Boolean Algebra


   
 
Boolean Algebra as an Algebraic Structure
Boolean Algebra is an algebraic structure defined by a set of elements B, together with two operations, + and . satisfying the following axioms (Hunington postulates) .
 
Axiom 1:
 
(a) Closure with respect to +, that is if x, y B, x + y B.
 
(b) Closure with respect to, that is x, y B x . y B.
 
Axiom 2:
 
(a) There exists element 0B such that: x + 0 = 0 + x = x, x B.
 
(0 is the identity for +)
 
(b) There exists element 1 B such that: x . 1 = 1 . x = x.
 
(1 is the identity for ' . ')
 
For all x, y, z B.
 
Axiom 3:
 
(a) Commutative w.r.t + : x + y = y + x
 
(b) Commutative w.r.t . : x . y = y . x
 
Axiom 4:
 
(a) . is distributive over + : x . (y+z) = x . y + x . z
 
(b) + is distributive over . : x + (y . z) = (x+y) . (x+z)
 
Axiom 5:
 
" x B, $ x' B such that (a) x + x' = 1 and (b) x.x' = 0.
 
x' is called the complement or negation of x.
 
Axiom 6:
 
There exists at least two elements x and y such that x y.
 
Note:
 
To prove a set to be a Boolean algebra, we have to prove all the above six properties to be true.
 
Whenever we say B is a Boolean algebra, it should be understood that B is accompanied with two operations satisfying all the above six properties.
 
Two-valued Boolean Algebra
 
A two valued Boolean algebra is defined on a set of two elements B = {0, 1}, with rules for + and . as shown in the following tables.
 
 
 
 
These rules are exactly the same as the AND, OR and NOT operations.
 
Note:
 
Closure is obvious from the tables, since the results of each operation is either 1 or 0.
 
Also observe from the tables:
 
(a) 0 + 0 = 0, 0 + 1 = 1 + 0 = 1 0 is the identity w.r.t +.
 
(b) 1.1 = 1, 1.0 = 0.1 = 0 1 is the identity w.r.t .
 
For 0, 1B, 0 + 1 = 1 + 0
 
0 . 1 = 1 . 0
 
Commutative law is satisfied for both + and .
 
(a)
 
 
Since the last two columns are identical x.(y + z) = (x.y) + (x.z)
 
(b) Similarly it can be shown that
 
x + (y.z) = (x + y) . (x + z)
 
Also from the complement table
 
 
 
 
x . x' = 0
 
(a) x + x' = 1. since 0 + 0' = 0 + 1 = 1, 1 + 1' = 1 + 0 = 1
 
(b) x.x' = 0, since 0.0' = 0.1 = 0, 1.1' = 1.0 = 0
 
 
Thus, we have a two valued Boolean algebra having a set of two elements 1 and 0.
 
 
     
   

 

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