I call a set **PECULIAR**, if its elements are uncountable, pairwise disjoint subsets of R (the real number system). As for example, the set {(0,1),(3,5),[8,9]\Q},where Q denotes the set of rationals, is peculiar. My first question may have a very trivial answer that I cannot see immediately,that does there exist an uncountable, peculiar set? Next, I define the **STRENGTH** of a peculiar set to be the union of all its elements (which are sets). My question is that what are the subsets of R (if any), that are the strength of some uncountable peculiar set? In case complete specification seems impossible or meaningless, some examples will do.



Uncountability of the "Peculiar" sets: 
Since the OP asked about sets of reals, and since these cannot be amorphous (because no amorphous set admits a linear ordering), let me point out that it is consistent with ZF that there is a Dedekind finite (hence not countable, as in Joel's answer) set of reals that cannot be partitioned into uncountably many many infinite pieces. One model in which this happens is the basic Cohen model, as defined in Section 5.3 of Jech's book "The Axiom of Choice". The relevant Dedekind finite set is the set of Cohen reals explicitly added by the forcing (the set Jech calls $A$). The fact that it cannot be split into uncountably many uncountable sets follows from work of Halpern and Levy ("The Boolean prime ideal theorem does not imply the axiom of choice," Axiomatic Set Theory, Part I, Proc. Symp. Pure Math. XIII, Part I (1971) pp. 83-134).


