# Is this a valid partition?

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.

-
Hmm... $\{S_1,S_2\}$ would be a partition of $S$, but not $S_1 \cup S_2=S$. –  Douglas S. Stones Jan 31 at 0:39
@DouglasS.Stones Can you explain why the union is not $S$? –  Calvin Lin Jan 31 at 0:51
I'm saying the union is $S$, and that's the problem. To illustrate, $\{1\} \cup \{2,3\}=\{1,2,3\}$ is not a partition of $\{1,2,3\}$, but $\{\{1\},\{2,3\}\}$ is. –  Douglas S. Stones Jan 31 at 0:55
@DouglasS.Stones Ah. I caught that too, and was thinking that it's just strange language on OP's part. Though, isn't $S$ a partition of $S$? That's why I felt it would be yes off yes. –  Calvin Lin Jan 31 at 0:57
I'm guessing your answer is right, and it's a bug in the question. (I wouldn't say $S$ is a partition of $S$.) –  Douglas S. Stones Jan 31 at 1:00