Digital-electronics 简明教程
Digital Electronics - DeMorgan’s Theorem
在布尔代数中，定义了几条规则来执行数字逻辑电路中的运算。布尔代数是一种用于对二进制位（即 0 和 1）执行运算的工具。这两个二进制位 0 和 1 用于表示数字电路中输入和输出端的假和真状态。布尔代数由 George Boole 创建，使用 0 和 1 创建真值表以及数字电路的逻辑表达式（例如 AND、OR、NOT 等），用于分析和简化复杂电路。
In Boolean algebra, several rules are defined to perform operations in digital logic circuits. Boolean algebra is a tool to perform operation on binary digits, i.e. 0 and 1. These two binary digits 0 and 1 are used to denote FALSE and TRUE states of a digital circuit at input and output ends. Boolean algebra, developed by George Boole, uses 0s and 1s to create truth tables and logic expressions of digital circuits like AND, OR, NOT, etc. which are used to analyze and simplify the complex circuits.
还有另一位英国数学家 Augustus DeMorgan 解释说，与非运算 (NAND) 和或非运算 (NOR) 分别对应于非与运算 (NOT AND) 和非或运算 (NOT OR)。此解释被称为德摩根定理。本教程将详细讨论 DeMorgan’s theorem 。
There were another English mathematician Augustus DeMorgan who explained the NAND and NOR operations as NOT AND and NOT OR operations respectively. This explanation was named De Morgan’s Theorem. In this tutorial, we will discuss the DeMorgan’s theorem in detail.
What is DeMorgan’s Theorem?
德摩根定理是布尔代数中一个有力的定理，它拥有一组两条规则或定律。这两条定律是为展示两个变量与运算、或运算和非运算之间的关系而制定的。这两条规则使变量能够被取反，即取其原始形式的反形式。因此，德摩根定理给出了逻辑函数的对偶。
DeMorgan’s Theorem is a powerful theorem in Boolean algebra which has a set of two rules or laws. These two laws were developed to show the relationship between two variable AND, OR, and NOT operations. These two rules enable the variables to be negated, i.e. opposite of their original form. Therefore, DeMorgan’s theorem gives the dual of a logic function.
现在，让我们讨论德摩根定理的两条定律。
Now, let us discuss the two laws of DeMorgan’s theorem.
DeMorgan’s First Theorem (Law 1)
德摩根第一定律指出，一堆变量的补集 (ORing) 等于它们的个体补集的乘积 (ANDing)。换言之，两个或更多个 OR 运算的变量的补集相当于每个个体变量补集的 AND 运算，即
DeMorgan’s First Law states that the complement of a sum (ORing) of variables is equal to the product (ANDing) of their individual complements. In other words, the complement of two or more ORed variables is equivalent to the AND of the complements of each of the individual variables, i.e.
\mathrm{\overline{A+B} \: = \: \bar{A} \cdot \bar{B}}
或者，也可以表示为
Or, it may also be represented as,
\mathrm{\lgroup A \: + \: B \rgroup' \: = \: A'\cdot B'}
此定律左右两侧的逻辑实现如图 1 所示。
The logic implementation of left side and right side of this law is shown in Figure 1.
因此，德摩根第一定理证明了或非门 (NOR) 等于一个带气泡的与门 (AND)。下方的真值表显示了此定律的证明。
Thus, DeMorgan’s first law proves that the NOR gate is equivalent to a bubbled AND gate. The following truth table shows the proof of this law.
Left Side |
Right Side |
Input |
Output |
Input |
Output |
A |
B |
(A + B)' |
A' |
B' |
A'· B' |
0 |
0 |
1 |
1 |
1 |
1 |
0 |
1 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
此真值表证明了左侧的布尔表达式等价于德摩根第一定律表达式的右侧。
This truth table proves that the Boolean expression on the left is equivalent to that on the right side of the expression of DeMorgan’s first law.
此外，德摩根定理的第一定律可以扩展到任意数量的变量或变量组合。
Also, the first law of DeMorgan’s theorem can be extended to any number of variables, or a combination of variables.
例如，
For example,
\mathrm{\overline{A \: + \: B \: + \: C \: + \: D \: + \: E \: + \: \dotso} \: = \: \bar{A} \: \bar{B} \: \bar{C} \: \bar{D} \: \bar{E} \: \dotso}
又
Also,
\mathrm{\overline{ABC \: + \: DE \: + \: FGH \: + \: \dotso}\: = \: \overline{\lgroup ABC \rgroup}.\overline{\lgroup DE \rgroup}.\overline{\lgroup FGH\rgroup}.\dotso}
从上述讨论中，我们可以得出结论：德摩根第一定律将一个非运算符号下的求和形式的表达式转换为乘积形式。
From the above discussion, we may conclude that the DeMorgan’s First Law converts an expression from a sum form under a NOT sign to a product form.
DeMorgan’s Second Theorem (Law 2)
德摩根第二定律指出，一堆变量的乘积 (ANDing) 的补集相当于它们的个体补集的和 (ORing)。
DeMorgan’s second law states that the complement of the product (ANDing) of variables is equivalent to the sum (ORing) of their individual complements.
