Page 51 - Computer science 868 Class 12
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2.  Consider the following propositions
                       P = “It is raining heavily.”
                        Q = “There is a prediction for cyclone.”
                        R = “There will be damage of life and property.”
                        Now, write the following statements in symbolic form:
                       a.  If there is a prediction for cyclone, then it is raining heavily and there will be damage to life and property.
                       b.  It is raining heavily and there is a prediction for cyclone.
                       c.  There is a prediction for cyclone if and only if it is raining heavily.
                       d.  It is not raining heavily or there is no prediction of cyclone.
                       e.  If there is a prediction for cyclone then there will be damage to life and property.
                    3.  Consider the following propositions
                       P = “It is raining heavily.”
                        Q = “There is a prediction for cyclone.”
                        R = “There will be damage to life and property.”
                        Now, express the following expressions in words:
                       a.  ∼P ∨ ∼Q
                      b.  ∼P ∨ (∼Q ∧ ∼R)
                       c.  ∼P ↔ ∼Q
                      d.  ∼(P ∧ Q ∧ R)
                      e.   ∼Q → ∼R
                    4.  Write the converse, inverse and contrapositive for the following propositions:
                       i.  If you work hard, then you will crack IIT.
                      ii.  If the flower has a beautiful smell, then it is a rose.
                      iii.  If I am invited to a party, then I will wear my red gown.
                      iv.  If India wins the World Cup cricket match, then we will celebrate.
                       v.  If a number is equal to its reverse, then it is a palindrome.
                    5.  Construct the truth tables and check if the following propositions are tautology, contradiction or contingency:
                       a.  (a ↔ b) → (a ∧ b)
                       b.  b → (∼a → b) ↔ a
                       c.  (a ∧ b) → (∼a ∧ ∼b)
                       d.  [(a → b) → (b → c)] → a
                       e.  (a ∧ 0) → (∼a ∧ b)
                       f.  a ∧ (a → b) → b
                       g.  (a ↔ b) ∧ (∼a ↔ ∼b)
                       h.  (a ∧ b) → (∼a ∧ b) → (∼b ∧ a)
                       i.  ∼(a ∨ b) ∧ (a ∨ b)
                       j.  ∼(a ∧ b ∧ c) ∨ (a ∧ b ∧ c)
                    6.  Complete the truth table given below.
                           P       Q       P → Q    (P → Q) → P'  ∼ ((P → Q) →  ∼P)


                    7.  Minimise the following Boolean expressions using the laws of Boolean Algebra:
                       a.  X.Y + X'.Z + Y.Z = X.Y + X'.Z
                       b.  A.(B + C.(A+B.C)')'
                       c.  (X'+Y'+Z).(X'+Y'+Z') + (X+Y+Z).(X+Y+Z')
                       d.  (X+X'.Y).(X'+Y')
                       e.  A.B.C + A.B.C' + A'.B.C + A.B'.C
                       f.  X'.Y + X'.Y' + (Z.Z)' + Z
                       g.  A.B + A.B' + A'.B'
                       h.  (X' + (X.Y)' + Z)'
                       i.  A.(A'+B).C.(A+B) = A.B.C                                                              [ISC 2018]
                       j.  F = (x'+z) + [(y'+z).(x'+y)]'                                                         [ISC 2017]



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