Page 49 - Computer science 868 Class 12
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=A'(1 + B + C'.D)          [Distributive law]
                          =A'.1                      [Properties of 0 and 1]
                          =A'
                      (b)  [(C.D)'+A] + A + C.D + A.B                                                            [ISC 2011]
                          =(C.D)' + C.D + A + A + A.B    [Associative law]
                          =1 + A + A + A.B           [De Morgan’s law]
                          =1                         [Properties of 0 and 1]
                      (c)  A.[B + C.(A.B + A.C)']                                                                [ISC 2011]
                          =A.[B + C.( (A.B)'+(A.C)')]     [De Morgan’s law]
                          =A.[B+C.( (A'+B').(A'+C'))]    [De Morgan’s law]
                          =A.[B+ C.( A'+B'.C'))]           [Distributive law]
                          =A. [ B + A'.C]            [Complement law]
                          =A.B + A.A'.C              [Distributive law]
                          =A.B                       [Complement law]
                      (d)  A'.B + A.C + B.C
                          =A'.B + A.C + B.C.1        [Properties of 0 and 1]
                          =A'.B + A.C + B.C.(A+A')     [Complement law]
                          =A'.B + A.C + A.B.C + A'.B.C   [Distributive law]
                          =A'.B + A'.B.C + A.C + A.B.C   [Associative law]
                          =A'B(1+C) + A.C(1+C)       [Distributive law]
                          =A'.B + A.C                [Properties of 0 and1]
                      (e)  A.B' + A'.B.C' + (A.C)' + B.C
                          =A.B' + A'.B.C' + A' + C' + B.C   [De Morgan’s law]
                          =A.B' + A'.(B.C'+1) + (B+C').(C+C')  [Distributive law]
                          =A.B' + A'+ B + C'         [Complement law]
                          =(A+A').(A'+B') + B + C'      [Distributive law]
                          =A' + B' + B + C'          [Complement law]
                          =A' + 1 + C'               [Complement law]
                          =1                         [Properties of 0 and 1]
                    7.  Write the cardinal form of the following canonical expressions:
                        (a)  A.B.C'.D + A.B'.C'.D + A'.B.C'.D + A.B.C.D' + A.B.C.D + A'.B'.C'D + A.B'.C'.D' [eliminating duplicates]
                   Ans.  F(A, B, C, D) = Σ(1101, 1001, 0101, 1110, 1111, 1000)
                       =Σ(13, 9, 5, 14, 15, 1, 8)
                       =Σ(1,5,8,9,13,14,15)
                        (b)  (A+B+C+D).(A'+B'+C'+D).(A+B'+C'+D').(A'+B+C'+D).(A+B'+C+D')
                   Ans.  F(A, B, C, D) = π(0000, 1110, 0111, 1010, 0101)
                       =π(0, 14, 7, 10, 5)
                       =π(0, 5, 7, 10, 14)
                    8.  Express X'+Y in canonical SOP.
                   Ans.  X'+Y
                         =X'.1 + Y.1
                         =X'.(Y+Y') + Y.(X+X')
                         =X'.Y + X'.Y' + X.Y + X'.Y
                        =X'.Y + X'.Y' + X.Y






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                                                                                                      Boolean Algebra   47
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