Editing Propositional formula (section)

==== Reduce minterms ====

Minterms of adjacent (abutting) 1-squares (T-squares) can be reduced with respect to the number of their [[literal (mathematical logic)|literal]]s, 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) lose 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) lose two literals, eight squares in a rectangle lose 3 literals, etc. (One seeks out the largest square or rectangles and ignores the smaller squares or rectangles 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 lose one literal (e.g. "p" from squares #3 and #7), four squares in a rectangle or square lose two literals, eight squares in a rectangle lose 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 &or; 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 [ and ] 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 ) &or; (p & d & c) &or; (p & d & ~c) &or; (p & ~d & ~c) )''' = ( #3 &or; #7 &or; #6 &or; #4 )
* Idempotency (absorption) [ A &or; A) = A:
::: ( #3 &or; [ #7 &or; #7 ] &or; [ #6 &or; #6 ] &or; #4 )
* Associative law (x &or; (y &or; z)) = ( (x &or; y) &or; z )
::: ( [ #3 &or; #7 ] &or; [ #7 &or; #6 ] &or; [ #6 &or; #4]  )
::: '''[''' (~p & d & c ) &or; (p & d & c) ''']''' &or; '''[''' (p & d & c) &or; (p & d & ~c) ''']''' &or; '''[''' (p & d & ~c) &or; (p & ~d & ~c) ''']'''.
* Distributive law ( x & (y &or; z) ) = ( (x & y) &or; (x & z) ) :
::: ( [ (d & c) &or; (~p & p) ]  &or; [ (p & d) &or; (~c & c) ] &or; [ (p & ~c) &or; (c & ~c) ] )
* Commutative law and law of contradiction (x & ~x) = (~x & x) = 0:
::: ( [ (d & c) &or; (0) ] &or; [ (p & d) &or; (0) ] &or; [ (p & ~c) &or; (0) ] )
* Law of identity ( x &or; 0 ) = x leading to the reduced form of the formula:
::: '''q = ( (d & c) &or; (p & d)  &or; (p & ~c) )'''