Page 59 - Computer science 868 Class 12
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obj.day_to_date();
                             obj.display();
                          }
                       }
                    32.   Using a truth table, state whether the following proposition is a Tautology, Contradiction or Contingency:
                       ∼(P → Q) ↔ (∼P ∨ Q)                                                                       [ISC 2019]
                   Ans.
                           P       Q       P → Q  ∼(P → Q)    ∼P       ∼P ∨ Q
                           0        0        1        0        1         1
                           0        1        1        0        1         1
                           1        0        0        1        0         1
                           1        1        1        0        0         1
                         The final column has all 0's. Hence, it is a contradiction.
                   33.  (a)   Given the Boolean function F(A, B, C, D) = Σ(0, 2, 3, 4, 5, 8, 10, 11, 12, 13).       [ISC 2019]
                             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, 2, 3, 4, 5, 8, 10, 11, 12, 13)
                         CD
                       AB     C'.D'   C'.D     C.D   C.D'
                             0       1       3       2
                       A'.B'    1       0       1       1

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

                       Quad 1: (m +m +m +m ) = B'D'
                                0
                                      8
                                         10
                                   2
                       Quad 2: (m +m +m +m ) = C'D'
                                0
                                          8
                                      12
                                   4
                       Quad 3: (m +m +m +m ) = B'C
                                      11
                                          10
                                3
                                   2
                       Hence, F(A, B, C, D) = B'D' + BC' + B'C
                        (b)  Given the Boolean function F(P, Q, R, S) = π(0, 1, 2, 8, 9, 11, 13, 15).            [ISC 2019]
                             Reduce the above expression by using a 4-variable Karnaugh map, showing the various groups (i.e., octals, quads and pairs).
                   Ans.       R+S    R+S'     R'+S'   R'+S
                            0       1       3       2
                       P+Q     0       0       0       0
                            4       5       7       6
                       P+Q'    1       1       1       1
                            12      13      15      14
                       P'+Q'   1       0       0       1
                            8       9       11      10
                       P'+Q    0       0       0       0
                       Quad 1: m .m .m .m  = Q+R
                               0
                                  1
                                    8
                                       9
                       Quad 2: (m .M .M .M ) = P'+S'
                                         11
                                      9
                                13
                                   15
                       Pair 1:   (m .m ) = P+Q+S
                                  2
                               0
                       Hence, F(P, Q, R, S) = (Q + R).(P' + S').(P + Q + S)
                    34.  The owner of a company pays the bonus to his salesmen as per the criteria are given below:    [ISC 2019]
                        If the salesman works overtime for more than 4 hours but does not work on off days/ holidays.
                                                                    OR
                        If the salesman works when festival sales are on and updates showroom arrangements.
                                                                    OR
                        If the salesman works on an off day/holiday when the festival sales are on.
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                                                                                                      Boolean Algebra   57
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