Let A=\1,2,3,4,5\. The number of equivalence relations on A that contain (1,2) but do not contain (1,3) is

An equivalence relation on $A$ corresponds to a partition of $A$. Since $(1,2)$ is present, $1$ and $2$ lie in the same block, so treat them as one object $X$. Equivalence relations containing $(1,2)$ then correspond to partitions of $\{X,3,4,5\}$, which number the Bell number $B_4=15$. Those that also contain $(1,3)$ put $X$ and $3$ in one block, i.e. partitions of a $3$-object set, $B_3=5$. Hence the required number is $15-5=10$.