How to prove family of rectangles to be semiring?

Question

Prove that $I_2 = \{[a,b) \times [c,d) \mid a,b,c,d\in\Bbb{R}\}$ is a semiring over $\Bbb{R}^2$.

Answer 1

Let $A = \{[a_1,b_1) \times [c_1,d_1) \mid a_1,b_1,c_1,d_1\in\Bbb{R}\}$ and $B = \{[a_2,b_2) \times [c_2,d_2) \mid a_2,b_2,c_2,d_2\in\Bbb{R}\}$. Then $$ A\cap B=\begin{cases}(\max(a_1,a_2), \min(b_1, b_2))\times (\max(c_1,c_2), \min(d_1, d_2)), \\ \quad\quad\quad\max(a_1,a_2)\leqslant\min(b_1, b_2) \text{ and } \max(c_1,c_2)\leqslant\min(d_1, d_2) \\ \varnothing, \quad\text{otherwise} \end{cases} $$ So clearly $A\cap B$ is a rectangle. Also $A\cup B$ is the union of at most $5$ rectangles.

Thus $I_2$ is semiring.