Wednesday, 8 February 2012

Boolean Algebra

  •   In 1854 George Boole developed Boolean algebra
  • Switching algebra was introduced by C.E. Shannon
Huntington postulates: 
1. A. Closure with respect to ‘+’
    B. Closure with respect to ‘.’
   Note: a set is said to be closed with respect to a operator if, for every pair of elements of s, the result of operation is also a unique element of S. For example N = {1, 2, 3…} is closure with respect to operator ‘+’ because 2+4=6 is also element of N.

2. Commutative with respect to + and .
      a + b = b + a
      a . b = b . a 
3. A. identity element with respect to ‘.’ is ‘1’
      a . 1 = 1. a = a
     B. identity element with respect to ‘+’ is ‘0’
      a + 0 = 0 + a = a
       4. Distributive over + and .
     a + ( b . c ) = ( a + b ) . ( a + c ) with respect to +
     a . ( b + c ) = ( a . b ) + ( a . c ) with respect to .

      5. For every element ‘a’ there exists a unique element called a’
      a+a’ = 1
      a.a’=0 
      Note:  Huntington postulates doesn’t include associative law
      Associative with respect to + and .
           a + ( b + c ) = ( a + b ) + c
           a . ( b . c ) = (a . b ) . c
Differences between normal and Boolean algebra:
  • Complement of a number doesn’t exist in normal algebra
  • Distributive law over + is valid for Boolean algebra but not normal algebra
  • Boolean algebra doesn’t have additive and multiplicative inverse
Duality:
  •  Dual of an expression can be obtained by replacing all ‘+’ with ‘.’, all ‘0’ with ‘1’, all ‘1’ with ‘0’, all ‘.’ With ‘+’.
  • If Boolean expression is valid it’s dual is also  valid
  • Example: 
                  a.b+c=(a+c).(b+c) 
                  (a+b).c=(a.c)+(b.c) – is dual of above expression 


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