(3) Reduce Minterms
Minterms of adjacent (abutting) 1squares (Tsquares) can be reduced with respect to the number of their literals, and the number terms also will be reduced in the process. Two abutting squares (2 x 1 horizontal or 1 x 2 vertical, even the edges represent abutting squares) loose one literal, four squares in a 4 x 1 rectangle (horizontal or vertical) or 2 x 2 square (even the four corners represent abutting squares) loose two literals, eight squares in a rectangle loose 3 literals, etc. (One seeks out the largest square or rectangles and ignores the smaller squares or rectanges contained totally within it. ) This process continues until all abutting squares are accounted for, at which point the propositional formula is minimized.
For example, squares #3 and #7 abut. These two abutting squares can loose one literal (e.g. "p" from squares #3 and #7), four squares in a rectangle or square loose two literals, eight squares in a rectangle loose 3 literals, etc. (One seeks out the largest square or rectangles.) This process continues until all abutting squares are accounted for, at which point the propositional formula is said to be minimized.
Example: The map method usually is done by inspection. The following example expands the algebraic method to show the "trick" behind the combining of terms on a Karnaugh map:
 Minterms #3 and #7 abut, #7 and #6 abut, and #4 and #6 abut (because the table's edges wrap around). So each of these pairs can be reduced.
Observe that by the Idempotency law (A V A) = A, we can create more terms. Then by association and distributive laws the variables to disappear can be paired, and then "disappeared" with the Law of contradiction (x & ~x)=0. The following uses brackets only to keep track of the terms; they have no special significance:
 Put the formula in conjunctive normal form with the formula to be reduced:


 q = ( (~p & d & c ) V (p & d & c) V (p & d & ~c) V (p & ~d & ~c) ) = ( #3 V #7 V #6 V #4 )

 Idempotency (absorption) [ A V A) = A:


 ( #3 V V V #4 )

 Associative law (x V (y V z)) = ( (x V y) V z )


 ( V V )
 V V .

 Distributive law ( x & (y V z) ) = ( (x & y) V (x & z) ) :


 ( V V )

 Commutative law and law of contradiction (x & ~x) = (~x & x) = 0:


 ( V V )

 Law of identity ( x V 0 ) = x leading to the reduced form of the formula:


 q = ( (d & c) V (p & d) V (p & ~c) )

Read more about this topic: Propositional Formula, Normal Forms, Reduction By Use of The Map Method (Veitch, Karnaugh)
Famous quotes containing the word reduce:
“The more we reduce ourselves to machines in the lower things, the more force we shall set free to use in the higher.”
—Anna C. Brackett (1836–1911)