计算机概论第四章

Answers are in blue, except for circuit diagrams.

For Exercises 1–17, mark the answers true and false as follows:

1. True
2. False
   1. Logic diagrams and truth tables are equally powerful in expressing the processing of gates and circuits.

A

1. 1. Boolean expressions are more powerful than logic diagrams in expressing the processing of gates and circuits.

B

1. 1. A NOT gate accepts two inputs. B
   2. The output value of an AND gate when both inputs are 1 is 1. A
   3. The AND and OR gates produce opposite results for the same input

B

1. 1. The output value of an OR gate when both inputs are 1 is 1. A
   2. The output of an OR gate when one input is 0 and one input is 1 is 0.

B

1. 1. The output value of an XOR gate is 0 unless both inputs are 1. B
   2. The NOR gate produces the opposite results of the XOR gate. B
   3. A gate can be designed to accept more than two inputs. A
   4. A transistor is made of semiconductor material. A
   5. Inverting the output of an AND gate is equivalent to invert- ing the individual signals first, then passing them through an OR gate.

A (DeMorgan‘s law)

1. Integrated circuits are classified by the number of gates con- tained in them.

A

1. A CPU is an integrated circuit. A

For Exercises 18–29, match the gate with the diagram or description of the operation.

1. 1. AND
   2. NAND
   3. XOR
   4. OR
   5. NOR
   6. NOT
2. Inverts its input. F
3. Produces a 1 only if all its inputs are 1 and a 0 otherwise. A
4. Produces a 0 only if all its inputs are 0 and a 1 otherwise. D
5. Produces a 0 only of its inputs are the same and a 1 otherwise. C
6. Produces a 0 of all its inputs are all 1 and a 1 otherwise. B
7. Produces a 1 if all its inputs are 0 and a 0 otherwise. E

Exercises 30–73 are short answer or design questions.

1. How is voltage level used to distinguish between binary digits?

A voltage level in the range of 0 to 2 volts is interpreted as a binary 0. A voltage level in the range of 2+ to 5 volts is inter- preted as a binary 1.

0-2为0.2-5为1

1. Distinguish between a gate and a circuit.

A gate accepts one or more input signals and produces an output signal. Each type of gate performs one logical func- tion. A circuit is a combination of gates designed to accom- plish a more-complex logical function.

门接受一个或多个，产生一个输出

电路时门的组合

1. What are the three notational methods for describing the behavior of gates and circuits?

Boolean expressions, logic diagrams, and truth tables

1. Characterize the notations asked for in Exercise 32.

Boolean expressions use the operations of Boolean algebra to describe the behavior of gates and circuits. Logic dia- grams use a graphical representation to describe the behav- ior of gates and circuits. Truth tables define the behavior of gates and circuits by showing all possible input and output combinations of the gates and circuits.

布尔表达式使用布尔代数的操作来描述门和电路的行为。逻辑图使用图形表示法来描述门和电路的行为。真值表通过显示门和电路的所有可能输入和输出组合来定义它们的行为。

1. How many input signals can a gate receive and output sig- nals can a gate produce?

A gate can accept one or more input signals, but can pro- duce only a single output value.

1. Name six types of gates.

NOT, AND, OR, XOR, NAND, and NOR

1. Give the three representations of a NOT gate and say in words what NOT means.

A is the input signal and X is the output signal. Boolean expression: X = A’

Truth table:

NOT takes a binary input value and inverts it.

1. Give the three representations of an AND gate and say in words what AND means.

A and B are the input signals and X is the output signal.

Boolean expression: A ∙ B (A AND B) Logic diagram:

A                           X

B

Truth table:

If both input values are 1, AND returns a 1; otherwise, AND returns a 0.

1. Give the three representations of an OR gate and say in words what OR means.

A and B are the input signals and X is the output signal. Boolean expression: A + B (A OR B)

Logic diagram:

A                           X

B

Truth table

If both input values are 0, OR returns 0; otherwise, OR returns a 1.

1. Give the three representations of an XOR gate and say in words what XOR means.

A and B are the input signals and X is the output signal. Boolean expression: A Å B (A XOR B)

Logic diagram:

A                           X

B

Truth table

If both inputs are the same value, XOR returns a 0; otherwise, XOR returns a 1.

1. Give the three representations of a NAND gate and say in words what NAND means.

A and B are the input signals and X is the output signal.

Boolean expression: (A × B)’ (NOT (A AND B)) Logic diagram:

A                           X

B

Truth table

If the inputs are different or both 0, NAND returns a 1; if both are 1, it returns a 0.

1. Give the three representations of a NOR gate and say in words what NOR means.

A and B are the input signals and X is the output signal. Boolean expression: (A + B)’ (NOT (A AND B))

Logic diagram:

1.        X

B

Truth table

If the inputs are both 0, NOR returns a 1; otherwise, NOR returns a 0.

1. Compare and contrast the AND gate and the NOR gate.

An AND gate produces a 1 as output only if both inputs are 1, whereas a NAND gate produces a 1 as output in all cases / except/ when both inputs are 1. That is, the AND and NAND gates produce opposite results. The values produced by one of these gates can be replicated by inverting the results pro- duced by the other.

1. Draw and label the symbol for a three-input AND gate, then show its behavior with a truth table.

A       

1.        X

C

X = A • B • C

1. Draw and label the symbol for a three-input OR gate, then show its behavior with a truth table.

X = A + B + C

1. What is used in a gate to establish how the input values map to the output value?

A transistor

在门中，用什么来确定输入值如何映射到输出值？

一个晶体管

1. How does a transistor behave?

Depending on the voltage of an input signal, a transistor either acts as a wire that conducts electricity or as a resister that blocks the flow of electricity.

取决于输入信号的电压，晶体管要么充当导电的导线，要么充当阻挡电流流动的电阻。

1. Of what is a transistor made?

Transistors are made of semiconductor material, which is nei- ther a good conductor of electricity nor a particularly good insulator. Transistors are usually made from silicon.

晶体管由半导体材料制成，这既不是良好的电导体，也不是特别好的绝缘体。晶体管通常是由硅制成的。

1. What happens when an electric signal is grounded?

If an electric signal is grounded, the signal flows through an alternative route to the ground where it can do no harm. When a signal is grounded, it is pulled down to 0 volts.

如果电信号接地，信号将通过一条替代路径流向地，从而不会造成损害。当信号接地时，它被拉低到0伏特。

1. What are the three terminals in a transistor and how do they operate?

The source is an electric signal. The base value regulates a gate that determines whether the connection between the source and the ground (emitter) is made. An output line is usually connected to the source. If the base value is high, the source is grounded and the output is low (representing 0). If the base value is low, the gate is closed and the source is not

grounded and the output is high (representing 1).

源是一个电信号。基值调节一个门，确定是否建立源和地（发射极）之间的连接。通常将输出线连接到源。如果基值高，源接地，输出低（表示0）。如果基值低，门关闭，源不接地，输出高（表示1）。

1. How many transistors does it take for each of these gates?

1. NOT 1
2. AND 2

1. NOR 2
2. OR 2
3. XOR 8

1. Draw a transistor diagram for an AND gate. Explain the processing.

source

1. Draw a circuit diagram corresponding to the following Bool- ean expression:

(AB + C)D

Vout                                                                                                                                                                                                                                     

B

V1

V2

ground

C D

1. Draw a circuit diagram corresponding to the following Bool-

ean expression:

The NAND gate is the inverse of the AND gate, and the inverse of the inverse is the original. Thus, the output from the NAND gate is input to a NOT gate, giving us the AND.

Vout

V1

1. Draw a circuit diagram corresponding to the following Bool- ean expression:

(AB)’ + (CD)’

ground                                                                             C                          

The NOR gate is the inverse of the OR gate, and the inverse of the inverse is the original. Thus, the output from the NOR gate is input to a NOT gate, giving us the NOR.

1. 1. How can gates be combined into circuits?

Gates are combined into circuits by using the output of one gate as the input for another. Also, the same input value can be used as input to two different gates.

1. 1. What are the two general categories of circuits, and how do they differ?

Combinational circuits are circuits in which the input values explicitly determine the output. Sequential circuits are cir- cuits in which the output is a function of input values and the current state of the circuit.

53. 门如何组合成电路？

通过将一个门的输出用作另一个门的输入，可以将门组合成电路。此外，相同的输入值可以用作两个不同门的输入。

54. 电路有哪两个一般类别，它们有什么区别？

组合电路是输入值明确决定输出的电路。时序电路是输出是输入值和电路的当前状态的函数的电路。

1. 1. Draw a circuit diagram corresponding to the following Bool- ean expression:

(A + B)(B + C)

D

1. Show the behavior of the following circuit with a truth table:

A B

Show the behavior of the following circuit with a truth table:

A

1. Show the behavior of the following circuit with a truth table:

1. Show the behavior of the following circuit with a truth table:

A                     B

C

1. What is circuit equivalence?

Circuit equivalence is when two circuits produce the same output from the same input value combination.

