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CS8351
BOOLEAN ALGEBRA AND MINIMIZATION
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1.1 Introduction:
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UNIT 1
The English mathematician George Boole (1815-1864) sought to give symbolic form to
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Aristotle‘s system of logic. Boole wrote a treatise on the subject in 1854, titled An Investigation
of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and
Probabilities, which codified several rules of relationship between mathematical quantities
limited to one of two possible values: true or false, 1 or 0. His mathematical system became
known as Boolean algebra.
All arithmetic operations performed with Boolean quantities have but one of two possible
Outcomes: either 1 or 0. There is no such thing as ‖2‖ or ‖-1‖ or ‖1/2‖ in the Boolean world. It is a
world in which all other possibilities are invalid by fiat. As one might guess, this is not the kind
of math you want to use when balancing a checkbook or calculating current through a resistor.
However, Claude Shannon of MIT fame recognized how Boolean algebra could be applied to
on-and-off circuits, where all signals are characterized as either ‖high‖ (1) or ‖low‖ (0). His1938
thesis, titled A Symbolic Analysis of Relay and Switching Circuits, put Boole‘s theoretical work
to use in a way Boole never could have imagined, giving us a powerful mathematical tool for
designing and analyzing digital circuits.
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Like ‖normal‖ algebra, Boolean algebra uses alphabetical letters to denote variables.
Unlike ‖normal‖ algebra, though, Boolean variables are always CAPITAL letters, never
lowercase.
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Because they are allowed to possess only one of two possible values, either 1 or 0, each and
every variable has a complement: the opposite of its value. For example, if variable ‖A‖ has a
value of 0, then the complement of A has a value of 1. Boolean notation uses a bar above the
variable character to denote complementation, like this:
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CS8351
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1.2 Boolean Arithmetic:
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In written form, the complement of ‖A‖ denoted as ‖A-not‖ or ‖A-bar‖. Sometimes a ‖prime‖
symbol is used to represent complementation. For example, A‘ would be the complement of A,
much the same as using a prime symbol to denote differentiation in calculus rather than the
fractional notation dot. Usually, though, the ‖bar‖ symbol finds more widespread use than the
‖prime‖ symbol, for reasons that will become more apparent later in this chapter.
Let us begin our exploration of Boolean algebra by adding numbers together:
0+0=0
0+1=1
1+0=1
1+1=1
The first three sums make perfect sense to anyone familiar with elementary addition. The
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Last sum, though, is quite possibly responsible for more confusion than any other single
statement in digital electronics, because it seems to run contrary to the basic principles of
mathematics.
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Well, it does contradict principles of addition for real numbers, but not for Boolean numbers.
Remember that in the world of Boolean algebra, there are only two possible values for any
quantity and for any arithmetic operation: 1 or 0. There is no such thing as ‖2‖ within the scope of
Boolean values. Since the sum ‖1 + 1‖ certainly isn‘t 0, it must be 1 by process of elimination.
1.2.1 Addition – OR Gate Logic:
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Boolean addition corresponds to the logical function of an ‖OR‖ gate,
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as well as to parallel switch contacts:
There is no such thing as subtraction in the realm of Boolean mathematics. Subtraction
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Implies the existence of negative numbers: 5 - 3 is the same thing as 5 + (-3), and in Boolean
algebra negative quantities are forbidden. There is no such thing as division in Boolean
mathematics, either, since division is really nothing more than compounded subtraction, in the
same way that multiplication is compounded addition.
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1.2.2 Multiplication – AND Gate logic
Multiplication is valid in Boolean algebra, and thankfully it is the same as in real-number
algebra: anything multiplied by 0 is 0, and anything multiplied by 1 remains unchanged:
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0×0=0
0×1=0
1×1=1
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1×0=0
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This set of equations should also look familiar to you: it is the same pattern found in the truth
table for an AND gate. In other words, Boolean multiplication corresponds to the logical
function of an ‖AND‖ gate, as well as to series switch contacts:
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