Question

Let $f:N\to N$ ($N$ being the set of positive integers) be a function defined by $f\left(x\right)=$ the biggest positive integer obtained by reshuffling the digits of $x$. For example, $f\left(296\right)=962$
$f$ is

• A. One-one and onto
• B. One-one and into
• C. Many-one and onto
• D. Many-one and into

Answer 1

Answer: (D)

Many-one and into

$f\left(x\right)$ is many on one function.
for example
$f\left(296\right)$
$=f\left(269\right)$
$=f\left(629\right)$
$=f\left(692\right)$
$=f\left(962\right)$
$=f\left(926\right)$
$=962$
Hence considering a 3-digit number in the domain (without repeated digits), there would be 3! numbers in the domain which correspond to a single number in the co-domain of $f\left(x\right)$
Similarly for a 4-digit number (without repeated digits), there would be 4! numbers in the domain corresponding to a single number in the co-domain of domain to some elements of the co-domain.
Hence, the function $f\left(x\right)$ is many one and into function.