\newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} \colMM{red}{{\bf 和集合} } & \colMM{red}{\ は A たす B ひく {\bf 共通部分}}\\ n(A \cup B) &= n(A)+n(B)-n(A \cap B)\\ \\ とくに, \ \\ n(A \cup B) =&\varnothing のとき\colMM{red}{ n(A \cap B)=0}\\ n(A \cup B) &= n(A)+n(B)\colNS{lightgray}{-n(A \cap B)} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
\newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{\normalsize#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} \colMM{red}{{\bf 補集合}\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく\ {\bf 元の集合}}\\ n(\overline{\,A\,}) &= n(U)-n(A)\\ & \colMM{red}{ バーを取る!\Uarr}\\ \\ \colMM{red}{{\bf 補集合 }\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく {\bf 元の集合}}\\ n(\overline{\,\colBX{bisque}{集合の式}\,}) &= n(U)-n(\colBX{bisque}{集合の式})\\ & \colMM{red}{ バーを取る!\Uarr} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
補集合の線が切れているのをつなぐ
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} \colMM{red}{\bf 線が切れてる} & \colMM{red}{\bf ①つなぐと}\\ \large\overline{\,A\,} \colBX{mistyrose}{$\cap$} \overline{\,B\,} &\large = \overline{A \colBX{mistyrose}{$\cup$} B}\\ \colMM{red}{\bf \Uarr ②記} & \colMM{red}{\bf 号が逆に \Uarr} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} \colMM{red}{\bf 線が切れてる} & \colMM{red}{\bf ①つなぐと}\\ \large\overline{\,A\,} \colBX{mistyrose}{$\cup$} \overline{\,B\,} &\large = \overline{A \colBX{mistyrose}{$\cap$} B}\\ \colMM{red}{\bf \Uarr ②記} & \colMM{red}{\bf 号が逆に \Uarr} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
補集合の線を切る
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} \colMM{red}{\bf つながってる} & \colMM{red}{\bf ①切ると}\\ \large\overline{\,A \colBX{mistyrose}{$\cap$} B\,} &\large = \overline{\,A\,} \colBX{mistyrose}{$\cup$} \overline{\,B\,}\\ \colMM{red}{\bf \Uarr ②記} & \colMM{red}{\bf 号が逆に \Uarr} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
\newcommand\colNS[2]{\color{#1}#2\color{black}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} \begin{align*} \colMM{red}{\bf つながってる} & \colMM{red}{\bf ①切ると}\\ \large\overline{\,A \colBX{mistyrose}{$\cup$} B\,} &\large = \overline{\,A\,} \colBX{mistyrose}{$\cap$} \overline{\,B\,}\\ \colMM{red}{\bf \Uarr ②記} & \colMM{red}{\bf 号が逆に \Uarr} \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
集合 A,\ B が全体集合 U の部分集合で,
n(U)=40,\ n(A)=23,
n(B)=15,\ n(A \cap B)=3
であるとき,次の集合の要素の個数を求めよ。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{40} \def\nA{23} \def\nB{15} \def\nAcapB{3} \def\Kotae{35} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 和集合} } & \colMM{red}{\ \ は A \ \ たす B ひく {\bf 共通部分}}\\ n(A \cup B) &= \colBX{bisque}{$n(A)$}+\colBX{palegreen}{$n(B)$}-\colBX{violet}{$n(A \cap B)$}\\ \\ &= \colBX{bisque}{$\nA$}+\colBX{palegreen}{$\nB$}-\colBX{violet}{$\nAcapB$}\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{40} \def\nA{23} \def\nB{15} \def\nAcapB{3} \def\Kotae{17} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 補集合}\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく\ {\bf 元の集合}}\\ n(\overline{\,A\,}) &= n(U) - \colBX{bisque}{$n(A)$}\\ & \colMM{red}{ \ バーを取る!\Uarr}\\ &= \nU-\colBX{bisque}{$\nA$}\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{40} \def\nA{23} \def\nB{15} \def\nAcapB{3} \def\Kotae{25} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 補集合}\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく\ {\bf 元の集合}}\\ n(\overline{\,B\,}) &= n(U) - \colBX{bisque}{$n(B)$}\\ & \colMM{red}{ \ バーを取る!