I would like to use set builder notation for the following text:
The functions maps a series of numbers (L) (assume all positive) to a list of sub-partitions (P). Each sub-partition contains some elements of the L. The union of all sub-partitions is L and there is no common element between two sub-partitions. Also each sub-partition has at least one element.
$f: L \rightarrow P$
$P=\{p: |p_i| \geq 1, \cup p_i=U, \cap p_i= \phi\}$
$p_i=[L_{ab}], 1 \leq a,b \leq m$ (where $m$ is the number of elements in L)
For example, if $L=\{2,4,5,10\}$, then $p_1=[2,4,5]$ (or $L_{13}$). Therefore $P=\{(2,4,5),(10)\}$. I would like to know if the notations are correct in the set-builder rules and reader understands that. If you think another representation is more meaningful, please let me know.