鐃緒申鐃旬語技法 No.6

$\forall$ 鐃緒申 $\exists$ (鐃緒申鐃緒申3)

鐃緒申鐃緒申肋鐃緒申謄鐃緒申箸任鐃緒申襦Ｎ�申未鐃緒申鐃緒申鐃緒申鬚い董鐃緒申鐃緒申鐃緒申兒鐃緒申鐃緒申鐃緒申鐃緒申鐃暑こ鐃夙￥申

鐃淑わ申鐃緒申

鐃旬わ申痢鐃緒申鐃緒申鐃緒申鮖仮箸鐃緒申討鐃緒申匹鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申聾鐃緒申悗箸鐃緒申襦�
鐃緒申鐃瞬わ申1鐃緒申鐃渋緒申了鐃緒申鐃殉ででわ申鐃暑。
鐃緒申鐃緒申鐃熟簡件申覆鐃塾でわ申鐃暑こ鐃夙に越わ申鐃緒申鐃緒申鐃夙はなわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申砲匹鐃緒申鐃緒申討鐃緒申鐃殉わ申鐃准わ申覆鐃緒申鐃緒申砲鐃重�申鐃緒申鐃緒申鐃緒申覆匹鐃� 鐃縦わ申鐃緒申裏鐃緒申鐃緒申鐃緒申鐃春わ申足鐃緒申鐃銃わ申鐃宿わ申鐃緒申
鐃淑駕申鐃塾駕申鐃緒申盪駕申佑砲鐃緒申鐃緒申鐃宿わ申鐃緒申鐃緒申鐃塾わ申覆鐃緒申鐃�

$\displaystyle \exists x \forall y P(x,y)$
鐃緒申鐃緒申鐃緒申鐃緒申鐃祝わ申鐃緒申鐃銃わ申鐃祝つわ申鐃銃逸申鐃銃わ申 P(x,y) 鐃緒申鐃淑りた鐃縦よう鐃緒申鐃緒申鐃緒申鐃藷げわ申鐃緒申匹鐃緒申鐃� 鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申覆鐃緒申鐃緒申鐃淑�申鐃� 鐃緒申鐃緒申鐃緒申鵑押鐃緒申鐃緒申鐃� 鐃祝関わ申鐃緒申 (鐃緒申鐃縮には駕申鐃初か鐃塾刻申妙鐃淑￥申多鐃緒申鐃塾ワ申里鬚気个鐃緒申廛謄鐃緒申縫奪鐃緒申鐃緒申僂鐃緒申鐃�) 鐃緒申鐃銃わ申鐃祝つわ申鐃銃逸申鐃銃わ申鐃緒申鐃淑りた鐃縦わ申鐃夙を示わ申鐃緒申鐃夙になる。

鐃緒申: $\exists x\in$ $\mbox{${\mathbb{R}}$}$ $\forall y \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1273 $ (y^2\geq x)$$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃淑わ申鐃淑ら、鐃夙わ申鐃緒申鐃緒申鐃緒申个鐃緒申鐃�

$\displaystyle \forall y \in$ $\displaystyle \mbox{${\mathbb{R}}$}$ $% latex2html id marker 1279 $\displaystyle (y^2\geq 0) $$

鐃緒申鐃淑りた鐃縦わ申鐃緒申任鐃緒申襦�

$\displaystyle \forall y \exists x P(x,y)$
鐃緒申鐃緒申鐃緒申鐃緒申鐃祝は￥申鐃銃￥申鐃緒申鐃緒申鐃出わ申鐃銃わ申鐃曙ぞ鐃緒申鐃緒申鐃緒申鐃藷げわ申个茲わ申鐃�

鐃緒申:

