图论

需积分: 18 73 浏览量更新于2024-08-02 收藏 3.69MB PDF 举报

"Comprehensive Mathematics for Computer Scientists 2nd ed [Vol 1]" 本书是《计算机科学家的全面数学》第二版的第一卷，涵盖了基础数学概念、逻辑、集合论、图论以及初等代数等内容，旨在为计算机科学的学习提供坚实的数学基础。 1. 基本概念与逻辑 - 推理逻辑：介绍了命题逻辑，这是理解更复杂逻辑结构的基础。 - 概念架构：探讨了构建和理解数学概念的基本原理。 2. 形式集合理论 - 公理系统：阐述了集合理论的公理化方法，这些公理是数学体系的基础。 - 基本概念与结果：讨论了集合的基本操作和性质。 3. 布尔集合代数 - 子集的布尔代数：讲解了如何将集合论与布尔代数相结合，用于处理集合的交、并、补等运算。 4. 函数与关系 - 图与函数：定义了图的概念，以及它们与函数的关系。 - 关系：讨论了关系的性质，如对称性、传递性和反自反性等。 5. 序数与自然数 - 序数：引入了序数，用于表示有限和无限序列的顺序。 - 自然数：定义了自然数集合，并讨论其基本算术运算。 6. 归纳定理与普遍性质 - 归纳定理：介绍了通过归纳法构造数学对象的原理。 - 普遍性质：在数据库理论中展示了如何应用普遍性质。 7. 自然数算术 - 自然数运算：包括加、减、乘、除等运算的性质。 - 欧几里得算法与正规形式：讲述了如何通过欧几里得算法求最大公约数。 8. 无穷 - 对角线化过程：利用对角线论证法探讨无穷概念。 9. 古典数域：整数、有理数、实数和复数 - 整数Z、有理数Q、实数R和复数C的定义和性质。 10. 图的分类 - 有向和无向图：区分了图的不同类型及其性质。 - 图的同态：研究了图之间的结构保持映射。 11. 图的构造 - 介绍如何构造不同的图。 12. 特殊图 - n-元树：讨论了一类特殊的图结构，适用于树形数据结构。 - 摩尔图：与电路理论相关的图型。 13. 平面性 - 多面体的欧拉公式：平面图形的基本拓扑性质。 - 库拉托夫斯基平面性定理：决定了一个图是否可以平面绘制。 14. 首个高级主题 - 浮点计算：深入讨论浮点数表示和运算，包括误差分析。第二部分涉及代数、形式逻辑和线性几何： 15. 单群、群、环与域 - 介绍了代数学的基本结构，如单群、群、环和域。 16. 质数 - 质因数分解：质数在数论中的重要角色及应用。 - 多项式根与插值：质数与多项式理论的联系。 17. 形式命题逻辑 - 逻辑语言的语法和语义：形式逻辑的基本构成元素。 18. 形式谓词逻辑 - 第一阶语言的语法和语义：深入到更高阶的逻辑表达。 19. 语言、文法与自动机 - 语言与文法：形式语言的定义和分类。 - 自动机和接受器：自动计算模型。 20. 矩阵的类别 - 矩阵的定义和基本运算：矩阵在数学和计算机科学中的应用。 21. 模块与向量空间 - 引入了线性代数的重要概念。 22. 线性相关、基与维度 - 向量空间的基：定义了向量空间中一组基的重要性。这本书不仅适合计算机科学家，也适合所有希望深入理解计算机科学背后数学原理的读者。通过这些概念，读者可以更好地理解和解决计算机科学中的问题，例如算法设计、数据结构和计算复杂性。

6 Fundamentals—Concepts and Logic

In practice it is cumbersome to say: “The proposition It rains is true”.

