ic
Section 8.1 Boolean Algebra Structure
629
We already know (it was one of our original inspirations) that for any set
S, `(S ) under the operations of union, intersection, and complementation constitutes a Boolean algebra. Example 3 talks about this type of Boolean algebra where
S = 5a, b, c6. If we pick S = 51, 26, then the elements of `(S ) are [, 516, 526,
and 51, 26. The operations are given by the following tables.
c
[
51, 26
516
526
[
[
51,26
516
526
51, 26
51,26
51, 26
51, 26
51, 26
516
516
51, 26
516
51, 26
526
526
51, 26
51, 26
526
d
[
51, 26
516
526
′
[
[
[
[
[
[
51, 26
51, 26 [
51, 26
516
526
51, 26 [
516
[
516
516
[
516
526
526
[
526
[
526
526
516
A Boolean algebra can be defined on the set B = 50, 1, a, a′6 where the operations of + , # , and ′ are defined by the following tables (see Exercise 1).
+
0
1
a
a′
#
0
1
a
a′
′
0
0
1
a
a′
0
0
0
0
0
0
1
1
1
1
1
1
1
0
1
a
a′
1
0
a
a
1
a
1
a
0
a
a
0
a
a′
a′
a′
1
1
a′
a′
0
a′
0
a′
a′
a
Here is the definition of an isomorphism for Boolean algebras.
Definition IsomorphIsm for Boolean algeBras
Let 3B, +, # , ′, 0, 14 and 3b, &, * , ″, f, 14 be Boolean algebras. A function B S b
is an isomorphism from 3B, +, # , ′, 0, 14 to 3b, &, * , ″, f, 14 if
1. f is a bijection
2. f (x + y) = f (x) & f ( y)
3. f (x # y) = f (x) * f ( y)
4. f (x′) = ( f (x))
′ ′
■
Practice 6 Illustrate properties 2, 3, and 4 in the definition by commutative diagrams.