Calculate how many equivalence relations there are on $\{1,2,3,4,5,6,7 \}$ that include the set $\{(2,2),(1,3),(3,6),(7,5)\}$ and are foreign to the set $\{(1,7),(4,7),(4,3)\}$.
Well I first drew a matrix, highlighted the wanted pairs and crossed the unwanted ones, I realise for example that if we want $(1,3)$ then $(3,1)$ must be included as well. Now partitioning them by hand is extremely tedious and I doubt that's what I'm supposed to do.
I also found out about Bell numbers from here which we don't even cover in this course but I'm not sure how to use it since there are six pairs that must always be excluded and eight that must be included.
How am I supposed to approach this ?
The inclusion condition implies there is an equivalence class $A$ containing $\{1,3,6\}$ and a class $B$ containing $\{5,7\}$. The fact that $1$ and $7$ are not equivalent means $A\not=B$. Furthermore, the fact that $4$ is not equivalent to either $7$ or $3$ means there is a third equivalence class $C$ containing $\{4\}$. The remaining element, $2$, can be in any of these three classes, or could constitute its own class, $D$. Thus there are four different equivalence relations satisfying the two conditions.
Note that the inclusion condition on $(2,2)$ is irrelevant, since equivalence requires each number to be equivalent to itself.