  For positive integers $n$ and $r$ such that $r \leq \lfloor n/2\rfloor$, let
$X$ be a set of $n$ elements and let $\binom{X}{r}$ be the family of all
$r$-subsets of $X$. Two sub-families $\mathcal{A}$ and $\mathcal{B}$ of
$\binom{X}{r}$ are called cross intersecting if $A \cap B \neq \emptyset$ for
all $A \in \mathcal{A}$ and $B \in \mathcal{B}$. One of main tools in the study
of extremal set theory, and cross intersecting families in particular, is
compression operation. In this paper, we give an example of cross intersecting
families $\mathcal{A}$ and $\mathcal{B}$ that the compression operation is not
applicable when $\mathcal{A}$ and $\mathcal{B}$ are disjoint. We develop new
technique to prove that, for disjoint cross intersecting families $\mathcal{A}$
and $\mathcal{B}$ of $\binom{X}{r}$, $|\mathcal{A}| + |\mathcal{B}| \leq
\binom{n}{r} - \binom{l}{p}$ where $n = 2r + l$ and $p = min\{r, \lceil
\frac{l}{2}\rceil\}$. This bound is asymptotically sharp.
