JEE Main 2025 — Sets And Relations Question with Solution

To find the number of elements in set $B$, we consider pairs $\left(\frac{m}{n}\right)$ where $m,n\in A$ with $m<n$ and $\text{gcd}(m,n)=1$.

Here's the breakdown for each possible $m$:

For $m=1$:

Possible values for $n$ are $2,3,4,5,6,7,8,9,10$.

Total pairs: $9$.

For $m=2$:

Possible values for $n$ are $3,5,7,9$ (since these have $\text{gcd}(2,n)=1$).

Total pairs: $4$.

For $m=3$:

Possible values for $n$ are $4,5,7,8,10$.

Total pairs: $5$.

For $m=4$:

Possible values for $n$ are $5,7,9$.

Total pairs: $3$.

For $m=5$:

Possible values for $n$ are $6,7,8,9$.

Total pairs: $4$.

For $m=6$:

Possible value for $n$ is $7$.

Total pairs: $1$.

For $m=7$:

Possible values for $n$ are $8,9,10$.

Total pairs: $3$.

For $m=8$:

Possible value for $n$ is $9$.

Total pairs: $1$.

For $m=9$:

Possible value for $n$ is $10$.

Total pairs: $1$.

Adding all these up, the total number of elements in set $B$ is:

$9+4+5+3+4+1+3+1+1=31$