sep-2-14_54

LaTeX/Mängder 2.md

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

LaTeX/Mängder 3.md

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+- Kartesisk produkt:
+	- $A\cdot\text{X} \overset{\mathrm{def}}{=} \{(a,b): a \in A, b \in B\}$,
+	-