Question

Let $P$ be the set of all $45$ students of class X in a school. Let $f:P\to N$ be a function defined by $f\left(x\right)=$ roll number of student $x$.
State whether the function $f$ is bijective or not.

Answer 1

Here, $f\left(x\right)$ is defined as roll number of student $x$.
Let $f\left(x_1\right)$ and $f\left(x_2\right)$ be the roll number of student $x_1$ and $x_2$ respectively.
Now, putting $f\left(x_1\right)=f\left(x_2\right)$
Since, no two students can have the same roll number.
$f\left(x_1\right)=f\left(x_2\right)\Rightarrow$ $x_1=x_2$
So, $f\left(x\right)$ is one-one.

For onto,
$f:P\to N$ and since, there are only $45$ students in the class.
So, $f\left(x\right)$ can take the numbers from $1$ to $45$ only.
Thus, there is no pre-image of $46,47,...\in N$.
So, it does not fulfill the condition for onto function.
Therefore, the function $f$ is not bijective.