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4.  Verify the following proposition with the help of a truth table:
                        (P ∧ Q) ∨ (P ∧ ∼Q) = P                                                                [ISC 2018]
                  Ans.    P      Q     P ∧ Q   P ∧ ∼Q     (P ∧ Q) ∨ (P ∧ ∼Q)

                          0      0       0       0              0
                          0      1       0       0              0
                          1      0       0       1              1
                          1      1       1       0              1

                         We derive that column 1 and column 5 have the same combinations of 0 and 1. Hence proved.
                    5.  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)   (∼A → B)    ∼(A ∧ B) ∨ (∼A → B)
                         0    0     1         0              1
                         0    1     1         1              1
                         1    0     1         1              1
                         1    1     0         1              1
                         The final column has a value 1. Hence, it is a tautology.
                    6.  Differentiate between half adder and full adder. Draw the logic circuit diagram for a full adder.   [ISC 2018]
                   Ans.             Half Adder                          Full Adder

                        Half adder performs the addition of 2 bits.  Full adder performs the addition of 3 bits.
                         The logic circuit diagram of a full adder circuit is:

                                         A                   Sum = A ⊕ B ⊕ C
                                         B                                in
                                         C in



                                                                A.B


                                                                                 A.B+B.C +C .A
                                                                                          in
                                                                                       in
                                                                B.C in




                                                               C .A
                                                                in


                    7.  Verify using truth table F = (P → Q) ∨ (Q → P').                                      [ISC 2020]
                   Ans.    P       Q     (P → Q)   (Q → P')    (P → Q) ∨ (Q → P')

                           0       0        1         1              1
                           0       1        1         1              1
                           1       0        0         1              1
                           1       1        1         0              1
                       The final column has a value 1. Hence, it is a tautology.







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                                                           Propositional Logic, Hardware Implementation, Arithmetic Operations  71
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