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Let $n$ be any positive integer and $\mathcal{F}$ be a family of subsets of$[n]$.  A family $\mathcal{F}'$ is said to be $D$-\emph{secting} for $\mathcal{F}$ iffor every $A \in \mathcal{F}$, there exists a subset $A' \in \mathcal{F}'$ suchthat $|A \cap A'| - |A \cap ([n] \setminus A')|=i$, where $i \in D$, $D\subseteq \{-n,-n+1,\ldots,0,\ldots,n\}$.  A $D$-\emph{secting} family $\mathcal{F}'$ of $\mathcal{F}$, where$D=\{-1,0,1\}$, is a \emph{bisecting} family ensuring the existence of a subset$A' \in \mathcal{F}'$ such that $|A \cap A'| \in \{\lceil\frac{|A|}{2}\rceil,\lfloor \frac{|A|}{2}\rfloor\}$, for each $A \in\mathcal{F}$.  In this paper, we study $D$-secting families for $\mathcal{F}$ withrestrictions on $D$, and the cardinalities of $\mathcal{F}$ and the subsets of$\mathcal{F}$.

Bisecting and D-secting families for set systems