I need help with the following question. I just can't seem to make any sense out of it. Any productive help will be appreciated!
Let $A_k$ be a set for each positive integer $k$. Define another collection of sets $B_k$:
$B_1 = A_1$
$B_2 = A_2 \setminus A_1$
and so on:
$B_k = A_k \setminus \cup_{t<k}A_t$
Prove that
- The sets $B_k$ are pairwise disjoint
- $\cup_{t \leq k} A_t = \cup_{t\leq k}B_t$ for each positive integer $k$
- $\cup_{t \geq 1} A_t = \cup_{t \geq 1}B_t$
To be sure I'm understanding you we have $B_k=A_k \setminus (\bigcup_{t<k} A_t)$? If this is the case, I notice that $B_k$ and $\bigcup_{t<k} A_t$ are disjoint and $B_{i}\subset \bigcup_{t<k} A_k$ (def. of $B_i$) for all $i<k$. Also, we certainly have $\bigcup_{t<k} B_t \subset \bigcup_{t \le k} A_t$. We just need to show that nothing from $A_t$, $t\le k$ is being left out.