63. 电路等效是指两个电路在相同的输入值组合下产生相同的输出。

1. Name six properties of Boolean algebra and explain what each means.

Commutative: The commutative property says that binary operations AND and OR may be applied left to right or right to left. (A AND B is the same as B AND A; A OR B is the same as B OR A)

Associative: The associative property says that given three Boolean variables, they may be ANDed or ORed right to left or left to right. ((A AND B) AND C is the same as A AND (B AND C); (A OR B) OR C is the same as A OR (B OR C))

Distributive: The distributive property says that given three Boolean variables, the first AND the result of the second OR the third is the same as the first AND the second OR the first AND the third. (A AND (B OR C) = (A AND B) OR (A AND C))

Also, the first OR the result of second AND the third is the same as the first OR the second AND the result of the first OR the third. (A OR (B AND C) = (A OR B) AND (A OR C))

Identity: The identity property says that any value A AND the OR identity always returns A and that any value A OR the AND identity always returns A. (A AND 1 = A; A OR 0 = A)

Compliment: The compliment property says that any value AND the compliment of that value equals the OR identity and that any value OR the compliment of that value equals the OR identity. (A AND (A') = 0; A OR (A') = 1)

DeMorgan's law: DeMorgan‘s law says that the compliment of A AND B is the same as the compliment of A OR the com- pliment of B and the compliment of A OR B is the same as the compliment of B AND the compliment of A. ((A AND B)'

= A' OR B'; (A OR B)' = A' AND B')

64. 布尔代数的六个性质及其含义：

1. **交换律（Commutative）：** 交换律表示二进制操作 AND 和 OR 可以从左到右或从右到左应用。（A AND B 等同于 B AND A；A OR B 等同于 B OR A）

2. **结合律（Associative）：** 结合律表示对于给定的三个布尔变量，它们可以从右到左或从左到右进行 AND 或 OR 操作。（(A AND B) AND C 等同于 A AND (B AND C)；(A OR B) OR C 等同于 A OR (B OR C)）

3. **分配律（Distributive）：** 分配律表示对于给定的三个布尔变量，第一个 AND 第二个 OR 第三个的结果等同于第一个 AND 第二个 OR 第一个 AND 第三个。（A AND (B OR C) = (A AND B) OR (A AND C)）

   同样，第一个 OR 第二个 AND 第三个的结果等同于第一个 OR 第二个 AND 第一个 OR 第三个。（A OR (B AND C) = (A OR B) AND (A OR C)）

4. **恒等律（Identity）：** 恒等律表示任何值 A AND 1 总是返回 A，任何值 A OR 0 总是返回 A。（A AND 1 = A; A OR 0 = A）

5. **补充律（Complement）：** 补充律表示任何值 AND 其补集等于 OR 恒等元，以及任何值 OR 其补集等于 OR 恒等元。（A AND (A') = 0; A OR (A') = 1）

6. **德摩根定律（DeMorgan's Law）：** 德摩根定律表示 A AND B 的补集等于 A 的补集 OR B 的补集，以及 A OR B 的补集等于 B 的补集 AND A 的补集。((A AND B)' = A' OR B'; (A OR B)' = A' AND B')

1. Differentiate between a half adder and a full adder.

A half adder is a circuit that computes the sum of two bits and produces the appropriate carry bit. A full adder is a cir- cuit that computes the sum of two bits, taking into account the carry bit.

65. **半加器与全加器的区别：**

- **半加器（Half Adder）：** 半加器是一种电路，用于计算两个比特的和并生成适当的进位比特。它只考虑输入比特的和，而不考虑来自其他电路的进位。

- **全加器（Full Adder）：** 全加器是一种电路，用于计算两个比特的和，同时考虑进位比特。它不仅计算输入比特的和，还考虑来自其他电路的进位。因此，全加器可以接收两个输入比特和一个进位比特，并生成和输出以及可能的进位输出。

1. What is the Boolean expression for a full adder? C is the carry in.

Sum is (A Å B) Å C)

Carry out is (A AND B) OR ((A Å B) AND C)

1. What is a multiplexer?

1. A. Circuits used for memory are what type of circuits? Memory circuits are sequential circuits because they are dependent on the existing state of the circuit as well as input to the circuit.

1. How many digits does an S-R latch store? One binary digit
2. The design for an S-R latch shown in Figure 4.12 guaran- tees what about the outputs X and Y?

The values of X and Y are always compliments.

1. What is an integrated circuit or chip?