Question

If $m=$ number of distinct rational numbers $\frac{p}{q}\in (0,1)$ such that $p,q$ $\in$ $\{1,2,3,4,5\}$ and $n=$ number of onto mappings from$\{1,2,3\}$ onto$\{1,2\}$, then $m-n$ is

• A. $1$
• B. $-1$
• C. $0$
• D. $3$

Answer 1

The correct option is D $3$

Rational number $\frac{p}{q}ϵ(0,1)$, so numerator has to be less than denominator. To satisfy this condition we have the following :

for $p=1;$ number of numbers $\frac{p}{q}$ are 4.
for $p=2$, number of rational numbers are 3.
for $p=3;$ there are 2 numbers
for $p=4,$ there is only one number
$\therefore m=4+3+2+1=10$
Number of onto functions from $(1,2,3)$ to $(1,2)$
are total number of functions - number of into functions
$n=3^2-2=9-2=7$
$\therefore m-n=10-7=3$