Neutral Functions

Calculating Number of Minterms and Maxterms-

 

For any function consisting of n Boolean variables,

Number of Minterms possible = Number of Maxterms possible = 2n

 

Example-01:

 

For any function consisting of 2 Boolean variables A and B, we have the following 22 =  4 minterms and 4 maxterms possible-

 

ABMintermsMaxterms
00A’B’A + B
01A’BA + B’
10AB’A’ + B
11ABA’ + B’

 

Example-02:

 

For any function consisting of 3 Boolean variables A, B and C, we have the following 23 =  8 minterms and 8 maxterms possible-

 

ABCMintermsMaxterms
000A’B’C’A + B + C
001A’B’CA + B + C’
010A’BC’A + B’ + C
011A’BCA + B’ + C’
100AB’C’A’ + B + C
101AB’CA’ + B + C’
110ABC’A’ + B’ + C
111ABCA’ + B’ + C’

 

Neutral Functions-

 

A neutral function is a function for which-

Number of Minterms = Number of Maxterms

 

Example-

 

The following function is an example of a neutral function since it has equal number of minterms and maxterms-

f( A, B) = A ⊕ B 

 

ABA ⊕ B
000
011
101
110

 

Number of Neutral Functions possible-

 

where n = number of Boolean variables in the function

 

Explanation-

 

We know,

For n variables,

Total number of terms possible = Number of combinations of n variables = 2n

Since, maximum number of terms possible = 2n, so we choose half of the terms i.e 2n / 2 = 2n-1 number of terms and assign them the output logic ‘1’ and rest half of the terms are assigned ‘0’.

Thus,

Number of Neutral functions possible with n Boolean variables = C ( 2n , 2n-1 )

 

PRACTICE PROBLEM BASED ON NEUTRAL FUNCTION-

 

Problem-

 

Consider any function consisting of 2 Boolean variables A and B-

  1. Calculate the total number of neutral functions that are possible.
  2. Write all the neutral functions.

 

Solution-

 

We know,

Number of Neutral functions possible with n Boolean variables = C ( 2n , 2n-1 )

 

Here, n = 2. Substituting the values, we get-

Number of Neutral functions possible with 2 Boolean variables

= C ( 22 , 22-1 )

= C ( 4 , 2 )

= 6

Thus,

Total 6 Boolean functions are possible with 2 Boolean variables.

 

Those 6 Boolean Functions are-

  1. f ( A , B ) = A
  2. f ( A , B ) = A’
  3. f ( A , B ) = B
  4. f ( A , B ) = B’
  5. f ( A , B ) = A ⊙ B
  6. f ( A , B ) = A ⊕ B

 

VariablesBoolean Functions
ABAA’BB’A ⊙ BA ⊕ B
00010110
01011001
10100101
11101010

 

The above table clearly shows that for each function, number of minterms = number of maxterms. So, they all are neutral functions.

 

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Summary
Neutral Functions
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Neutral Functions
Description
A neutral function is a function for which- Number of Minterms = Number of Maxterms. Number of Neutral functions possible with n Boolean variables = C ( 2n , 2n-1 )
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Gate Vidyalay
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