How to test one directional correlations? Let's say, I have two math problems (puzzles), like:
ProblemA) Solve $ax+b=0$ and
ProblemB) Solve $ax^2+bx+c=0$
and would like to check how similar they are.
I ask 1000 people to solve these two problems, and learn that k people solved both, m - solved only P1, n - solved only P2 and l - solved neither.
Now I can calculate correlation coefficient, which would be:
$\sigma=\frac{kl-mn}{\sqrt{(k+m)(n+l)(k+n)(m+l)}}$
Now, if $\sigma \approx 1$, then the problems are almost identical, and if $\sigma \approx 0$, then the problems are different and independent on each other.  
My issue with that is $\sigma$ tests only the hypothesis that the problems are identical.
But problems, like in the example above, can be different and still similar in the sense that everyone who solves P2 can solve P1 (though not everyone who solves P1 can solve P2). In this case $\sigma$ doesn't show clear correlation (it can be far different from 1), and, most importantly it doesn't show, which problem is the "consequence" of which (if I swap problems $\sigma$ won't change at all).
I wonder is there a criteria, which can show how probable each of the following cases are:
1) problems are independent;
2) all people who solve P1 can solve P2, but not vice versa;
3) all people who solve P2 can solve P1, but not vice versa;
4) problems are identical and everyone who can solve P1 can solve P2 and vice versa.
 A: Suppose $A$ is the set of people who solve one problem, and $B$ is the set of people who solve another problem. 

Define
$$ D(A, B) \triangleq \frac{ \left| A \setminus B\right| - \left| B \setminus A\right|  }{\text{max}\left(\left| A \setminus B\right|, \left| B \setminus A\right|\right)} \; (1)
$$
as the degree of asymmetrical inclusiveness, and
$$
T(A, B) \triangleq \frac{\left| A \bigcap B \right|}{\text{min}(|A|, |B|)}
\; (2)
$$
as the degree of intersection.
In the above diagram, $D(A, B)$ is the difference between the yellow and pink areas divided by the largest among them, and $T(A, B)$ is the blue area divided by the area of the smaller circle.
At the limit of $A$ and $B$ overlapping each other exactly, $D(A, B) = \frac{0}{0}$ which we'll define as 0 for this case. There are other cases for which the fractions are $\frac{0}{0}$, and we'll take them as 0 as well.
Following that, it's easy to see that
$$
-1 \leq D(A, B) \leq 1 \; (3),
$$
and 
$$
-1 \leq T(A, B) \leq 1 \; (4).
$$

Now define the directional correlation as
$$\sigma_{\Rightarrow}(A, B) \triangleq D(A, B) T(A, B).
$$
It has the following properties:
1) $A = B \Rightarrow \sigma_{\Rightarrow}(A, B) = 0$
In this case, $D(A, B) = 0$.
2) $A \bigcap B = \emptyset \Rightarrow  \sigma_{\Rightarrow}(A, B) = 0$
In this case, $T(A, B) = 0$.
3) $A \subsetneq B \Rightarrow \sigma_{\Rightarrow}(A, B) = 1$
In this case, $D(A, B) = 1 = T(A, B)$.
4) $B \subsetneq A \Rightarrow \sigma_{\Rightarrow}(A, B) = -1$
In this case, $D(A, B) = 1 = -T(A, B)$.
5) $\forall_{A, B}-1 \leq \sigma_{\Rightarrow}(A, B) \leq 1$
This follows from (3) and (4).

