数学代写|离散数学作业代写discrete mathematics代考|CS3653

2023年4月3日

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数学代写|离散数学作业代写discrete mathematics代考|Representations of Relations

There are various ways to represent a binary relation between two finite sets. Suppose that the relation is from the set $A$ to the set $B$, where the elements of $A$ and $B$ have been listed in some arbitrary order. A set of ordered pairs reflecting a binary relation from $A$ to $B$ can be represented by tables, arrow diagrams, digraphs, and matrices.

Tables can be used to represent binary relations on the same set or on two different sets. In a table, columns are labeled by the elements of the finite set $A$, and rows are labeled by the elements of the finite set $B$. Only the entries of the table that show the set of the ordered pairs are marked. In other words, if a certain entry in the table highlights an ordered pair that is not in the set of ordered pairs reflecting the relation of interest, it is then left unmarked.

Arrow diagrams can show binary relations on the same set or on two different sets using two disjoint disks. In an arrow diagram, the elements of the finite set $A$ (the domain of the relation) are shown in the left-hand disk and the elements of the finite set $B$ (the range of the relation) are shown in the right-hand disk; then arrows from the elements in the left-hand disk to the elements in the right-hand disk are drawn to represent all ordered pairs reflecting the relation of interest.

Digraphs, also known as directed graphs, will be extensively discussed in the chapter on graphs. However, we briefly mention it here in the context of representation of relations. To draw the digraph of a binary relation on a set $A$, points, vertices, or nodes, representing the elements of $A$ are drawn. Each ordered pair is represented using an arc, a link, or an edge, with its direction indicated by an arrow. A directed graph, also known as digraph, consists of a set $V$ of vertices together with a set $E$ of edges. In the edge $\left(a_1, a_2\right), a_1$ is called the initial vertex, and $a_2$ is called the terminal vertex. Note that when the initial vertex is the same as the terminal vertex, the edge is called a loop.

Note that the digraph representing a relation can be used to determine the relation properties in an insightful way. The digraph of a reflexive relation has a loop at every vertex of the digraph. The digraph of a symmetric relation has the property that whenever there is a direct edge from one vertex to another, there is also a direct edge in the opposite direction. The digraph of an antisymmetric relation has the property that between any two distinct vertices, there is at most one direct edge. The digraph of a transitive relation has the property that whenever there are directed edges from, say, the first node to the second node and from the second node to the third node, there is also a directed edge from the first node to the third node.

数学代写|离散数学作业代写discrete mathematics代考|Operations on Relations

Relations can be combined to produce new relations. Operations on relations may include union, intersection, difference, and composition.

Let $R$ and $S$ be any two relations from $A$ to $B$. The union of two relations $R$ and $S$ is defined as $R \cup S={(a, b) \mid(a, b) \in R$ and /or $(a, b) \in S}$; the intersection of two relations $R$ and $S$ is defined as $R \cap S={(a, b) \mid(a, b) \in R$ and $(a, b) \in S}$; and the difference of two relations $R$ and $S$ is defined as $R-S={(a, b) \mid(a, b) \in R$ and $(a, b) \notin S}$. Graphically (i.e., in terms of digraphs), $R \cup S$ consists of all edges in $R$ together with those in $S$, $R \cap S$ consists of all common edges in $R$ and $S$, and $R-S$ consists of all edges in $R$ that are not in $S$.

Suppose the zero-one matrices for the relations $R$ and $S$ are represented by $M_R$ and $M_S$, respectively. The zero-one matrix representing the union of these relations, denoted by $M_R \cup S$, has a 1 in the position where either $M_R$ or $M_S$ has a 1 or both of them have a 1. The zero-one matrix representing the intersection of these relations, denoted by $M_R \cap S$, has a 1 in the position where both $M_R$ and $M_S$ have a 1 . The zero-one matrix representing the difference between the relations $R$ and $S$, denoted by $M_{R-S}$, has a 1 in the position where $M_R$ has a 1 but $M_S$ does not have a 1 .

Let $A, B$, and $C$ be sets, $R$ be a relation from $A$ to $B, S$ be a relation from $B$ to $C$, with $a \in A, b \in B$, and $c \in C$, while noting that $A, B$, and $C$ have $m, n$, and $p$ elements, respectively. Then $R$ and $S$ give rise to a relation from $A$ to $C$, denoted by $S \circ R$, called the composition of two relations $R$ and $S$, and defined by $(a, c) \in(S \circ R)$ if there exists an element $b$ in $B$ such that $(a, b) \in R$ and $(b, c) \in S$. Note that the composition of relations $R$ and $S$ is denoted by $S \circ R$ rather than $R \circ S$. This is done in order to conform with the usual use of $g \circ f$ to denote the composition of $f$ and $g$, where $f$ and $g$ are functions. However, when a relation $R$ is composed with itself, then the meaning of $R \circ R$ is unambiguous.

Suppose $R$ is a relation on a set $A$, that is $R$ is a relation from a set $A$ to itself. The powers of a relation $R$ can be recursively defined from the composition of two relations. Therefore $R \circ R=R^2$ is always defined, and similarly, $R^n=R^{n-1} \circ R$ is defined for all integers $n \geq 2$.

离散数学代写

数学代写|离散数学作业代写discrete mathematics代考|Operations on Relations

$R \cap S=(a, b) \mid(a, b) \in R \$ a n d \$(a, b) \in S$; 以及 两种关系的区别 $R$ 和 $S$ 定义为
$R-S=(a, b) \mid(a, b) \in R \$ a n d \$(a, b) \notin S$. 以图 形方式 (即，在有向字母方面)， $R \cup S$ 由所有边组成 $R$ 与那些在 $S, R \cap S$ 由所有公共边组成 $R$ 和 $S$ ，和 $R-S$ 由所有边组成 $R$ 不在 $S$.

有限元方法代写

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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。