\Uarr}\\ &= \nU-\colBX{bisque}{$\nB$}\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{40} \def\nA{23} \def\nB{15} \def\nAcapB{3} \def\nAcupB{35} \def\Kotae{5} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 補集合 }\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく {\bf 元の集合}}\\ n(\overline{\,A \cup B\,}) &= n(U) - \colBX{bisque}{$n(A \cup B)$} \colMM{orange}{{\bf 和集合}は}\\ & \colMM{orange}{ A \ たす B \ ひく {\bf 共通部分}}\\ &= \nU-\left\{\colBX{bisque}{$n(A)+n(B)-n(A \cap B)$}\right\}\\ \\ &= \nU-\{\nA+\nB-\nAcapB\}\\ \\ &= \nU-\nAcupB\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{40} \def\nA{23} \def\nB{15} \def\nAcapB{3} \def\nAcupB{35} \def\Kotae{37} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{\bf 線が切れてる} & \colMM{red}{\bf ①つなぐと}\\ n(\overline{\,A\,} \cup \overline{\,B\,}) &= n(\overline{\,A \cap B\,}) \colMM{orange}{{\bf 補集合}は}\\ \colMM{red}{\bf \Uarr ②\ 記} & \colMM{red}{\bf 号が逆に\ \Uarr}\\ &= n(U) - \colBX{bisque}{$n(A \cap B)$}\\ & \colMM{orange}{ 全体 ひく {\bf 元の集合}}\\ &= \nU-\nAcapB\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{40} \def\nA{23} \def\nB{15} \def\nAcapB{3} \def\nAcupB{35} \def\Kotae{37} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{\bf 線が切れてる} & \colMM{red}{\bf ①つなぐと}\\ n(\overline{\,A\,} \cup \overline{\,B\,}) &= n(\overline{\,A \cap B\,}) \colMM{orange}{{\bf 補集合}は}\\ \colMM{red}{\bf \Uarr ②\ 記} & \colMM{red}{\bf 号が逆に\ \Uarr}\\ &= n(U) - \colBX{bisque}{$n(A \cap B)$}\\ & \colMM{orange}{ 全体 ひく {\bf 元の集合}}\\ &= \nU-\nAcapB\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
集合 A,\ B が全体集合 U の部分集合で,
n(U)=20,\ n(A)=13,
n(B)=9,\ n(A \cap B)=5
であるとき,次の集合の要素の個数を求めよ。
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{20} \def\nA{13} \def\nB{9} \def\nAcapB{5} \def\Kotae{17} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 和集合} } & \colMM{red}{\ \ は A \ \ たす B ひく {\bf 共通部分}}\\ n(A \cup B) &= \colBX{bisque}{$n(A)$}+\colBX{palegreen}{$n(B)$}-\colBX{violet}{$n(A \cap B)$}\\ \\ &= \colBX{bisque}{$\nA$}+\colBX{palegreen}{$\nB$}-\colBX{violet}{$\nAcapB$}\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{20} \def\nA{13} \def\nB{9} \def\nAcapB{5} \def\Kotae{7} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 補集合}\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく\ {\bf 元の集合}}\\ n(\overline{\,A\,}) &= n(U) - \colBX{bisque}{$n(A)$}\\ & \colMM{red}{ \ バーを取る!\Uarr}\\ &= \nU-\colBX{bisque}{$\nA$}\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{20} \def\nA{13} \def\nB{9} \def\nAcapB{5} \def\Kotae{11} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 補集合}\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく\ {\bf 元の集合}}\\ n(\overline{\,B\,}) &= n(U) - \colBX{bisque}{$n(B)$}\\ & \colMM{red}{ \ バーを取る!\Uarr}\\ &= \nU-\colBX{bisque}{$\nB$}\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{20} \def\nA{13} \def\nB{9} \def\nAcapB{5} \def\nAcupB{17} \def\Kotae{3} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 補集合 }\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく {\bf 元の集合}}\\ n(\overline{\,A \cup B\,}) &= n(U) - \colBX{bisque}{$n(A \cup B)$} \colMM{orange}{{\bf 和集合}は}\\ & \colMM{orange}{ A \ たす B \ ひく {\bf 共通部分}}\\ &= \nU-\left\{\colBX{bisque}{$n(A)+n(B)-n(A \cap B)$}\right\}\\ \\ &= \nU-\{\nA+\nB-\nAcapB\}\\ \\ &= \nU-\nAcupB\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{20} \def\nA{13} \def\nB{9} \def\nAcapB{5} \def\nAcupB{17} \def\Kotae{15} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{\bf 線が切れてる} & \colMM{red}{\bf ①つなぐと}\\ n(\overline{\,A\,} \cup \overline{\,B\,}) &= n(\overline{\,A \cap B\,}) \colMM{orange}{{\bf 補集合}は}\\ \colMM{red}{\bf \Uarr ②\ 記} & \colMM{red}{\bf 号が逆に\ \Uarr}\\ &= n(U) - \colBX{bisque}{$n(A \cap B)$}\\ & \colMM{orange}{ 全体 ひく {\bf 元の集合}}\\ &= \nU-\nAcapB\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan
↓この問題へのリンクはこちら(右クリックで保存)
【解答】
\def\nU{20} \def\nA{13} \def\nB{9} \def\nAcapB{5} \def\nAcupB{17} \def\Kotae{15} % \newcommand\colNS[2]{\textcolor{#1}{#2}} \newcommand\colUL[2]{\textcolor{#1}{\underline{\color{black}#2}}} \newcommand\colMM[2]{\color{#1}\scriptsize #2\color{black}} \newcommand\colBX[2]{\colorbox{#1}{#2}} \newcommand\colFR[2]{\textcolor{#1}{\fbox{\color{black}#2}}} \newcommand\colFB[2]{\textcolor{#1}{\fbox{\scriptsize\bf\color{#1}#2}}} % \begin{align*} \colMM{red}{{\bf 補集合 }\ } & \colMM{red}{\ \ は\ \ \ 全体\ \ ひく {\bf 元の集合}}\\ n(\overline{\,A \cap B\,}) &= n(U) - \colBX{bisque}{$n(A \cap B)$}\\ \\ &= \nU-\nAcapB\\ \\ &= \Kotae \end{align*} %1 orange,bisque %2 green,palegreen %3 magenta, violet %4 deepskyblue, lightcyan