Digital-electronics 简明教程

Laws of Boolean Algebra

Boolean algebra 是一种处理逻辑运算和二进制数字系统的数学工具。它构建了数字电子器件和计算机科学的基础。

Boolean algebra is a mathematical tool that deals with logical operations and binary number system. It builds the foundation of digital electronics and computer science.

布尔代数中的定律和规则是对所有逻辑表达式构建的一组逻辑语句或表达式。布尔代数的每项定律都可以解释为由逻辑门等逻辑电路执行的操作。

The laws and rules in Boolean algebra are the sets of logical statements or expressions upon which all the logical expressions are built. Each law of the Boolean algebra can be interpreted as an operation performed by a logic circuit like a logic gate.

在本章，我们将学习用于简化逻辑函数和布尔表达式的 laws and rules of Boolean algebra 。这些定律和规则是布尔代数中的基本工具，有助于降低复杂性及优化数字电路和系统。

In this chapter, we will learn about laws and rules of Boolean algebra that are used to simplify the logical functions and Boolean expressions. These laws and rules are essential tools in Boolean algebra that help to reduce the complexity and optimize the digital circuits and systems.

让我们详细学习用于执行逻辑运算的布尔代数的主要定律和规则。

Let us learn the primary laws and rules of Boolean algebra in detail that are used to perform logical operations.

Laws of Boolean Algebra

下文中解释了布尔代数的所有重要定律和法则 -

All the important laws and rules of Boolean algebra are explained below −

Rules of Logical Operations

有三个基本的逻辑运算，即 AND、OR 和 NOT。下表突出显示了与这三个逻辑运算相关联的法则 -

There are three basic logical operations namely, AND, OR, and NOT. The following table highlights the rules associated with these three logical operations −

AND Operation

OR Operation

NOT Operation

0 AND 0 = 0

0 OR 0 = 0

NOT of 0 = 1

0 AND 1 = 0

0 OR 1 = 1

NOT of 1 = 0

1 AND 0 = 0

1 OR 0 = 1

1 AND 1 = 1

1 OR 1 = 1

可以使用逻辑门实现这些布尔代数定律。

These rules of Boolean algebra can be implemented using logic gates.

AND Laws

在布尔代数中，有如下四个 AND 定律 -

In Boolean algebra, there are four AND laws given below −

Law 1 − A · 0 = 0 (This law is called null law).
Law 2 − A · 1 = A (This law is called identity law).
Law 3 − A · A = A
Law 4 − A · A' = 0

OR Laws

下面描述了四个 OR 定律 -

There are four OR laws described below −

Law 1 − A + 0 = A (This law is called null law).
Law 2 − A + 1 = 1 (This law is called identity law).
Law 3 − A + A = A
Law 4 − A + A' = 1

Complementation Laws

布尔代数中有以下五个补码定律

There are following five complementation laws in Boolean algebra −

Law 1 − 0' = 1
Law 2 − 1' = 0
Law 3 − If A = 0, Then A' = 1
Law 4 − If A = 1, Then A' = 0
Law 5 − (A')' = A (This is called double complementation law)

Commutative Laws

布尔代数中有以下两个交换律

There are following two commutative laws in Boolean algebra −

Law 1 − 根据此定律，A OR B 运算产生的输出与 B OR A 运算产生的输出相同，即

Law 1 − According to this law, the operation A OR B produces the same output as the operation B OR A, i.e.,

A + B = B + A

因此，变量的顺序不影响 OR 运算。

Hence, the order of the variables does not affect the OR operation.

此定律可以扩展到任意数量的变量。例如，对于三个变量，它将变为

This law can be extended to any number of variables. For example, for three variables, it will be,

A + B + C = C + B + A = B + C + A = C + A + B

Law 2 − 根据此定律，A AND B 运算的输出与 B AND A 运算的输出相同，即

Law 2 − According to this law, the output of the A AND B operation is same as that of the B AND A operation, i.e.,

A · B = B · A

此定律指出，ANDed 变量的顺序不影响结果。

This law states that the order in which the variables are ANDed does not affect the result.

