39 lines
2 KiB
Markdown
39 lines
2 KiB
Markdown
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# Číselné množiny
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$$\emptyset \subset \mathbb{N} \subset \mathbb{N}_{0} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{R}^* \quad \mathbb{R}^* = \mathbb{R} \, \cup \{ -\infty, +\infty \}$$
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Mějme neprázdnou množinu $A \subset \mathbb{R}$.
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### Omezenost
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| značka | typ | podmínka |
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| ------ | ------------- | ----------------------------------------------------------------- |
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| **OZ** | omezená zdola | $\exists \, d \in \mathbb{R} \quad \forall \, x \in A : d \leq x$ |
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| **OS** | omezená shora | $\exists \, h \in \mathbb{R} \quad \forall \, x \in A : x \leq h$ |
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| **O** | omezená | omezená shora i zdola |
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### Minimum, maximum
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| typ | podmínka | zápis |
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| ------- | -------------------------------------------------------- | ------------- |
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| minimum | $\exists \, a \in A \quad \forall \, x \in A : a \leq x$ | $a = \min(A)$ |
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| maximum | $\exists \, b \in A \quad \forall \, x \in A : x \leq b$ | $b = \max(A)$ |
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### Infimum, supremum
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Množina $A$ má **infimum**, pokud existuje $i \in \mathbb{R}^*$ takové, že platí
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1) $\forall \, x \in A : i \leq x$,
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2) $\forall \, x_{1} \in \mathbb{R} : i < x_{1} \implies (\exists \, x_{2} \in A : x_{2} < x_{1})$,
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- píšeme $i = \inf(A)$.
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Množina $A$ má **supremum**, pokud existuje $s \in \mathbb{R}^*$ takové, že platí
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1) $\forall \, x \in A : x \leq s$,
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2) $\forall \, x_{1} \in \mathbb{R} : x_{1} < s \implies (\exists \, x_{2} \in A : x_{1} < x_{2})$,
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- značíme $s = \sup(A)$.
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Pro každou neprázdnou množinu $A \subset \mathbb{R}$ platí
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1) $\exists! \, \inf A, \quad \exists! \, \sup A$,
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2) $\inf A \leq \sup A$,
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3) $\exists \, \min A \implies \inf A = \min A$,
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4) $\exists \, \max A \implies \sup A = \max A$,
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5) $A$ není omezená zdola $\Leftrightarrow \inf A = -\infty$,
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6) $A$ není omezená shora $\Leftrightarrow \sup A = +\infty$.
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