Let $S_{n}$ be the set of all n-element sets in some universe.
Given that $R_{k}$ is the relation on $S_{n}$ which is defined as x$R_k$y if and only if the intersection of x and y has at least k elements. We need to find the composition of $R_p$ and $R_q$ where p,q are some natural numbers.
At first I drew a Venn diagram of them and got a system of inequalities, but then I found that the inequalities cannot be solved since the rank is not enough. PS: My method is as following picture:
So I reached someone who has already solved it for help, but it turned out to be that he just guessed the answer so that he cannot explained how to get it.
The answer he told me is $R_{p+q-n}$ where n is the index of $S_n$ as mentioned above.
Could you please give me a hint? I’ve already been stuck on this problem for almost a month.