2022 AMC 10B Problem 14

Suppose that S is a subset of \left\{ 1, 2, 3, \ldots , 25 \right\} such that the sum of any two (not necessarily distinct) elements of S is never an element of S. What is the maximum number of elements S may contain?

\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.