Self Dual functions in Digital Logic

A function is said to be Self dual if and only if its dual is equivalent to the given function, i.e., if a given function is f(X, Y, Z) = (XY + YZ + ZX) then its dual is, fd(X, Y, Z) = (X + Y).(Y + Z).(Z + X) (fd = dual of the given function) = (XY + YZ + ZX), it is equivalent to the given function. So function is self dual. 

In a dual function: 

1. AND operator of a given function is changed to OR operator and vice-versa.
2. A constant 1 (or true) of a given function is changed to a constant 0 (or false) and vice-versa.

A Switching function or Boolean function is said to be Self-dual if:  

1. The given function is neutral i.e., (the number of minterms is equal to the number of max terms). For more about minterm and max term (see Canonical and Standard Form).
2. The function does not contain two mutually exclusive terms.

Note: Mutually exclusive term of XYZ is (X'Y'Z') i.e, the complement of XYZ. So, two mutually exclusive terms are the complement each other. 

Example: 

SL NO.	X	Y	Z
0	0	0	0
1	0	0	1
2	0	1	0
3	0	1	1
4	1	0	0
5	1	0	1
6	1	1	0
7	1	1	1

In the above table, Mutually exclusive terms are:  

(0,7), (1,6), (2,5), (3,4) 

Explanation:  

• Complement of (000) i.e, 0 is (111) i.e, 7 so, (0, 7 are mutually exclusive to each other.)
• Complement of (001) i.e, 1 is (110) i.e, 6 so, (1, 6 are mutually exclusive to each other.)
• Complement of (010) i.e, 2 is (101) i.e, 5 so, (2, 5 are mutually exclusive to each other.)
• Complement of (011) i.e, 3 is (100) i.e, 4 so, (3, 4 are mutually exclusive to each other.)

Now, let us check the number of Self-dual functions possible for a given function. 
Let, a function has n variables then,

 Number of pairs possible = 2n/2 = 2(n-1)

Therefore, the number of Self-dual functions is possible with n variables  

= 22^(n-1) 

There are 2 possibilities for each pair. 

Example: What is the total number of self-duals of a function which has 3 variables X, Y, and Z?  

= 22^(3-1) 
= 22^2
= 24
= 16 

Note:  

1. Every Self-dual function is neutral but every neutral function is not Self-dual.
2. Self-duality is closed under complement i.e, the complement of a Self-dual function is also Self-dual.