我们可以将这个定理扩展到任何数量的变量。例如，对于三个变量，我们得到，

We can extend this law to any number of variables. For example, for three variables, we get,

A · B · C = A · C · B = C · B · A = C · A · B

Associative Laws

结合律定义了组合变量的方式。如下所述，有两种结合律。

Associative laws define the ways of grouping the variables. There are two associative laws as described below.

Law 1 − 表达式 A OR B ORed 与 C 的结果与 A Ored 与 B OR C 的结果相同，即

Law 1 − The expression A OR B ORed with C results the same as the A Ored with B OR C, i.e.,

(A + B) + C = A + (B + C)

这个定理可以扩展到任何数量的变量。例如，对于 4 个变量，我们得到，

This law can be extended to any number of variables. For example, for 4 variables, we get,

(A + B + C) + D = A + (B + C + D) = (A + B) + (C + D)

Law 2 − 表达式 A AND B ANDed 与 C 的结果与表达式 A ANDed 与 B AND C 的结果相同，即

Law 2 − The expression A AND B ANDed with C results the same as the expression A ANDed with B AND C, i.e.,

(A · B) · C = A · (B · C)

我们可以将这个定理扩展到任何数量的变量。例如，如果我们有 4 个变量，则

We can extend this law to any number of variables. For example, if we have 4 variables, then

(ABC)D = A(BCD) = (AB)·(CD)

Distributive Laws

在布尔代数中，以下两个分配律允许乘法或对表达式进行因式分解。

In Boolean algebra, there are the following two distributive laws that allow for multiplying or factoring out of expressions.

Law 1 − 根据这个定理，我们对多个变量进行 OR 操作，然后对结果与单个变量进行 AND 操作。

Law 1 − According to this law, we OR several variables and then AND the result with a single variable.

它与将单个变量与每个多个变量进行 AND 操作，然后对乘积项进行 OR 操作的表达式产生相同结果，即，

It gives the same result as the expression in which the single variable is ANDed with each of the several variables and then ORed the product terms, i.e.,

A · (B + C) = AB + AC

我们可以将这个定理扩展到任何数量的变量。例如，

We can extend this law to any number of variables. For example,

A(BC + DE) = ABC + ADE

AB(CD + EF) = ABCD + ABEF

Law 2 − 根据这个定理，如果我们对多个变量进行 AND 操作，然后对结果与单个变量进行 OR 操作。

Law 2 − According to this law, if we AND several variables and then the result is ORed with a single variable.

它得到与我们对单个变量与多个变量中的每个变量取“或”操作并计算和运算的结果相同，即

It gives the same result as we OR the single variable with each of the several variables and then the sum terms are ANDed together, i.e.,

A + BC = (A + B)(A + C)

Proof − 此定律的证明在此处说明，

Proof − The proof of this law is explained here,

右侧 = (A + B)(A + C)

RHS = (A + B)(A + C)

AA + AB + AC + BC

A + AB + AC + BC

A (1 + B + C) + BC

因为

Since,

1 + B + C = 1 + C = 1

因此，

Therefore,

A · 1 + BC = A + BC = 左侧

A · 1 + BC = A + BC = LHS

Redundant Literal Rule (RLR)

根据此规则，布尔代数有两条定律，在此处进行说明。

Under this rule, there are two laws in Boolean algebra, which are explained here.

Law 1 − 根据此定律，如果我们将某个变量与变量的补集与另一个变量的与运算的结果进行“或”运算，则与将这两个变量进行“或”运算结果相同，即

Law 1 − According to this law, if we OR a variable with the AND of the complement of the variable and another variable. Then, it is same as the OR of the two variables, i.e.,

A + A’B = A + B

Proof − 此定律的证明在此处说明，

Proof − The proof of this law is explained here,

左侧 = A + A’B = (A + A’)(A + B)

LHS = A + A’B = (A + A’)(A + B)

