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  "body": "\n## Definition\n\nLet $\\mathbb{IR}$ be the set of closed bounded [[intervals|0Q1D85B8G6]] $[a, b] = \\set{x \\in \\R : a \\le x \\le b}$ with real $a \\le b$.\nFor $X, Y \\in \\mathbb{IR}$ and an operation $\\circ \\in \\set{+, -, \\times, \\div}$, set\n$$X \\circ Y = \\set{x \\circ y : x \\in X, \\ y \\in Y},$$\nwhere for $\\div$ one requires $0 \\notin Y$.\n*Interval arithmetic* is $\\mathbb{IR}$ with these four operations.\nFor $X = [a, b]$ and $Y = [c, d]$ they are given by:\n\n1. $X + Y = [a + c, \\ b + d]$;\n2. $X - Y = [a - d, \\ b - c]$;\n3. $X \\times Y = [\\min(ac, ad, bc, bd), \\ \\max(ac, ad, bc, bd)]$;\n4. $X \\div Y = X \\times [1/d, \\ 1/c]$ when $0 \\notin Y$.\n\n## Remarks\n\nEach 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.\nExample: $[1, 2] \\times [-1, 3]$ 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 $y \\in [-1, 3]$.\nNon-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.\nAddition 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.\nFor example, $[-1, 1] \\times ([1, 1] + [-1, -1]) = [0, 0]$, while $[-1, 1] \\times [1, 1] + [-1, 1] \\times [-1, -1] = [-2, 2]$.\n",
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