An equivalence relation on a finite set is uniquely determined by its partition into equivalence classes. Counting the number of ways to partition the set ${1, 2, 3}$:

1. Three blocks: Each element in its own block. There is only one way: ${{1}, {2}, {3}}$.

2. Two blocks: We can have ${{1, 2}, {3}}$, ${{1, 3}, {2}}$, or ${{2, 3}, {1}}$. There are 3 ways.

3. One block: All elements together. There is only one way: ${{1, 2, 3}}$.

In total, there are $1 + 3 + 1 = 5$ distinct partitions, which means there are 5 equivalence relations on the set ${1, 2, 3}$.