Question

Let $A=\{1,3,5,7\},B=\{2,4,6,8\}$ and $f:A\to B,$ then number of function(s) $f$ that can be defined from $A$ to $B$ is

• A. $24$ if $i+f\left(i\right)<10,\forall i=\{1,3,5,7\}$
• B. $12$ if $f\left(i\right)-i>2,\forall i=\{1,3,5,7\}$
• C. $12$ if $i+f\left(i\right)<10,\forall i=\{1,3,5,7\}$
• D. $0$ if $f\left(i\right)-i>2,\forall i=\{1,3,5,7\}$

Answer 1

Answer: (A), (D)

$0$ if $f\left(i\right)-i>2,\forall i=\{1,3,5,7\}$

If $i+f\left(i\right)<10,\forall i=\{1,3,5,7\}$
$\Rightarrow f\left(1\right)<9,f\left(3\right)<7,f\left(5\right)<5,f\left(7\right)<3$
So, in $B=\{2,4,6,8\}$
$1$ has $4$ choices
$3$ has $3$ choices
$5$ has $2$ choices
$7$ has $1$ choice
$\therefore$ Number of required functions $=4\cdot 3\cdot 2\cdot 1=24$

If $f\left(i\right)-i>2,\forall i=\{1,3,5,7\}$
$\Rightarrow f\left(1\right)>3,f\left(3\right)>5,f\left(5\right)>7,f\left(7\right)>9$
Element $7$ in $A$ have no choice in $B.$
$\therefore$ Number of required functions is $0.$