Need explanation for generalization of the associative law for sets

Question

I am currently reading Naive Set Theory by P. R. Halmos and I don't seem to understand the notation he uses to generalize the associative law for sets: ∪_k∈KA_k = ∪_j∈J(∪_i∈IjA_i). I understand the left-hand side, and of the right-hand side only all thats included in the brackets. I don't seem to connect what the union outside the brackets of the RHS means. Perhaps I have met the notation before, and maybe I'm just not seeing the picture, but I've been looking at this formula for a long time now. Thank you in advance. ​

Answer 1

An example could clarify how does it work: Take

$K = \{1,2,3,4,5\}$, $J=\{1,2\}$, $I_1 = \{1,2,3\}$, $I_2 = \{4,5\}$.

(The set $\{I_j\mid j\in J\}$ is supposed to be a partition of $K$, in this case $\{I_1,I_2\}$ is a partition of $K$)

Then $$\bigcup_{k\in K}A_k = A_1\cup A_2\cup A_3\cup A_4\cup A_5$$

and $$\bigcup_{j\in J} \bigcup_{i\in I_j}A_i = (\bigcup_{i\in I_1}A_i)\cup(\bigcup_{i\in I_2}A_i) = (A_1\cup A_2\cup A_3)\cup (A_4 \cup A_5)$$