I want to count the equivalence relations on the set ${\{1,2,3,4,5,6}\}$ with the conditions that the relation includes the tuples ${(1,2)}$ and ${(2,3)}$ but it does not include the tuple ${(3,4)}$ I tried reducing the problem to this :
The relation should have the tuple ${(1,2)}$ and ${(2,1)}$ ( Mirror principle ), likewise with ${(3,2)}$ and ${(2,3)}$, and it should not include ${(4,3)}$ if it does not include ${(3,4)}$. It should also include ${(2,2)}$ and ${(3,3)}$ and other elements likewise.
Other similar questions were associated with bell number and bijections with set divisions , if we want to follow the similar solution how should our set be divided in order not to include or include the tuples provided?
We will use the division association but we only form divisions that have 1 2 and 3 in the same set and exclude 4 from those . counting the divisions we conclude that 10 ways are possible