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  "body": "\n## Definition\n\nLet $X$ be a [[set|0SJ3RX9TXA]].\nAn *algebra of sets* on $X$ is a subset $\\mathcal{A}$ of the [[power set|0BC7KX51Z8]] of $X$ such that:\n\n1. $X \\in \\mathcal{A}$.\n2. If $A \\in \\mathcal{A}$, then its [[complement|0B1FSS1VZW]] $X \\setminus A$ is in $\\mathcal{A}$.\n3. If $A, B \\in \\mathcal{A}$, then the [[union|07KVB4JSC7]] $A \\cup B$ is in $\\mathcal{A}$.\n\n## Remarks\n\nOn $X = \\N$, the family $\\mathcal{A}$ of the subsets $A$ such that $A$ is finite or $\\N \\setminus A$ is finite is an algebra.\nClause 1 holds since $\\N \\setminus \\N = \\varnothing$ is finite.\nClause 2 holds since the condition is symmetric in $A$ and $\\N \\setminus A$.\nClause 3 holds since a union of two finite sets is finite, and a union that contains a set with finite complement has finite complement.\nThis algebra is not a [[σ-algebra|04WSW8T3SY]]: the set of even numbers is a countable union of singletons, but neither it nor its complement is finite.\nThe family of all finite subsets of $\\N$ is not an algebra, since clause 1 fails: $\\N$ is not finite.\nAn algebra contains $\\varnothing$ and is closed under finite intersections and differences, by De Morgan's laws.\nSo an algebra of sets is exactly a [[ring of sets|KWE1DBMN9B]] that contains $X$.\nThe name *field of sets* is also common.\n",
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