Definitions of algebra on a set and sigma algebra on a set

Let $X$ be an infinite set. And let $\mathcal{A}$ be the collection of all subsets $A$ of $X$ such that either $A$ or $A^{c} ( = X \setminus A)$ is finite. Prove that $\mathcal{A}$ is an algebra on $X$ , but not a sigma algebra on $X$ .

Proof

A related exercise :
Let $X$ be an uncountable set. Let $\mathcal{A}$ be the collection of all subsets $A$ of $X$ such that either $A$ is countable or $A^{c} ( = X \setminus A )$ is countable.
Is $\mathcal{A}$ a $\sigma$ algebra on $X$ ?

Answer
Hint to sketch

An Introduction to the Theory of Mathematics