Page 74 - Computer science 868 Class 12
P. 74

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

              Referring C  as C the Boolean expression of Sum term = A'.B'.C+A'.B.C'+A.B'.C'+A.B.C which is equivalent to three
                        in
              variable XOR gate. The proof is given below:
              A'.B'.C+A'.B.C'+A.B'.C'+A.B.C

              = A'.(B'.C+B.C')+A.(B'.C'+B.C)             [Distributive Law]
              = A'.(B⊕C) + A.(B⊕C)'

              = A⊕ B⊕C
              Boolean expression for carry term = A'.B.C+A.B'.C+A.B.C'+A.B.C can be further minimised to A.B+B.C+C.A as done
              below:

              = A'.B.C+A.B'.C+A.B.C'+A.B.C+A.B.C+A.B.C     [Idempotent Law]
              = A'.B.C+A.B.C+A.B'.C+A.B.C+A.B.C'+A.B.C      [Associative Law]
              = B.C.(A'+A)+A.C.(B'+B)+A.B.(C+C')         [Distributive Law]

              = B.C+A.C+A.B                              [Complement Law]
              = A.B+B.C+C.A

              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



              The full adder circuit can also be represented as two half adders connected by OR gate as follows:
              Sum of full adder  = A'.B'.C+A'.B.C'+A.B'.C'+A.B.C
                              = A'.(B'.C+B.C')+A.(B'.C'+B.C)            [Distributive Law]
                              = A'.(B⊕C)+A.(B⊕C)'
                              = A⊕B⊕C

              Carry of full adder  = A'.B.C+A.B'.C+A.B.C'+A.B.C
                               = C.(A'.B+A.B')+A.B.(C'+C)               [Distributive Law]
                               = C.(A⊕B)+A.B                            [Complement Law]



                7272  Touchpad Computer Science-XII
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