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
Example01:
 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)
Example02:
 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
SelfDual 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
F_{d} (A , B , C)
= (A + B)(B + C)(C + A)
= AB + BC + CA
Clearly, F (A , B , C) = F_{d} (A , B , C)
∴ F (A , B , C) is a selfdual function.
Conditions For SelfDual Function
The necessary and sufficient conditions for any function to be a selfdual 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

Number of SelfDual Functions
Here n = number of Boolean variables in the function.
Explanation
 For a function to be a selfdual 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. 2^{n} / 2 = 2^{n1 }terms.
 Now, for each of these terms, we have two choices whether to include it or not in the selfdual function.
So, possible number of selfdual functions
= 2 x 2 x 2 x ……. x 2^{n1}
= 2^{2}^{^(n1)}
Relationship Between Neutral Functions & Selfdual Functions
 Every selfdual function is surely a neutral function.
 But every neutral function need not be a selfdual function.
Important Property of SelfDual Functions
Selfduality is closed under complementation. 
Example
 If the function F (A , B , C) = ∑ (0 , 1 , 2 , 4) is a selfdual function.
 Then, its complement function F’ (A , B , C) = ∑ (3 , 5 , 6 , 7) will also be a selfdual function.
PRACTICE PROBLEM BASED ON SELFDUAL 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 selfdual functions?
 Only (iii)
 Only (ii)
 Only (iii) and (iv)
 All are selfdual functions
Solution
Condition01:
According to condition01, for a function to be a selfdual 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 2^{n1} = 2^{31} = 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 selfdual function and it gets eliminated.
 We are now left with three other functions which satisfies condition01 and are all neutral functions.
 We will now use 2^{nd} condition to eliminate the incorrect option(s).
Condition02:
According to condition02, a selfdual 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 selfdual 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 selfdual function.
 But functions (iii) and (iv) do not contain any mutually exclusive terms.
 Therefore, functions (iii) and (iv) are selfdual functions.
Thus, Option (C) is correct.
NOTE
 Functions (iii) and (iv) are complementary functions.
 So, if one function is a selfdual function, the other function will also be a selfdual function.
 This is because selfdual functions are closed under complementation.
To gain better understanding about SelfDual Functions,
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