Recently i was thinking about reprecentation of ternary conditional operator in programming:
"a ? b : c" means "if (a) then b else c".
Obviously it's equivalent to $(a \land b) \lor (\lnot a \land c)$.
Can we go more and represent any boolen f(x,y,z) as follows?
let $f(x,y,0) = g(x,y)$; and $f(x,y,1) = h(x,y)$;
$f(x,y,z) = z?f(x,y,1):f(x,y,0) = (g(x,y) \land \lnot z)) \lor (h(x,y) \land z) $
Can we go slightly more and get this representaion?
$f(x_1,...,x_n) = \bigvee \limits _{a_1,...a_n = \{0,1\}} {(f(a_1,...a_n) \land (x_1 \leftrightarrow a_1) \land ... \land (x_n \leftrightarrow a_n))} $
So, since $x \leftrightarrow y = (x \land y) \lor (\lnot x \land \lnot y)$, we can representi any *-ary function, given by it's truth table $f(a_1,...a_n)$, as the composition of classical binary functions.
Where i went wrong?