Question

How many different Boolean functions are there of degree \(7\)\({\bf{?}}\)

Short answer

The Boolean function of degree 7 will be

\(340,282,336,920,938,463,463,374,607,431,768,211,456\)

Step-by-step solution

Definition

Product rule: If one event can occur in m ways AND a second event can occur in n ways, then the number of ways that the two events can occur in sequence is then \(m \times n\)

Using the Boolean product

A Boolean function of degree n is a function from \({{\bf{B}}_{\bf{n}}}{\bf{ = }}\left\{ {\left( {{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{, \ldots ,}}{{\bf{x}}_{\bf{n}}}} \right)\mid {{\bf{x}}_{\bf{i}}} \in {\bf{B}}} \right\}\) to \({\bf{B = }}\left\{ {{\bf{0,1}}} \right\}\), \(n = 7\). A Boolean function of degree 7 consists of 7 Boolean variables. Since each Boolean variable can take on 2 values (0 or 1, you can determine the number of 7 tuples for the 7 Boolean variables by using the product rule:\(\underbrace {{\bf{2 \times 2 \times \ldots \times 2}}}_{{\bf{7 times }}}{\bf{ = }}{{\bf{2}}^{\bf{7}}}{\bf{ = 128}}\) 7- tuples. A Boolean function is the assignment of 0 or 1 to each of the 128 7-tuples.

Thus, by the product rule, you can then determine the number of Boolean functions of

degree 7: \(\underbrace {{\bf{2 \times 2 \times \ldots \times 2}}}_{{{\bf{2}}^{\bf{7}}}{\bf{ = 128 times }}}{\bf{ = }}\)

\({2^{{2^7}}} = {2^{128}} = 340,282,336,920,938,463,463,374,607,431,768,211,456\).

Therefore, the Boolean function of degree 7 will be

\(340,282,336,920,938,463,463,374,607,431,768,211,456\)