Functions Previous Year Questions (PYQs) for CMI M.Sc. and Ph.D. Computer Science: 4+ Solved Questions with Step-by-Step Solutions

The number of one-to-one functions is $3^4$.

There are as many one-to-one functions as onto functions.

The number of onto functions is strictly less than the number of one-to-one functions.

The number of one-to-one functions is 24.

The number of functions in $F$ equals $n^2$.

The number of functions in $F$ equals $2^n$.

The number of functions in $G$ equals $n^2-2$.

The number of functions in $G$ equals $0$.

$2^n$

$2^n-1$

$2^n-2$

$2^{n/2}$

Let $f:X\to Y$, $g:Y\to Z$ be functions such that the composite function $g\circ f:X\to Z$ is onto. Then both $g$, $f$ have to be onto.

Assume the number of elements of $X$, denoted as $|X|$, is strictly less than $|Y|$ and also assume that $|X|\le |Z|$. There exists a one-one function $f:X\to Y$ and a $g:Y\to Z$ which is not one-one such that $g\circ f:X\to Z$ is one-one.

Let $i{d}_X$ be the identity function on $X$, i.e. $i{d}_X(x)=x$ for all $x\in X$. Let $f:X\to Y$ be a function. Then there exists a function $g:Y\to X$ such that $g\circ f=i{d}_X$ exists.

Denote by $i{d}_Y$ the identity function on $Y$, $i{d}_Y(y)=y$ for all $y\in Y$. Then there exists $g:Y\to X$ such that $f\circ g=i{d}_Y$ if and only if $f$ is onto.