Boolean Algebra
Reference: Introduction to Digital SystemsMiloš Ercegovac, Tomás Lang, Jaime H. MorenoPages: 480-487
��������������� ������������
����������
A Boolean Algebra is a 3-tuple { B , + , · } , where
• B is a set of at least 2 elements
• ( + ) and ( · ) are binary operations (i.e. functions )
satisfying the following axioms:
BBB ��
A1. Commutative laws: For every a, b � B
I. a + b = b + a
II. a · b = b · a
“times” / ”AND”“plus” / ”OR”
A2. Distributive laws: For every a, b, c � B
I. a + (b · c) = (a+ b) ·(a + c)
II. a · (b + c) = (a · b) + (a · c)
A3. Existence of identity elements: The set B has two distinct identity elements, denoted as 0 and 1, such that for every element a � B
I. a + 0 = 0 + a = a
II. a · 1 = 1 · a = a
multiplicative identity element
additive identity element
A4. Existence of a complement: For every element a � B there exists an element a’ such that
I. a + a’ = 1
II. a · a’ = 0 the complement of a
Precedence ordering: · before +
For example:a + (b · c) = a + bc
Switching Algebra
101
000
10ANDB = { 0 , 1 }
111
100
10OR
Theorem 1: The switching algebra is a Boolean algebra.
Proof:
By satisfying the axioms of Boolean algebra:
• B is a set of at least two elements
B = { 0 , 1} , 0 � 1 and |B| � 2.
101
000
10AND
111
100
10OR
closure
• Closure of (+) and (·) over B (functions ) . BBB ��
101
000
10AND
111
100
10OR
A1. Cummutativity of ( + ) and ( · ).
Symmetric about the main diagonal
A2. Distributivity of ( + ) and ( · ).
11111
11110
11101
11100
11011
00010
00001
00000
(a + b )(a + c)a + bcabc
11111
11110
11101
00100
00011
00010
00001
00000
ab + aca(b + c)abc
* Alternative proof of the distributive laws:
Claim: (follow directly from operators table)
• AND( 0 , x ) = 0 AND( 1 , x ) = x
• OR( 1 , x ) = 1 OR( 0 , x ) = x
Consider the distributive law of ( · ):
AND( a , OR( b , c ) ) = OR( AND( a , b ) , AND( a , c ) )
a = 0 : AND( 0 , OR( b , c ) ) = OR( AND( 0 , b ) , AND( 0 , c ) )
0 0
0
0
a = 1 : AND( 1 , OR( b , c ) ) = OR( AND( 1 , b ) , AND( 1 , c ) )
b c
OR( b , c )
OR( b , c )
Consider the distributive law of ( + ):
OR( a , AND( b , c ) ) = AND( OR( a , b ) , OR( a , c ) )
a = 0 : OR( 0 , AND( b , c ) ) = AND( OR( 0 , b ) , OR( 0 , c ) )
b c
AND( b , c )
AND( b , c )
a = 1 : OR( 1 , AND( b , c ) ) = AND( OR( 1 , b ) , OR( 1 , c ) )
1 1
1
1
Why have we done that?!
For complex expressions truth tables are not an option.
A3. Existence of additive and multiplicative identity element.
0 + 1 = 1 + 0 = 1
0 · 1 = 1 · 0 = 0
0 – additive identity
1 – multiplicative identity
A4. Existence of the complement.
0110
0101
a · a’a + a’a’a
All axioms are satisfied Switching algebra is Boolean algebra.
Theorems in Boolean Algebra Theorem 2:
Every element in B has a unique complement.
Proof:
Let a � B. Assume that a1’ and a2’ are both complements of a, (i.e. ai’ + a = 1 & ai’ · a = 0), we show that a1’ = a2’ .
111 ���� aa
� �21 aaa ����
211 aaaa ������
211 aaaa ������
210 aa ����
21 aa ����
Identity
a2’ is the complement of a
distributivity
commutativity
a1’ is the complement of a
Identity
We swap a1’ and a2’ to obtain,
21
122
aa
aaa
����
�����
21 aa ���
’ can be considered as a unary operation
called complementation
Complement uniqueness
BB�
Boolean expression - Recursive definition:base: 0 , 1 , a � B – expressions.recursion step: Let E1 and E2 be Boolean expressions.
Then,E1’( E1 + E2 ) ( E1 · E2 )
expressions
Example:
c
b aa’ 0
( a’ + 0 )
( ( a’ + 0 ) · c )
( b + a )
( ( a’ + 0 ) · c ) + ( b + a )’ )
( b + a )’
a
Dual transformation - Recursive definition:Dual: expressions expressionsbase: 0 1
1 0a a , a � B
recursion step: Let E1 and E2 be Boolean expressions.Then,
E1’ [dual(E1)]’( E1 + E2 ) [ dual(E1) · dual(E2) ] ( E1 · E2 ) [ dual(E1) + dual(E2) ]
Example:( ( a + b ) + ( a’ · b’ ) ) · 1
( ( a · b ) · ( a’ + b’ ) ) + 0
A1. Commutative laws: For every a, b � B
I. a + b = b + a
II. a · b = b · a
A2. Distributive laws: For every a, b, c � B
I. a + (b · c) = (a+ b) ·(a + c)
II. a · (b + c) = (a · b) + (a · c)
A3. Existence of identity elements: The set B has two distinct identity elements, denoted as 0 and 1, such that for every element a � B
I. a + 0 = 0 + a = a
II. a · 1 = 1 · a = a
A4. Existence of a complement: For every element a � B there exists an element a’ such that
I. a + a’ = 1
II. a · a’ = 0
The axioms of Boolean algebra are in dual pairs.
