Science Metro

集合の演算

$集合の演算1$

共通集合 : A \cap B := {x ∣ x \in A \land x \in B} 和集合 : A \cup B := {x ∣ x \in A \lor x \in B} 差集合 : A ∖ B := {x ∣ x \in A \land x \in / B} 直積集合 : A \times B := {(a, b) ∣ a \in A \land b \in B} 補集合 : A^{c} := U \ A := {x ∣ x \in U \land x \in / A}

$n個$

k = 1 ⋃ n A_{k} := A_{1} \cup A_{2} \cup ... \cup A_{n} = {x ∣\exists k \in {1, 2, ... n} (x \in A_{k})}

k = 1 ⋂ n A_{k} := A_{1} \cap A_{2} \cap ... \cap A_{n} := {x ∣\forall k \in {1, 2, ... n} (x \in A_{k})}

$無限個$

n = 1 ⋃ \infty A_{n} := A_{1} \cup A_{2} \cup ... := {x ∣\exists n \in N (x \in A_{n})}

n = 1 ⋂ \infty A_{n} := A_{1} \cap A_{2} \cap ... := {x ∣\forall n \in N (x \in A_{n})}

$基本法則1$

\overline{U} = ϕ, \overline{ϕ} = U

$基本法則2$

A \cap U = A A \cap ϕ = ϕ A \cup U = U, A \cup ϕ = A \overline{U} = ϕ, \overline{ϕ} = U

$排中律,矛盾律から導かれる法則$

A \cup A^{c} = U

A \cap A^{c} = ϕ

$二重否定の法則$

(A^{c})^{c} = A

$巾等法則$

A \cap A = A

A \cup A = A

$交換法則$

A \cap B = B \cap A

A \cup B = B \cup A

$結合法則$

(A \cap B) \cap C = A \cap (B \cap C)

(A \cup B) \cup C = A \cup (B \cup C)

$分配法則$

A \cap (B \cup C) = (A \cap B) \cup (B \cap C)

A \cup (B \cap C) = (A \cup B) \cap (B \cup C)

$分配法則一般形$

A \cap (i \in I ⋃ B_{i}) = i \in I ⋃ (A \cap B_{i})

A \cup (i \in I ⋂ B_{i}) = i \in I ⋂ (A \cup B_{i})

$吸収法則$

A \cap (A \cup B) = A

A \cup (A \cap B) = A

$ドモルガンの法則$

(A \cap B)^{c} = A^{c} \cup B^{c} (A \cup B)^{c} = (A)^{c} \cap (B)^{c}

$ドモルガンの法則一般形$

(k = 1 ⋃ n A_{k})^{c} = k = 1 ⋂ n A_{k}^{c}

(k = 1 ⋂ n A_{k})^{c} = k = 1 ⋃ n A_{k}^{c}

$部分集合の性質$

A \subset B \Leftrightarrow B^{c} \subset A^{c} \Leftrightarrow A ∖ B = ϕ \Leftrightarrow A \cap B = A \Leftrightarrow A \cup B = B

$差集合の性質$

B ∖ A = A^{c} \cap B

$直積集合の相等$

(a, b) = (a^{'}, b^{'}) : a = a^{'} \land b = b^{'}

$直積集合の性質$

A \times (B \cap C) = (A \times B) \cap (A \times C)

A \times (B \cup C) = (A \times B) \cup (A \times C)

(B \cap C) \times A = (B \times A) \cap (C \times A)

(B \cup C) \times A = (B \times A) \cup (C \times A)

$集合族の和集合と共通部分$

λ \in Λ ⋃ A_{λ} = {x ∣\exists λ \in Λ (x \in A_{λ})}

λ \in Λ ⋂ A_{λ} = {x ∣\forall λ \in Λ (x \in A_{λ})}

$集合族の結合法則$

(λ \in Λ ⋃ A_{λ}) \cup B = λ \in Λ ⋃ (A_{λ} \cup B)

(λ \in Λ ⋂ A_{λ}) \cap B = λ \in Λ ⋂ (A_{λ} \cap B)

$集合族の分配法則$

(λ \in Λ ⋃ A_{λ}) \cap B = λ \in Λ ⋃ (A_{λ} \cap B)

(λ \in Λ ⋂ A_{λ}) \cup B = λ \in Λ ⋂ (A_{λ} \cup B)

$集合族のドモルガンの法則$

(λ \in Λ ⋃ A_{λ})^{c} = λ \in Λ ⋂ A_{λ}^{c}

(λ \in Λ ⋂ A_{λ})^{c} = λ \in Λ ⋃ A_{λ}^{c}