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. 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.

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-

1. 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