# how many boolean functions exist that satisfy the condition

how many boolean functions exist that satisfy the condition : $$Not[f(x_1,x_2,x_3,....,x_n)]$$ = $$f(Not(x_1), Not(x_2),...,Not(x_n))$$

• Are the $x_i$ fixed? – Kai Nov 8 at 21:48
• I'm pretty sure they aren't fixed – Gabi G Nov 8 at 21:55
• It's a counting exercise. Realise that whenever you choose $f(x_1,\ldots , x_n) = b$ for fixed $x_1,\ldots , x_n$, you fix $f(\neg x_1,\ldots , \neg x_n) = \neg b$. So how many choices do you have to make to fully determine $f$? – Kai Nov 8 at 22:06
• @Kai This is an answer to the question. Post it as such? – Yuval Filmus Nov 8 at 22:18
• aww, did I give away too much? – Kai Nov 9 at 10:43