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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]].\nA *ring of sets* on $X$ is a nonempty family $\\mathcal{R}$ of subsets of $X$ such that, for all $A, B \\in \\mathcal{R}$:\n\n1. the [[union|07KVB4JSC7]] $A \\cup B$ is in $\\mathcal{R}$;\n2. the difference $A \\setminus B$ is in $\\mathcal{R}$.\n\n## Remarks\n\nThe finite subsets of $\\N$ form a ring of sets: the union and the difference of two finite sets are finite.\nThis ring does not contain $\\N$, so it is not an algebra of sets.\nThe bounded intervals of $\\R$ do not form a ring of sets, since clause 2 fails: $[0, 3] \\setminus [1, 2] = [0, 1) \\cup (2, 3]$ is not an interval.\nA ring contains $\\varnothing = A \\setminus A$ and the [[intersection|0HBRW4E6G1]] $A \\cap B = A \\setminus (A \\setminus B)$.\nIt also contains the symmetric difference $A \\mathbin{\\triangle} B = (A \\setminus B) \\cup (B \\setminus A)$.\nWith $\\triangle$ as addition and $\\cap$ as multiplication, a ring of sets is a commutative ring in the sense of algebra, possibly without identity.\nEvery ring of sets is a [[semiring of sets|PYH82FMXQ2]].\n",
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