【数学I】例題2.1.13：対偶を用いた証明1（One More）★★

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% 例題I2.1.13：対偶を用いた証明1（One More）★★
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次の命題を証明せよ．ただし，$m,n$を整数とする．

(1) $n^2$が偶数ならば，$n$は偶数である．

(2) $mn$が偶数ならば，$m$または$n$は偶数である．

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% 例題I2.1.13の解答（One More）★★
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(1) もとの命題の対偶「$n$が奇数ならば，$n^2$は奇数である」を証明する．

$n$は奇数であるとき，整数$k$を用いて$n=2k+1$と表せるから，

\[
n^2=(2k+1)^2=2(2k^2+2k)+1
\]

$k$は整数より，$2k^2+2k$も整数であるから，$n^2$は奇数である．

よって，対偶が証明されたから，もとの命題も成り立つ．$\blacksquare$

(2) もとの命題の対偶「$m,n$がともに奇数ならば，$mn$は奇数である」を証明する．

$m,n$は奇数であるとき，整数$k,l$を用いて$m=2k+1,n=2l+1$と表せるから，

\[
mn=(2k+1)(2l+1)=2(2kl+k+l)+1
\]

$k,l$は整数より，$2kl+k+l$も整数であるから，$mn$は奇数である．

よって，対偶が証明されたから，もとの命題も成り立つ．$\blacksquare$

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% 問題I2.1.13：対偶を用いた証明1（One More）★★
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次の命題を証明せよ．ただし，$a,b$を整数とする．

(1) $a^3$が偶数ならば，$a$は偶数である．

(2) $a^2+b^2$が奇数ならば，積$ab$は偶数である．

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% 問題I2.1.13の解答（One More）★★
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(1) もとの命題の対偶「$a$が奇数ならば，$a^3$は奇数である」を証明する．

$a$は奇数であるとき，整数$k$を用いて$a=2k+1$と表せるから，

\[
a^3=(2k+1)^3=8k^3+12k^2+6k+1=2(4k^3+6k^2+3k)+1
\]

$k$は整数より，$4k^3+6k^2+3k$も整数であるから，$a^3$は奇数である．

よって，対偶が証明されたから，もとの命題も成り立つ．$\blacksquare$

(2) もとの命題の対偶「積$ab$が奇数ならば，$a^2+b^2$は偶数である」を証明する．

$ab$は奇数であるとき，$a,b$はともに奇数であり，整数$k,l$を用いて$a=2k+1,b=2l+1$と表せるから，

\begin{align*}
a^2+b^2&=(2k+1)^2+(2l+1)^2\\
&=4k^2+4k+1+4l^2+4l+1\\
&=2(2k^2+2k+2l^2+2l+1)
\end{align*}

$k,l$は整数より，$2k^2+2l^2+2k+2l+1$も整数であるから，$a^2+b^2$は偶数である．

よって，対偶が証明されたから，もとの命題も成り立つ．$\blacksquare$

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% 例題I2.1.13の側注：対偶を用いた証明1（One More）★★
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% 側注1
$n=2k-1$とおいてもよい．

% 側注2
$2k^2+2k$が整数であることを記すように注意すること．

% 側注3
「$m$が偶数または$n$が偶数」の否定は，「$m$は奇数かつ$n$は奇数」，すなわち，「$m,n$はともに奇数」である．

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% 問題I2.1.13の側注：対偶を用いた証明1（One More）★★
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% 側注1
$a=2k-1$としてもよい．

% 側注2
$4k^3+6k^2+3k$が整数であることを記すように注意すること．

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% 例題I2.1.13の考え方・補足など：対偶を用いた証明1（One More）★★
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% 考え方
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(1) 「$n^2$が偶数$\Longrightarrow$ $n$は偶数」を直接証明するのは，$n^2$が扱いにくいので難しい（素因数分解の一意性を用いれば証明できるが，敷居が高い）．そこで，このように直接証明することが難しいときは，対偶を用いて証明することを考える．対偶を考えると，次数の低い$n$を含む条件が仮定となり，扱いやすくなる．

(2) $mn$が偶数であることから，「$m$が偶数または$n$が偶数$\cdots(\mathrm{i})$」であることを直接証明するのは，3つの場合があり扱いにくい．そこで，対偶を考えると，(i)は「$m$が奇数かつ$n$が奇数」となり，扱いやすくなる．

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

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% 対偶証明法
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直接証明するのが難しいときは，対偶を用いて証明する．

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$\blacksquare$はハルモス記号や墓石記号といい，証明の終わりを示す（$\square$などでも表される）．なお，証明の終わりに必ずしも記す必要はない．

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% 余談
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例えば，ある整数$k$を用いると，整数$n$は次のように表すことができる．

(i) $2k,2k+1$（偶数，奇数）

(ii) $3k,3k+1,3k+2$（3で割った余りが$0,1,2$）

(iii) $pk,pk+1,pk+2,\ldots,pk+(p-1)$($p$で割った余りが$0,1,2,\ldots,p-1$)

これらを用いると，(1)と同様の手法で，「$n^2$が3の倍数ならば，$n$は3の倍数である」や，「$n^2$が5の倍数ならば，$n$は$5$の倍数である」などを示すことができる．