Hi ever body,
could you please let me know how I can prove the following problem.

for some $\displaystyle $A,B \subseteq [n]$ $ such that for any $\displaystyle $x\in\{0,1\}$$ we have two function s.t. $\displaystyle $r(x)=(-1)^{\varSigma_{i \in A}x_i}$$ and $\displaystyle $t(x)=(-1)^{\varSigma_{i \in B}x_i}$$ and suppose that $\displaystyle $f(x)$$ is function depends on r and
$\displaystyle f^\prime(x) = \left \{\begin{array}{l}+\infty \hspace*{0.8cm} if \hspace*{0.3cm} wt(x) \geq n/2 + 4\delta \\ f(x) \hspace*{0.8cm} if \hspace*{0.3cm} n/2 - 4\delta \leq wt(x) \leq n/2 + 4\delta \\-\infty \hspace*{0.8cm} if \hspace*{0.3cm} wt(x) \leq n/2 - 4\delta \end{array} \right . \begin{array}{l} \end{array} $

$\displaystyle wt(x)$ is the bitstrings hamming weight of size n $\displaystyle \in \{0,1\}^n$

How we could prove that if $\displaystyle $A \cap B = \emptyset $$ then $\displaystyle $f^{\prime}$$ is monotone.