Which of the following is NOT true about self-dual function?

Question

Prepp Team · Accepted Answer

In digital logic and Boolean algebra, understanding the characteristics of different types of Boolean functions is crucial. This question focuses on the properties of a self-dual function. A self-dual function possesses unique attributes that distinguish it from other Boolean functions.

Key Concept	Description
Boolean Function	A mathematical function that maps input Boolean values (true/false, 1/0) to a single Boolean output value.
Dual Function (\(F^D\))	Obtained from a Boolean function \(F\) by interchanging logical AND (·) with OR (+), OR (+) with AND (·), 0 with 1, and 1 with 0. Variables are kept as they are.
Self-Dual Function	A Boolean function \(F\) is said to be self-dual if it is equal to its dual, i.e., \(F = F^D\).

Understanding Self-Dual Function Properties

Let's analyze each option to determine which statement is NOT true about a self-dual function.

Self-Dual Function Property 1: The function must be a neutral function.

A neutral function is a Boolean function where the number of minterms (combinations for which the function output is 1) is equal to the number of maxterms (combinations for which the function output is 0).
For a Boolean function of 'n' variables, there are \(2^n\) possible input combinations. A neutral function will have \(2^{n-1}\) minterms and \(2^{n-1}\) maxterms.
If a function \(F\) is self-dual, then \(F = F^D\). This means that for every input combination where \(F\) evaluates to 1, its dual \(F^D\) also evaluates to 1. Since \(F^D\) is obtained by complementing the output of \(F\) when applying complemented inputs (using De Morgan's theorem properties), if \(F\) maps to 1, then the complementary input maps to 0. This implies that the truth table of a self-dual function must have an equal number of 0s and 1s.
Therefore, a self-dual function must indeed be a neutral function. This statement is TRUE.

Self-Dual Function Property 2: The function must be equal to its dual.

This is the fundamental definition of a self-dual function. By definition, a Boolean function \(F\) is self-dual if and only if \(F = F^D\).
This statement is therefore TRUE.

Self-Dual Function Property 3: The function must not contain any mutually exclusive terms.

In Boolean algebra, "mutually exclusive terms" typically refer to product terms (like minterms) that cannot be true simultaneously. For instance, in a sum-of-products (SOP) expression, all minterms are inherently mutually exclusive.
Any Boolean function can be expressed as a sum of its minterms (canonical SOP form), and minterms are, by definition, mutually exclusive.
For example, consider a self-dual function of 2 variables: \(F(A, B) = A\overline{B} + \overline{A}B\). This function is a sum of two minterms, \(A\overline{B}\) and \(\overline{A}B\), which are mutually exclusive.
Therefore, the statement "The function must not contain any mutually exclusive terms" is generally incorrect for any non-trivial Boolean function, including self-dual functions.
While this statement is likely false, let's compare it with option 4 which states a specific numerical relationship.

Self-Dual Function Property 4: The function must contain ‘n’ mutually exclusive terms.

As discussed, a self-dual function of 'n' variables is also a neutral function. This means it has \(2^{n-1}\) minterms (which are mutually exclusive terms) and \(2^{n-1}\) maxterms.
The statement claims that a self-dual function must contain 'n' mutually exclusive terms. For this statement to be true, it would require that the number of minterms, \(2^{n-1}\), is equal to 'n'. Let's check this condition for different values of 'n':
- For \(n=1\): A self-dual function has \(2^{1-1} = 2^0 = 1\) minterm. Here, \(n=1\), so \(2^{n-1} = n\) holds (1=1). Example: \(F(x) = x\). It has 1 minterm (\(x\)).
- For \(n=2\): A self-dual function has \(2^{2-1} = 2^1 = 2\) minterms. Here, \(n=2\), so \(2^{n-1} = n\) holds (2=2). Example: \(F(x_1, x_2) = x_1 \oplus x_2 = x_1\overline{x_2} + \overline{x_1}x_2\). It has 2 minterms.
- For \(n=3\): A self-dual function has \(2^{3-1} = 2^2 = 4\) minterms. Here, \(n=3\), so \(2^{n-1} \ne n\) (4 \ne 3).
- For \(n=4\): A self-dual function has \(2^{4-1} = 2^3 = 8\) minterms. Here, \(n=4\), so \(2^{n-1} \ne n\) (8 \ne 4).
Since the condition \(2^{n-1} = n\) is only true for \(n=1\) and \(n=2\), and not for all values of 'n', the general statement "The function must contain 'n' mutually exclusive terms" is NOT TRUE for all self-dual functions.

Conclusion on Self-Dual Function Properties

Based on the analysis:

The statement "The function must be a neutral function" is TRUE.
The statement "The function must be equal to its dual" is TRUE.
The statement "The function must not contain any mutually exclusive terms" is generally false, as any function can be expressed using mutually exclusive minterms.
The statement "The function must contain ‘n’ mutually exclusive terms" is only true for specific values of 'n' (n=1 and n=2) but not universally true for all 'n'. Therefore, this statement is considered NOT TRUE in the general context of self-dual functions for any number of variables.

The question asks which of the given statements is NOT true. Option 4 presents a condition that is not universally applicable to all self-dual functions.