布尔代数_布尔代数简介

布尔代数

A Boolean Algebra is an algebra(set, operations, elements) consisting of a set B with >=2 elements, together with three operations- the AND operation(Boolean Product), the OR operation(Boolean Sum), and the NOT operation(Complement) - defined on the set, such that for any element a, b, c…. of set B, a*b, a+b and a' are in B.

布尔代数是由具有> = 2个元素的集合B以及三个运算组成的代数(集合，运算，元素)，这三个运算分别是AND运算(布尔乘积)， OR运算(布尔和)和NOT运算(补码)-在集合上定义，使得对于任何元素a，b，c…。集合B中的a * b，a + b和a'在B中。

For Example:

例如：

Consider the four-element Boolean algebra B4 = ( {0, x, y, 1}; *, +, '; 0, 1). The AND, OR and NOT operations are described by the following tables:

考虑四元素布尔代数B4 =({0，x，y，1}; *，+，'; 0，1) 。下表描述了AND ， OR和NOT运算：
Consider the two-element Boolean algebra B2 = ({0,1}; *, +, '; 0, 1). The three operations *(AND), +(OR) and '(NOT) are defined as follows:

考虑二元布尔代数B2 =({0,1}; *，+，'; 0，1) 。定义三个操作*(AND) ， +(OR)和'(NOT)如下：

一些关键术语 (Some Key Terms)

Here are some key terms of the Boolean Algebra with a brief description about them:

以下是布尔代数的一些关键术语，并对其进行了简要说明：

布尔函数 (Boolean Function)

Boolean algebra is a switching algebra that deals with binary variables and logic operations. The variables are designated by letters such as A, B, x, and y. The three basic logic operations are AND, OR and NOT. A Boolean function can be expressed algebraically with binary variables, the logic operation symbols, parentheses and equal sign. For a given combination of values of the variables, the Boolean function can be either 1 or 0. Consider for example, the Boolean Function:

布尔代数是处理二进制变量和逻辑运算的交换代数。变量由字母(例如A，B，x和y)指定。三个基本逻辑运算是AND，OR和NOT。布尔函数可以用二进制变量，逻辑运算符号，括号和等号代数表示。对于给定的变量值组合，布尔函数可以为1或0。例如，考虑布尔函数：

F = x + y'z

The Function F is equal to 1 if x is 1 or if both y' and z are equal to 1; F is equal to 0 otherwise.

如果x为1或y'和z都等于1，则函数F等于1；否则F等于0。

真相表 (Truth Table)

The relationship between a function and its binary variables can be represented in a truth table. To represent a function in a truth table we need a list of the 2ⁿ combinations of the n binary variables.

函数及其二进制变量之间的关系可以在真值表中表示。为了在真值表中表示一个函数，我们需要ⁿ个二进制变量的2 ^n个组合的列表。

逻辑图 (Logic Diagram)

A Boolean function can be transformed from an algebraic expression into a logic diagram composed of AND, OR and NOT gates.

布尔函数可以从代数表达式转换为由AND，OR和NOT门组成的逻辑图。

The purpose of Boolean algebra is to facilitate the analysis and design of digital circuits. It provides a convenient tool to:

布尔代数的目的是促进数字电路的分析和设计。它提供了一个方便的工具来：

Express in algebraic form a truth table relationship between binary variables.

以代数形式表示二进制变量之间的真值表关系。
Express in algebraic for the input-output relationship of logic diagrams.

以代数形式表示逻辑图的输入输出关系。
Find simpler circuits for the same function.

查找具有相同功能的更简单的电路。

A Boolean function specified by a truth table can be expressed algebraically in many different ways. Two ways of forming Boolean expressions are Canonical and Non-Canonical forms.

由真值表指定的布尔函数可以用许多不同的方式代数表示。形成布尔表达式的两种方法是经典和非规范的形式。

规范形式 (Canonical Form)

It expresses all binary variables in every product(AND) or sum(OR) term of the Boolean function. To determine the canonical sum-of-products form for a Boolean function F(A, B, C) = A'B + C' + ABC, which is in non-canonical form, the following steps are used:

它表示布尔函数的每个乘积(AND)或sum(OR)项中的所有二进制变量。要确定非规范形式的布尔函数F(A，B，C)= A'B + C'+ ABC的规范乘积和形式，请使用以下步骤：

F = A'B + C' + ABC= A'B(C + C') + (A + A')(B + B')C' + ABC

where x + x' = 1 is a basic identity of Boolean algebra

其中x + x'= 1是布尔代数的基本恒等式

= A'BC + A'BC' + ABC' + AB'C' + A'BC' + A'B'C' + ABC
= A'BC + A'BC' + ABC' + AB'C' + A'B'C' + ABC

By manipulating a Boolean expression according to Boolean algebra rules, one may obtain a simpler expression that will require fewer gates.

通过根据布尔代数规则操纵布尔表达式，可以得到需要较少门的较简单表达式。

The below table lists the most basic identities of Boolean algebra. All the identities in the table can be proven by means of truth tables.

下表列出了布尔代数的最基本身份。表格中的所有身份都可以通过真值表来证明。

翻译自: https://www.studytonight.com/computer-architecture/boolean-algebra

布尔代数