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Unformatted text preview: c A.H.Dixon CMPT 150: Week 4 (Jan 28 - Feb 1, 2008) 26 10 PRODUCT OF MAXTERMS CANONICAL FORM For each row of the function table where the value of the function is 0, construct a sum term with the property that all variables are included, and the value of the sum term is 0 for the assign of values to the variables indicated in the row. Example: x y z f sum term x + y + z 1 x + y + z 1 x + y + z 1 1 1 1 x + y + z 1 1 1 1 1 1 1 1 1 1 Then the expression: ( x + y + z )( x + y + z )(( x + y + z )( x + y + z ) is 0 only when any one of the sum-terms is 0. Therefore it must have the same truth table as the function f . That is, f ( x,y,z ) = ( x + y + z )( x + y + z )(( x + y + z )( x + y + z ) Since all variables occur as literals in each sum term, each is a called a max- term of the function f , and therefore the expression defines f as a product of maxterms. Like the sum of minterms, the product of maxterms representation is also a canon- ical form, and a notation system exists to represent it ”numerically” similar to the Σ notation. In this case we list all the rows in which the function is 0, in a notation called the Π Notation, with the understanding that corresponding maxterms will be ”anded” together: f ( x,y,z ) = Π M (0 , 1 , 2 , 4) Just as every function has a least one sum-of-products representation, so every function has at least one product-of-sums representation. Other product-of-sums c A.H.Dixon CMPT 150: Week 4 (Jan 28 - Feb 1, 2008) 27 representations can be found by applying the distributive, identity, and comple- ment laws in a manner similar to the simplification of sums-of-products. For example, consider the following function: f ( x,y,z ) = Π M (0 , 1 , 2 , 7) = ( x + y + z )( x + y + z )( x + y + z )( x + y + z ) = ( x + y + z )( x +...
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- Spring '08
- Boolean Algebra, Karnaugh map, σ, sum-of-products rerepsentation, ANDing