Page 89 - Cs_withBlue_J_C11_Flipbook
P. 89

Referring C  as C, the Boolean expression of the sum term:
                          in
                 = A'.B'.C + A'.B.C' + A.B'.C' + A.B.C

                 which is equivalent to a three variables XOR gate.
                 Boolean expression for carry term = A'.B.C + A.B'.C + A.B.C' + A.B.C which on minimisation becomes A.B + B.C + C.A
                 [Minimisation will be covered in the class XII book]

                 The logic circuit diagram of a full adder circuit is:

                                           A                     Sum = A⊕B⊕C in
                                           B
                                          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.(B9C)
                                 = 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]



                                  HALF ADDER 1

                               A                A ⊕ B
                               B



                                                 A.B             HALF ADDER 2
                                                                                                  C.(A ⊕ B) + A.B

                                                                          A ⊕ B ⊕ C
                              C



                                                                          C.(A ⊕ B)





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                                                           Propositional Logic, Hardware Implementation, Arithmetic Operations  87
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