【数学I】例題2.1.3：不等式で表される集合（One More）★★

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% 例題I2.1.3：不等式で表される集合（One More）★★
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実数全体を全体集合とし，その2つの部分集合を$A=\{x\mid x-1>0\}$,$B=\{x\mid|x-1|\leqq 2\}$とするとき，次の集合を求めよ．

(1) $A\cap B$

(2) $A\cup\overline{B}$

(3) $\overline{A\cup B}$

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% 例題I2.1.3の解答（One More）★★
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$x-1>0$より，$x>1$

よって，$A=\{x\lvert x>1\}$

$|x-1|\leqq 2$より，$-2\leqq x-1\leqq 2$

すなわち，$-1\leqq x\leqq 3$

よって，$B=\{x\mid-1\leqq x\leqq 3\}$

(1) 右の数直線より，

$A\cap B=\{x\lvert 1<x\leqq 3\}$

(2) $\overline{B}=\{x\mid x<-1,3<x\}$であるから，右の数直線より，

$A\cup\overline{B}=\{x\mid x<-1,1<x\}$

(3) 右の数直線より，$A\cup B=\{x\mid x\geqq-1\}$であるから，

$\overline{A\cup B}=\{x\mid x<-1\}$

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% 問題I2.1.3：不等式で表される集合（One More）★★
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実数全体を全体集合とし，その2つの部分集合を$A=\{x\mid x+3<0\}$,$B=\{x\mid|x+3|\leqq 1\}$とするとき，次の集合を求めよ．

(1) $A\cup B$

(2) $A\cap\overline{B}$

(3) $\overline{A\cap B}$

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% 問題I2.1.3の解答（One More）★★
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$x+3<0$より，$x<-3$

よって，$A=\{x\lvert x<-3\}$

$|x+3|\leqq 1$より，$-1\leqq x+3\leqq 1$

すなわち，$-4\leqq x\leqq-2$

よって，$B=\{x\mid-4\leqq x\leqq-2\}$

(1) 右の数直線より，

$A\cup B=\{x\lvert x\leqq-2\}$

(2) $\overline{B}=\{x\mid x<-4,-2<x\}$であるから，右の数直線より，

$A\cap\overline{B}=\{x\mid x<-4\}$

(3) 右の数直線より，$A\cap B=\{x\mid-4\leqq x<-3\}$であるから，

$\overline{A\cap B}=\{x\mid x<-4,-3\leqq x\}$

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% 例題I2.1.3の側注：不等式で表される集合（One More）★★
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% 側注1
$A\cap B$は共通部分である．

% 側注2
端点を含むか否かをよく考える（解を$x\leqq-1,3\leqq x$とするのは誤りであるので注意すること）．

% 側注3
集合$A,B$の条件を表す不等式を解く．

% 側注4
$|x|\leqq a$（$a>0$）の解は，$-a\leqq x\leqq a$

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% 問題I2.1.3の側注：不等式で表される集合（One More）★★
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% 側注1
$A\cup B$は和集合である．

% 側注2
端点を含むか否かをよく考える．

% 側注3
集合$A,B$の条件を表す不等式を解く．

% 側注4
$|x|\leqq a$（$a>0$）の解は，$-a\leqq x\leqq a$

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% 例題I2.1.3の考え方・補足など：不等式で表される集合（One More）★★
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% 考え方
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全体集合が実数全体の場合，要素を書き並べて表すことは難しいので，要素の満たす条件を述べて表す．また，集合を視覚化して，数直線で表すとよい．なお，端点を含むか否かにはよく注意すること．