##### Self Dual function and Neutral function

**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)

**example1:**

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)

**Example-02:**

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.

**Example-**

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.

Example-

**(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

**Explanation-**

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. 2
^{n}/ 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.

therefore,

Possible number of self-dual functions

= 2 x 2 x 2 x ……. x 2^{n-1}

= 2^2^{n-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.

**Example-**

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-

Problem-

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?

(a)Only 3

(b)Only 2

(c)Only 3 and 4

(d)All are self-dual functions

**Solution-**

Condition-01:

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).

Condition-02:

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.

Thus,

Option (C) is correct.

**Note-**

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