Teoria dos conjuntos

petras escreveu:A\Delta (B\Delta C) = (A\Delta B)\Delta C 


\\ \text{Propriedades Utilizadas:}~ \\ A \Delta B = \underbrace{(A\cap \overline{B})}\cup \underbrace{(\overline{A}\cap B)}\\ \\ \overline{A\cap B}=\overline{A}\cup \overline{B} \\ \\ A\cap(B\cup C) = (A\cap B)\cup(A\cap C) \\ \\\text{Lado Esquerdo} ~A\Delta(B\Delta C) = (A \cap\overline{B\Delta C})\cup(\overline{A}\cap((B\cap\overline{C})\cup(\overline{B}\cap C)))=\\\\ (A \cap \overline{((B\cup C)\cap(\overline{B}\cup \overline{C})}))\cup (\overline{A}\cap((B\cap\overline{C})\cup(\overline{B}\cap C)))=\\\\ (A\cap((\overline{B\cup C})\cup\overline{(\overline{B}\cup\overline{C}})))\cup(\overline{A}\cap B\cap\overline{C})\cup(\overline{A}\cap \overline{B}\cap C)\\\\ (A\cap ((\overline{B}\cap\overline{C})\cup(B\cap C)))\cup (\overline{A}\cap B\cap\overline{C})\cup(\overline{A}\cap \overline{B}\cap C)\\\\ \boxed{(A\cap\overline{B}\cap\overline{C})\cup(A\cap B\cap C)\cup (\overline{A}\cap B\cap\overline{C})\cup(\overline{A}\cap \overline{B}\cap C)}


\text{Lado Direito}~(A\Delta B)\Delta C=\\\\ (((A\cap\overline{B})\cup(\overline{A}\cap B))\cap \overline{C}))\cup(\overline{A\Delta B}\cap C)=\\\\ (((A\cap \overline{B})\cup(\overline{A}\cap B))\cap\overline{C})\cup(\overline{((A \cup B)\cap(\overline{A}\cup \overline{B}))}\cap C)=\\\\ (A\cap\overline{B}\cap\overline{C})\cup(\overline{A}\cap B\cap\overline{C})\cup((\overline{A\cup B})\cup\overline{(\overline{A}\cup\overline{B}}))\cap C)=\\\\ (A\cap\overline{B}\cap\overline{C})\cup(\overline{A}\cap B\cap\overline{C})\cup(((\overline{A}\cap\overline{B})\cup(A\cap B)\cap C)=\\\\ \boxed{(A\cap\overline{B}\cap\overline{C})\cup(\overline{A}\cap B\cap\overline{C})\cup(\overline{A}\cap\overline{B}\cap C)\cup(A\cap B\cap C)}