Let $(A_i)_{i=1}^{n}$ be $n$ different sets. Say $Z := \bigcup_{i=0}^{n}A_i$.  
**Q1**: Is it true that if $|Z| \gt n$ then you can find $x \in Z$ such that the $(A_i-{x)}$ are still all different?  
**Q2**: If $|Z|\gt n-1$ is a sufficient hypothesis, is this tight? 

**Remark** : if Q1 is true then of course you can find $X	\subset Z $ such that $|Z-X|=n$ and all $A_i-X$ are different.  
An alternative formulation is:  
If you have $n$ words of size $n+1$ on alphabet ${ \{ 0,1 \}}$. 
Can you find $i \in [1,n=1]$ so that the $i$th letter of each words still produce $n+1$ different words?