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倍数の集合の個数の問題の解法 | 夢を叶える塾
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倍数の集合の個数の問題の解法

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倍数の集合の個数に関する問題です。

基本問題

(1)100以下の自然数のうち、「4かつ7の倍数」、「4または7の倍数」、「4と7の少なくとも一方で割り切れない整数」の個数を求めなさい。
(2)100より大きく200以下の自然数のうち、「4かつ7の倍数」、「4または7の倍数」、「4と7の少なくとも一方で割り切れない整数」の個数を求めなさい。

解き方

集合の要素の個数の問題ではn(A\cup B)=n(A)+n(B)-n(A\cap B)を使いましょう。
ベン図を描くとこの公式も明らかでわかりやすいと思います
n(A\cap B)=n(A)+n(B)-n(A\cup B)と変形もできますね。

補集合の個数n(\overline{A})場合は、全体集合の個数からn(A)を引いても良いですね。
また、ド・モルガンの法則も有効です。

解説

A=\{4n|4n \leq 100,nは自然数\}B=\{7n|7n \leq 100,nは自然数\}C=\{4n|4n \leq 200,nは自然数\}D=\{7n|7n \leq 200,nは自然数\}とします。

(1)100以下の自然数のうち、「4かつ7の倍数」、「4または7の倍数」、「4と7の少なくとも一方で割り切れない整数」の個数を求めなさい。
まず全体集合が100以下の自然数になりますからA,Bの全体集合Uを100以下の自然数とします。

「4かつ7の倍数」の個数
AとBの共通部分の集合の個数n(A\cap B)になります。

n(A\cap B)は100以下の28の倍数の個数ですから、100\div 28=3.5..よりn(A\cap B)=3

「4または7の倍数」の個数
AとBの和集合の個数n(A\cup B)になります。

n(A)は100以下の4の倍数の個数ですから、100\div 4=25よりn(A)=25
n(B)は100以下の7の倍数の個数ですから、100\div 7=14.2..よりn(B)=14
n(A\cap B)=3でした。
n(A\cup B)=n(A)+n(B)-n(A\cap B)=25+14-3=36
※最後の足し算引き算の誤りを指摘いただいて、2018/06/03修正しました!ありがとうございました!

「4と7の少なくとも一方で割り切れない整数」の個数
4で割り切れない、もしくは7で割り切れないという事ですから、個数n(\overline{A} \cup \overline{B})になります。
4と7の最小公倍数の28の倍数の集合Eの補集合\overline{E}が問題の集合になるという事を使ってn(\overline{E})を求めても良いですが、集合が等しくなることは証明して求めてくださいね。
マーク式で時間が足りない場合は「感覚」で進めることも時には必要ですが、「感覚」で進めてしまうと間違えたり、証明が不足していれば減点されてしまいますから注意しましょう
今回はEのような集合を使わず、既存の集合だけを使って進めていきたいと思います。

n(\overline{A} \cup \overline{B})=n(\overline{A})+n(\overline{B})-n(\overline{A} \cap \overline{B})
で求める事もできますが、\overline{A} \cap \overline{B}は求めていません。
求めるのが大変そうですね。

ド・モルガンの法則を使って、\overline{A} \cup \overline{B}=\overline{A\cap B}になります。
もちろん集合が同じわけですから、n(\overline{A} \cup \overline{B})=n(\overline{A\cap B})になります。
n(\overline{A} \cup \overline{B})=n(\overline{A\cap B})=n(U)-n(A\cap B)
これで求める事ができそうですね。

n(\overline{A} \cup \overline{B})=n(\overline{A\cap B})=n(U)-n(A\cap B)=100-3=97

(2)100より大きく200以下の自然数のうち、「4かつ7の倍数」、「4または7の倍数」、「4と7の少なくとも一方で割り切れない整数」の個数を求めなさい。
C,Dの全体集合U’を200以下の自然数としてそれぞれ200以下の個数を求め、(1)で求めた個数を引いてみましょう。

「4かつ7の倍数」の個数
CとDの共有部分の集合の個数からAとBの共通部分の集合の個数を引いたn(C\cap D)-n(A\cap B)になります。

n(C\cap D)は200以下の28の倍数の個数ですから、200\div 28=7.1..よりn(C\cap D)=7
n(C\cap D)-n(A\cap B)=7-3=4

「4または7の倍数」の個数
CとDの和集合の個数からAとBの和集合の個数を引いたn(C\cup D)-n(A\cup B)になります。

n(C)は100以下の4の倍数の個数ですから、200\div 4=50よりn(C)=50
n(D)は100以下の7の倍数の個数ですから、200\div 7=28.5..よりn(D)=28
n(C\cap D)=7でした。
n(C\cup D)=n(C)+n(D)-n(C\cap D)=50+28-7=71
n(C\cup D)-n(A\cup B)=71-35=36

「4と7の少なくとも一方で割り切れない整数」の個数
200以下のものから100以下のものを引くのでn(\overline{C} \cup \overline{D})-n(\overline{A} \cup \overline{B})です。

100以下の計算と同様にド・モルガンの法則を使ってn(\overline{C} \cup \overline{D})を求めます。
n(\overline{C} \cup \overline{D})=n(\overline{C\cap D})=n(U')-n(C\cap D)=200-7=193
n(\overline{C} \cup \overline{D})-n(\overline{A} \cup \overline{B})=193-97=96

終わりに

公式を覚えなくても、ベン図さえかければ公式は直ぐに出てきますね。
補集合が出てくるケースはド・モルガンを使えると間違いも減って楽もできると思います。

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