can real line be written as a disjoint unions of set with cardinality 5 Can the real line be written as a disjoint union of sets with cardinality 5?
I tried using to write it as a set of sets.. but I didn't end up to a good position.

If we allow the axiom of Choice, then one can show that the cardinality of $\mathbb R \times 5$ is equal to the cardinality of $\mathbb R$.
Now, $card(\mathbb R \times 5)$ is the same as $card(\sum_{i \in \mathbb R}5)  $. This indicates that there exists a correspondence between $\mathbb R$ and $\mathbb R$ disjoint sets of cardinality $5$. 
Using this correspondence we can thus find an infinite collection of sets of cardinality 5, whose union is $\mathbb R$ and pairwise disjoint.
References: Naive Set Theory, by P. Halmos.
