disM/LaTeX/Mängder 2.md

• 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}\} = (A\cup{B})\setminus(A\cap{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}