$\displaystyle \forall y \exists x (x>y)$

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃淑わ申鐃淑ら、鐃銃￥申鐃塾実随申

鐃緒申鐃出わ申鐃銃￥申

鐃緒申鐃緒申箸鐃緒申鐃�

鐃緒申箸鐃緒申鐃処い鐃緒申鐃緒申任鐃緒申襦�

鐃緒申鐃緒申 5.1 鐃緒申鐃緒申:(1),(2)鐃夙わ申傍鐃緒申任鐃緒申襦ｏ申覆鐃緒申覆蕁￢随申鐃�5.1 鐃緒申鐃術わ申鐃緒申(1),(2) 鐃緒申鐃緒申鐃曙ぞ鐃緒申

() $\exists y\in$ $\mbox{${\mathbb{Q}}$}$ $_{>0} (y^2=3)$

() $\forall x\in$ $\mbox{${\mathbb{Q}}$}$ $_{>0}\exists y\in$ $\mbox{${\mathbb{Q}}$}$ $_{>0} (y^2=x)$

鐃緒申同鐃粛でわ申鐃暑。平鐃緒申鐃緒申鐃緒申鐃祝なわ申鐃緒申鐃� $% latex2html id marker 1312 $ \pm \sqrt{3}$$ 鐃祝限わ申譟� 鐃緒申鐃緒申鐃緒申有鐃緒申鐃緒申鐃叔はなわ申鐃緒申鐃初、鐃熟居申鐃叔わ申鐃暑。鐃殉わ申鐃緒申

$\displaystyle \exists y\in$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $\displaystyle _{>0} (y^2=x)$

鐃夙わ申鐃緒申

鐃祝つわ申鐃銃わ申命鐃緒申鐃�

鐃夙書くと￥申

鐃緒申

鐃緒申同鐃緒申鐃緒申鐃夙でわ申鐃所、

鐃緒申

$\displaystyle \forall x\in$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $\displaystyle _{>0} (P(x))$

鐃夙わ申鐃緒申命鐃緒申砲曚鐃緒申覆鐃淑わ申鐃緒申反鐃緒申

鐃緒申鐃緒申鐃緒申里鐃�

鐃盾偽鐃叔わ申鐃暑。

$\fbox{{\Large 問題}}$

鐃緒申鐃緒申 A-I 鐃緒申鐃淑駕申鐃塾よう鐃淑種申張鐃薯してわ申鐃暑。鐃緒申鐃緒申鐃塾種申張鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� 鐃緒申鐃塾随申鐃緒申鐃緒申鐃緒申由鐃緒申鵑欧峠劼戮覆鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃�A,C,D 鐃緒申鐃緒申鐃緒申箸鐃緒申鐃� 鐃熟わ申鐃潤か鐃緒申鐃緒申鐃緒申鐃緒申颪い討鐃緒申襦�

$\fbox{レベル1}$

(鐃緒申鐃緒申)鐃緒申鐃緒申 A 鐃熟￥申鐃瞬実随申鐃熟わ申鐃曙が鐃宿わ申覆鐃塾でわ申鐃獣てわ申鐃淑常申任鐃緒申鐃竣と醐申鐃獣わ申鐃緒申

鐃緒申鐃緒申 B 鐃緒申鐃緒申鐃淑常申亮多鐃� 鐃緒申存鐃淳わ申鐃緒申廚噺鐃緒申辰鐃緒申鐃�

$\fbox{レベル2}$

(鐃緒申鐃緒申)鐃緒申鐃緒申 C 鐃緒申鐃瞬わ申鐃淑わ申鐃緒申鐃宿わ申兵多鐃� 鐃緒申鐃縦誌申鐃獣てわ申鐃銃も、鐃緒た鐃緒申鐃熟わ申鐃緒申鐃緒申鐃淑常申鐃順き鐃緒申鐃渋随申鐃緒申鵑欧襪鰹申箸鐃緒申任鐃緒申鐃竣と醐申鐃獣わ申鐃緒申

(鐃緒申鐃緒申)鐃緒申鐃緒申 D 鐃緒申鐃瞬わた鐃緒申鐃緒申鐃塾ってわ申鐃暑あ鐃緒申多鐃� 鐃熟どわ申兵多鐃� 鐃緒申鐃� 鐃淑常申鐃順き鐃緒申鐃緒申鐃夙醐申鐃獣わ申鐃緒申

