第五次作业

一-简答

1. Give the three representations of an AND gate and say in your words what AND means.
   Boolean expressions: X = A·B；
   truth tables：

   A	B	X
   0	0	0
   0	1	0
   1	0	0
   1	1	1

   logic diagrams:
   
   (图片来自亿图)
   AND指当且仅当两个输入为1时，输出为1，否则为0。

2. Give the three representations of an XOR gate and say in your words what XOR means.
   Boolean expressions: X = A + B；
   truth tables：

   A	B	X
   0	0	0
   0	1	1
   1	0	1
   1	1	0

   logic diagrams:
   
   (图片来自亿图)
   XOR若两个输入端相同则为0，否则为1。

3. Draw a circuit diagram corresponding to the following Boolean expression: (A + B)(B + C)
   
   (使用亿图绘制)

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

   A	B	A’	A·B	A’⊕(A·B)
   0	0	1	0	1
   0	1	1	0	1
   1	0	0	0	0
   1	1	0	1	1

5. What is circuit equivalence? Use truth table to prove the following formula.
   (AB)’ = A’ + B’

   A	B	A·B	(A·B)’
   0	0	0	1
   0	1	0	1
   1	0	0	1
   1	1	1	0

   A	B	A’	B’	A’ + B’
   0	0	1	1	1
   0	1	1	0	1
   1	0	0	1	1
   1	1	0	0	0

所以(AB)’ = A’ + B’

1. There are eight 1bit full adder integrated circuits. Combine them to 8bit adder circuit using the following box diagram.
   图片如下 (未知bug，亿图图片无法显示 只好将原图片裁剪，有点不清晰)
   

2. Logical binary operations can be used to modify bit pattern. Such as
   (X8X7X6X5X4X3X2X1)2 and (00001111)2 = (0000X4X3X2X1)2
   We called that (00001111)2 is a mask which only makes low 4 bits to work.
   Fill the follow expression
   (1) (X8X7X6X5X4X3X2X1)2 or (00001111)2 = (X8X7X6X50000)2
   (2) (X8X7X6X5X4X3X2X1)2 xor (00001111)2 = (X8X7X6X5X4’X3’X2’X1’)2
   (3) ((X8X7X6X5X4X3X2X1)2 and (11110000)2 ) or (not (X8X7X6X5X4X3X2X1)2 and (00001111)2) = (00001111)2

二-名词解释

1. Logic gate:
   In electronics, a logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more binary inputs and produces a single binary output. Depending on the context, the term may refer to an ideal logic gate, one that has for instance zero rise time and unlimited fan-out, or it may refer to a non-ideal physical device[1] (see Ideal and real op-amps for comparison).
2. Boolean algebra: In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction and denoted as ∧, the disjunction or denoted as ∨, and the negation not denoted as ¬. It is thus a formalism for describing logical relations in the same way that elementary algebra describes numeric relations.
3. Flip-flop
   (1) 中文翻译：触发器
   (2) 1