Page 21 - Computer science 868 Class 12
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•  Inverse: The inverse of a conditional is obtained by complementing the antecedent and the consequent of that
                   conditional.
                   For any conditional a → b, its inverse will be a' → b'.
                 •  Contrapositive: The contrapositive of a conditional is obtained by interchanging the complemented antecedent with
                   the complemented consequent of that conditional. It is equivalent to the converse of the inverse of that conditional.
                   For any conditional a → b, its contrapositive will be b' → a'.
                 Let us see the following example.

                 Example 3: Consider the following propositions:

                 p = “MADAM reads same from both sides”
                 q = “It is a palindrome”
                 Write the (i) converse (ii) inverse (iii) contrapositive of the conditional p → q.

                 Solution: The conditional of the given proposition is:

                 p → q = “If MADAM reads same from both sides then it is a palindrome”
                 i.  Converse q → p = “If it is a palindrome then MADAM reads the same from both sides”
                 ii.  Inverse p' → q' = “If MADAM does not read the same from both sides then it is not a palindrome”
                 iii. Contrapositive q' → p' = “If it is not a palindrome then MADAM does not read the same from both sides”


                     1.6 PRECEDENCE OF THE CONNECTIVES                                   1     Brackets
                 The propositions  may contain  two or more connectives together.  As in       Negation
                 mathematics, multiple  operators present in  an  expression  follow the   2     Conjunction
                 precedence  rule, similarly,  the precedence  rule is  followed in  Boolean   3
                 algebra. They determine the order of evaluation of operators or connectives.   4  Disjunction
                 The precedence of the connectives in decreasing order is shown here.    5     Implication
                                                                                         6     Bi-conditional
                     1.7 EQUIVALENCE PROPOSITIONAL LAWS
                 Two statements are said to be equivalent if they have the same truth value for all possible combinations of their
                 variables. Thus, if the value of a and b is true in the same set of models then they are said to be logically equivalent.

                 Some commonly used equivalence laws are given below. These laws can be proved using the truth table:

                                                  Name                           Expression
                                                                      a.  a ∨ 0 = a
                                      1.  Properties of 0
                                                                      b.  a ∧ 0 = 0
                                                                      a.  a ∨ 1 = 1
                                      2.  Properties of 1
                                                                      b.  a ∧ 1 = a
                                      3.  Involution law              a.  (a')'  = a
                                                                      a.  a ∨ a = a
                                      4.  Idempotent law
                                                                      b.  a ∧ a = a
                                                                      a.  a ∨ a' = 1
                                      5.  Complementarity law
                                                                      b.  a ∧ a' = 0
                                                                      a.  a ∨ b = b ∨ a
                                      6.  Commutative law
                                                                      b.  a ∧ b = b ∧ a




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                                                                                                      Boolean Algebra   19
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