Page 63 - Computer science 868 Class 12
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SOP Expression for X (A, F, C, I) : X (A ,F, C, I) = Σ (1, 2, 3, 5, 6, 7, 8, 9, 10, 11)
                        A'F'C'I + A'F'CI' + A'F'CI + A'FC'I + A'FCI' + A'FCI + AF'C'I' + AF'C'I +AF'CI' + AF'CI
                        (b)  Using the truth table, state whether the following proposition is a tautology, contingency or a contradiction:
                           ∼(A ∧ B) ∨ ( ∼A => B)                                                                 [ISC 2018]
                   Ans.  ∼(A∧B) ∨ (∼A=>B) is a Tautology, Contradiction or Contingency
                           A        B       ∼A       ∼B     ∼(A ∧ B)  (∼A => B) ∼(A ∧ B) ∨ (∼A => B)
                           0        0        1        1        1        0            1
                           0        1        1        0        1        1            1
                           1        0        0        1        1        1            1
                           1        1        0        0        0        1            1
                        Hence, it is a Tautology.
                        (c)  Simplify the following expression, using Boolean laws:                              [ISC 2018]
                          A  • (A' + B)  • C  • (A + B)
                   Ans.  A  • (A' + B)  • C  • (A + B)
                        = (AA' + AB).(AC + BC)
                        = (0 + AB).(AC + BC)
                        = ABC + ABC
                        = ABC
                    42.  State the law represented by the following proposition and prove it with the help of a truth table: P ∨ P = P   [ISC 2017]
                   Ans.  The law represented by the given proposition is Idempotent law.
                                                           Truth Table
                            P         P       P∨P    P∨P⇔P
                            T         T        T        T
                            F         F        F        F
                    43.  State the Principle of Duality.                                                         [ISC 2017]
                   Ans.  Principal of duality states that each Boolean expression has its dual expression which can be obtained by replacing:
                        •  AND (.) with OR (+) or OR (+) with AND (.)
                        •  0 with 1 or 1 with 0
                        •  Complements remain unchanged
                    44.  Find the complement of the following Boolean expression using De Morgan’s law:          [ISC 2017]
                       F(a, b, c) = (b' + c) + a
                   Ans.  ((b' + c) + a)'
                               = (b'+c)'. a'
                                = ((b')' . c') . a'
                                = bc'a'
                    45.  If (∼P => Q) then write its:                                                            [ISC 2017]
                         (i)   Inverse       (ii)  Converse
                   Ans.   (i)  Inverse of the given statement is P => ∼Q
                        (ii)  Converse of the given statement is Q >= ∼P
                    46.  Given the Boolean function F(A, B, C, D) = Σ (2, 3, 4, 5, 6, 7, 8, 10, 11).
                        Reduce the above expression by using 4-variable Karnaugh map, showing the various groups (i.e. octal, quads and pairs).  [ISC 2017]
                   Ans.  F(A, B, C, D) = Σ (2, 3, 4, 5, 6, 7, 8, 10, 11)
                          CD
                        AB     C'D'   C'D   CD     CD'
                        A'B'              0              1     1  3     1  2

                         A'B    1  4   1   5    1  7     1  6

                         AB
                                 12     13     15    14
                         AB'     1                 1         1
                                  8     9     11     10



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