Geoscience Reference
In-Depth Information
TABLE 9.1
Boolean Postulates
Postulate 1
X
= 0 or
X
= 1
Postulate 2 0•0=0
Postulate 3 1 + 1 = 1
Postulate 4 0 + 0 = 0
Postulate 5 1•1=1
Postulate 6 1•0=0•1=0
Postulate 7
TABLE 9.2
Boolean Laws
Commutative law
A
+
B
=
B
+
A
AB
=
BA
Distributive law
A
(
B
+
C
) =
AB
+
AC
A
+ (
BC
) = (
A
+
B
)(
A
+
C
)
Identity and inverse laws
A
+
A
=
A
AA
=
A
A
+ not (
A
) = 1
A
•not(
A
) = 0
1 + 0 = 0 + 1 = 1
Note:
In algebra, the commutative laws for addition and multiplication are similar to the commu-
tative law for Boolean variable. The
commutative law for addition
(algebra) states that the
sum of two quantities is the same in whatever order they are added. The
commutative law
for multiplication
(algebra) states that the product of two quantities is the same whatever the
order of multiplication.
9.2.2 d
istributive
l
aW
The equation
A
(
B
+
C
) = (
AB
) + (
AC
) is the first distributive law for Boolean variables, which, again,
is not that different from the distributive law for numeric algebra. This simply states that
A
AND
with the results of
B
OR
C
is the same as the results of
A
AND
B
and
A
AND
C
undergoing OR. On
the other hand, the equation
A
+ (
BC
) = (
A
+
B
)(
A
+
C
) is the second distributive law for Boolean
logic, which is not so logical from real algebra. It states that
A
OR with the results of
B
AND
C
is
the same as the results of
A
OR
B
and
A
OR
C
undergoing AND.
Note:
In algebra, the distributive law states that the product of an expression of two or more terms
multiplied by a single factor is equal to the sum of the products of each term of the expression
multiplied by the single factor.
9.2.3 i
dentity
and
i
nverse
v
ariables
Equations
A
+
A
=
A
,
AA
=
A
,
A
+ not (
A
) = 1, and
A
•not(
A
) = 0 are statements for the existence of
an identity and inverse element for a given Boolean variable. Similar to addition for real algebra, the
identity for the OR operator is 0. For the AND operator it is 1, similar to multiplication in real algebra.
Note:
Because the
bit
of a Boolean variable (similar to the digit of a real number variable) is either
0 or 1 by definition, then the inverse is found by simply switching all the bits to the other
Boolean value (e.g., 0 becomes 1 and 1 becomes 0). Unlike numeric algebra the inverse of
Boolean variable AND with the Boolean variable is 0. If the Boolean variable undergoes the
OR operation with its inverse then the result is 1. Also, if a Boolean variable undergoes either
the OR or AND operation with itself, the result is simply the original Boolean variable.
9.3 BOOLEAN SYNTHESIS
To this point we have focused briefly on those Boolean laws and variables important to the environ-
mental practitioner. There is, however, much more to Boolean algebra, but the additional concepts
are beyond the scope of this text. Notwithstanding the brevity of the overview of Boolean principles
provided to this point, we can now provide a basic illustration of how these Boolean principles can
be applied in the real world of environmental safety and health.
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