Semiring of sets

Remarks

The half-open intervals $[a, b)$ with $a, b \in \R$ form a semiring on $\R$, where $[a, b) = \varnothing$ when $b \le a$. For clause 2, $[a, b) \cap [c, d) = [\max(a, c), \min(b, d))$. For clause 3, when $c < d$, $[a, b) \setminus [c, d) = [a, \min(b, c)) \cup [\max(a, d), b)$, a union of two disjoint half-open intervals. The open intervals $(a, b)$, together with $\varnothing$, do not form a semiring, since clause 3 fails. Indeed $(0, 3) \setminus (1, 2) = (0, 1] \cup [2, 3)$ is not open, while a finite union of open intervals is open. By clause 2 a semiring is a π-system. The finite disjoint unions of members of a semiring form the smallest ring of sets that contains it.