ISI2014-PCB-A-1a

Question

Let $x=(x_1, x_2, \dots x_n) \in \{0,1\}^n$ By $H(x)$ we mean the number of 1's in $(x_1, x_2, \dots x_n)$. Prove that $H(x) = \frac{1}{2} (n-\Sigma^n_{i=1} (-1)^{x_i})$.

Comments

What do we mean by 'x'? Is it the set of n, n-bit binary numbers? How do we build set x? How do we consider the elements in set x?