AMC 12 2025a Problem 15

A set of numbers is called sum-free if whenever $x$ and $y$ are (not necessarily distinct) elements of the set, $x+y$ is not an element of the set. For example, $\{1,4,6\}$ and the empty set are sum-free, but $\{2,4,5\}$ is not. What is the greatest possible number of elements in a sum-free subset of $\{1,2,3,...,20\}$?

$\textbf{(A) } 8 \textbf{(B) } 9 \textbf{(C) } 10 \textbf{(D) } 11 \textbf{(E) } 12$