(5 pts) Exercise B-1
•
Show the truth table for NAND and NOR gates
NOR
NAND
A
0
0
1
1
B
0
1
0
1
x
1
1
1
0
A
0
0
1
1
B
0
1
0
1
x
1
0
0
0
(5 pts) Exercise B-2
•
A.) Show the truth table for the following logic circuit
A
A
C
y
000
0
001
0
010
1
011
0
100
B.) Write the Boolean equation for this circuit.
1
101
0
110
0
111
0
B
y
C
•
B
y = (A xor B) C
(5 pts) Exercise B-3
•
Draw a circuit for the following formula:
F = ( (A + B) C ) + D
A
B
C
D
(2 pts EXTRA CREDIT) Exercise B-4
•
•
Recall – how many entries are in a truth table for a function with n
inputs?
Consider – how many different truth tables are possible for a function
with n inputs?
The truth table has 2^n entries.
For each entry, I can choose a 0 or a 1.
Thus there are 2^n choices to make, and each choice
has 2 options, so total number of possibilities is:
2^(2^n)
(5 pts) Exercise B-11
•
Show the sum of products for the following truth table.
A
B
C
f
0
0
0
1
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
0
1
1
1
0
z  ( A  B  C)  ( A  B  C)  ( A  B  C)  ( A  B  C)
(5 pts) Exercise B-12
•
Simplify the following equations (use Boolean laws discussed earlier)
B( A  0) 
AB
B( AA ) 
0
( A  B )( A  B) 
AB  A B
( A  B)  ( A  B  C ) 
ABC
Is
AB
No
the same as
AB
?
(5 pts) Exercise B-13
•
Use bubble pushing to simplify this circuit
(10 pts) Exercise B-14
•
A) Show the sum of products for the following truth table.
A
B
C
f
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
1
0
0
1
1
0
1
1
1
1
0
1
1
1
1
1
z  ( A  B  C)  ( A  B  C)  ( A  B  C)  ( A  B  C)  ( A  B  C)  ( A  B  C)
•
B) Simplify this equation
z  ( A  B  C)  ( A  B  C)  A
z  (B  C)  (B  C)  A
(8 pts) Exercise B-15
•
Simplify the following equations
C ( A  1) 
C(1) = C
AB( A  C ) 
ABA + ABC = AB + ABC = AB(1+C) = AB
( A  B )( A  C ) 
AA  A B  AC  B C  A B  AC  B C
( B  0)(C  D  1)
(B)(1) = B
(5 pts) Exercise B-21
•
•
1. Fill in the following K-Map based on the truth
table at right
2. Minimize the function using the K-map
A
A
BC
1
1
f  AB  AC
BC
0
1
A
B
C
f
0
0
0
1
0
0
1
1
0
1
0
0
0
1
1
0
1
0
0
1
1
0
1
0
1
1
0
1
1
1
1
0
BC
0
0
BC
1
0
(5 pts) Exercise B-22
•
Draw the two-level circuit for the function from Exercise B-21
A
B
C
(5 pts) Exercise B-23
•
Suppose we already have this k-Map. Minimize the function.
CD
AB
1
AB
1
AB
1
AB
0
C D CD CD
0
0
1
1
1
1
1
0
0
0
0
1
f  AB  AD  BC  BC D
(3 pts) Exercise B-24
•
Consider your answer to Exercise B-23. Using a K-map, you found some
particular two-level, minimal circuit.
a.) Is your answer unique? In other words, is there only one possible twolevel circuit for that K-Map that is minimal, or is there another one that is
logically different but still correct and just as small? Circle one:
UNIQUE
NOT UNIQUE
b.) Will this always be the case, or could a different K-map change your
answer? If you use a K-map and reduce as much as possible, then this will
find a two-level circuit of minimal size. However, is this “unique”
– or is there more than one possible solution?
a.) For the function above, only one way to minimize, so YES, it is
unique.
b.) NO, this is not always true. For example, look at the solution to
B-25. There are three different solutions that are equally minimal,
but different.
(10 pts) Exercise B-25
•
Suppose we already have this k-Map. Minimize the function.
See note
below on
how to group
this one
CD
AB
1
AB
0
AB
1
AB
1
AC D
C D CD CD
0
1
1 AC
1
1
1
1
1
0
0
1
0
BD
CD
Note: one more grouping is not shown – the “1” at top left must combine with either the one
at top right OR the one at bottom left – that’s why there are two answers.
OR
f  AC D  BD  CD  AC  AB D
f  AC D  BD  CD  AC  BC D
OR a third way – handling the “groups of two” differently from how circled above:
f  ABC  BD  CD  AC  BC D