# Countably Infinite Collections of Sets

I need to find examples of:

(a) A countably infinite collection of pairwise disjoint finite sets whose union is countably infinite (b) A countably infinite collection of nonempty sets whose union is finite. (c) A countably infinite collection of pairwise disjoint nonempty sets whose union is finite.

For (a), I'm wondering if the collection of sets each containing one of the integers will work. I know that $\mathbb{Z}$$\approx$$\mathbb{N}$, so it is countably infinite, and if each only contains one integer it is only pairwise disjoint finite sets, with their union is countably infinite. Does this work?

I'm having trouble having any ideas for (b) and (c).

• Your answer for $(a)$ is correct. – Studzinski Apr 17 '15 at 16:20
• You example for (a) is perfectly fine. In other words, you could have stated your collection $\mathcal{A} = \{A_1,A_2,\dots\}$ where each $A_n = \{n\}$. As for (b), if their union is finite, what does that imply about them being disjoint? What is the easiest example of a collection which is not disjoint? (Hint, what is $\{5\}\cup \{5\}$? What is $\{5\}\cup\{5\}\cup\{5\}$?) For (c), think about what it means for them to be disjoint. Look at the union of the first $n$ sets: $\mathcal{A}_n=\bigcup_{i=0}^n A_i$ and its size. Compare that to $\mathcal{A}_{n+1}$. Is it always bigger? – JMoravitz Apr 17 '15 at 16:20
• In $(b)$ you do not need the sets to be disjoint. – Studzinski Apr 17 '15 at 16:20

Your idea for (a) is fine: For $n\in\mathbb N$ let $A_n=\{n\}$. Then $A_n\cap A_m=\emptyset$ for $n\ne m$ as required, and $\bigcup_{n\in\mathbb N} A_n=\mathbb N$
For (b), try something like $A_n=\{1\}$. Then $\bigcup A_n=\{1\}$ is finite.
For (c), no example exists: If $I$ is an index set and for each $i\in I$, $A_i$ is a nonempty set, then by picking an element $a_i\in A_i$, we obtain a map $I\to\bigcup_{i\in I} A_i$. If the $A_i$ are pairwise disjoint, this map is certainly injective, hence $|\bigcup_{i\in I}A_i|\ge |I|$