Sum of Product And Product of Sum

Sum of Product and Product of Sum by: izzatul arifah bt ismail

Two level representation: 

• Sum of Product

• Product of Sum 

Sum of Product expansion

•    Sum of product for Boolean function is a Boolean expression constructed by:

•  Where each function result is a 1
•  Form product of all variables 
•  Sum each product 

  
Rules for Grouping: 

•  The number of squares in a grouping is 2i for some i such that 1 = i = k.   
•  There are exactly k-i variables that have constant value for all squares in the grouping. 

Resulting Product Terms:

•  If X is a variable that has value 0 in all of the squares in the grouping, then the literal X' is in the product term.   
•  If X is a variable that has value 1 in all of the squares in the grouping, then the literal X is in the product term.   
•  If X is a variable that has value 0 for some squares in the grouping and value 1 for others, then neither X' nor X are in the product term. 

 

Product of Sum expansion

•       Product of sum for Boolean function is Boolean expression constructed by:

•  Where each function result is 0   
•  Form sum of all variables   
•  Product of each sum  

Rules for Grouping:

•   Same as for sum-of-products, except that zero's are grouped instead of ones. 

Resulting Sum Terms:

 

•  If variable X has value 0 for all squares in the group, thenthe literal X is in the sum term. 
•  If variable X has value 1 for all squares in the group, thenthe literal X' is in the sum term. 
•  If variable X has value 0 for some squares in the group and value 1 for the others, then that variable does not appear in the sum term.

 Prime Implicate: 

• Maximal grouping of zeros .