Boolean algebra and logic simplification
Because binary logic is used in all of today’s digital computers
and devices, the cost of the circuits that implement it is an important factor
addressed by designers—be they computer engineers, electrical engineers, or
computer scientists. Finding simpler and cheaper, but equivalent, realizations
of a circuit can reap huge payoffs in reducing the overall cost of the design.
Mathematical methods that simplify circuits rely primarily on Boolean algebra.
Closure
Associative law
Commutative law
Identity element
Inverse
Distributive law
BASIC DEFINITIONS ( Be Engineer )
Boolean algebra, like any other deductive mathematical system, may
be defined with a set of elements, a set of operators, and a number of unproved
axioms or postulates.
Closure. A set S is
closed with respect to a binary operator if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique
element of S.
For example, the set of natural numbers N = {1, 2, 3, 4, c} is closed with respect to the binary operator +
by the rules of arithmetic addition, since, for any a, b ∈ N,
there is a unique c ∈ N such that a + b = c. The set of natural numbers is not closed
with respect to the binary operator - by the rules of arithmetic subtraction,
because 2 - 3 = -1 and 2, 3 ∈ N,
but (-1) x N.
Associative law. A
binary operator * on a set S is said to be associative whenever
(x * y) * z = x * (y * z) for all x, y, z,
(x * y) * z = x * (y * z) for all x, y, z, ∈ S
Commutative law. A
binary operator * on a set S is said to be commutative whenever
x * y = y * x for all x, y
x * y = y * x for all x, y ∈ S
Identity element. A
set S is said to have an identity element with respect to a binary operation
* on S if there exists an element e H S with the property that
e * x = x * e = x for every x
e * x = x * e = x for every x ∈ S
Example: The element 0 is an identity element with respect to the binary
operator + on the set of integers I = {c, -3, -2, -1, 0, 1, 2, 3,c}, since x +
0 = 0 + x = x for any x ∈ I
The set of natural numbers, N, has no identity element, since 0 is excluded
from the set.
Inverse. A
set S having the identity element e with respect to a binary operator * is said
to have an inverse whenever, for every x H S, there exists an element y ∈ S
such that.
x * y = e
Example: In the set of integers, I, and the operator +, with e = 0, the inverse
of an element a is (-a), since a + (-a) = 0.
x * y = e
Example: In the set of integers, I, and the operator +, with e = 0, the inverse
of an element a is (-a), since a + (-a) = 0.
Distributive law. If
* and . are two binary operators on a set S, * is said to be distributive over
. whenever
x * (y . z) = (x * y) . (x * z)
x * (y . z) = (x * y) . (x * z)
A field is an example of an algebraic structure. A field is a set
of elements, together with two binary operators, each having properties 1
through 5 and both operators combining to give property 6. The set of real
numbers, together with the binary operators + and . forms the field of real
numbers. The field of real numbers is the basis for arithmetic and ordinary
algebra. The operators and postulates have the following meanings:
·
The
binary operator + defines addition.
·
The
additive identity is 0.
·
The
additive inverse defines subtraction.
·
The
binary operator . defines multiplication
·
The
multiplicative identity is 1.
·
For
a _ 0, the multiplicative inverse of a = 1>a defines division (i.e., a .
1>a = 1 ).
·
The
only distributive law applicable is that of . over +:
a . (b + c) = (a . b) + (a . c)
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