question_answer

A function f form the set of natural numbers to integers defined by  \[f(n)=\left\{ \begin{matrix}    \frac{n-1}{2},\,\,\,\text{when}\,\,n\,\,\text{odd}  \\    -\frac{n}{2}\,,\,\,\,\text{when}\,\,n\,\,\text{is}\,\,\text{odd}  \\ \end{matrix}\,\,\,\text{is} \right.\]

A) one-one but not onto.

B) onto but not one-one

C) one-one and onto both.

D) neither one-one nor onto.

Correct Answer: C

Solution :

[c] \[f:N\to I\] \[f\left( 1 \right)=0,\,\,f\left( 2 \right)=-1,\] \[f\left( 3 \right)=1,\,\,f\left( 4 \right)=-\,2,\] \[f\left( 5 \right)=2,\,\,f\left( 6 \right)=-\,3,\]and so on. In this function, every element of set A has unique image in set B and there is no element left in set B. Here f is a one-one and onto function.