Advanced Mathematics 1

集合

交并补

Latex Math symbols
交: $A \cap B={x| x \in A\ and\ x \in B }$
并: $A \cup B={ x| x \in A\ or\ x \in B }$
补: $A-B={x|x \in A\ and\ x \notin B } = A \cap \overline{B} $

基础运算

1. 交换律: $A \cap B = B \cap A; A \cup B = B \cup A$
2. 结合律: $A \cap (B \cap C) = (A \cap B) \cap C; A \cup (B \cup C) = (A \cup B) \cup C$
3. 分配率: $A\cap(B \cup C)=(A \cap B) \cup (A \cap C); A \cup (B \cap C)=(A \cup B)\cap(A \cup C)$
4. 对偶律, De Morgan’s laws: $\overline{ A \cap B} = \overline{A} \cup \overline{B}; \overline{ A \cup B} = \overline{A} \cap \overline{B}$
5. 吸收率: $A\cup(A\cap B) = A; A \cap(A \cup B) = A$
6. 幂等律: $A\cup A = A; A \cap A = A$
7. $0-1$律: $A \cup \varnothing = A; A \cap \varnothing = \varnothing; A \cup X = X; A \cap X = A$
8. 对合律: $\overline{\varnothing}=X; \overline{X} = \varnothing; A \cap \overline{A} = \varnothing; \overline{\overline{A}} = A$