Boolean algebra has three rules of combination, as any algebra must have: the associative, the commutative, and the distributive rules. To show the features of the algebra we use the variables A, B, C, and so on. To write relations between variables each one of which may take the value 0 or l, we use to mean “not A,” so if A = l , then = 0. Thecomplement of every variable is expressed by placing a bar over the variable; the complement of
= \ I = l; the other is null, null = 0
布尔代数与任何代数一样具有结合律、交换律和分配律。为了表示代数的特性我们使用变量A,B和C以及诸如此类的变量。为了写出这些可能取值为0或1的各个变量之间的相互关系，我们采用来ā表示“非A”，因此如果A＝1，那么ā＝0。每个变量的补码用每个变量上方加一横线来表示，B的补码就是ā也即“非B”。同时还存在两个固定的量。第一个量是单位量，即I＝1，另外一个量是零，即null＝0。 Boolean algebra applies to the arithmetic of three basic types of gates: an OR-gate, an AND-gate and the inverter. The symbol and the truth tables for the logic gates are shown in Fig.2-3, the truth table illustrate that the AND-gate corresponds to multiplication, the OR-gate corresponds to addition, and the inverter yield the complement of its input variable. 布尔代数应用于三种基本类型的逻辑门的运算：一种是或门，一种是与门，还有一种是反相器（非门）。逻辑门的符号和真值表如图2－3所
21
示，真值表显示与门对应于乘，或门对应于加，而反相器产生其输入变量的补码
We have already found that AB = \ for the AND-gate and A + B = \
for the OR-gate我们已经算出对于与门来说 AB=“A AND B”而对与或门来说 A+B=“A OR B”
The AND, or conjunctive, algebraic form and the OR, or disjunctive, algebraic form must each obey the three rules of algebraic combination. In the equations that follow, the reader may use the two possible values 0 and l for the variables A, B, and Cto verify the correctness of each expression. Use A = 0, B = 0, C = 0; A = l, B = 0, C = 0; and so on, in each expression. The associative rules state how variables may be grouped.对于“与”，即逻辑乘，以及“或”，即析取，它们的代数形式必须遵循代数组合的三个法则。在接下来的等式中，读者可以把变量A,B,C设为两个可能的值0和1来证明每个表达式的正确性。例如采用A=0,B=0,C=0，或A=1, B=0,C=0等等，在每个表达式中，结合律表明如何把变量进行重组
For AND (AB)C = A(BC) = (AC)B,
and for OR (A + B) + C = A + (B + C) = (A + C) + B 对于“与”有（AB）C＝A（BC）＝（AC）B而对于“或”有（A＋B）＋C＝A＋（B＋C）＝（A＋C）＋B
the rules indicate that different groupings of variables may be used
22
without altering the validity of the algebraic expression这个法则表明我们可以采用变量的不同组合而不改变代数表达式的正确性。交换率表明了变量的顺序
The commutative rules state the order of variables. For AND AB = BA and for OR A+B = B+A
the rules indicate that the operations can be grouped and expanded as shown
对于“与”有AB＝BA，而对于“或”有A＋B＝B＋A。这个法则表明了可以如上式所示进行运算的组合和展开
Before we show the remaining rules of Boolean algebra for digital devices, let us confirm the distributive rule for AND by writing the truth table, Table 2-l. We will discover soon how we knew that we could write AB + C = (A + C)(B + C), which is proved by the truth table to be a proper expansion. 在我们展示数字设备布尔代数的剩下的那个法则之前，让我们通过写出真值表的方式即真值表2－1来验证对于“与”的分配律。我们将很快发现如何写出等式AB＋C＝（A＋C）（B＋C），这一等式由真值表证明了是一个正确的展开式。
The more complex expression and its simpler form yield identical values. Because binary logic is dominated by an algebra in which a sum of ones equals one, the truth table permits us to identify the equivalence among algebraic expressions. A truth table may be used to find a simpler
23
equivalent to a more complex relation among variables, if such an equivalent exists. We will see shortly how the reduction of complexity may be achieved in a systematic manner with truth tables and other techniques更为复杂的表达式和它的一次式产生了相等的值。由于二进制逻辑取决于某一代数，其单个变量之和等于一个变量，所以真值表允许我们在代数表达式中找出等效值，我们可以使用真值表来求出一个等效于变量之间较复杂的关系式的一次表达式。如果这样的等效关系存在，我们将很快看到利用真值表以及其它方法以一种系统性的方式如何完成这样一个复杂步骤的简化工作。
Some additional relations in the algebra, which use identity and null, are worth nothing. Here we illustrate properties of the AND and OR operations that use the distributive rules and the fact that I is always l and null is always 0. AND
AI = A or A1 = AOR
A = null
A+ null = A
A + 0 = AAND A = 0OR A + A null = null A0 =
AA = AOR
A + A
= I A + =1 0OR = A
AND
A + I = I A + 1 = 1AND
The relation points out an important fact, that is, that I, the identity, is the universal set. Null is called the empty set.代数中另外的一些关系式，这些式子中使用单位一和零，是没有意义的，这里我们列举了运用分配律后“与”和“或”运算的性质，结果是1永远是1而
24
零永远是0。
与：AI＝A即A1＝A或：A＋null＝A即A＋0＝A与： 即或： 即 与：Anull＝0即A0＝0或：A＋I＝I即A＋1＝与：AA＝A或：A＋A＝A关系式A＋A＝I指出了一个重要事实，即I，也就是单位量，是全集，而零被称为空集。
We have considered several logical relations. For the two-value Boolean algebra of digital electronics, the choice of the technique depends upon the nature of the function whose reduction is desired. Some simple functions may be easily reduced by examining their truth table; others require the manipulation of Boolean algebra to reveal the relationship . When we consider the circuit foradding binary numbers, we see that Boolean algebra is required to discover a simplification in that particular application我们已经研究了几种逻辑关系。对于电子学的二值布尔代数来说，选择何种方法取决于我们所期望的简化函数的性质。一些简单的函数可以通过观察它们的真值表很容易进行简化；而另一些函数需要通过计算布尔代数来揭示它们的关系。当我们研究有关二进制数相加的电路时，我们将看到需要布尔代数来揭示该特定应用中的简化过程
Semiconductor switches are very important and crucial components in power electronic systems.these switches are meant to be the substitutions of the mechanical switches,but they are severely limited by the properties of the
25