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Number of Self-Dual Function

Before seeing this chapter please follows previous chapters: 

Neutral and Mutually Exclusive Function,  Self Dual Expression

As we know if a function satisfies two conditions then this function is called the self-dual function.

The two conditions are:

(i) It is neutral (no. of minterms = no. of maxterms).

(ii) The function does not contain two mutually exclusive terms.

See Neutral and Mutually Exclusive Function

Now, we can calculate with 3 variables how many self-dual functions are possible.

Let consider a truth table with 3 variables:

As we know with 3 variables, maximum possible minterms or maxterms are 

For details please follow: 

Number of minterm, maxterm, and Logical Expression

Now among maximum 8 possible minterms or maxterms, how many are mutually exclusive terms:

(0, 7), (1, 6), (2, 5) and (3, 4)

So, 4 pairs (half of maximum possible minterms and maxterms (8)) mutually exclusive terms are possible.

Now as we know a self-dual function does not contain two mutually exclusive terms.

So, if we want to form a self-dual function, we have to take any of one term from a pair of mutually exclusive terms.

Like we can take terms to form a self-dual function:

(0, 7) – Either 0 or 7

(1, 6) – Either 1 or 6

(2, 5) – Either 2 or 5

(3, 4) – Either 3 or 4

For example: 

Now, how many self-dual functions are possible? 

Every mutually exclusive pair has two choices to form a self-dual function either 0 or 7 same way. 

So, the total self-dual function is possible with 3 variables

For n variable how many self-dual functions possible:

So, the number of mutually exclusive pairs are exactly half of the maximum possible minterms or maxterms i.e. 

Now as we know from above every mutually exclusive pair has two choices to form a self-dual function.

So, total self-dual functions are possible 

For n variables, maximum numbers of possible Self-dual expressions are: