First question:
Is it true or false that the countable union of disjointed finite sets is always infinite?
In symbols: let $\{A_n\}_{n\in\mathbb{N}}$ a sequence of sets such that $A_i\cap A_j=\emptyset$ for $i\ne j$, and $|A_n|<+\infty$ for all $n\in\mathbb{N}$. Then
$$
\bigg|\bigcup_n A_n\bigg|=+\infty.
$$
For me it is true.
My problem is if this thing always happens.
Second question
Is it true or false that the countable union of a finite number of finite sets, where the remaining ones are empty, has finite cardinality?
In symbols: let $\{A_n\}_{n\in\mathbb{N}}$ a sequence of sets such that $A_i\cap A_j=\emptyset$ for $i\ne j$, and exists $\overline{n}\in\mathbb{N}$ such that $|A_n|<+\infty$ for $n=1,2,\dots,\overline{n}$ and $A_{\overline{n}+1}=\cdot\cdot\cdot=A_m=\emptyset=\cdot\cdot\cdot$, then
$$
\bigg|\bigcup_n A_n\bigg|<+\infty.
$$
This obviously seems true to me, but in mathematics the obvious should also be shown, because I do not understand why it is important that the sets should be disjointed.
In my opinion, it proceeds in this way:
$$
\bigcup_n A_n =\bigcup_{n=1}^\overline{n} A_n,
$$
then
$$
\bigg |\bigcup_n A_n\bigg|=\bigg |\bigcup_{n=1}^{\overline{n}} A_n\bigg|=\sum_{n=1}^{\overline{n}}|A_n|<+\infty\quad(\text{here we use the hypotheses that are disjointed)}
$$
Thanks!
Best Answer
Yes for the first question.
At most, only one of the A's are disjoint.
So assume WLOG, they are all not empty.
For each i in N, pick some a$_i$ in A$_i$.
Show the map i -> a$_i$ is an injection from N into the union of the A's.
Thus the union is infinite.