Say we have the $S$ which is the set of all compositions of n >= 0 with an odd num. parts.
Define $S_1$ to be the set of all compositions of n >= 0 with an odd num. of parts where at least one part is <= 9
Define $S_2$ to be the set of all compositions of n >= 0 with an odd num. of parts where each part is >= 10
Is $S_1$ union $S_2$ a partition of $S$? I can't seem to wrap my head around it...
Yes they are. These sets are complements of each other.
If not all of the elements are greater than or equal to 10, then at least 1 element is less than or equal to 9.
If none of the elements are at most 9, then all of the elements are at least 10.