# 数学代写|离散数学作业代写discrete mathematics代考|MATH200

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## 数学代写|离散数学作业代写discrete mathematics代考|Minimization of Combinational Circuits

It is important to note that combinational circuits are equivalent if and only if their corresponding Boolean expressions are equal or their Boolean tables are identical. In order to design a combinational circuit, we need to have a table specifying the output for each combination of input values. We then determine the sum-of-product expansion to find a set of logic gates that can implement the combinational circuit. However, the sumof-product expansion generally contains more terms than necessary. The Boolean identities along with the binary expression simplification rule, which states $e f+\bar{e} f=f$,

where $e$ and $f$ are binary expressions, can be used iteratively to reduce an expression into a simpler, but equivalent, expression. In order to minimize the number of logic gates, it is important to produce Boolean sums of products that represent a Boolean function with the fewest products of literals such that these products contain the fewest literals possible among all sums of products. This process is called the minimization of the Boolean function, by which a circuit with the fewest gates and fewest inputs can be constructed.
Example 8.13
Simplify the following Boolean expression so as to be able to obtain a simpler combinational circuit:
$$F(x, y, z)=x y \bar{z}+x \bar{y} z+x \bar{y} \bar{z}+x y z+\bar{x} y z$$
Solution
Using idempotent law $(x+x=x)$, we add the $x y z$ term to the expression, which is already in the expression. Using the unity property $(x+\bar{x}=1)$, the identity law $(x \cdot 1=x)$, and the binary expression simplification rule law, we can then simplify the expression:
\begin{aligned} F(x, y, z) & =x y \bar{z}+x \bar{\gamma} z+x \bar{\gamma} \bar{z}+x y z+\bar{x} y z \ & =x y \bar{z}+x \bar{y} z+x \bar{y} \bar{z}+x y z+(x y z+\bar{x} y z) \ & =x(y \bar{z}+\bar{\gamma} z+\bar{\gamma} \bar{z}+y z)+y z(x+\bar{x})=x(y+\bar{y})(z+\bar{z})+y z \ & =x \cdot 1 \cdot 1+y z=x \cdot 1+\gamma z=x+y z . \end{aligned}
If we do not simplify the original expression, we need three inverters, five AND gates, and four OR gates, whereas after simplification, we need only one AND gate and one OR gate.
Reducing the number of gates on a chip can lead to an increase in circuit reliability, a decrease in cost production, an increase in the number of circuits on a chip, and a reduction in processing time required by a circuit. However, simplification of a Boolean algebra to reduce the number of logic gates may be a very difficult task because grouping various terms and applying the laws of Boolean algebra may not always be quite straightforward.

Minimizing Boolean functions with many variables is a computationally intensive problem, but there are methods that can significantly simplify, but not necessarily minimize, Boolean expressions with a large number of literals. One such method is the Karnaugh map, which is an effective graphical method involving just a few variables, as it becomes significantly more difficult when the number of variables is beyond a few.

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

In the context of mathematics of relations, relationships between two sets are often based on ordered pairs made up of two related elements, each belonging to a set. An ordered pair of elements $a$ and $b$ is denoted by $(a, b)$, while noting that $(a, b) \neq(b, a)$ unless $a=b$. The sets of ordered pairs are called binary relations. The binary relations are in contrast to $\boldsymbol{n}$-ary relations, which express relationships among elements of $n$ sets with $n>2$ being an integer and thus involve ordered $n$-tuples. Such a relation is the fundamental structure used in relational databases. The term relation by itself generally refers to a binary relation unless otherwise stated or implied.

A relation between the $\operatorname{sets} A$ and $B$ is a subset $R$ of the Cartesian product $A \times B$, where the Cartesian product is defined as $A \times B={(a, b) \mid a \in A$ and $b \in B}$. If $(a, b) \in R$, it is then read as $a$ is related to $b$. The set $A$ is called the domain of the relation, and the set $B$ is called the range of the relation. If $(a, b) \notin R$, it is then read as $a$ is not related to $b$. If $A=B$, the relation is said to be a relation on $A$.

The relation $R$ is a one-to-one relation if no element of $B$ appears as a second coordinate in more than one ordered pair in $R$, and the relation $R$ is an onto relation if every element of $B$ appears as a second coordinate in at least one ordered pair in $R$.

Unlike functions, every relation has an inverse. If $R$ is a relation between the sets $A$ and $B$, then the inverse relation of $R$, denoted by $R^{-1}$, is a subset of the Cartesian product $B \times A$. In other words, the inverse relation is defined as follows: $R^{-1}=$ ${(b, a) \mid(a, b) \in R}$. The domain and range of $R^{-1}$ are equal, respectively, to the range and domain of $R$. Moreover, if $R$ is a relation on $A$, then $R^{-1}$ is also a relation on $A$. The complementary relation $\bar{R}$ is the set of ordered pairs, which is defined as follows: $\bar{R}=$ ${(a, b) \mid(a, b) \notin R}$.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|Minimization of Combinational Circuits

$F(x, y, z)=x y \bar{z}+x \bar{y} z+x \bar{y} \bar{z}+x y z+\bar{x} y z$

$$F(x, y, z)=x y \bar{z}+x \bar{\gamma} z+x \bar{\gamma} \bar{z}+x y z+\bar{x} y z$$

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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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