Relating variables: $A_{i}$ with $|A_{i}|=p$ and $B_{j}$ with $|B_{j}|=q$ such that $\cup_{i=1}^{n}A_{i}=\cup_{j=1}^{m}B_{j}=S$ with constraint on $S$

Question

I am supposed to relate variables in the following problem. It would be great if someone could explain what the problem statement means and also how to tackle such a problem.

I do understand the notation, also, in the question image $O(X)$ denotes the cardinality of set $X$. I have used $|X|$ in the title. At present, I have no idea on how to proceed. Any hints are appreciated. Thanks.

Answer 1

$A_i$'s are collection of $m$ sets such that there are $p$ elements in each of them. $B_j$'s are collection of $n$ sets such that there are $q$ elements in each of them.

Any element in a given $A_i$ also occurs in exactly $\alpha -1$ other $A_i$'s $(\alpha \le m)$. Similarly any element in a given $B_j$ also occurs in exactly $\beta -1$ other $A_i$'s $(\beta \le n)$.

Also union of all $A_i$'s is identical with union of all $B_j$'s. This means same total objects make up both the collections, $\{A_i\}$ and $\{B_j\}$. These distinct objects are elements of (universal) set $S$.

Now there are $mp$ elements in total in $\{A_i\}$, each repeated exactly $\alpha$ times. Hence number of distinct elements in $\{A_i\}$ is only $$\dfrac{mp}{\alpha} $$

Similarly, number of distinct elements in $\{B_i\}$ is only $$\dfrac{nq}{\beta} $$

And given is $$|S| = \dfrac{mp}{\alpha} = \dfrac{nq}{\beta}$$