19. $A_{1}, A_{2}, \cdots, A_{t}$ are all $r$-sets, $X=\bigcup_{i=1}^{t} A_{i}$, find $\min |X|$. Here the minimum is over all $A_{1}, A_{2}, \cdots, A_{i}$ of $|X|$.
19. Let $n$ be the smallest integer satisfying $C_{n}^{r} \geqslant t$. On one hand, $A_{1}, A_{2}, \cdots, A_{4}$ are all $r$-element subsets of $X$, so $t \leqslant C_{|X|}^{r}$. Therefore, $|X| \geqslant n$.

On the other hand, for any $n$-element set $X$, there are $C_{n} \geqslant t$ $r$-element subsets. Choose any $t$ of them. Let their union be $Y$, then by the above, $|Y| \geqslant n$, hence $Y=X$. Therefore, $X$ is the union of the $t$ chosen $r$-element subsets.
Thus, the required minimum value is $n$.