鐃緒申鐃緒申 E 鐃緒申鐃瞬わ申鐃淑わ申鐃緒申鐃宿わ申兵多鐃� 鐃緒申鐃縦誌申鐃獣てわ申鐃銃も、鐃緒た鐃緒申鐃熟実随申鐃緒申鐃緒申鐃術意わ申鐃銃￥申鐃緒申弔凌鐃緒申鐃緒申造鐃� 鐃祝でわ申鐃暑。鐃竣と醐申鐃獣わ申鐃緒申

鐃緒申鐃緒申 F 鐃緒申鐃瞬わた鐃緒申鐃熟実随申鐃緒申鐃緒申鐃術意わ申鐃銃￥申鐃緒申鐃淑わ申鐃緒申鐃宿わ申兵多鐃� 鐃緒申鐃縦誌申鐃獣てわ申鐃銃も、鐃緒申弔凌鐃緒申鐃緒申造鐃� 鐃叔わ申鐃緒申茲�申砲任鐃緒申襦ｏ申廚噺鐃緒申辰鐃緒申鐃�

鐃緒申鐃緒申 G 鐃緒申鐃瞬わた鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒申弔亮多鐃緒申鐃緒申儖佞鐃緒申董鐃� 鐃緒申鐃淑わ申鐃緒申鐃宿わ申兵多鐃� 鐃緒申鐃縦誌申鐃獣てわ申鐃銃も、鐃叔わ申鐃緒申茲�申砲任鐃緒申鐃竣と醐申鐃獣わ申鐃緒申

$\fbox{レベル 3}$ (鐃緒申鐃緒申泙任麓多鐃緒申魄靴辰討鐃緒申鐃緒申鐃緒申焚鐃緒申鐃緒申鐃緒申鐃叔わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒申)

鐃緒申鐃塾よう鐃淑ワ申鐃緒申鐃緒申鐃粛わ申鐃処う鐃緒申

鐃緒申鐃緒申鐃緒申鐃緒申鐃所、鐃緒申鐃緒申鐃緒申佑納鐃緒申里茲�申兵鐃緒申嚢圓覆鐃緒申襦�

(鐃緒申鐃�1).

鐃緒申蠅�申鐃緒申鐃� 鐃緒申鐃縦醐申鐃緒申鐃緒申

(鐃緒申鐃�2).

鐃緒申蠅�申鐃緒申鐃� 鐃緒申鐃縦醐申鐃緒申鐃緒申

(鐃緒申鐃�3).

鐃緒申蠅�申鐃緒申鐃� 鐃緒申鐃縦醐申鐃緒申鐃緒申

鐃緒申鐃塾とわ申鐃緒申

鐃緒申鐃緒申鐃緒申平鐃緒申鐃緒申( $\frac{x+y}{2}$ )鐃淑わ申鐃緒申鐃緒申両鐃緒申鐃緒申鐃�

鐃緒申鐃緒申奮鐃緒申覆鐃出醐申鐃塾常申鐃緒申鐃緒申

鐃緒申鐃緒申 H 鐃熟￥申鐃瞬わ申鐃塾ワ申鐃緒申鐃緒申如鐃緒申鐃所が1鐃緒申鐃旬にどわ申鐃緒申鐃緒申鐃� 鐃緒申鐃緒申辰討癲� 鐃緒申蠅�申悗泙鬚垢鐃出￥申鐃緒申蠅�申鐃緒申弔鐃緒申箸鐃緒申鐃緒申鐃緒申鐃緒申泙鐃緒申鐃緒申鐃� 鐃夙醐申鐃獣わ申鐃緒申

