Q:

Let $F$ be a finite subset of the natural numbers and consider the sum

$$\sum(-1)^l\tag{1}$$
Define $F_{\text{even}} = \{n \in F : n\text{ is even}\}$  and  $F_{\text{odd}} = \{n \in F : n\text{ is odd}\}$
(a) Suppose that $\#F$ is odd. Show that $\#F_{\text{even}}\ne \#F_{\text{odd}}$
(b) Suppose that $\#F$ is odd. Show that $(1)$ is nonzero

A:

HINT: $(-1)^\ell$ is $1$ when $\ell$ is even and $-1$ when $\ell$ is odd. When you take the sum of all $(-1)^\ell$ with $\ell\in F$, you’re adding one for each even number in $F$ and subtracting $1$ for each odd number. Under what circumstances can the result of that operation possibly be $0$?

