5. The number of non-empty subsets of the set $\{1,2, \cdots, 2016\}$ whose elements sum to an odd number is $\qquad$.
Since the number of subsets containing 1 and the number of subsets not containing 1 in the set $\{1,2, \cdots, 2016\}$ are equal, we can pair subsets that have the same elements except for 1. In each pair, exactly one subset has an odd sum of elements. Additionally, the subset $\{1\}$ is paired with the empty set $\varnothing$, so the number of non-empty subsets with an odd sum of elements is $\frac{1}{2} \cdot 2^{2016}=2^{2015}$.