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., (number of min terms is equal to the number of max terms).For more about min term 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, compliment of XYZ. So, two mutually exclusive terms are compliment of 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:

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

Now, lets check 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, number of Self dual functions possible with n variables

`= 22(n-1) `

There are 2 possibilities for each pair.

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

```= 22(3-1)
= 222
= 24
= 16 ```

Note:

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

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