ELECTRONIC DESIGN NOTES #10 - BOOLEAN ALGEBRA
a beautiful name! Algebra was brought to us by the Arab migratory people in the seventh
century. More specifically, the name "Algebra" is derived from
[Al-Jabar means "reunion"
in Persian] the title of a book written by the great mathematician Muhammad Ibn Musa Al-Khwarizmi in 820.
However, Algebra was first used during the great--and the most
mysterious--Sumerian Civilization about 7000 years
Now, the true beauty about Algebra is, it is a logic mind-game! In fact, there are two logic systems that could help us
gain the most in terms of brain-power:
1. Analytic Plane Geometry
Both of them are branches of Mathematics, and we hope we shall start one good
sunny day to write a few
"Math Reference Notes" in
Now, hardware, firmware and software are related, because they all work with the same mathematical model: the binary code--this
is, a special
interpretation model built on the Base 2 numerical system. Another common name used instead of
binary code is "digital". Frankly, we have absolutely no idea why this name, "digits", sounded
that impressive during the golden years 1950s and 1960s. Anyway, digital/logic electronics means electronic applications
of the Base 2 numerical system. Therefore, in digital electronics we work
with only two states:
"One" in hardware is a saturated transistor; "zero" represents its cutoff state
[or backwards, for negative logic]. In firmware one is one set bit, and zero is the bit cleared. Of course, in software things
behave exactly as they do in firmware, because software is translated into firmware and, eventually, into machine
The power of the Base 2 Numerical System is enormous, particularly because it is the simplest,
mathematical model. Although the Base 10 numerical system we currently use in our life appears to be more complex, and
elegant, Base 2 is far more powerful. In fact the Base 2 numerical system is the
one and only UNIVERSAL MATHEMATICAL MODEL.
In order to present Boolean Algebra in this page, the following simplified structure was employed:
1. Basic notions in Boolean Algebra
2. Boolean Theorems
3. Equivalent switching circuits
The basic notions highlighted in this page are related to
a few electronic design topics presented in the first part, Hardware Design, of
FIRMWARE AND SOFTWARE DESIGN.
1. BASIC NOTIONS IN BOOLEAN ALGEBRA
Boolean Algebra is a mathematical model developed for the TRUE and FALSE states by Mr. George Boole
(1815-1864). Unfortunately, throughout his life Mr. Boole's work was not
much appreciated by ordinary engineers; he was
"discovered" very late, in 1930, when Boolean Algebra started being used to simulate switching circuits.
Boolean Algebra is a particular application of the Base 2 numerical system, therefore it also deals with 2 states:
1. True, commonly related to the mathematical "one"
2. False, commonly related to the mathematical "zero"
Only two operations have been defined for Boolean Algebra; they are:
1. Addition (+)
2. Multiplication (*)
The addition and multiplication operations of Boolean Algebra
ARE NOT mathematical addition and
multiplication. These two logic operations use the signs (+) and (*) in order to facilitate understanding Boolean Algebra,
but never forget that they are not the mathematical addition and multiplication. The rules governing Boolean addition
and multiplication follow next.
The sign "=" used in Boolean Algebra
the mathematical "=". In Boolean Algebra
"=" means "may be substituted with/for" not "equal to".
1.1 BASIC BOOLEAN OPERATIONS
A + 0 = A
A * 0 = 0
A + 1 = 1
A * 1 = A
A = 0
A = 1
A * A = A + A = A
The meaning of
A is, complement of A. For example: if A is
0, then A is one; however, A may be 1, and its complement
A is zero in this case. Another example is:
1 = 0, and
0 = 1.
1.2 ASSOCIATIVE RULES
"Associative" in mathematics is the quality of
1.2.1 BOOLEAN MULTIPLICATION
A * (B * C) = (A * B) * C
1.2.2 BOOLEAN ADDITION
A + (B + C) = (A + B) + C
1.3 COMMUTATIVE RULES
"Commutative" is a term used in mathematics with the meaning: "it may change places".
A + B = B + A
The above expression is valid in Boolean Algebra, and its meaning is:
the Boolean operation of addition is
A * B = B * A
The above expression it is also valid in Boolean Algebra, and its meaning is:
the Boolean operation of
multiplication is commutative.
1.4. DISTRIBUTIVE RULES
The "distributive" term is used to
mark particular properties of mathematical operations. In our
case, Boolean Algebra, the meaning is, the operation of addition (+) is distributive when it is related to the
operation of multiplication (*), and backwards.
A * (B + C) = A * B + A * C
A + B * C = (A + B) * (A + C)
The above relations are very important, and they are used a lot.
1.5. SIMILARITY OF BOOLEAN ALGEBRA TO THE MATHEMATICAL
LOGIC "AND" AND
The AND logic operation used in firmware/software is expressed in Boolean Algebra as:
A (AND) B = A * B
The OR logic operation is expressed in Boolean Algebra as:
A (OR) B = A + B
The last two logic equations above exemplify perfectly clear that the Boolean (+) and (*) operations are not the
mathematical addition and multiplication.
2. BOOLEAN THEOREMS
The English mathematician Mr. Augustus de Morgan (1806 - 1871)--he worked together with
Mr. George Boole--is
one of the founders of the Boolean Algebra. He left us a few nice theorems as follows.
2.1. DE MORGAN THEOREMS
A * B =
A + B =
2.2. OTHER BOOLEAN THEOREMS
A * (A + B) = A
A + A * B = A
A * (A + B) = A * B (also noted as AB for simplification)
A + A * B = A + B
A * B + A * B = A
(A + B) * (A +
B) = A
2.3. INCLUDED TERM AND INCLUDED FACTOR THEOREMS
A * B + B * C + C *
A = A * B + C *
(A + B) * (B + C) * (C +
A) = (A + B) * (C +A)
The Boolean theorems presented in this page are the most common constructions used to simplify Boolean logic. In the next
section you will see a few practical implementations of the Boolean logic functions.
3. EQUIVALENT SWITCHING HARDWARE CIRCUITS
The simple switching circuits presented further down should be sufficient to understand practical implementation
in hardware of Boolean
Algebra. For clarity, we have attached the correspondent firmware/software logic equivalent formulas.
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