Let $\oplus$ mean the exclusive OR operator, I have this boolean formula $F$ which takes inputs $x$, where given $|x|=n$, we have:
$$x:\{a_1,a_2,a_3,...,a_{n/2},b_1,b_2,b_3,...,b_{n/2}\}$$
and the $a_i \in \{0,1\}$ and $b_i \in \{0,1\}$, i.e. $a_i$ and $b_i$ are boolean variables for $1 \leq i \leq n/2$.
Now, $F$ is defined as:
$$ F = \oplus_i^{n/2}(a_i \wedge b_i) $$
I read from my textbook that given this input $x'$ to $F$, where for ease of notation, $1^2=11,1^3=111,$etc. (and the same for $0$) and $\delta_i \in \{0,1\}$ for $1 \leq i \leq n/4$:
$$ x' = 1^{n/4}\delta_1,\delta_2,...,\delta_{n/4}0^{n/2} $$
We have that $F$ always evaluates to $0$, for whatever values the string $\delta_1,\delta_2,...,\delta_{n/4}$ may have ... but I couldn't figure out why. Any help?
From the definition of $x'$, we have $b_k=0$ for every $k$. It follows that $a_k\wedge b_k=0$ for all $k$, and hence $F=\bigoplus_{k=1}^{\frac{n}{2}}0=0$.