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命題の逆、裏、対偶の問題の解法 | 夢を叶える塾
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命題の逆、裏、対偶の問題の解法

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命題と対偶の真偽の性質を使った問題です。

基本問題

整数aに対し、a^2が31の倍数でないとき、aは31の倍数でないことを証明しなさい。

解き方

まず条件の否定から。
「x=1」という条件の否定はx \ne 1です。
x\leq 1」という条件の否定はx > 1です。
数直線上で表される点のような物であれば、その数直線の塗りつぶした部分を足すと数直線に戻るような関係になっています。

「自然数nに対してnが3以上のとき・・・」のときに否定を考える場合、気を付けなければなりません。
「自然数でないnに対してnが3未満・・・」としたくなるかもしれませんが「自然数n」は否定しちゃいけないのです。
「正の整数pに対してpが偶数ならば・・・」みたいな時も、pはあくまで正の整数です。
全体集合を変えてはいけない

次に逆、裏、対偶について。
命題p⇒qに対して、
q⇒pを逆
pの否定⇒qの否定を裏
qの否定⇒pの否定を対偶
と言います。
ちなみに逆の対偶は裏です。

逆、裏はややこしいかもしれませんが、
対偶の真偽が元の命題の真偽に一致する
という、何とも使い勝手の良さそうな性質があります。

p⇒qというのは集合としてとらえるとP\subset Qでしたね。
(pははみ出ると反例となり、はみ出ちゃいけないけど、qははみ出ていい)
対偶は\overline{Q}\subset \overline{P}です。
p⇒qが真ならPがすっぽりQに含まれている。
これを外側を見てみると、Qの外側はPの外側に覆われていますね。
故にp⇒qが真なら、qの否定⇒Pの否定も真です。
一方、Qの外側はPの外側に覆われていれば、PはQに覆われています。
故にqの否定⇒Pの否定が真なら、p⇒qも真です。

解説

整数aに対し、a^2が31の倍数でないとき、aは31の倍数でないことを証明しなさい。

命題
a^2が31の倍数でないとき、aは31の倍数でない」
とその対偶
「aが31の倍数のとき、a^2が31の倍数になる」
の真偽が一致します。
対偶が真であることを証明できれば元の命題が真であることを証明することができます

aが31の倍数であるならば、a=31n(nは整数)という形で表すことができます。
これを2乗して
(31n)^2=31(31n^2)
という事で31と整数の積ですから31の倍数です。
よってこの命題
「aが31の倍数のとき、a^2が31の倍数になる」
が真になり、その対偶である、
a^2が31の倍数でないとき、aは31の倍数でない」
も真になります。

終わりに

命題の真偽と対偶の真偽が一致するという性質は証明問題を解く際のテクニックとして使えるようになりましょう。
そのままでは中々証明できないが、対偶の証明は簡単だったという事もあります。
そのためにはまずは正しく否定を取れる必要があります。

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