## Monotonicity

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
t
$\displaystyle f^\prime(x) = \left \{\begin{array}{l}+\infty \hspace*{0.8cm} if \hspace*{0.3cm} wt(x) \geq n/2 + 4\delta \\ f(x) \hspace*{0.8cm} if \hspace*{0.3cm} n/2 - 4\delta \leq wt(x) \leq n/2 + 4\delta \\-\infty \hspace*{0.8cm} if \hspace*{0.3cm} wt(x) \leq n/2 - 4\delta \end{array} \right . \begin{array}{l} \end{array}$

$\displaystyle wt(x)$ 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.

Cheers,