Families of sets集合族しゅうごうぞく and index sets添字集合そえじしゅうごう

date2026-07-14document_iddoc_f3c01f19fca940a03f101b9bd4fd4d07description集合族と添字集合を、同じ種類の集合をまとめて扱う記法として導入し、添字付き和集合・共通部分を丁寧に整理する講義である。prerequisites集合の基本 / 集合演算と包含関係type講義content_typelecturestatusactive

mathdiscrete-mathfamily-of-setslecture

1Introduction

A family of sets集合族しゅうごうぞく is introduced to handle many sets集合しゅうごう at once, not only two or three. For example, for each $n = 1, 2, 3, [PARSE ERROR: Undefined("Command(\"dots\")")]$ , let $A_{n} = {x \in R ∣ - 1 / n < x < 1 / n}$ . We can then treat all the $A_{n}$ as one indexed collection.

The important point is that an index添字そえじ serves as a label specifying $A_{i}$ and need not be an element of $A_{i}$ .

2Terms and definitions

Fix a universal set全体集合ぜんたいしゅうごう $U$ . Given an index set添字集合そえじしゅうごう $I$ , if each $i \in I$ is assigned a set $A_{i} \subseteq U$ , then $(A_{i})_{i \in I}$ is called a family of sets集合族しゅうごうぞく.

Distinct indices $i \neq j$ may label the same set, so $A_{i} = A_{j}$ is allowed. A family records not only the sets but also the indices by which they are labeled.

The union和集合わしゅうごう of the family is defined by

⋃_{i \in I} A_{i} = {x ∣ thereexists i \in I suchthat x \in A_{i}} .

The intersection共通部分きょうつうぶぶん of the family is defined by

⋂_{i \in I} A_{i} = {x \in U ∣ forevery i \in I, x \in A_{i}} .

3Method

For an indexed union添字付き和集合, look for “there exists an index.” For an indexed intersection添字付き共通部分, check “for every index.” This difference between existence and universality determines the proof strategy.

For example, if $x \in ⋃_{i \in I} A_{i}$ , then there exists $i_{0} \in I$ such that $x \in A_{i_{0}}$ . In contrast, if $x \in ⋂_{i \in I} A_{i}$ , then $x \in A_{i}$ for every $i \in I$ .

4Intuitive explanation

A family of sets集合族しゅうごうぞく can be imagined as numbered drawers, each containing a set集合しゅうごう. The union和集合わしゅうごう accepts an element if it appears in at least one drawer. The intersection共通部分きょうつうぶぶん accepts an element only if it appears in every drawer.

In this view, union weakens the condition, while intersection strengthens it.

The empty index set needs special care. The union over an empty family is the empty set空集合くうしゅうごう because there is no element to collect. The intersection over an empty family is taken to be the ambient universal set全体集合ぜんたいしゅうごう, when that universe has been fixed.

5Precise point: the empty index set添字集合そえじしゅうごう

The case $I = [PARSE ERROR: Undefined("Command(\"varnothing\")")]$ is a boundary case. The union和集合わしゅうごう $⋃_{i \in [PARSE ERROR: Undefined("Command(\"varnothing\")")]} A_{i}$ is the empty set空集合くうしゅうごう, because there is no index $i \in [PARSE ERROR: Undefined("Command(\"varnothing\")")]$ that can witness membership.

Under the definition in this lecture, the intersection共通部分きょうつうぶぶん $⋂_{i \in [PARSE ERROR: Undefined("Command(\"varnothing\")")]} A_{i}$ equals the fixed set $U$ . For every $x \in U$ , the condition “for every $i \in [PARSE ERROR: Undefined("Command(\"varnothing\")")]$ , $x \in A_{i}$ ” holds because there is no index to check. A universal condition that is true because there is no possible counterexample is called a vacuous truth空虚真くうきょしん.

6Worked example: indexed intersection添字付き共通部分

6.1Problem

Let $A_{n} = {m \in Z ∣ n divides m}$ . Describe $⋂_{n = 1}^{3} A_{n}$ in words.

6.2Explanation

The statement $m \in ⋂_{n = 1}^{3} A_{n}$ means that all of $m \in A_{1}$ , $m \in A_{2}$ , and $m \in A_{3}$ hold. In other words, $m$ is an integer整数せいすう divisible by all of $1, 2, 3$ .

Therefore $⋂_{n = 1}^{3} A_{n}$ is the set of all multiples of $6$ . This example shows that intersection共通部分きょうつうぶぶん extracts elements元げん satisfying several conditions at the same time.

7How to identify it and related links

If many sets集合しゅうごう with the same form appear, organize them as a family of sets集合族しゅうごうぞく.
Use $⋃$ for “belongs to at least one.”
Use $⋂$ for “belongs to all.”
When the index set添字集合そえじしゅうごう is empty, union和集合わしゅうごう and intersection共通部分きょうつうぶぶん behave differently.

data/lecture/math/discrete-math/set-operations-and-inclusion.lecture.n.md data/lecture/math/discrete-math/cartesian-product-basics.lecture.n.md

8Exercise links

data/exercise/math/discrete-math/sets-and-set-operations.exercise.n.md