8 The number of non-empty subsets of the set $\{1,2,3, \cdots, 2009\}$ whose elements sum to an odd number is $\qquad$ .
$82^{2008}$ Hint: Method 1: Let
$$
f(x)=(1+x)\left(1+x^{2}\right)\left(1+x^{3}\right) \cdots\left(1+x^{2009}\right),
$$

then the answer required in the problem is the sum of the coefficients of the odd powers of $x$ in the expansion of $f(x)$. Therefore, the answer is
$$
\frac{f(1)-f(-1)}{2}=\frac{2^{2009}-0}{2}=2^{2008} .
$$

Method 2: Discuss the subsets $A$ of the set $\{1,2,3, \cdots, 2009\}$ that do not contain 2009. If the sum of the elements in $A$ is even, then add 2009; otherwise, do not add it. Hence, there are $2^{2008}$ non-empty subsets with an odd sum of elements.