Discrete Mathematics..

Minimization of Boolean Expressions 477expression in the two variables x 1 and x 2 are shown in the diagram below.x 2 ¯x 2x 1 x 1 x 2 x 1 ¯x 2¯x 1 ¯x 1 x 2 ¯x 1 ¯x 2Written in each cell is the corresponding minterm. Notice that the requirementthat adjacent cells differ by just one literal also applies at the edges of the map ifwe view the rightmost column of cells as being adjacent to the left-hand columnand also the top and bottom rows as being adjacent. In using the map it isimportant to realize that the right and left edges are to be regarded as contiguousand so are the top and bottom edges.The following is a layout for a Karnaugh map for three variables x 1 , x 2 and x 3 .x 2 x 3 ¯x 2 x 3 ¯x 2 ¯x 3 x 2 ¯x 3x 1 a b c d¯x 1 e f g hThe cell labelled a represents the minterm x 1 x 2 x 3 , cell c represents x 1 ¯x 2 ¯x 3 , frepresents ¯x 1 ¯x 2 x 3 , etc. There is more than one way of constructing a Karnaughmap so that the necessary criteria are satisfied. An alternative for three variablesis given below.x 3 ¯x 3x 1 x 2 a dx 1 ¯x 2 b c¯x 1 ¯x 2 f g¯x 1 x 2 e hA Boolean expression given as the sum of minterms (i.e. in disjunctive normalform) is represented on the Karnaugh map by placing a one in each cellcorresponding to a minterm which is present. For example, the Booleanexpression x 1 x 2 ⊕¯x 1 x 2 in the two variables x 1 and x 2 is represented byx 2 ¯x 2x 1 1¯x 1 1