鐃緒申鐃緒申 I 鐃熟￥申鐃瞬わ申鐃塾ワ申鐃緒申鐃緒申聾鐃所が鐃緒申鐃殉わ申鐃緒申鐃緒申必鐃緒申鐃緒申鐃塾常申鐃緒申鐃叔わ申鐃粛￥申鐃緒申鐃夙醐申鐃獣わ申鐃緒申

$\fbox{宿題}$

鐃緒申僚鐃緒申鐃緒申鐃熟若干鐃緒申鐃緒申鐃緒申鐃緒申鐃銃わ申鐃淑わ申鐃緒申鐃盾す鐃暑。鐃緒申鐃緒申鐃叔￥申鐃銃種申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $\exists$ , $\forall$ 鐃祝関わ申鐃緒申鐃駿ワ申2鐃緒申鐃緒申戰鐃�3鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃緒申辰討澆襪鰹申函鐃� 鐃緒申鐃緒申麓鐃緒申鐃塾常申鐃初習鐃塾駕申鐃緒申箸鐃緒申鐃粛緒申鐃叔わ申鐃暑が鐃緒申鐃緒申鐃緒申鐃淑りそ鐃塾常申任六鐃緒申屬鐃緒申鐃緒申覆鐃緒申萄鐃緒申覆鐃緒申鐃緒申鐃緒申里鐃淑わ申鐃塾で￥申鐃緒申鐃緒申箸鐃緒申銅鐃緒申鐃殉でに案わ申鐃緒申鐃獣てわ申鐃緒申鐃緒申鐃夙をお器申鐃潤す鐃暑。

鐃緒申鐃緒申鐃銃随申鐃緒申鐃緒申鐃緒申辰鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申襪鰹申箸鐃緒申鐃峻わ申鐃銃わ申鐃暑。

鐃緒申鐃緒申鐃術誌申

鐃緒申鐃緒申鐃瞬号、名鐃緒申鐃緒申

鐃緒申張鐃緒申鐃緒申鐃緒申鐃祝わ申鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申由

A(鐃緒申) $\forall x\in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1406 $ (x\geq 3)$$ 鐃緒申鐃緒申多鐃緒申鐃緒申鐃� 鐃淑常申任呂覆鐃緒申鐃�

B

C(鐃緒申) $\forall x \exists y \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1414 $ (y\geq x+5)$$ 鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申鐃出わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃術意わ申鐃緒申鐃緒申匹鐃緒申鐃�

D(鐃緒申) $\exists x\in$ $\mbox{${\mathbb{R}}$}$ $\forall y \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1426 $ (x\geq y+5)$$ 鐃緒申鐃緒申鐃塾よう鐃緒申鐃緒申鐃緒申鐃獣わ申鐃夙わ申鐃緒申函鐃緒申鐃緒申覆鐃緒申鐃� 鐃夙わ申鐃緒申鐃緒申箸鐃緒申矛鐃盾が鐃緒申鐃緒申鐃暑。

E

F

G

H

I

鐃緒申鐃緒申鐃瞬号、名鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申0000 鐃緒申鐃緒申鐃緒申鐃� 鐃緒申郎鐃緒申鐃緒申鐃緒申

鐃緒申張鐃緒申鐃緒申鐃緒申鐃祝わ申鐃緒申鐃緒申鐃� 鐃緒申鐃緒申鐃緒申由

A(鐃緒申) $\forall x\in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1478 $ (x\geq 3)$$ 鐃緒申鐃緒申多鐃緒申鐃緒申鐃� 鐃淑常申任呂覆鐃緒申鐃�

B $\exists x\in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1486 $ (x\geq 5)$$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任鐃緒申襦�

C(鐃緒申) $\forall x \exists y \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1492 $ (y\geq x+5)$$ 鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申鐃出わ申鐃緒申鐃緒申鐃緒申 C 鐃緒申鐃緒申鐃術意わ申鐃緒申鐃緒申匹鐃緒申鐃�

D(鐃緒申) $\exists x\in$ $\mbox{${\mathbb{R}}$}$ $\forall y \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1502 $ (x\geq y+5)$$ 鐃緒申鐃緒申鐃塾よう鐃緒申鐃緒申鐃緒申鐃獣わ申鐃夙わ申鐃緒申函鐃緒申鐃緒申覆鐃緒申鐃� 鐃夙わ申鐃緒申鐃緒申箸鐃緒申矛鐃盾が鐃緒申鐃緒申鐃暑。

