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데코수학/ 집합론/ 곱집합
(유튜브 동영상인데 현재는 삭제되어서 내용만 남김)
1
개념
•
순서쌍
◦
(
a
,
b
)
=
{
{
a
}
,
{
a
,
b
}
}
(a, b) = \{ \{a\}, \{a,b\}\}
(
a
,
b
)
=
{{
a
}
,
{
a
,
b
}}
•
카테시안(곱집합)
◦
데카르트의 이름을 따서 지은 이름
◦
A
×
B
=
{
(
a
,
b
)
∣
a
∈
A
,
b
∈
B
}
A \times B = \{ (a, b) | a \in A, b \in B \}
A
×
B
=
{(
a
,
b
)
∣
a
∈
A
,
b
∈
B
}
•
카테시안은 순서쌍 개념이기 때문에, 교환법칙과 결합법칙이 성립하지 않는다.
◦
A
×
B
≠
B
×
A
A \times B \neq B \times A
A
×
B
=
B
×
A
◦
A
×
(
B
×
C
)
≠
(
A
×
B
)
×
C
A \times (B \times C) \neq (A \times B) \times C
A
×
(
B
×
C
)
=
(
A
×
B
)
×
C
•
직선을
R
\mathbb{R}
R
, 원을
S
1
S^{1}
S
1
이라 할 때
◦
R
×
R
\mathbb{R} \times \mathbb{R}
R
×
R
: 평면
◦
S
1
×
R
S^{1} \times \mathbb{R}
S
1
×
R
: 원주면
◦
R
×
R
×
R
\mathbb{R} \times \mathbb{R} \times \mathbb{R}
R
×
R
×
R
: 3차원 유클리드 공간
◦
S
1
×
S
1
S^{1} \times S^{1}
S
1
×
S
1
: 도넛면, 토러스
카테시안식
•
A
×
(
B
∪
C
)
=
(
A
×
B
)
∪
(
A
×
C
)
A \times (B \cup C) = (A \times B) \cup (A \times C)
A
×
(
B
∪
C
)
=
(
A
×
B
)
∪
(
A
×
C
)
•
A
×
(
B
∩
C
)
=
(
A
×
B
)
∩
(
A
×
C
)
A \times (B \cap C) = (A \times B) \cap (A \times C)
A
×
(
B
∩
C
)
=
(
A
×
B
)
∩
(
A
×
C
)
•
A
×
(
B
∖
C
)
=
(
A
×
B
)
∖
(
A
×
C
)
A \times (B \setminus C) = (A \times B) \setminus (A \times C)
A
×
(
B
∖
C
)
=
(
A
×
B
)
∖
(
A
×
C
)
•
(
A
×
B
)
∩
(
C
×
D
)
=
(
A
∩
C
)
×
(
B
∩
D
)
(A \times B) \cap (C \times D) = (A \cap C) \times (B \cap D)
(
A
×
B
)
∩
(
C
×
D
)
=
(
A
∩
C
)
×
(
B
∩
D
)
•
(
A
×
B
)
∪
(
C
×
D
)
⊆
(
A
∪
C
)
×
(
B
∪
D
)
(A \times B) \cup (C \times D) \subseteq (A \cup C) \times (B \cup D)
(
A
×
B
)
∪
(
C
×
D
)
⊆
(
A
∪
C
)
×
(
B
∪
D
)
•
(
A
×
B
)
c
=
(
A
c
×
B
c
)
∪
(
A
c
×
B
)
∪
(
A
×
B
c
)
(A \times B)^{c} = (A^{c} \times B^{c}) \cup (A^{c} \times B) \cup (A \times B^{c})
(
A
×
B
)
c
=
(
A
c
×
B
c
)
∪
(
A
c
×
B
)
∪
(
A
×
B
c
)
•
A
⊆
B
⇒
A
×
C
⊆
B
×
C
A \subseteq B \Rightarrow A \times C \subseteq B \times C
A
⊆
B
⇒
A
×
C
⊆
B
×
C
•
(
A
,
B
≠
∅
)
⇔
A
⊆
C
∧
B
⊆
D
(A, B \neq \emptyset) \Leftrightarrow A \subseteq C \wedge B \subseteq D
(
A
,
B
=
∅
)
⇔
A
⊆
C
∧
B
⊆
D
