Let $j$ be the index of a minimum cardinality set $A_j$, and $k$ be the index of a minimum cardinality set $B_k$. Then $|A_j| \leq \dfrac {|M|}{n}$ and also $|B_k| \leq \dfrac {|M|}{n}$. Since $n\leq |A_j\cup B_k| = |A_j|+|B_k| - |A_j\cap B_k| \leq |A_j|+|B_k| \leq \dfrac {2|M|}{n}$, it follows $|M| \geq n^2/2$.