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# 数学代写|离散数学代写Discrete Mathematics代考|MA210 Properties of Relations

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## 数学代写|离散数学代写Discrete Mathematics代考|Properties of Relations

A binary relation on a set $A$ is a binary relation from $A$ to $A$ that is a subset of $A \times A$. There are various ways to classify relations on a set. In this section, we focus on the most important properties that a relation $R$ on a set $A$ can have, namely, reflexive, symmetric, and transitive. In order to prove a relation has one of these properties, the method of exhaustion or the method of generalization needs to be employed.

A relation $R$ on a set $A$ is reflexive if and only if $(a, a) \in R$ for every element $a \in A$. In informal terms, in a reflexive relation, each element is related to itself. An example of a reflexive relation is the equality relation on the set of real numbers, as every real number is equal to itself, another example of a reflexive relation is the divides relation on the set of positive integers, as every positive integer divides itself. Using quantifiers, the relation $R$ on the set $A$ is reflexive if $\forall a((a, a) \in R)$.

A relation $R$ on a set $A$ is antireflexive, also known as irreflexive, if and only if $(a, a) \notin$ $R$ for every element $a \in A$. In informal terms, no element in $A$ is related to itself. An example of an antireflexive relation is the greater-than relation on the real numbers. Note that antireflexive does not mean not reflexive, as it is possible to define a relation where some elements are related to themselves, but others are not (i.e., neither all nor none is).

## 数学代写|离散数学代写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.

## 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。