Problem 1
From Brookshear, Chapter One Review Problem 41b:
Solve by translating into two's complement notation
(using patterns of 5 bits). Check your work by converting your answer
into base ten notation: 7 + (-1)
Convert 7 into base-2: 00111
Convert -1 into base-2: 11111 (since 11111 + 00001 = 00000, in 5-bit addition)
Add the two values:
00111
+ 11111
-----
00110 (carry bits were 11110)
Check work: 00110 has a 1 in the 2's place and the 4's place, so it equals 2+4 in base-ten, or 6. Since 7+(-1) = 6, we got the right answer.
Problem 2
From Brookshear, Chapter One Review Problem 41d:
Solve by translating into two's complement notation
(using patterns of 5 bits). Check your work by converting your answer
into base ten notation: 8 + (-7)
Convert 8 into base-2: 01000
Convert -7 into base-2: 11001 (since 11001 + 00111 = 00000, in 5-bit addition)
Add the two values:
01000
+ 11001
-----
00001 (carry bits were 10000)
Check work: 00001 has a 1 in the 1's place and zeroes everywhere else, so it equals 1 in base ten. Since 8+(-7) = 1, we got the right answer.
Problem 3
From Brookshear, Chapter One Review Problem 41f:
Solve by translating into two's complement notation
(using patterns of 5 bits). Check your work by converting your answer
into base ten notation: 4 + 11
Convert 4 into base-2: 00100
Convert 11 into base-2: 01011
Add the two values:
00100
+ 01011
-----
01111 (carry bits were 00000)
Check work: 01111 has ones in the 1's, 2's, 4's, and 8's places, so it equals 1+2+4+8 in base-ten, or 15. Since 4+11 = 15, we got the right answer.
Problem 4
Give the truth table for the following logical expression
not ((not x) or (not y))
Do you recognize this function?
Here is the truth table:
x | y | not(x) | not(y) | (not x) or (not y) | not ((not x) or (not y)) ----------------------------------------------------------------------- 0 | 0 | 1 | 1 | 1 | 0 0 | 1 | 1 | 0 | 1 | 0 1 | 0 | 0 | 1 | 1 | 0 1 | 1 | 0 | 0 | 0 | 1
This function is the same as (x and y); you can verifying this by comparing the truth table of this function to the truth table for (x and y).
Problem 5
The truth table below shows the value of the carry bit resulting from the
addition of three input bits: x, y, and z.
Write an expression (using AND, OR, and NOT), or, if you prefer,
draw a circuit (using AND, OR, and NOT gates) to represent the carry bit
function.
x y z | carry(x,y,z)
-------|-------------
0 0 0 | 0
0 1 0 | 0
1 0 0 | 0
1 1 0 | 1
0 0 1 | 0
0 1 1 | 1
1 0 1 | 1
1 1 1 | 1
There are several ways to write this function. Here is one:
(x and y and not z) or (y and z and not x) or (x and z and not y) or (x and y and z)
Here is a simpler one:
(x and y) or (x and z) or (y and z)
You can verify that it is the same function by comparing truth tables.