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Updated Mar 29, 2026

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集合と命題の基礎

集合と命題は数学の基礎中の基礎だよ！集合はものを分類して整理する考え方で、命題は論理的に考える力を身につけるためのツールなんだ。この単元をマスターすれば、数学全体がもっと理解しやすくなるよ。

# 集合と命題 (Sets and Propositions)

1. 概要

2. Key definitions and concepts

* **集合とは**
ある条件を満たすものの集まりのこと。集まっている一つ一つのものを**要素**という。

*

集合と命題の基礎

集合って聞くと難しそうだけど、実は君たちが普段やってることと同じなんだ。クラスメイトを「野球部の人」「バスケ部の人」って分けるのも集合の考え方だよ。

集合とは、ある条件を満たすものの集まりのこと。その一つ一つを要素って呼ぶんだ。例えば、10以下の素数なら{2, 3, 5, 7}になる。要素が集合Aに含まれるときは $a \in A$ 、含まれないときは $a \notin A$ と書くよ。

集合の表し方は2つある。{1, 2, 3}みたいに要素を直接書く方法と、{x | xは偶数}みたいに条件で表す方法だ。どちらも使いこなせるようになろう。

部分集合は、集合Aのすべての要素が集合Bにも含まれるとき、AはBの部分集合って言う。 $A \subset B$ と書くんだ。これは「AはBに含まれる」って意味だよ。

ポイント！ 集合は身の回りの分類と同じ。難しく考えすぎないで！

共通部分 $A \cap B$ は両方の集合に入ってる要素、和集合 $A \cup B$ は少なくともどちらかに入ってる要素の集合だ。補集合 $\overline{A}$ は全体から集合Aを除いた残りの部分だよ。

命題は真偽がはっきり決まる文のこと。「1+1=3」は偽の命題だし、「x>0」みたいに変数があるものは条件って呼ぶんだ。条件pの否定は $\overline{p}$ で表すよ。

集合の要素の個数は $n(A)$ で表して、 $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ って公式がある。これはテストによく出るから覚えておこう！

ド・モルガンの法則も重要だ。 $\overline{A \cap B} = \overline{A} \cup \overline{B}$ と $\overline{A \cup B} = \overline{A} \cap \overline{B}$ 。「かつ」と「または」が入れ替わるのがポイントだよ。

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Updated Mar 29, 2026

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集合と命題の基礎

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集合と命題の基礎

ポイント！ 集合は身の回りの分類と同じ。難しく考えすぎないで！

集合の要素の個数は $n(A)$ で表して、 $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ って公式がある。これはテストによく出るから覚えておこう！