换句话说，两个或更多 AND 变量的补等于每个单个变量的补的和，即，
In other words, the complement of two or more ANDed variables is equal to the sum of the complement of each of the individual variables, i.e.,
\(\mathrm{\overline{AB} \: = \: \overline{A} \: + \: \overline{B}}\)
\mathrm{\overline{AB} \: = \: \overline{A} \: + \: \overline{B}}
也可以表示为，
It may also be represented as,
\(\mathrm{\lgroup AB \rgroup' \: = \: A' \: + \: B'}\)
\mathrm{\lgroup AB \rgroup' \: = \: A' \: + \: B'}
此表达式的左右两边的逻辑实现如图 2 所示。
The logic implementation of left and right sides of this expression is shown in Figure 2.
因此，德·摩根的第二定律证明了 NAND 门等效于加圈的 OR 门。下表显示了此定律的证明。
Hence, DeMorgan’s second law proves that the NAND gate is equivalent to a bubbled OR gate. The following truth table shows the proof of this law.
Left Side |
Right Side |
Input |
Output |
Input |
Output |
A |
B |
AB |
A' |
B' |
A' + B' |
0 |
0 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
1 |
0 |
1 |
1 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
1 |
0 |
0 |
0 |
此真值表证明了德·摩根第二定律表达式的左侧的布尔表达式等效于右侧的表达式。
This truth table proves that the Boolean expression on the left side is equivalent to that on the right side of the expression of DeMorgan’s second law.
类似于第一定律，我们可以将德·摩根第二定律扩展到任意数量的变量或变量组合。
Similar to the first law, we may extend the DeMorgan’s second law for any number of variables or combination of variables.
例如，
For example,
\(\mathrm{\overline{ABCDE \dotso} \: = \: \overline{A} \: + \: \overline{B} \: + \: \overline{C} \: + \: \overline{D} \: + \: \overline{E} \: + \: \dotso}\)
\mathrm{\overline{ABCDE \dotso} \: = \: \overline{A} \: + \: \overline{B} \: + \: \overline{C} \: + \: \overline{D} \: + \: \overline{E} \: + \: \dotso}
而对于变量组合，
And, for a combination of variables,
\(\mathrm{\overline{\lgroup ABC \rgroup} \overline{\lgroup DE \rgroup} \overline{\lgroup FG \rgroup \dotso} \: = \: \overline{ABC} \: + \: \overline{DE} \: + \: \overline{FG}}\)
\mathrm{\overline{\lgroup ABC \rgroup} \overline{\lgroup DE \rgroup} \overline{\lgroup FG \rgroup \dotso} \: = \: \overline{ABC} \: + \: \overline{DE} \: + \: \overline{FG}}
因此，从上面的讨论中，我们可以得出结论，即德·摩根第二定律将变量的乘积形式或变量组合在 NOT 符号下转换为加和形式。
Hence, from the above discussion, we can conclude that DeMorgan’s second law transforms a product form of variables or combination of variables under a NOT sign into a sum form.
因此，德·摩根定律将 AND 运算转换为 OR 运算，将 OR 运算转换为 AND 运算。此原理称为对偶性。
Therefore, DeMorgan’s laws transforms an AND operation into an OR operation, and an OR operation into an AND operation. This principle is called duality.
Example 1
将德·摩根定理应用于以下布尔表达式，
Apply DeMorgan’s theorem to the following Boolean expression,
\(\mathrm{F \: = \: \overline{AB \overline{ \lgroup C \: + \: D \rgroup}EF}}\)
\mathrm{F \: = \: \overline{AB \overline{ \lgroup C \: + \: D \rgroup}EF}}
Solution
给定表达式为，
Given expression is,
\(\mathrm{F \: = \: \overline{AB \overline{ \lgroup C \: + \: D \rgroup}EF}}\)
\mathrm{F \: = \: \overline{AB \overline{ \lgroup C \: + \: D \rgroup}EF}}
由于给定表达式在 NOT 符号下有 AND 运算，因此应用德·摩根第二定律，得到，
As the given expression has AND operation under a NOT sign, thus on applying DeMorgan’s second law, we get,
\(\mathrm{F \: = \: \overline{AB} \: + \: \lgroup C \: + \: D \rgroup \: + \: \overline{EF}}\)
\mathrm{F \: = \: \overline{AB} \: + \: \lgroup C \: + \: D \rgroup \: + \: \overline{EF}}
这是给定表达式的等价或对偶形式。
This is the equivalent or the dual of the given expression.
Example 2
将德·摩根定理应用于以下布尔表达式，
Apply DeMorgan’s theorem to the following Boolean expression,
\mathrm{F \: = \: \overline{AB \: + \: \overline{CD}}}
Solution
给定表达式为，
Given expression is,
\mathrm{F \: = \: \overline{AB \: + \: \overline{CD}}}
给定的表达式是符号 NOT 下变量的和形式，因此在应用 DeMorgan 定律时，我们可以获得这个表达式的对偶。
The given expression is in the form of a sum of variables under a NOT sign, thus on applying DeMorgan’s first law, we get the dual of this expression.
\mathrm{F \: = \: \overline{AB} \cdot \overline{\overline{CD}} \: = \: \overline{AB} \cdot CD}
在本章中，我们解释了 DeMorgan 定理的两个定律，并展示了它们如何在数字逻辑电路中执行不同的操作中提供帮助。
In this chapter, we explained the two laws of DeMorgan’s Theorem and showed how they are helpful in performing different operations in digital logic circuits.