Let $N$ be the set of all natural numbers, $Z$ be the set of all integers and $\sigma: N \rightarrow Z$ be…
Let $N$ be the set of all natural numbers, $Z$ be the set of all integers and $\sigma: N \rightarrow Z$ be defined by
$
\sigma(n)=\left\{\begin{array}{ccc}
\frac{n}{2}, & \text { if } & n \text { is even } \\
-\frac{n-1}{2}, & \text { if } & n \text { is odd }
\end{array}\right. \text {. Then, }
$
$\sigma$ is onto but not one-one
$\sigma$ is one-one but not onto
$\sigma$ is neither one-one nor onto
$\sigma$ is one-one and onto
Solution
Given that,
$
\sigma(n)=\left\{\begin{array}{cl}
\frac{n}{2} & \text { if } n \text { is even } \\
-\frac{(n-1)}{2} & \text { if } n \text { is odd }
\end{array}\right.
$
Case-I : If $n$ is even
$
\sigma(n)=1,2,3,4,5, \ldots
$
Case-II: If $n$ is odd
$
\sigma(n)=0,-1,-2,-3,-4, \ldots
$
Thus, for every value of $n \sigma(n)$ has an unique image.
$\therefore \quad \sigma(n)$ is one-one function.
So, range of $\sigma(n)=Z$
$\therefore \quad$ Range of $\sigma(n)=$ codomain of $\sigma(n)=Z$.
Hence, $\sigma(n)$ is one-one and onto function