The number of equivalence relations on the set \1,2,3\ is

An equivalence relation corresponds to a partition of the set into classes. The partitions of $\{1,2,3\}$ are: one class of three, three partitions of type $\{a,b\},\{c\}$, and the partition into three singletons. That is $1+3+1=5$. The value $3$ counts only the two-and-one partitions and $8$ is the number of subsets.