E $\forall x\in$ $\mbox{${\mathbb{R}}$}$ $\exists y\in$ $\mbox{${\mathbb{R}}$}$ 鐃緒申鐃緒申鐃淑わ申鐃緒申鐃緒申鐃出わ申鐃緒申鐃緒申鐃緒申 E 鐃緒申鐃夙わ申鐃緒申鐃緒申匹鐃緒申鐃�

F $\exists x\in$ $\mbox{${\mathbb{R}}$}$ $\forall y \in$ $\mbox{${\mathbb{R}}$}$ 鐃緒申鐃緒申鐃塾よう鐃緒申鐃緒申鐃緒申鐃獣わ申鐃夙わ申鐃緒申箸鐃緒申覆鐃緒申鐃� 鐃祝とわ申鐃縮件申發�申鐃緒申鐃緒申襦�

G $\exists x\in$ $\mbox{${\mathbb{R}}$}$ $\exists y\in$ $\mbox{${\mathbb{R}}$}$ $\forall z\in$ $\mbox{${\mathbb{R}}$}$ 鐃緒申 G 鐃緒申鐃夙件申鐃緒申弌鐃� 鐃宿のよう鐃緒申 $z\in$ $\mbox{${\mathbb{R}}$}$ 鐃緒申鐃出わ申鐃銃わ申鐃叔わ申鐃暑。

H $\forall x\in{\mbox{${\mathbb{Z}}$}}\exists y\in{\mbox{${\mathbb{Z}}$}}\exists z\in{\mbox{${\mathbb{Z}}$}}(z=\frac{x+y}{2})$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃出わ申鐃銃醐申蠅� 鐃緒申鐃緒申鐃緒申鐃緒申箸鐃緒申茲�申鐃� 鐃緒申鐃塾とわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃駿わ申鐃宿わ申鐃緒申

I $% latex2html id marker 1553 $ \forall x\in{\mbox{${\mathbb{Z}}$}}\exists y\in{\mbox{${\mathbb{Z}}$}}\forall z\in{\mbox{${\mathbb{Z}}$}}(z\neq\frac{x+y}{2})$$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃出わ申鐃銃醐申蠅� 鐃緒申鐃緒申鐃駿わ申 $\frac{x+y}{2}=\frac{1}{2}$ 鐃緒申鐃緒申鐃緒申鐃祝ならず鐃緒申蠅� 鐃夙わ申鐃銃どわ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃叔わ申鐃初け鐃叔わ申鐃暑。

: 鐃緒申鐃緒申文鐃緒申砲弔鐃緒申鐃�...

平鐃緒申19年11鐃緒申29鐃緒申

鐃緒申張鐃緒申	鐃緒申鐃緒申鐃祝わ申鐃緒申鐃緒申鐃�	鐃緒申鐃緒申	鐃緒申由
A(鐃緒申)	$\forall x\in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1406 $ (x\geq 3)$$	鐃緒申	鐃緒申多鐃緒申鐃緒申鐃� 鐃淑常申任呂覆鐃緒申鐃�
B
C(鐃緒申)	$\forall x \exists y \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1414 $ (y\geq x+5)$$	鐃緒申	鐃緒申鐃淑わ申鐃緒申鐃緒申鐃出わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃術意わ申鐃緒申鐃緒申匹鐃緒申鐃�
D(鐃緒申)	$\exists x\in$ $\mbox{${\mathbb{R}}$}$ $\forall y \in$ $\mbox{${\mathbb{R}}$}$ $% latex2html id marker 1426 $ (x\geq y+5)$$	鐃緒申	鐃緒申鐃塾よう鐃緒申鐃緒申鐃緒申鐃獣わ申鐃夙わ申鐃緒申函鐃緒申鐃緒申覆鐃緒申鐃� 鐃夙わ申鐃緒申鐃緒申箸鐃緒申矛鐃盾が鐃緒申鐃緒申鐃暑。
E
F
G
H
I