question_answer

A function f from the set of natural numbers to integers defined by \[f(n)=\left\{ \begin{align}   & \frac{n-1}{2},\ \text{when}\ n\ \text{is}\ \text{odd} \\  & -\frac{n}{2},\ \text{when }n\text{ is even} \\ \end{align} \right.\], is [AIEEE 2003]

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 :

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