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

Question

$$\sum(-1)^l\tag{1}$$

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

Do you mean $(1)$ to be the $\sum{(-1)^\ell:\ell\in F}$? – Brian M. Scott Jun 10 '13 at 22:14 — Brian M. Scott, Jun 10 '13 at 22:14

score 0 · Accepted Answer · answered Jun 10 '13 at 22:21

0

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 $0$?

answered Jun 10 '13 at 22:21

Brian M. Scott

616,228

when #F${even}$ = #F${odd}$ – Felix Montoya Jun 11 '13 at 16:04
@Felix: Exactly. And that’s the only time. Now just use the fact that $#F=#F_{\text{even}}+#F_{\text{odd}}$ to do (a), and then use (a) to do (b). – Brian M. Scott Jun 11 '13 at 18:09

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

1 Answers1