Page 48 - Computer science 868 Class 12
P. 48

2.  Prove ∼(a → b) ∨ (∼a ∨ (a ∧ b)) is a tautology.
                Ans.
                        a        b        ∼a     a → b   ∼(a → b)    a ∧  b  ∼a∨ (a ∧ b)  ∼(a → b) ∨ (∼a∨ (a ∧ b))
                        0        0        1        1        0        0         1               1
                        0        1        1        1        0        0         1               1
                        1        0        0        0        1        0         0               1
                        1        1        0        1        0        1         1               1

                  3.  If,  A = “Taj Mahal is one of the seven wonders of the world.” and
                            B = “It is a favourite tourist destination.”
                     Then, write statements for the following propositions.
                     (i)  A ∧ B     (ii)  A →B    (iii)  A ↔ B   (iv)  A' ∨ B'   (v) B' → A
                Ans.  (i)  Taj Mahal is one of the seven wonders of the world and it is a favourite tourist destination.
                     (ii)  If, the Taj Mahal is one of the seven wonders of the world then it is a favourite tourist destination.
                     (iii)  Taj Mahal is one of the seven wonders of the world if and only if it is a favourite tourist destination.
                     (iv)  Taj Mahal is not one of the seven wonders of the world or it is not a favourite tourist destination.
                     (v)  If it is not a favourite tourist destination, then the Taj Mahal is one of the seven wonders of the world.
                  4.  Draw the truth table of (A9B) 5 C.
                Ans.    A        B        C      A 9 B   (A 9 B) 5 C

                        0        0        0        1          1
                        0        0        1        1          0
                        0        1        0        0          0
                        0        1        1        0          1
                        1        0        0        0          0
                        1        0        1        0          1
                        1        1        0        1          1
                        1        1        1        1          0
                  5.  Prove (A 5 B) 5 C = A 5 (B 5 C) using truth table.
                Ans.
                        A        B        C      A 5 B   (A 5 B) 5 C     A 9 B       (A 9 B) 5 C
                        0        0        0        0          0            0             0
                        0        0        1        0          1            1             1
                        0        1        0        1          1            1             1
                        0        1        1        1          0            0             0
                        1        0        0        1          1            0             1
                        1        0        1        1          0            1             0
                        1        1        0        0          0            1             0
                        1        1        1        0          1            0             1
                     We find that the fifth column and the last column are identical. Hence proved the given equation is true.
                  6.  Minimise the following using Boolean laws:
                    (a)  (A'+C).(A'+C').(A'+B+C'.D)                                                            [ISC 2014]
                        =(A'+ C.C').(A'+ B + C'.D)     [Distributive law]
                        =(A'+ 0).(A'+ B + C'.D)      [Complement law]
                        =A'(A' + B + C'.D)        [Properties of 0 and 1]
                        =A'.A' + A'.B + A'.C'.D      [Distributive law]
                        =A' + A'.B + A'.C'.D      [Idempotent law]





                4646  Touchpad Computer Science-XII
   43   44   45   46   47   48   49   50   51   52   53