Dual of a Boolean Expression-
To get a dual of any Boolean Expression, replace-
AND with OR i.e. . with +
OR with AND i.e. + with.
0 with 1 and 1 with 0
Consensus theorem says if in the expression
1. Three variables are there
2.One of the variable must contain complement term
3.every variable paired with every other variable
Resultant form=the variable which is in complement form will be in result(minimized form)
Ex . AB + A'C + BC =AB+A'C
We know, Consensus theorem is-
AB + A'C + BC =AB+A'C
The dual of Consensus theorem is-
(A + B)(A’ + C)(B + C) = (A + B)(A’ + C)
Consider any Boolean Expression-
xyz + x’yz’ + y’z = 1
The dual of this Boolean expression is-
(x + y + z)(x’ + y + z’)(y’ + z) = 0
SELF DUAL FUNCTION:
- When a function is equal to its dual, it is called as a self dual function.
- Self-duality is closed under complementation.
Consider the function-
F (X, Y ,Z) = XY + YZ + ZX
The dual of this function is-
Fd (X , Y , Z)
= (X+ Y)(Y + Z)(Z + X)
= XY + YZ + ZX
Since, F (X , Y , Z) = Fd (X , Y , Z)
∴ F (X , Y , Z) is a self-dual function.
Conditions for a self-dual function-
The necessary and sufficient conditions for a function to be a self-dual function are-
- The function must not contain mutually exclusive terms.
- The function must be a Neutral function.
What is the meaning of mutually exclusive terms?
Consider we have any term 'A' consisting of some variables, then a term obtained by complementing each variable of term 'A' is called as its mutually exclusive term.
(XYZ , X’Y’Z’) are mutually exclusive terms.
(XY’Z , X’YZ’) are mutually exclusive terms.
Number of Self-dual Functions-
where 'n' is the number of variables in boolean function
There are certain condition , for a function to be a self-dual function
- The function must be a neutral function.
- number of minterms must be equal to number of maxterms for a function to be neutral
- therefore, we choose half of the terms i.e. 2n / 2 = 2^n-1 terms.
- For each of these terms, we have two choices whether to include it or not in the self-dual function.
Possible number of self-dual functions
= 2 x 2 x 2 x ……. x 2n-1
Relationship between Neutral Functions and Self-dual Functions-
- Every self-dual function is surely a neutral function but every neutral function need not be a self-dual function.
If the function F (A , B , C) = ∑ (3 , 5 , 6 , 7) is a self-dual function, then-
Its complement function F' (A , B , C) = ∑ (0,1,2,4) will also be a self-dual function.
PRACTICE PROBLEM BASED ON SELF-DUAL FUNCTIONS-
Consider the following functions-
(1)F (A , B , C) = ∑ (0 , 2 , 3)
(2)F (A , B , C) = ∑ (0 , 1 , 6 , 7)
(3)F (A , B , C) = ∑ (0 , 1 , 2 , 4)
(4)F (A , B , C) = ∑ (3 , 5 , 6 , 7)
Which of the above functions are self-dual functions?
(c)Only 3 and 4
(d)All are self-dual functions
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.
For a function with 3 Boolean variables, neutral function must contain exactly = 2^3-1 = 4 minterms and 4 maxterms.
But, Function-(i) contains only 3 minterms. So, it is not a neutral function and therefore it can’t be a self-dual function. So, Function-(i) 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).
According to condition-02, a self-dual function must not contain mutually exclusive terms.
First, let us find which terms are mutually exclusive-
(0,7) , (1,6) , (2,5) , (3,4) pairs of mutually exclusive terms. since 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-(2). So, it can not be a self-dual function.
But functions (3) and (4) do not contain any mutually exclusive terms.
∴ Functions (3) and (4) are self-dual functions.
Option (C) is correct.
Functions (3) and (4) are complementary functions. So, if one function is a self-dual function, the other function will also be a self-dual function because self-dual functions are closed under complementation