$A_{2}=\left \{ 1,2 \right \}$

$A_{3}=\left \{ 1,2,3 \right \}$

Now, given that $|A_{i}|$ is odd number

So, value of $i$ is 1 or 3

Now we know , a set can contain even number of elements, if an odd number set is subtracted from another odd number set

or,

an even number set is subtracted from another even number set

Now, $||A_{i}|\cap|A_{j}| |=$even number

that means j value could be 1 or 3

Here n is largest number of that set

So, if n=3, m could be 1 or 3

So, $m\leq n$