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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]].\nA *semiring of sets* on $X$ is a family $\\mathcal{S}$ of subsets of $X$ such that:\n\n1. $\\varnothing \\in \\mathcal{S}$.\n2. If $A, B \\in \\mathcal{S}$, then the [[intersection|0HBRW4E6G1]] $A \\cap B$ is in $\\mathcal{S}$.\n3. If $A, B \\in \\mathcal{S}$, then there are finitely many pairwise disjoint sets $C_1, \\ldots, C_n \\in \\mathcal{S}$ such that $A \\setminus B$ is their [[union|07KVB4JSC7]] $C_1 \\cup \\cdots \\cup C_n$.\n\n## Remarks\n\nThe half-open intervals $[a, b)$ with $a, b \\in \\R$ form a semiring on $\\R$, where $[a, b) = \\varnothing$ when $b \\le a$.\nFor clause 2, $[a, b) \\cap [c, d) = [\\max(a, c), \\min(b, d))$.\nFor clause 3, when $c \u003c d$, $[a, b) \\setminus [c, d) = [a, \\min(b, c)) \\cup [\\max(a, d), b)$, a union of two disjoint half-open intervals.\nThe open intervals $(a, b)$, together with $\\varnothing$, do not form a semiring, since clause 3 fails.\nIndeed $(0, 3) \\setminus (1, 2) = (0, 1] \\cup [2, 3)$ is not open, while a finite union of open intervals is open.\nBy clause 2 a semiring is a [[π-system|4W9NM803BB]].\nThe finite disjoint unions of members of a semiring form the smallest ring of sets that contains it.\n",
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