Example of an algebra which is not a σ-algebra.
Solution 1:
Let $X$ be an infinite set, and $\mathcal A$ be the collection of all subsets of $X$ which are finite or have finite complement. Then $\mathcal A$ is an algebra of sets which is not a $\sigma$-algebra.
Solution 2:
Let $L$ be the collection of all finite disjoint unions of all intervals of the form:
$(−\infty, a], (a, b], (b, \infty), \emptyset, \mathbf{R}$.
Then $L$ is an algebra over $\mathbf{R}$, but not a σ-algebra because
union of sets $\left\{(0,\frac{i-1}{i}]\right\}$ for all $i \ge 1 = (0, 1) \notin L $.