The question is:
Let $A = \{1,2,3,4,5,6,7,8\}$. If five integers are selected from $A$, must at least one pair of the integers have a sum of $9$?
The book explains the solution by dividing the number into $4$ disjoint subsets and pointing out that if the numbers in each set are added together, they result in a sum of $9$: $\{1,8\}, \{2,7\}, \{3,6\}, \{4,5\}$.
Thus, if you pick $5$ numbers, you will inevitably find a pair, whose sum will equal to $9$.
I don't get it. When the book asks to select $5$ numbers from $A$, I can pick $\{1,2,3,4,5\}$. Then from there, I can pick two integers like $1$ and $2$, whose sum does not equal to $9$. What am I not getting here?
Thanks for any help.