Question

State, whether each of the following sets is a finite set or an infinite set:

$\{x:x=\frac{n-2}{n+1},n\in W\}$

Answer 1

Step: Check whether the given set is a finite set or an infinite set
We know that,
Finite sets are the sets having a finite/countable number of members.

Infinite sets are the sets having an infinite/uncountable number of members.

Given: $\{x:x=\frac{n-2}{n+1},n\in W\}$

For $n\in W,n=0,1,2,.....$

So, there are an infinite number of values $n$.

Corresponding values of $x$ are

$n=0\Rightarrow x=\frac{0-2}{0+1}=-2$
$n=1\Rightarrow x=\frac{1-2}{1+1}=-\frac{1}{2}$
$n=2\Rightarrow x=\frac{2-2}{2+1}=\frac{0}{3}=0$ and so on.

So, for infinite value of $n$, we will get infinite values of $x$.

Therefore, the given set is an infinite set.

Hence, the set $\{x:x=\frac{n-2}{n+1},n\in W\}$ is an infinite set.