Let $(V,\preceq )$ be a poset or a strictly ordered set and let $S\subseteq V$.
An element $m\in S$ is called:
| maximal in $S$ | no $x\in S$ is greater, formally $\not\exists x\in S\; x\succ m$ |
| minimal in $S$ | no $x\in S$ is smaller, formally $\not\exists x\in S\; x\prec m$ |
| maximum of (greatest in) $S$ | all $x\in S$ are smaller or equal, formally $\forall x\in S\; x\preceq m$ |
| minimum of (smallest in) $S$ | all $x\in S$ are greater or equal, formally $\forall x\in S\; x\succeq m$ |
An element $m\in V$ is called:
| upper bound of $S$ | all $x\in S$ are smaller or equal, formally $\forall x\in S\; x\preceq m$ |
| lower bound of $S$ | all $x\in S$ are greater or equal, formally $\forall x\in S\; x\succeq m$ |
| supremum of $S$ | $m$ is minimum of all upper bounds of $S$, formally $m=\sup(S):=\min(\{n\in V\mid \forall x\in S\; x\preceq n\})$ |
| infimum of $S$ | $m$ is maximum of all lower bounds of $S$, formally $m=\inf(S):=\max(\{n\in V\mid \forall x\in S\; x\succeq n\})$ |
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| created: 2018-12-16 23:14:40 | modified: 2020-07-06 20:53:35 | by: bookofproofs | references: [577], [979]
[577] Knauer Ulrich: “Diskrete Strukturen - kurz gefasst”, Spektrum Akademischer Verlag, 2001
[979] Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10
