Digital Electronics: Principles, Devices and Applications

226 Digital ElectronicsFigure 6.18Three-function Karnaugh map.The formation of groups begins with the largest multifunction map, which is nothing but theintersection of maps of all individual functions. Then we move to the Karnaugh maps one step downthe order. The process continues until we reach the maps corresponding to individual functions. Thegroups in all the Karnaugh maps other than the largest map are formed subject to the condition that,once a group is identified in a certain function, then the same cannot be identified in any map of asubset of that function. For example, a group identified in a four-function map cannot be identifiedin a three-, two- or one-function map. With the formation of groups, prime implicants are identified.These prime implicants can be compiled in the form of a table along with input combinations ofdifferent output functions in the same way as for the tabular method to write minimized expressions. Ifthe expressions corresponding to different output functions are not very complex, then the minimizedexpressions can even be written directly from the set of maps.Example 6.12Using Karnaugh maps, write the minimized Boolean expressions for the output functions of a two-outputlogic system whose outputs Y 1 and Y 2 are given by the following Boolean functions:Y 1 = ABC + ABC + ABC + ABC (6.62)Y 2 = ABC + ABC + ABC + ABC + ABC (6.63)SolutionThe individual Karnaugh maps and the two-function map are shown in Fig. 6.19 along with theformation of groups. The prime implicant table along with the input combinations for the two outputfunctions is given below:Y 1 Y 2 Prime implicants000 010 100 111 000 001 101 110 111̌ ̌ 0 0 0̌ ̌ 1 1 1̌ ̌ 0 − 0̌ ̌ − 0 0̌ ̌ 1 1 −̌ ̌ − 0 1