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
x
y
x
or x'
x+y
xy
Traditional Representation
~x
x|y
x&y
Bitwise HDL Verilog
0
0
1
0
0
0
1
1
0
1
0
0
1
0
1
1
1
1
Basic Laws and Theorems of Boolean Algebra
Law
Dual (D)
1
x x
Involution
OR Laws
AND Laws
2
xx 0
xx 1
Identity element under addition is 0
and under multiplication it is 1
3
11x
00 x
Dominance
4
x x x
x x x
Idempotent
5
1 xx
x x 0
Complements
Commutative
6
x y y x
x y y x
Associative
7
zyxzyx )()(
xyz xy z( ) ( )
Distributive
8
x y z xy xz( )
xyz x y x z ( )( )
Theorem
Simplification
9
xxy x
x x y x( )
Absorption
10
x xy x y
x x y xy( )
Degenerate-Reflect Law
De Morgan’s
11
x y x y
x y x y
