37 lines
1.5 KiB
Markdown
37 lines
1.5 KiB
Markdown
- Mängdoperation:
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- Differens: $A\setminus\text{B} = \{x\in\mathcal{U}:x\in\text{A}\land\text{x}\not\in\text{B}\}$
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- Symetrisk Differens $A\triangle\text{B}=\{x\in\mathcal{U}:x\in\text{A}\cup\text{B}\land\text{x}\not\in\text{A}\cap\text{B}\}$ = $(A\cup{B})\setminus(A\cap{B})$
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- Räkne regler:
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- Dublekomplement: $\overline{\overline{A}} = A$
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- Idempotens: $A\cup\text{A}=A$, $A\cap\text{A}=A$
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- Identitet: $A\cup\emptyset=A$, $A\cap\emptyset=\emptyset$
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- Dominans: $A\cup\mathcal{U}=\mathcal{U}$, $A\cap\emptyset=\emptyset$
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- Kommutativ: $A\cup\text{B}\iff\text{B}\cup\text{A}$, $A\cap\text{B}\iff\text{B}\cap\text{A}$
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- Associativ: $$
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\begin{align}
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(A\cup\text{B})\cup\text{C}\iff\text{A}\cup(B\cup\text{C})\iff\text{A}\cup\text{B}\cup\text{C} \\ (A\cap\text{B})\cap\text{C}\iff\text{A}\cap(B\cup\text{C})\iff\text{A}\cap\text{B}\cap\text{C}
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\end{align}
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$$
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- Paranteser: om alla oprationer är $\cup$ eller $\cap$ spelar årdning ingen roll
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- De Morgand: $\overline{(A\cup\text{B})}\iff\cap{A}\cap\overline{B}$, $\overline{(A\cap\text{B})}\iff\overline{A}\cup\overline{B}$
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- Bevis:
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$$
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\begin{align}
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x\in\overline{(A\cup\text{B})}\Rightarrow\text{x}\not\in\text{A}\cup\text{B}\Rightarrow
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\left\{
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\begin{aligned}
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& \text{x}\notin\text{A} \\
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& \text{x}\notin\text{B}
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\end{aligned}
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\right.
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\Rightarrow\text{x}\notin\overline{A}\cap\overline{B}\Rightarrow
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\left\{
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\begin{aligned}
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& \text{x}\in\overline{A} \\
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& \text{x}\in\overline{B}
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\end{aligned}
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\right.
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\Rightarrow
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x\in\overline{(A\cup\text{B})}\text{. Vissar att }\overline{}
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\end{align}
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$$ |