Instead, one just says: “It rains.” Also, since the formal combination of

propositions by the above operators is often an overhead, we mostly use

the common language denotation, such as: “2 = 3 is false” instead of

“NOT (2 = 3)” or: “It’s raining or/and I am tired.” instead of “It’s rain-

ing OR/AND I am tired”, or: “If it’s raining, then I am tired” instead of

“It’s raining IMPLIES I am tired.” Moreover, we use the mathematical ab-

breviation “A iﬀ B” for “(A IMPLIES B) AND (B IMPLIES A)”. Observe that

brackets (. . .) are used in order to make the grouping of symbols clear if

necessary.

The operations NOT, AND, OR, and IMPLIES have a number of properties

which are very useful for simplifying complex combinations of these op-

erations. Let P , Q, and R be truth values. Then the following properties

hold:

Commutativity of AND

P AND Q is the same as Q AND P.

Commutativity of OR

P OR Q is the same as Q OR P .

Associativity of AND

(P AND Q) AND R is the same as P AND (Q AND R).

One usually omits the parentheses and writes P AND Q AND

Associativity of OR

(P OR Q) OR R is the same as P OR (Q OR R).

One usually omits the parentheses and writes P OR Q OR R.

De Morgan’s Law for AND

NOT (P AND Q) is the same as (NOT P) OR (NOT Q).

De Morgan’s Law for OR

NOT (P OR Q) is the same as (NOT P) AND (NOT Q).

Distributivity of AND over OR

P AND (Q OR R) is the same as (P AND Q) OR (P AND R).

Distributivity of OR over AND

P OR (Q AND R) is the same as (P OR Q) AND (P OR R).

Contraposition

P IMPLIES Q is the same as (NOT Q) IMPLIES (NOT P).

1.2 Architecture of Concepts 9

The value of an instance of a concept constructed as a conjunction is the

sequence of the references to the instances of each component.

The value of an instance of a concept constructed as a disjunction is a

reference to an instance of one of the components.

Perhaps some examples will clarify those three construction principles.

A selection is really a selection in its common sense meaning: You point

at a thing and say, “I select this”, you point at another thing and say “I

select this, too” and so on.

One example for a conjunction are persons’ names which (at least in the

western world) always consists of a ﬁrst name and a family name. An-

other example is given by the points in the plane: every point is deﬁned

by an x- and a y-coordinate.

A disjunction is a simple kind of “addition”: An instance of the disjunc-

tion of all fruits and all animals is either a fruit or an animal.

Notation

If we want to write about concepts and instances, we need an expressive

and precise notation.

Concept: ConceptName.ConstructionPrinciple(Component(s))

This means that we ﬁrst write the concept’s name followed by a dot.

After the dot we write the construction principle (Selection, Conjunc-

tion, or Disjunction) used to construct the concept. Finally we add

the component or components which were used for the construction

enclosed in brackets.

Instance: InstanceName@ConceptName(Value)

In order to write down an instance, we write the instance’s name fol-

lowed by an ‘@’. After this, the name of the concept is added, followed

by a value enclosed in brackets. In the case of a disjunction, a semi-

colon directly following the value denotes the ﬁrst component, and a

semicolon preceding the value denotes the second component.

Very often it is not possible to write down the entire information needed

to deﬁne a concept. In most cases one cannot write down components

and values explicitly. Therefore, instead of writing the concept or in-

stance, one only writes its name. Of course, this presupposes that these

剩余386页未读，继续阅读

点点吃得太多了

粉丝: 180
资源: 681

计算机科学家的全面数学基础：集合、数论与图论

Comprehensive Mathematics for Computer Scientists 2nd ed [Vol 2]

Mathematics for Computer Graphics(2nd)

Springer.Comprehensive.Mathematics.for.Computer.Scientists.1

Mathematics for Natural Scientists

Springer.Comprehensive.Mathematics.for.Computer.Scientists.2

Mathematics for Computer Science

mathematics for computer science

mathematics for computer 中文

vb定时显示报警系统设计(论文+源代码)(2024a7).7z

Java毕设项目：基于spring+mybatis+maven+mysql实现的会员积分管理系统【含源码+数据库+毕业论文】

最新资源