Interval arithmetic

Remarks

Each set $X \circ Y$ is again a closed bounded interval, because it is the image of the compact connected set $X \times Y$ under a continuous map. Example: $[1, 2] \times [-1, 3]$ has the products $-1, 3, -2, 6$ of the end points, so it is $[-2, 6]$; indeed $x y$ takes every value between $-2$ and $6$ for $x \in [1, 2]$ and $y \in [-1, 3]$. Non-example: $[1, 2] \div [-1, 1]$ is not defined, because $0 \in [-1, 1]$ and the set $\set{x / y : y \ne 0}$ is unbounded. Addition and multiplication are associative and commutative, but $X - X \ne [0, 0]$ unless $X$ has one point, and only the subdistributive law $X \times (Y + Z) \subseteq X \times Y + X \times Z$ holds in general. For example, $[-1, 1] \times ([1, 1] + [-1, -1]) = [0, 0]$, while $[-1, 1] \times [1, 1] + [-1, 1] \times [-1, -1] = [-2, 2]$.