3 Derivative Operations for Classes C N of Boolean Functions
57
C N = C N
f
re (x), f
id (x)
.
(3.3)
The specification of a class C N by means of (3.3) seems to be laborious for the
classes of Boolean functions of a single variable; however, this presentation of
classes can be used for Boolean functions of any number n of variables. The larger
the number of variables, the larger is the benefit of the representation (3.3) for a class
C N ; instead of the enumeration of up to 2 n functions of the class the two function
f re (x) and f id (x) completely specify the class.
Example 3.3 We demonstrate that the representation (3.3) can also be used for the
classes C N of B 1 :
C N 0 :
f
re (x 1 ) = 0(x 1 )
f
id (x 1 ) = der
x 1
f (x 1 ),
C N 1 :
f
re (x 1 ) = x 1
f
id (x 1 ) = 0,
C N 2 :
f
re (x 1 ) = 1(x 1 )
f
id (x 1 ) = der
x 1
f (x 1 ).
The Boolean functions of a single variable do not reveal all properties of classes
C N . Therefore, we explore all functions of two variables and generate all classes C N
of B 2 :
Example 3.4 There are 2 2 2 = 2 4 = 16 Boolean functions of two variables x 1 , x 2 :
f 0 (x 1 , x 2 ) = x 1 ∧ x 1 ∧ x 2 ∧ x 2
= 0(x 1 , x 2 ) ,
f 1 (x 1 , x 2 ) = x 1 ∧ x 2 ,
f 2 (x 1 , x 2 ) = x 1 ∧ x 2 ,
f 3 (x 1 , x 2 ) = x 1 ,
f 4 (x 1 , x 2 ) = x 1 ∧ x 2 ,
f 5 (x 1 , x 2 ) = x 2 ,
f 6 (x 1 , x 2 ) = x 1 ⊕ x 2 ,
f 7 (x 1 , x 2 ) = x 1 ∨ x 2 ,
f 8 (x 1 , x 2 ) = x 1 ∧ x 2 ,
f 9 (x 1 , x 2 ) = x 1 x 2 ,
f 10 (x 1 , x 2 ) = x 2 ,
f 11 (x 1 , x 2 ) = x 1 ∨ x 2 ,
f 12 (x 1 , x 2 ) = x 1 ,
f 13 (x 1 , x 2 ) = x 1 ∨ x 2 ,
f 14 (x 1 , x 2 ) = x 1 ∨ x 2 ,
f 15 (x 1 , x 2 ) = x 1 ∨ x 1 ∨ x 2 ∨ x 2
= 1(x 1 , x 2 ) .