lowest upper bound

Let $S$ be a set with a partial ordering $\\leq$ , and let $T$ be a subset of $S$ . A lowest upper bound, or supremum, of $T$ is an upper bound $x$ of $T$ with the property that $x\\leq y$ for every upper bound $y$ of $T$ . The lowest upper bound of $T$ , when it exists, is denoted $\\operatorname{sup}(T)$ .

A lowest upper bound of $T$ , when it exists, is unique.

Greatest lower bound is defined similarly: a greatest lower bound, or infimum, of $T$ is a lower bound $x$ of $T$ with the property that $x\\geq y$ for every lower bound $y$ of $T$ . The greatest lower bound of $T$ , when it exists, is denoted $\\operatorname{inf}(T)$ .

If $A=\\{a_{1},a_{2},\\ldots,a_{n}\\}$ is a finite set, then the supremum of $A$ is simply $\\max(A)$ , and the infimum of $A$ is equal to $\\min(A)$ .

单词	LowestUpperBound
释义	lowest upper bound Let $S$ be a set with a partial ordering $\\leq$ , and let $T$ be a subset of $S$ . A lowest upper bound, or supremum, of $T$ is an upper bound $x$ of $T$ with the property that $x\\leq y$ for every upper bound $y$ of $T$ . The lowest upper bound of $T$ , when it exists, is denoted $\\operatorname{sup}(T)$ . A lowest upper bound of $T$ , when it exists, is unique. Greatest lower bound is defined similarly: a greatest lower bound, or infimum, of $T$ is a lower bound $x$ of $T$ with the property that $x\\geq y$ for every lower bound $y$ of $T$ . The greatest lower bound of $T$ , when it exists, is denoted $\\operatorname{inf}(T)$ . If $A=\\{a_{1},a_{2},\\ldots,a_{n}\\}$ is a finite set, then the supremum of $A$ is simply $\\max(A)$ , and the infimum of $A$ is equal to $\\min(A)$ .
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