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NPP               Number System, Boolean Algebra and Logic Circuits              227


                      Four quads are made with the help of over-  Mma ŠdmS> AmodaboqnJ go ~Zo h¢ VWm Xmo noAa amoqbJ
                  lapping and two pairs are made with the help  go ~Zr h¡Ÿ& gab ì`§OH$ Bg àH$ma Hw$b N>… nXm| H$m `moJ
                  of rolling. Rolling can be done when the 1’s are
                  in the same row or same column and at the cor-  hmoJm:
                  ners. The simplified expression is:

                                           F =  B . A  +  D . A  +  C . B  +  D . C  +  B .  C  . D +  D . B . A
                       Problem 3.67                                àíZ 3.67
                      Simplify the following Boolean expres-      {ZåZ ~y{b`Z ì`§OH$m| H$mo K- _on go hb H$amo:
                  sion using K-map method:
                                               (a)  f = Σm (0, 1, 6, 7)
                                               (b)  f = Σm (2, 3, 5, 9, 10, 11, 14, 15)
                  Solution:                                   hc:

                      (a) The given expression is Σm (0, 1, 6, 7)  (a)  Cnamoº$ ì`§OH$ f = Σm (0, 1, 6, 7) hoVw K-_on
                  The K-map can be drawn and grouping can be  VWm {d{^ÝZ J«wn Bg àH$ma ~Z|Jo:
                  done-               NPP
                                                 A  B C  00  01    11    10



                                                 0    1      1     0      0


                                                 1    0      0     1      1


                      There are two pairs in the K-map. The sim-  Xmo noAam| Ho$ H$maU gab ì`§OH$ Bg àH$ma hmoJm:
                  plified expression is:
                                                        f =   B . A  +  B . A
                      (b)  f = Σm (2, 3, 5, 9, 10, 11, 14, 15)    (b)  f = Σm (2, 3, 5, 9, 10, 11, 14, 15)
                      This is a four variable problem because the  `h EH$ Mma am{e`m| dmbr g_ñ`m h¡ Š`m|{H$ g~go
                  largest number is 15 (1111). The K-map can be  ~‹S>r g§»`m  15(1111) h¡ Ÿ& AmR> {_ÝQ>_m] hoVw AmR> 1
                  drawn by filling eight 1’s corresponding to eight  aIH$a VWm J«wn ~ZmZo na K-_on {ZåZmZwgma hmoJm:
                  minterms-
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