Page 56 - Computer science 868 Class 12
P. 56

24.  Find the dual of: (A' + 0) • (B' + 1) = A'                                               [ISC 2020]
                Ans.  Dual = A'•1 + B'• 0 = A'
                 25.  State whether the following proposition is a tautology, contradiction or a contingency:    [ISC 2020]
                     F = (P => Q) V (Q => ∼P)
                Ans.  F = (P => Q) V (Q => ∼P)
                        P        Q        ∼P     P => Q   Q => ∼P    (P => Q) ∨ (Q => ∼P)
                        0        0        1        1         1              1
                        0        1        1        1         1              1
                        1        0        0        0         1              1
                        1        1        0        1         0              1
                     (using laws): (P' + Q) + (Q' + P') = 1 ( as Q + Q' = 1)
                     Hence, it is a TAUTOLOGY
                 26.  Given the Boolean function F(A, B, C, D) = Σ(0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14).       [ISC 2020]
                      Reduce the above expression by using a 4-variable Karnaugh map, showing the various groups (i.e., octals, quads and pairs).
                Ans.  F(A, B, C, D) = Σ (0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 14)

                           C'.D'  C'.D  C.D   C.D'
                           0     1     3     2
                     A'.B'        1  1    1     1

                           4     5     7     6
                     A'.B    1      1     0     0
                           12    13    15    14
                     A.B     1      1     0     1
                           8     9     11    10
                     A.B'    1      1     1     1


                     There are two octets and one quads:
                     Quad 1: (m +m +m +m +m +m +m +m ) = B'
                                1
                             0
                                   2
                                                10
                                                    11
                                             9
                                      3
                                         8
                     Quad 2: (m +m +m +m +m +m +m +m ) = C'
                                                12
                                         8
                                1
                                      5
                                             9
                                   4
                                                    13
                             0
                     Quad 3: (m +m +m +m ) = AD'
                                        14
                             8
                                10
                                    12
                     Hence, F(A, B, C, D) = B' + C' + AD'
                 27.  Given the Boolean function F(A, B, C, D) = Σ(3, 4, 6, 9, 11, 12, 13, 14, 15).            [ISC 2020]
                      Reduce the above expression by using a 4-variable Karnaugh map, showing the various groups (i.e., octals, quads and pairs).
                Ans.       C+D   C+D'  C'+D'  C'+D
                           0     1     3     2
                     A+B         1  1     0     1
                           4     5     7     6
                     A+B'    0      1     1     0
                           12    13    15    14
                     A'+B'   0      0     0     0
                           8     9     11    10
                     A'+B    1      0     0     1
                     There are two quads and one pair:
                    Quad 1: (M M M M ) = B' + D
                                     14
                               6
                             4
                                  12
                    Quad 2: (M M M M ) = A' + D'
                                  13
                             9
                                     15
                               11
                    Pair: (M M ) = B + C' + D'
                             11
                          3
                     Hence, F(A, B, C, D) = ( B' + D) • (A' + D') • (B + C' + D')
                 28.   Convert the following expression to its cardinal SOP form:                              [ISC 2020]
                     F(P,Q,R) = P'Q'R + P'QR + PQ'R' + PQR'
                5454  Touchpad Computer Science-XII
   51   52   53   54   55   56   57   58   59   60   61