Definition (of Algebra on a set )
Let be an arbitrary set. Then a collection of subsets of is called an algebra on if and only if all the following are true :

for each the set
for every finite sequence of elements each the union

Definition (of sigma algebra on a set )
Let be an arbitrary set. Then a collection of subsets of is called an algebra on if and only if all the following are true :

for each the set
for every infinite sequence of elements each the union


Note that : since , so we conclude that every algebra on a set is also an algebra on .

Note that given a set then is infinite if and only if there is a one one function from to .
Thus, since given that is an infinite set so, there must be a one one function from to . Let be such a function.
Then, note that since each of the sets () is finite so they all belong to .
But, if be the union then is an infinite set and also is an infinite set. Thus, is not an element of .
So, (as per definitions), is not a algebra on .
However, since finite union of finite sets is finite and by de Moivre’s law, we get that, has to be an algebra on .
This completes the proof.

a set is called to be uncountable if and only if there is no one one function from to

Yes.

Countable union of countable sets is countable. Show this ! and then use this to conclude the result.