【数学I】例題2.1.12：逆・裏・対偶（One More）★

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% 例題I2.1.12：逆・裏・対偶（One More）★
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次の命題の逆，裏，対偶を述べよ．また，それらの真偽を調べよ．ただし，$x,a,b$は実数とする．

(1) $x=4$ならば，$x^2=16$

(2) $a+b\geqq 0$ならば，$a\geqq 0$かつ$b\geqq 0$

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% 例題I2.1.12の解答（One More）★
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(1) 逆は，「$x^2=16$ならば，$x=4$」

これは，偽である．反例は，$x=-4$

裏は，「$x\neq 4$ならば，$x^2\neq 16$」

これは，偽である．反例は，$x=-4$

対偶は，「$x^2\neq 16$ならば，$x\neq 4$」

これは，もとの命題が真$(x=4$ならば，$x^2=16)$であるから，真である．

(2) 逆は，「$a\geqq 0$かつ$b\geqq 0$ならば，$a+b\geqq 0$」

これは，明らかに成り立つから，真である．

裏は，「$a+b<0$ならば，$a<0$または$b<0$」

これは，裏の対偶，すなわち，逆が真であるから，真である．

対偶は，「$a<0$または$b<0$ならば，$a+b<0$」

これは，偽である．反例は，$a=-1,b=2$

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% 問題I2.1.12：逆・裏・対偶（One More）★
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次の命題の逆，裏，対偶を述べよ．また，それらの真偽を調べよ．ただし，$x,y$は実数とする．

(1) $x=-3$ならば，$x^2=9$

(2) $x+y>2$ならば，$x,y$の少なくとも1つは1より大きい．

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(1) 逆は，「$x^2=9$ならば，$x=-3$」

これは，偽である．反例は，$x=3$

裏は，「$x\neq-3$ならば，$x^2\neq 9$」

これは，偽である．反例は，$x=3$

対偶は，「$x^2\neq 9$ならば，$x\neq-3$」

これは，もとの命題が真$(x=-3$ならば，$x^2=9)$であるから，真である．

(2) 逆は，「$x>1$または$y>1$ならば，$x+y>2$」

これは，偽である．反例は，$x=2,y=-1$

裏は，「$x+y\leqq 2$ならば，$x\leqq 1$かつ$y\leqq 1$」

これは，偽である．反例は，$x=2,y=-1$

対偶は，「$x\leqq 1$かつ$y\leqq 1$ならば，$x+y\leqq 2$」

これは，明らかに成り立つから，真である．

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% 例題I2.1.12の側注：逆・裏・対偶（One More）★
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% 側注1
裏の反例は，逆の反例と同じものでよい．

% 側注2
$a\geqq 0$かつ$b\geqq 0$より，辺々を足し合わせると，$a+b\geqq 0$

% 側注3
逆と裏の真偽は一致する．

% 側注4
なお，もとの命題は偽である．

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% 問題I2.1.12の側注：逆・裏・対偶（One More）★
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% 側注1
裏の反例は，逆の反例と同じものでよい．

% 側注2
$x\leqq 1$かつ$y\leqq 1$より，辺々を足し合わせると，$x+y\leqq 2$

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% 例題I2.1.12の考え方・補足など：逆・裏・対偶（One More）★
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命題「$p\Longrightarrow q$」とその逆，裏，対偶との関係は，右の図のようになる．「$p\Longrightarrow q$」と「$\overline{q}\Longrightarrow\overline{p}$」，「$\overline{p}\Longrightarrow\overline{q}$」と「$q\Longrightarrow p$」は互いに対偶の関係にあり，その真偽は一致する．なお，真偽を答えるときは，明らかに真であることが成り立つのであれば証明を省略してもよいが，偽のときは反例を必ず示すように注意すること．

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命題とその対偶は，真偽が一致する．

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「$a<0$または$b<0$」は「$a,b$の少なくとも1つは$0$より小さい」などと表されることもある．