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:
- AND operator of a given function is changed to OR operator and vice-versa.
- 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 :
- The given function is neutral i.e., (number of minterms is equal to the number of max terms).For more about minterm and max term (see Canonical and standard Form).
- The function does not contain two mutually exclusive terms.
Note: Mutually exclusive term of XYZ is (X’Y’Z’) i.e, complement of XYZ. So, two mutually exclusive terms are the complement of each other.
In the above table, Mutually exclusive terms are:
(0,7), (1,6), (2,5), (3,4)
- 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
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
- Every Self-dual function is neutral but every neutral function is not Self-dual.
- Self-duality is closed under complement i.e, the complement of a Self-dual function is also Self-dual.
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