Question

A function $f$ from the set of natural numbers to integers defined by
$f\left(n\right)=\{\begin{array}{c}\frac{n-1}{2}\text{if n is odd}\\ \frac{n-1}{2}\text{if n is even}\end{array}$
is :

• A. Onto but not one-one.
• B. One-one and onto both.
• C. Neither one-one nor both.
• D. One-one but not onto.

Answer 1

Answer: (B)

One-one and onto both.

$f\left(n\right)=\{\begin{array}{c}\frac{n-1}{2}\text{if n is odd}\\ -\frac{n}{2}\text{if n is even}\end{array}$

$\therefore f:N⟶I$

$f\left(1\right)=0,f\left(2\right)=-1,f\left(3\right)=1,f\left(4\right)=-2,f\left(5\right)=2,f\left(6\right)=-3$ soon.

If this type of function every element of set $A$ has a unique image in set $B$ and there is no element left in set $B.$ Hence $f$ is one-one and onto function.