問題の解答
検索用コード（LaTeX）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I2.1.5：3つの集合の共通部分，和集合（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$U=\{1,2,3,4,5,6,7,8,9,10,11,12\}$を全体集合とする．$U$の部分集合$A,B,C$を$A=\{n\mid n\text{は12の正の約数}\}$，$B=\{n\mid n\text{は奇数}\}$，$C=\{n\mid n\text{は10の正の約数}\}$とするとき，次の集合を求めよ．
(1) $A\cap B\cap C$
(2) $(A\cup B)\cap C$
(3) $(A\cap C)\cup(B\cap C)$
(4) $\overline{A}\cap\overline{B}\cap\overline{C}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I2.1.5の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$A=\{1,2,3,4,6,12\},B=\{1,3,5,7,9,11\},C=\{1,2,5,10\}$であり，与えられた条件をもとに，$U,A,B,C$をベン図で表すと，下の図のようになる．
(1) $A\cap B\cap C$は，$A$，$B$，$C$の共通部分であるから，
\[
A\cap B\cap C=\{1\}
\]
(2) $A\cup B=\{1,2,3,4,5,6,7,9,11,12\}$より，
\[
(A\cup B)\cap C=\{1,2,5\}
\]
(3) $A\cap C=\{1,2\},B\cap C=\{1,5\}$より，
\[
(A\cap C)\cup(B\cap C)=\{1,2,5\}
\]
(4) ド・モルガンの法則より，
\[
\overline{A}\cap\overline{B}\cap\overline{C}=\overline{A\cup B\cup C}=\{8\}
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I2.1.5：3つの集合の共通部分，和集合（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$U=\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\}$を全体集合とする．$U$の部分集合$A,B,C$を$A=\{n\mid n\text{は10の正の約数}\}$，$B=\{n\mid n\text{は偶数}\}$，$C=\{n\mid n\text{は14の正の約数}\}$とするとき，次の集合を求めよ．
(1) $A\cap B\cap C$
(2) $(A\cup B)\cap C$
(3) $(A\cap C)\cup(B\cap C)$
(4) $\overline{(\overline{A}\cap\overline{C})\cup(\overline{B}\cap\overline{C})}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I2.1.5の解答（One More）★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$A=\{1,2,5,10\},B=\{2,4,6,8,10,12,14\},C=\{1,2,7,14\}$であり，与えられた条件をもとに，$U,A,B,C$をベン図で表すと，下の図のようになる．
(1) $A\cap B\cap C$は，$A$，$B$，$C$の共通部分であるから，
\[
A\cap B\cap C=\{2\}
\]
(2) $A\cup B=\{1,2,4,5,6,8,10,12,14\}$より，
\[
(A\cup B)\cap C=\{1,2,14\}
\]
(3) $A\cap C=\{1,2\},B\cap C=\{2,14\}$より，
\[
(A\cap C)\cup(B\cap C)=\{1,2,14\}
\]
(4) ド・モルガンの法則より，
\begin{align*}
\overline{(\overline{A}\cap\overline{C})\cup(\overline{B}\cap\overline{C})}&=\overline{(\overline{A}\cap\overline{C})}\cap\overline{(\overline{B}\cap\overline{C})}\\
&=(\overline{\overline{A}}\cup\overline{\overline{C}})\cap(\overline{\overline{B}}\cup\overline{\overline{C}})\\
&=(A\cup C)\cap(B\cup C)\\
&=(A\cap B)\cup C\\
&=\{1,2,7,10,14\}
\end{align*}
あわせて読みたい
【数学I】2章：集合と命題（基本事項）
検索用コード（LaTeX） 本文・解答側注 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 基本事項I2.1.1：集合（One More） %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%...
あわせて読みたい
【数学I】2章：集合と命題（節末問題・章末問題）
節末I2.1.1〜I2.1.6の解答 節末I2.1.1節末I2.1.2節末I2.1.3節末I2.1.4節末I2.1.5節末I2.1.6 リンク（関連例題） https://onemath.net/onemorei-reidai2-1-3 https://one...