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text: '[summary:\nA totally ordered set is a pair $(S, \\le)$ of a set $S$ and a *total order* $\\le$ on $S$, which is a [3nt binary relation] that satisfies the following properties:\n\n1. For all $a, b \\in S$, if $a \\le b$ and $b \\le a$, then $a = b$. (the [antisymmetric_relation antisymmetric] property)\n2. For all $a, b, c \\in S$, if $a \\le b$ and $b \\le c$, then $a \\le c$. (the [573 transitive] property)\n3. For all $a, b \\in S$, either $a \\le b$ or $b \\le a$, or both. (the [total_relation totality] property)]\n\nA **totally ordered set** is a pair $(S, \\le)$ of a set $S$ and a *total order* $\\le$ on $S$, which is a [-binary_relation] that satisfies the following properties:\n\n1. For all $a, b \\in S$, if $a \\le b$ and $b \\le a$, then $a = b$. (the [antisymmetric_relation antisymmetric] property)\n2. For all $a, b, c \\in S$, if $a \\le b$ and $b \\le c$, then $a \\le c$. (the [573 transitive] property)\n3. For all $a, b \\in S$, either $a \\le b$ or $b \\le a$, or both. (the [total_relation totality] property)\n\nA totally ordered set is a special type of [3rb partially ordered set] that satisfies the total property — in general, posets only satisfy the [5dy reflexive] property, which is that $a \\le a$ for all $a \\in S$.\n\n## Examples of totally ordered sets\n\nThe [4bc real numbers] are a totally ordered set. So are any of the subsets of the real numbers, such as the [4zq rational numbers] or the [48l integers].\n\n## Examples of not totally ordered sets\n\nThe [4zw complex numbers] do not have a canonical total ordering, and especially not a total ordering that preserves all the properties of the ordering of the real numbers, although one can define a total ordering on them quite easily.',
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