【数学I】例題2.1.10：条件の否定（One More）★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I2.1.10：条件の否定（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の条件の否定を述べよ．ただし，$a,b,c,x$を実数，$m,n$を整数とする．

(1) $1<x\leqq 2$

(2) $x<-2$または$x\geqq 3$

(3) $x=1$または$x=-1$

(4) $m,n$はともに偶数

(5) $a=b=c=0$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I2.1.10の解答（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 「$1<x\leqq 2$」は，「$x>1$かつ$x\leqq 2$」のことである．

よって，否定は，「$x\leqq 1$または$x>2$」

(2) 否定は，「$x\geqq-2$かつ$x<3$」

すなわち，「$-2\leqq x<3$」

(3) 否定は，「$x\neq 1$かつ$x\neq-1$」

(4) 「整数$m,n$はともに偶数」は，「整数$m$は偶数かつ整数$n$は偶数」のことである．

よって，否定は，「整数$m$は奇数または整数$n$は奇数」

(5) 「$a=b=c=0$」は，「$a=0$かつ$b=0$かつ$c=0$」のことである．

よって，否定は，$\text{「}a\neq 0\text{または}b\neq 0\text{または}c\neq 0\text{」}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I2.1.10：条件の否定（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

次の条件の否定を述べよ．ただし，$x,y$を実数とする．

(1) $0\leqq x<5$

(2) $x$は1でも2でもない

(3) $x=0$または$x=2$

(4) $x,y$の少なくとも一方は1である．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I2.1.10の解答（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) 「$0\leqq x<5$」は，「$x\geqq 0$かつ$x<5$」のことである．

よって，否定は，「$x<0$または$x\geqq 5$」

(2) 「$x$は1でも2でもない」は，「$x\neq 1$かつ$x\neq 2$」のことである．

よって，否定は，「$x=1$または$x=2$」

(3) 否定は，「$x\neq 0$かつ$x\neq 2$」

(4) 「$x,y$の少なくとも一方は1である」は「$x=1$または$y=1$」のことである．

よって，否定は，「$x\neq 1\text{かつ}y\neq 1$」

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I2.1.10の側注：条件の否定（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
「整数$m,n$の少なくとも1つは奇数」としてもよい．

% 側注2
「$a,b,c$のうち少なくとも1つは0ではない」としてもよい．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I2.1.10の側注：条件の否定（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$x$は1または2である．

% 側注2
$x$も$y$も1ではない．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I2.1.10の考え方・補足など：条件の否定（One More）★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 考え方
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

条件を，「かつ」「または」を用いて言い換えることができるかを考える．

(4) 「ともに」は「かつ」を用いて言い換えることができる．「整数$m,n$はともに偶数」$\Longleftrightarrow$「整数$m$が偶数かつ整数$n$が偶数」であり，その否定を考える．

\begin{tabular}{|c||c|c|c|c|}\hline
$m$&偶&偶&奇&奇\\\hline
$n$&偶&奇&偶&奇\\\hline
\end{tabular}

(5) $a=b=c=0$は，$a=0$かつ$b=0$かつ$c=0$であることから，次のド・モルガンの法則を用いる（ド・モルガンの法則は3つの集合や条件についても成り立つ）．

\[
\overline{p\text{かつ}q\text{かつ}r}\Longleftrightarrow\overline{p}\text{または}\overline{q}\text{または}\overline{r}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ド・モルガンの法則
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\[
\overline{p\text{かつ}q}\Longleftrightarrow\overline{p}\text{または}\overline{q},\overline{p\text{または}q}\Longleftrightarrow\overline{p}\text{かつ}\overline{q}
\]