Theorem 3:
For every a � B:
1. a + 1 = 1
2. a · 0 = 0
Proof:
(1)
� �111 �� aa
� � � �1��� aaa
� �1��� aa
aa ��
1�
Identity
a’ is the complement of a
distributivity
Identity
a’ is the complement of a
(2) we can do the same way:
� �000 ��� aa
� � � �0���� aaa
� �0��� aa
aa ���
0�
Identity
a’ is the complement of a
distributivity
Identity
a’ is the complement of a
Note that:• a · 0 , 0 are the dual of a + 1 , 1 respectively.• The proof of (2) follows the same steps exactly as the proof of (1) with
the same arguments, but applying the dual axiom in each step.
Theorem 4: Principle of Duality
Every algebraic identity deducible from the axioms of a Boolean algebra attains:
For example:( a + b ) + a’ · b’ = 1
( a · b ) · ( a’ + b’ ) = 0
Correctness by the fact that each axiom has a dual axiom as shown
� � � �2121 EdualEdualEE ��
Every theorem has its dual for “ free”
Theorem 5:
The complement of the element 1 is 0, and vice versa:
1. 0’ = 1
2. 1’ = 0
Proof:
By Theorem 3,
0 + 1 = 1 and
0 · 1 = 0
By the uniqueness of the complement, the Theorem follows.
Theorem 6: Idempotent Law
For every a � B
1. a + a = a
2. a · a = a
Proof:
(1)
� � 1�� aaaa
� � � �aaaa ���
� �� �aaa ���
0� aa�
Identity
a’ is the complement of a
distributivity
Identity
a’ is the complement of a
(2) duality.
Theorem 7: Involution Law
For every a � B
( a’ )’ = a
Proof:
( a’ )’ and a are both complements of a’ .
Uniqueness of the complement ( a’ )’ = a.
Theorem 8: Absorption Law
For every pair of elements a , b � B,
1. a + a · b = a
2. a · ( a + b ) = a
Proof: home assignment.
Theorem 9:
For every pair of elements a , b � B,
1. a + a’ · b = a + b
2. a · ( a’ + b ) = a·b
Proof:
(1)� �� �baaabaa �� '
� �ba�1
ba�
distributivity
a’ is the complement of a
Identity
(2) duality.
Theorem 10:
In a Boolean algebra, each of the binary operations ( + ) and ( · ) is associative. That is, for every a , b , c � B,
1. a + ( b + c ) = ( a + b ) + c
2. a · ( b · c ) = ( a · b ) · c
Proof: home assignment (hint: prove that both sides in (1) equal
[ ( a + b ) + c ] · [ a + ( b + c ) ] .)
Theorem 11: DeMorgan’s Law
For every pair of elements a , b � B,
1. ( a + b )’ = a’ · b’
2. ( a · b )’ = a’ + b’
Proof: home assignment.
Theorem 12: Generalized DeMorgan’s Law
Let { a , b , … , c , d} be a set of elements in a Boolean algebra. Then, the following identities hold:
1. (a + b + . . . + c + d)’ = a’ b’ . . .c’ d’
2. (a · b · . . . · c · d)’ = a’ + b’ + . . . + c’ + d’
� � � �� ����
dcbadcba ��
� � dcba ��
� �
dcba ����� �
Induction assumption � � cbacba �����
��
Proof: By induction.
Induction basis: follows from DeMorgan’sLaw
( a + b )’ = a’ · b’ .
Induction hypothesis: DeMorgan’s law is true for n elements.
Induction step: show that it is true for n+1 elements.
Let a , b , . . . , c be the n elements, and d be the (n+1)st element.
Associativity
DeMorgan’s Law
The symbols a, b, c, . . . appearing in theorems and axioms are gener ic var iables
Can be substituted by complemented variables or
expressions (formulas)
For example:
� � ''' baba �
� � abba ����
� �� � cbacba )'( ���
DeMorgan’sLaw� �
cb
baa
�
bb
aa
�
�
etc.
Other Examples of Boolean Algebras
Algebra of SetsConsider a set S.
B = all the subsets of S (denoted by P(S)).
� �� ���,,SPM �
“plus” � set-union U
“times” � set-intersection
Additive identity element – empty set Ø
Multiplicative identity element – the set S.
P(S) has 2|S| elements, where |S | is the number of elements of S
Algebra of Logic (Propositional Calculus)
Elements of B are T and F (true and false).
“plus” � Logical OR
“times” � Logical AND
�
�
� �� ���� ,,,FTM
Additive identity element – F
Multiplicative identity element – T