Self-Dual Functions | Dual Of Boolean Expression

Dual Of Boolean Expression-

 

To get a dual of any Boolean Expression, replace-

• OR with AND i.e. + with .
• AND with OR i.e. . with +
• 1 with 0
• 0 with 1

 

Dual of Boolean Expression Examples-

 

Following are examples of dual of Boolean Expressions-

 

Example-01:

 

• Consensus  theorem is xy + x’z + yz = xy + x’z
• Dual of Consensus theorem is (x + y)(x’ + z)(y + z) = (x + y)(x’ + z)

 

Example-02:

 

• Boolean expression is xyz + x’yz’ + y’z = 1
• Dual of the above Boolean expression is (x + y + z)(x’ + y + z’)(y’ + z) = 0

 

Self-Dual Functions-

 

When a function is equal to its dual, it is called as a Self dual function.

 

Example-

 

Consider the function : F (A , B , C) = AB + BC + CA

 

The dual of this function is-

F_d (A , B , C)

= (A + B)(B + C)(C + A)

= AB + BC + CA

 

Clearly, F (A , B , C) = F_d (A , B , C)

∴ F (A , B , C) is a self-dual function.

 

Conditions For Self-Dual Function-

 

The necessary and sufficient conditions for any function to be a self-dual function are-

• The function must be a Neutral Function.
• The function must not contain any mutually exclusive terms.

 

Mutually Exclusive Terms   Consider we have any term X consisting of some variables. Then, a term obtained by complementing each variable of term X is called as its mutually exclusive term.   Examples-   (ABC , A’B’C’) are mutually exclusive terms. (AB’C , A’BC’) are mutually exclusive terms.

 

Number of Self-Dual Functions-

 

 

Here n = number of Boolean variables in the function.

 

Explanation-

 

• For a function to be a self-dual function, the function must be a neutral function.
• For a function to be a neutral function, number of minterms must be equal to number of maxterms.
• So, we choose half of the terms i.e. 2ⁿ / 2 = 2^n-1 terms.
• Now, for each of these terms, we have two choices whether to include it or not in the self-dual function.

 

So, possible number of self-dual functions

= 2 x 2 x 2 x ……. x 2^n-1

= 2^{2^(n-1)}

 

Relationship Between Neutral Functions & Self-dual Functions-

 

• Every self-dual function is surely a neutral function.
• But every neutral function need not be a self-dual function.

 

 

Important Property of Self-Dual Functions-

 

Self-duality is closed under complementation.

 

Example-

 

• If the function F (A , B , C) = ∑ (0 , 1 , 2 , 4) is a self-dual function.
• Then, its complement function F’ (A , B , C) = ∑ (3 , 5 , 6 , 7) 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?

1. Only (iii)
2. Only (ii)
3. Only (iii) and (iv)
4. 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.
• So, Neutral function must contain exactly 2^n-1 = 2^3-1 = 4 minterms and 4 maxterms.
• But Function-(i) contains only 3 minterms. So, it is not a neutral function.
• Therefore, it can’t be a self-dual function and it gets eliminated.
• We are now left with three other functions which satisfies condition-01 and are all neutral functions.
• We will now use 2^nd 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-

 

	A	B	C	Minterms
0	0	0	0	A’B’C’
1	0	0	1	A’B’C
2	0	1	0	A’BC’
3	0	1	1	A’BC
4	1	0	0	AB’C’
5	1	0	1	AB’C
6	1	1	0	ABC’
7	1	1	1	ABC

 

• From here, pairs of mutually exclusive terms are (0,7) , (1,6) , (2,5) , (3,4).
• 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-(ii).
• So, it can not be a self-dual function.
• But functions (iii) and (iv) do not contain any mutually exclusive terms.
• Therefore, functions (iii) and (iv) are self-dual functions.

 

Thus, Option (C) is correct.

 

NOTE-

 

• Functions (iii) and (iv) are complementary functions.
• So, if one function is a self-dual function, the other function will also be a self-dual function.
• This is because self-dual functions are closed under complementation.

 

To gain better understanding about Self-Dual Functions,

Watch this Video Lecture

 

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