To express formula for $f_3$ via $f_2$ .

Question

Given:

$f_3 = f(x_1, x_2, x_3) = x_1 \equiv x_2 \equiv x_3$

$f_2 = f(x_1, x_2) = x_1 \oplus x_2$

To express formula for $f_3$ via $f_2$ .

I supposed

$x_1 \equiv x_2 = \overline {x_1 \oplus x_2} = \overline f_2$

$\overline x = x_1 \oplus 1 = f(x, 1)$

$f_3 = f_2(f_2(x_1, x_2), 1) \equiv x_3 = f_2(f_2(f_2(f_2(x_1, x_2), 1), x_3), 1)$

But this is not a correct decision

Any ideas?

Answer 1

Note that a generalized $\equiv$ is true if and only if an even number of its terms are false, while a generalized $\oplus$ is true if and only if an odd number of its terms are true. With three terms, that is in exactly the same cases. Hence:

$$x_1 \equiv x_2 \equiv x_3 \Leftrightarrow x_1 \oplus x_2 \oplus x_3$$

Or, in terms of your functions:

$$f_3(x_1,x_2,x_3) = f_2(x_1,f_2(x_2,x_3))$$