Karnaugh Maps:A Karnaugh map is a planar area subdivided into 2^{n} equal cells each representing a point for functions of n variables. Each variable x is used to split the area into two equal halves in a different way, i.e., one for x and other for x'. The cells corresponding to the arguments for which the function has the value 1 contains 1. Example1: When the number of variables n = 1, the karnaugh map is like as shown in fig: 2. When the number of variables n =2, the karnaugh map is like as shown in fig: 3. When the number of variables n =3, the karnaugh map is like as shown in fig: Simplification of Boolean Functions using KMap:Boolean Functions can be simplified with kmap. It is based on the principle of combining terms in adjacent cells. Two cells are said to be adjacent if they differ in only one variable. In adjacent cells, one of the variables is the same, whereas the other variable appears in the uncomplemented form in one and in the complemented form in the other cell. Minimization of SOP Form:The following algorithm can be used by which minimized expression can be obtained:
Example1: Minimize the following Boolean Expression using kmap: f(A, B) = A' B+BA Solution: First of all draw the 2variables kmap and insert 1's in the corresponding cells as shown in fig: The required minimized Boolean Expression is f=B. Example2: Minimize the following Boolean Expression using kmap: AB + A' B+BA' Solution: Draw the twovariable kmap and insert 1's in the corresponding cells as shown in fig: The required minimized Boolean Expression is f=A+B. Example3: Minimize the following Boolean Expression using kmap: f(A, B, C) = AB' C+A' BC+AB+A' B' C Solution: Draw the 3variable kmap and insert 1's in the corresponding cells as shown in fig: The required minimized Boolean Expression is f=AB+C' Minimisation of Boolean Functions not in Minterms/Maxterms:One way to minimize such functions is to convert them into standard forms i.e., SOP or POS, then make the kmap and obtain the minimized function. Another way is to prepare the kmap using the following algorithm directly
Example: Minimize the four variable logic function f (A, B, C, D) = A B C'D + A' BCD+A' B' C'+A' B' D'+AC'+AB' C+B' Solution: The kmap is obtained by following way (a)Enter 1 in the cell with A=1, B=1, C=0, D=1 corresponding to the minterm A B C'D (b) Enter 1 in the cell with A=0, B=1, C=1, D=1 corresponding to the minterm A' BCD (c) Enter 1's in the two cells with A=0, B=0, C=0 corresponding to the term A' B' C' (d) Enter 1's in the two cells with A=0, B=0, D=0 corresponding to the term A' B' D' (e) Enter 1's in the two cells with A=1, B=0, C=1 corresponding to the term AB' C (f) Enter 1's in the four cells with A=1,C=0 corresponding to the term AC' (g) Enter 1's in the eight cells with B=0 corresponding to the term B' The minimized expression is B'+ AC'+A' CD.
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