1 · (A + B) = A + B = RHS

Law 2 − 根据此定律，如果我们将某个变量与变量的补集与另一个变量的或运算的结果进行与运算，则与将这两个变量进行与运算的结果相同，即

Law 2 − According to this law, if we AND a variable with the OR of the complement of the variable and another variable, it is equivalent to when we AND the two variables, i.e.,

A(A’ + B) = AB

Proof − 此定律可通过以下方式证明，

Proof − This law can be proved as follows,

左侧 = A(A’ + B) = AA’ + AB

LHS = A(A’ + B) = AA’ + AB

0 + AB = AB = RHS

这两条定律表明出现在另一项中的项的补集是多余的。因此，此规则被命名为冗余文字规则。

Both these laws show that the complement of a term appearing in another term is redundant. Hence, the rule is named as Redundant Literal Rule.

Idempotence Laws

“幂等性”一词是“相同值”的同义词。布尔代数中有两条幂等性定律。它们是：

The term "idempotence" is a synonym for "same value". There are two idempotence laws in Boolean algebra. They are,

Law 1 − 根据此定律，将某个变量与其自身进行与运算等于该变量，即

Law 1 − According to this law, ANDing a variable with itself is equal to the variable, i.e.,

A · A = A

Law 2 - 按照此定律，变量自相或等于该变量，即

Law 2 − According to this law, ORing a variable with itself is equal to the variable, i.e.,

A + A = A

Absorption Laws

布尔代数中有两条吸收定律，具体说明如下。

There are two absorption laws in Boolean algebra and they are explained below.

Law 1 - 按照此定律，如果我们把一个变量与该变量和另一个变量的与或在一起，则等于变量本身，即

Law 1 − According to this law, if we OR a variable with the AND of the that variable and another variable, then it is equal to the variable itself, i.e.,

A + A · B = A

可按如下方式证明，

This can be proved as follows,

LHS = A + A · B = A · (1 + B)

A · 1 = A = RHS

Law 2 - 按照此定律，某个变量与该变量和另一个变量的或的与等于该变量本身，即

Law 2 − According to this law, the AND of a variable with the OR of that variable and another variable is equivalent to the variable itself i.e.,

A(A + B) = A

也可以按如下方式证明，

This can also be proved as follows,

LHS = A(A + B) = AA + AB

A + AB = A(1 + B) = A · 1 = A = RHS

因此，此定律证明，如果一个项出现在另一项中，那么后者将变得冗余，可以从该表达式中移除。

Hence, this law proves that if a term appears in another term, then the latter term will become redundant and can be removed from the expression.

DeMorgan’s Theorem

在布尔代数中，德摩根定理定义了两条定律，具体说明如下。

In Boolean algebra, DeMorgan’s theorem defines two laws which are explained below.

Law 1 - 按照此定律，若干个变量之和的补等于每个变量的补的乘积，即

Law 1 − According to this law, the complement of a sum of variables is equivalent to the product of complement of each of the variables, i.e.,

\mathrm{\overline{A+B} \: = \: \bar{A}\cdot\bar{B}}

此定律可推广至任意数量的变量。

This law can be extended to any number of variables.

Law 2 - 德摩根定理的第二条定律指出，若干个变量的乘积的补等于每个变量的补的和，即

Law 2 − The second law of DeMorgan’s theorem states that the complement of a product of variables is equivalent to the sum of complement of each of the variables, i.e.,

\mathrm{\overline{AB} \: = \: \bar{A}\: + \:\bar{B}}

此定律也可推广至任意数量的变量。

This law can also be extended to any number of variables.

Conclusion

在此章节中，我们解释了布尔代数中使用的一切重要定律、规则和定理。这些规则和定律被大量用于化简数码电子逻辑表达式。

In this chapter, we explained all the important laws, rules, and theorems used in Boolean algebra. These rules and laws are extensively used to simplify the logical expressions in digital electronics.

基本上，所有这些规则为我们提供了化简复杂布尔函数的一套工具，并能使数码电路变得更简单。

Basically, all these rules provide a set of tools for simplification of complex Boolean functions and make the digital circuits simpler.