Boolean Algebra
- Quick Reference
Boolean Algebra,
also known as the
'algebra of
logic',
is a branch of mathematics that is similar in form to algebra, but dealing
with
logical
instead of numerical relationships. It was invented by
George Boole,
after whom this system was named. Thus, instead of variables that
represent numerical quantities as in conventional algebra, Boolean algebra
handles variables that represent two types of logic propositions:
'true'
and
'false'.
Boolean algebra
has become the main cornerstone of digital electronics, since the latter
also operates with two logic states, '1' and '0', represented by two
distinct voltage levels. Boolean algebra's formal interpretation of
logical
operators
AND, OR, and NOT has allowed the systematic development of
complex
digital systems
from simple logic gates, that now not only include circuits that perform
mathematical operations, but intricate
data processing
as well. Tables 1 to 4 summarize the definitions of logical
operators and their basic mathematical properties as represented in
Boolean algebra.
Table 1.
Elementary Logic Gate Actions
|
OR
0+0=0
0+1=1
1+0=1
1+1=1 |
AND
0·0=0
0·1=0
1·0=0
1·1=1 |
NOT
0=1
1=0 |
NOR
0+0=1
0+1=0
1+0=0
1+1=0 |
NAND
0·0=1
0·1=1
1·0=1
1·1=0 |
Table 2.
Single-Variable Logic Gate Actions
|
OR
A+0=A
A+1=1
A+A=A
A+A=1 |
AND
A·0=0
A·1=A
A·A=A
A·A=0 |
NOT
A=NOT(A)
NOT(A)=A |
NOR
A+0=A
A+1=0
A+A=A
A
NOR A=0 |
NAND
A·0=1
A·1=A
A·A=A
A
NAND A=1 |
Table 3.
Multi-Variable Boolean Equalities
|
A+B=B+A
(A+B)+C=A+(B+C)
A·B=B·A
(A·B)·C=A·(B·C)
A·(B+C)=A·B+A·C
|
Table 4. De Morgan's
Theorem
See Also:
Logic Gates; What is a
Semiconductor?
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