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4.  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
                       The final column results in 1. So, it is a tautology. Hence proved.
                    5.  If, A = “The Taj Mahal is one of the seven wonders of the world” and
                             B = “It is a favourite tourist destination”
                        Then, write the statements for the following propositions:
                        (i) A ∧ B  (ii) A → B  (iii) A ↔ B  (iv) A' ∨ B'  (v) B' → A
                   Ans.  i.  The 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.  The Taj Mahal is one of the seven wonders of the world if and only if it is a favourite tourist destination.
                         iv.  The 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.
                    6.  Draw the truth table of (A ⊙ B) ⊕  C.
                   Ans.   A       B       A ⊙ B       C        (A ⊙ B) ⊕ C
                          0       0        1          0            1
                          0       0        1          1            0
                          0       1        0          0            0
                          0       1        0          1            1
                          1       0        0          0            0
                          1       0        0          1            1
                          1       1        1          0            1
                          1       1        1          1            0
                    7.  Prove using truth table (A ⊕ B) ⊕ C = A ⊕ (B ⊕ C).
                   Ans.   A       B       A ⊕ B       C       (A ⊕ B) ⊕ C          B ⊕ C        A ⊕ (B ⊕ C)
                          0       0         0         0           0                 0               0
                          0       0         0         1           1                 1               1
                          0       1         1         0           1                 1               1
                          0       1         1         1           0                 0               0
                          1       0         1         0           1                 0               1
                          1       0         1         1           0                 1               0
                          1       1         0         0           0                 1               0
                          1       1         0         1           1                 0               1

                       We find that the two columns are identical. Hence proved.


                      Unsolved Questions



                 A.  Tick ( ) the correct option:
                    1.  The propositional operator → represents ………………… .
                       a.  conjunction                                 b.  implication
                       c.  disjunction                                 d.  negation



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                                                           Propositional Logic, Hardware Implementation, Arithmetic Operations  67
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