8.4 Algebraic Manipulation of Boolean Expressions


8.4 Algebraic Manipulation of Boolean Expressions

You can transform one Boolean expression into an equivalent expression by applying the postulates and theorems of Boolean algebra. This is important if you want to convert a given expression to a canonical form (a standardized form) or if you want to minimize the number of literals or terms in an expression. (A literal is a primed or unprimed variable, and a term is a variable or a product [logical AND] of several different literals.) Minimizing the number of literals and terms can be important because electrical circuits often consist of individual components that implement each literal or term. Minimizing the number of literals and terms in an expression allows a circuit designer to use fewer electrical components and, therefore, to reduce the monetary cost of the system.

Unfortunately, there are no fixed rules you can apply to optimize a given expression. Much like constructing mathematical proofs, an individual's ability to easily do these transformations is usually a function of experience. Nevertheless, a few examples can show the possibilities:

 ab + ab' + a'b              =    a(b + b') + a'b             By P4                              =    a * 1 + a   b                 By P5                              =    a + a'b                     By Th4                              =    a + b                       By Th11  (a'b + a'b' + b')`          =    (a'(b + b') + b')'         By P4                              =    (a' * 1 + b   )   By P5                              =    (a' + b')                   By Th4                              =    ((ab)')'                  By Th8                              =    ab                          By definition of not  b(a + c) + ab' + bc' + c    =    ba + bc + ab' + bc' + c     By P4                              =    a(b + b') + b(c + c') + c   By P4                              =    a * 1 + b * 1 + c           By P5                              =    a + b + c                   By Th4 



Write Great Code. Understanding the Machine, Vol. 1
The Art of Assembly Language
ISBN: 1593270038
EAN: 2147483647
Year: 2003
Pages: 144
Authors: Randall Hyde

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