Page 28 - Computer science 868 Class 12
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A+A = A
                A.A = A
                The proof of the above expressions is illustrated in the truth table given below.

                                                      A          A+A         A.A
                                                      0           0           0
                                                      1           1           1
                The last two columns have the same combinations of 0 and 1 as that of the first column A. Hence proved.

              4.  Involution Law
                This law states that if a Boolean variable is complemented twice then it results in its original value.
                It can be expressed algebraically as:
                A" = A
                The proof is as follows:

                                                      A           A'          A"
                                                      0           1           0
                                                      1           0           1
                The first and the last columns have the same combinations of 0 and 1. Hence proved.
              5.  Commutative Law
                This law states that if we commute or switch the position of the Boolean variables, the result of its logical addition
                 or multiplication remains unchanged.
                It can be expressed algebraically as
                A+B = B+A

                A.B = B.A
                The proof is illustrated in the truth table given below.
                                   A           B           A+B         B+A         A.B          B.A

                                   0            0           0           0           0           0
                                   0            1           0           0           0           0
                                   1            0           1           1           0           0
                                   1            1           1           1           1           1
                Comparing the resultant columns, we can deduce that the law is proved.
              6.  Associative Law
                This law states that the association or order in which logical multiplication or logical addition is implemented in a
                 Boolean expression, is irrelevant, as they produce the same result.
                The Boolean expression can thus be represented as:
                (A.B).C = A.(B.C)
                (A+B) + C = A + (B+C)
                The proof using the truth table is shown below.

                          A      B       C    (A.B)   (A.B).C  (B.C)  A.(B.C)  (A+B)  (A+B)+C  (B+C)  A+(B+C)
                          0      0       0      0       0       0       0       0       0        0       0
                          0      0       1      0       0       0       0       0       1        1       1
                          0      1       0      0       0       0       0       1       1        1       1



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