Number of Equivalence Relations with Specific Cardinality

An equivalence relation $R$ on the set $S$ with $|S| = 10$ corresponds to a partition of $S$ into disjoint equivalence classes of sizes $n_1, n_2, \dots, n_k$ satisfying $\sum_{i=1}^k n_i = 10$ and $|R| = \sum_{i=1}^k n_i^2 = 42$.

The partitions of $10$ satisfying $\sum_{i=1}^k n_i^2 = 42$ are $(6, 2, 1, 1)$ and $(5, 4, 1)$.

1. The number of equivalence relations corresponding to class sizes $(6, 2, 1, 1)$ is: $\frac{10!}{6! \, 2! \, 1! \, 1! \, 2!} = 1260$

2. The number of equivalence relations corresponding to class sizes $(5, 4, 1)$ is: $\frac{10!}{5! \, 4! \, 1!} = 1260$

Thus, the total number of elements in $X$ is $1260 + 1260 = 2520$.