How to print all pairs of ID numbers such that the corresponding people don't belong to the same socieities
Firstly $\Pi_{i_1, i_2}(\sigma_{i_1\leq i_2}{(\rho_{c_1(i_1, s)}People\times \rho_{c_2(i_2, s)}People)})$ gives all possible permutations of ID pairs in the form $(a, b)$ with $a\leq b$.

By $\Pi_{i_1, i_2}(\sigma_{i_1\leq i_2}{(\rho_{c_1(i_1, s)}People\bowtie \rho_{c_2(i_2, s)}People)}$, we can gain all pairs of people who attend a same society. 

So the answer is 

$$\Pi_{i_1, i_2}\Big(\sigma_{i_1\leq i_2}{(\rho_{c_1(i_1,s)}People\times\rho_{c_2(i_2, s)}People)}-\sigma_{i_1\leq i_2}{(\rho_{c_1(i_1, s)}People\bowtie \rho_{c_2(i_2, s)}People)}\Big)$$