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Definition: Bounded Subsets of Ordered Sets

Let $(V,\preceq )$ be a poset or a strictly ordered set and let $S\subseteq V$ be its non-empty subset. 1. \(S\) is called bounded above if it has an upper bound, i.e. there is an element $B\in V$ with $x\preceq B$ for all $x\in S.$ 1. \(S\) is called bounded below if it has a lower bound, i.e. there is an element $B\in V$ with $x\succeq B$ for all $x\in S.$

If \(S\) is bounded above and bounded below, then we say that $S$ is bounded.

Examples and Notes

• Bounded subsets exist if we replace $V$ by the natural numbers $\mathbb N,$ the integers $\mathbb Z,$ the rational numbers $\mathbb Q,$ and the real numbers $\mathbb R,$ since they all are strictly ordered.
• If the underlying set is a poset, then only its chains can be bounded. If a subset of a poset is not a chain, then some of its elements might be not comparable with the bound in question, and then it does not make sense to speak about "the bound" of the subset.

Mentioned in:

Chapters: 1
Definitions: 2 3 4 5 6 7 8 9 10
Lemmas: 11 12
Proofs: 13 14 15 16 17 18 19 20
Subsections: 21
Theorems: 22

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983