Hardness of approximating Minimum Cardinality Exact Cover
Suppose that we are given an instance of EXACT 3-COVER (in which every set contains 3 elements) with $m$ sets on $n$ elements; EXACT 3-COVER is known to be NP-complete. Create a new instance in which each element is replicated $t$ times, each of the original $m$ sets contains each copy of each element, and additionally all $tn$ singletons are available. When the original EXACT 3-COVER instance is a YES instance, the new instance can be exactly covered using at most $m$ sets. When the original EXACT 3-COVER is a NO instance, every exact cover of the new instance requires at least $t$ sets (since every disjoint union of the original sets misses at least one original element, and so requires $t$ singletons in the new instance).

Now let us choose $t = mn^\theta$, for some constant $\theta$. The new instance contains $N = mn^{1+\theta} \leq n^{4+\theta}$ elements, and has a gap of $n^\theta \geq N^{\theta/(4+\theta)}$ between the YES and NO instances. As $\theta \to \infty$, the ratio $\theta/(4+\theta)$ approaches $1$. We conclude that your problem is NP-hard to approximate to within $n^{1-\epsilon}$ for any $\epsilon &gt; 0$. In contrast, there is a trivial $n$-approximation algorithm, which just takes all the singletons.