Boolean Algebra
Steps to Solution (SOP)
1.
From the problem statement a truth table is formed. The problem may be expressed in words, waveforms,
tables, Boolean expressions, or as a circuit. The problem statement must specify (a) the number of inputs,
and (b) the desired output for all input conditions. With this information in hand the problem is
synthesized into a set of input and corresponding output conditions in tabular form (a truth table).
2.
A column is added to the truth table and named product terms. For each row whose output is 1, a
product term is formed from the input columns.
3.
A sum-of-products (SOP) expression is built from these product terms.
4.
The algebraic expression is simplified
5.
The answer is checked.
6.
A logical circuit is designed.
Boolean Operators
INPUT
OUTPUT
NOT
OR
AND
Logical Operators
Traditional Representation
x
y
x or x'
~x
x+y
x|y
xy
x&y
0
0
1
1
0
1
0
1
1
0
1
1
1
0
0
0
1
0
Bitwise HDL Verilog
Basic Laws and Theorems of Boolean Algebra
Law
1
Dual (D)
xx
Involution
OR Laws
AND Laws
2
x0 x
x 1  x
3
x 1  1
xx  x
x  x 1
x0  0
xx  x
xx  0
4
5
Commutative
6
xy  yx
Identity element under addition is 0
and under multiplication it is 1
Dominance
Idempotent
Complements
x  y  y x
Associative
7
x  ( y  z )  ( x  y)  z
x( yz )  ( xy )z
Distributive
8
x( y  z)  xy  xz
Theorem
Simplification
9
10
x  xy  x
x  xy  x  y
x  yz  ( x  y)( x  z)
x ( x  y)  x
x( x  y)  xy
De Morgan’s
11
x y  xy
xy  x  y
Absorption
Degenerate-Reflect Law
ASCII Code
b6b5b4
b3b2b1b0
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
000
NUL
SOH
STX
EXT
EOT
ENQ
ACK
BEL
BS
HT
LF
VT
FF
CR
SO
SI
001
DLE
DC1
DC2
DC3
DC4
NAK
SYN
ETB
CAN
EM
SUB
ESC
FS
GS
RS
US
010
SP
!
″
#
$
%
&
’
(
)
*
+
,
–
.
/
011
0
1
2
3
4
5
6
7
8
9
:
;
<
=
>
?
100
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
101
P
Q
R
S
T
U
V
W
X
Y
Z
[
\
]
^
−
110
‛
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
111
p
q
r
s
t
u
v
w
x
y
z
{
¦
}
~
DEL