集合运算的基本法则 | 玄数

2012-03-03

集合的并、交、补运算满足下列定理给出的一些基本运算法则.

设A,B,C为任意三个集合,Ω与Ø分别表示全集和空集,则下面的运算法则成立:

 

1.  交换律(Commutative Laws):A ∪ B = B∪A, A ∩ B = B ∩ A

 

2.  结合律(Associative Laws):(A ∪ B) ∪ C = A ∪ (B∪C) = A ∪ B∪C

.                                                  (A ∩ B) ∩ C = A ∩ (B ∩ C) = A ∩ B ∩ C

Associative Laws

 

3.  分配律(Distributive Laws): (A ∩ B) ∪C = (A∪C) ∩ (B∪C)

.                                                  (A∪B) ∩ C = (A ∩ C) ∪(B ∩ C)

Distributive Laws

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Distributive Laws

 

4.  德摩根律(De Morgan’s Law):  德摩根律

De Morgan Law

 

5.  等幂律(Impotent laws):  A∪A = A,A∩A = A;

 

6.  吸收律(Absorption laws): (A∩B)∪A = A,(A∪B)∩A = A

 

7. 同一律(Domination laws):A∪Ø = A,A∩Ω= A

.                               A∪Ω=Ω,A∩Ø = Ø;

 

8.  互补律(Complement Laws):Complement Laws

 

 

练习:

1. 已知集合P = {x | x2 ≤ 1}, M = {a}, 若P∪M = P, 则a的取值范围是()
A. (-∞, -1]
B. [1, ∞)
C. [-1, 1]
D. (-∞, -1] ∪[1, ∞)

 
2. 已知集合A = {x | x2 – 2x – 8 = 0}, B = {x | x2 + ax + a2 – 12 = 0}.
求:满足A∪B = A的a值组成的集合.

 

 

集合运算的基本法则

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