Converted notes-sep-1 into Obsidian note 'Mängder 1.md'
This commit is contained in:
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|
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|
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||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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|
||||
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|
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|
||||
80
LaTeX/Mängder 1.md
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|
||||
- Grundbegrepp:
|
||||
- En Mänd är en lista av element
|
||||
- Listor betäknas med stora bokstäver som $A = \{1, 2, 3\}$, eller $A = \{x: x|120\}$ som betyder att alla $x$ som möter kravet att $x$ är jämt delbart med $120$ är med i mängden, och element betäknas med små bokstäver
|
||||
- Ej ordning, ej upp repning som $\{1, 2\}$ och $\{2, 1\}$ är samma sak
|
||||
- Betäkningar:
|
||||
- Tilhör: $\in$
|
||||
- Jämt delbart med $|$
|
||||
- Väldefinerad: $A=\{5,7,9\}$ eller $B=\{x : x|2, 0 \leq x \leq 10\}$
|
||||
- $5\in A$ kollar om elementet $5$ fins i mängden $A$
|
||||
- Alla mängden skall vara tydligt defineriade för att undlvika att ett element både är i och inte i mängden $(5\in A \;\wedge\; 5\notin A)$ vilket är en kontradiktion
|
||||
- Kardinalitet: $|A|$ betyder läng(antal element) i mängden $A$
|
||||
- $|A|$ är altid ett heltal eller $\infty$
|
||||
- Example $A = \{5, 7, 9\}$ så får vi att kardinaliteten $|A|=3$
|
||||
- Example $B=\{x:x|2\}$ så får vi att kardinaliteten $|B|=\infty$
|
||||
- Delmängd:
|
||||
- $A\subseteq B$ betyder att alla element i $A$ fins i $B$, samt att $A = B$
|
||||
- Exemple $A = \{1,2,3\}$ och $B=\{1,2,3\}$ då är $A \subseteq B$
|
||||
- Exemple $A = \{1,2\}$ och $B = \{1,2,3\}$ då ät $A \not\subseteq B$
|
||||
- Äkta Delmängd:
|
||||
- $A \subset B$ betyder att alla element i $A$ fins i $B$, men att åtminstånde ett element fins i $B$ men inte i $A$
|
||||
- Exemple $A = \{1,2,3\}$ och $B=\{1,2,3\}$ då är $A \not\subset B$
|
||||
- Exemple $A = \{1,2\}$ och $B = \{1,2,3\}$ då ät $A \subset B$
|
||||
- Rita mängder:
|
||||
- Venn Diagram används för att visa mängder, exemple ![[venn_diagram.png]]
|
||||
- etta diagramet vissar mängderna
|
||||
$$
|
||||
\begin{align*}
|
||||
A = \{5,7,9\} \\
|
||||
B = \{3,5,7,9\} \\
|
||||
C = \{5,7,9\} \\
|
||||
D = \{3,5,7\} \\
|
||||
\\
|
||||
C \subset B \\
|
||||
D \subset B \\
|
||||
C \cup D = B \\
|
||||
C \cap D = \{5,7\}
|
||||
\end{align*}
|
||||
$$
|
||||
- Null-set/Nollmängd/Empty set:
|
||||
- En tom mängs, där $|A| = 0$
|
||||
- $\emptyset = \{\}$
|
||||
- Där $\emptyset \subset A$ är alltid sant
|
||||
- $A = \{\{\}\}$ så är $A$ en mängd med elementet $\emptyset$ och $|A| = 1$ men $A \neq \emptyset$
|
||||
- Mängder är inte tal!
|
||||
- Skriv inte $A = 3$, om falletav kordinalitet skall det skrivas $|A| = 3$
|
||||
- Talmängder:
|
||||
- De natuliga talen: $\mathbb{N} = \{0, 1, 2, ...\}$
|
||||
- De hela talen: $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$
|
||||
- De rationella talen: $\mathbb{Q} = \{\frac{p}{q}: p, q \in \mathbb{Z}, q\neq0\}$
|
||||
- De reela talen: $\mathbb{R} = \mathbb{Q} \cup \{\text{de irratinela talen}\}$. Ex: irratinella tal $\pi$, $\mathrm{e}$, $\sqrt{2}$
|
||||
- De complexa talen: $\mathbb{C}=\{x+\mathrm{i}y:x,y\in\mathbb{R}, \mathrm{i}^2 = -1\}$
|
||||
- Där $\mathbb{N}\cup\mathbb{Z}\cup\mathbb{Q}\cup\mathbb{R}\cup\mathbb{C}$
|
||||
- $\cup$ - $\text{union/XOR}$, $\cap$ - $\text{och}$, $\subset$ - $\text{delmängd till}$, $\in$ - $\text{tillhör}$
|
||||
- Intervall:
|
||||
- Alla tal mellan två uppgivna tal
|
||||
- Slutet intervall: $[a, b] = \{x\in\mathbb{R}:a<=x<=b\}$
|
||||
- Öppen intervall:
|
||||
- $(a, b) =\;]a,b[\;= \{x\in\mathbb{R}:a<x<b\}$
|
||||
- $(-\infty, b) =\;]-\infty, b[\;= \{x\in\mathbb{R}:x<b\}$
|
||||
- $(a, \infty) = \;]a, -\infty[\;= \{x\in\mathbb{R}:a<x\}$
|
||||
- $(-\infty, \infty) = \;]-\infty, \infty[ = \mathbb{R}$
|
||||
- Halv öppen intervall
|
||||
- $(a, b] =\;]a, b]\;= \{x\in\mathbb{R}:a<x<=b\}$
|
||||
- $[a, b) =\;[a, b)\;= \{x\in\mathbb{R}:a<=x<b\}$
|
||||
- $(-infty, b] = \;]-\infty, b]\; = \{x\in\mathbb{R}:x<=b\}$
|
||||
- $[a, \infty) = [a, \infty) = \{x\in\mathbb{R}:a<=x\}$
|
||||
- Mängdoperationer
|
||||
- Grundmängd (Universual set): $\mathcal{U}$
|
||||
- tom mängd: $\emptyset = \{\}$, $\emptyset\subseteq A\subseteq\mathcal{U}$
|
||||
- Union: $A\cup B=\{x\in\mathcal{U}: x\in A\vee x\in B\}$
|
||||
- Snitt: $A\cap B=\{x\in\mathcal{U}: x\in A\wedge x\in B\}$
|
||||
- Komplement: $A^c = \bar A=\{x\in\mathcal{U}:x\not\in A\}$
|
||||
- ![[grundmänd_venn_diagram.png]]
|
||||
- $A=\{3,5,7,9\}$
|
||||
- $B=\{3,5,7\}$
|
||||
- $A\cup B=\{3,5,7,9\}$
|
||||
- $A\cap B = \{5,7\}$
|
||||
- $\mathcal{U} = \{1,3,5,7,9\}$
|
||||
- $A^c = \{1,2\}$
|
||||
- $B^c=\{1,9\}$
|
||||
Reference in New Issue
Block a user