Explanation:Dual Of Boolean Expression-   To get a dual of any Boolean Expression, replace- OR with AND i.e. + with . AND with OR i.e. . with + 1 with 0 0 with 1   Dual of Boolean Expression Examples-   Following are examples of dual of Boolean Expressions-   Example-01:   Consensus  theorem is XY + x’z + yz = xy + x’z Dual of Consensus theorem is (x + y)(x’ + z)(y + z) = (x + y)(x’ + z)   Example-02:   Boolean expression is xyz + x’yz’ + y’z = 1 Dual of the above Boolean expression is (x + y + z)(x’ + y + z’)(y’ + z) = 0   Self-Dual Functions-   When a function is equal to its dual, it is called as a Self dual function.   Example-   Consider the function : F (A , B , C) = AB + BC + CA   The dual of this function is- Fd (A , B , C) = (A + B)(B + C)(C + A) = AB + BC + CA   Clearly, F (A , B , C) = Fd (A , B , C) ∴ F (A , B , C) is a self-dual function.   Conditions For Self-Dual Function-   The necessary and sufficient conditions for any function to be a self-dual function are- The function must be a Neutral Function. The function must not contain any mutually exclusive TERMS.   Mutually Exclusive Terms   Consider we have any term X consisting of some VARIABLES. Then, a term obtained by complementing each variable of term X is called as its mutually exclusive term.   Examples-   (ABC , A’B’C’) are mutually exclusive terms. (AB’C , A’BC’) are mutually exclusive terms.   Number of Self-Dual Functions-     Here n = number of Boolean variables in the function.   Explanation-   For a function to be a self-dual function, the function must be a neutral function. For a function to be a neutral function, number of minterms must be equal to number of maxterms. So, we choose half of the terms i.e. 2n / 2 = 2n-1 terms. Now, for each of these terms, we have two choices whether to include it or not in the self-dual function.   So, possible number of self-dual functions = 2 x 2 x 2 x ……. x 2n-1 = 22^(n-1)   Relationship Between Neutral Functions & Self-dual Functions-   Every self-dual function is surely a neutral function. But every neutral function need not be a self-dual function.     Important Property of Self-Dual Functions-   Self-duality is closed under complementation.   Example-   If the function F (A , B , C) = ∑ (0 , 1 , 2 , 4) is a self-dual function. Then, its complement function F’ (A , B , C) = ∑ (3 , 5 , 6 , 7) will also be a self-dual function.   PRACTICE PROBLEM BASED ON SELF-DUAL FUNCTIONS-   Problem-   Consider the following functions- F (A , B , C) = ∑ (0 , 2 , 3) F (A , B , C) = ∑ (0 , 1 , 6 , 7) F (A , B , C) = ∑ (0 , 1 , 2 , 4) F (A , B , C) = ∑ (3 , 5 , 6 , 7)   Which of the above functions are self-dual functions? Only (iii) Only (ii) Only (iii) and (iv) All are self-dual functions   Solution-   Condition-01:   According to condition-01, for a function to be a self-dual function, the function must be a neutral function.   In all the given options, we have functions of 3 variables- A, B and C. So, Neutral function must contain exactly 2n-1 = 23-1 = 4 minterms and 4 maxterms. But Function-(i) contains only 3 minterms. So, it is not a neutral function. Therefore, it can’t be a self-dual function and it gets eliminated. We are now left with three other functions which satisfies condition-01 and are all neutral functions. We will now use 2nd condition to eliminate the incorrect option(s).   Condition-02:   According to condition-02, a self-dual function must not contain mutually exclusive terms. First, let us find which terms are mutually exclusive-   A B C Minterms 0 0 0 0 A’B’C’ 1 0 0 1 A’B’C 2 0 1 0 A’BC’ 3 0 1 1 A’BC 4 1 0 0 AB’C’ 5 1 0 1 AB’C 6 1 1 0 ABC’ 7 1 1 1 ABC   From here, pairs of mutually exclusive terms are (0,7) , (1,6) , (2,5) , (3,4). Mutually exclusive terms are not allowed in self-dual functions. Therefore, terms inside the pairs can not appear together. But terms 0 and 7 appear together in the function-(ii). So, it can not be a self-dual function. But functions (iii) and (iv) do not contain any mutually exclusive terms. Therefore, functions (iii) and (iv) are self-dual functions.   Thus, Option (C) is correct.   NOTE-   Functions (iii) and (iv) are complementary functions. So, if one function is a self-dual function, the other function will also be a self-dual function. This is because self-dual functions are closed under complementation.