1550251515-Classical_Complex_Analysis__Gonzalez_

Topology of Plane Sets of Points 79

00

H= n An

n=l

if I = J.

Definitions 2.5 If An B = 0, the sets.A and Bare said to be disjoint.

If An B -:/= 0, it is said that A and B intersect.

A collection { Aa} of sets is said to be a disjoint collection, or a collection

of pairwise disjoint sets, if any two sets in {Aa} are disjoint.

Definition 2.6 Let Ube the universal set and let A CU. The difference

U - A is called the absolute complement of A or, simply, the complement

of A. The complement of A will be denoted by A'. Other notations in use

are~ A, A, c A, and CA. The operation of taking the (absolute) complement

of a set is called complementation.

2.3 THE CALCULUS OF SETS. BOOLEAN ALGEBRA

The set operations of taking unions, intersections, and complements have
the properties listed in tlie following theorem.

Theorem 2.1 If U is the universal set and A, B, ... are subsets of U,

then:

1. A u 0 = A, A n 0 = 0


2. AU U = U, An U =A


3. A U A = A, A n A = A

4. Au (Bu C) =(Au B) u c, An (B n C) =(An B) n c

5. A u B = B u A, A n B = B n A

6. An (Bu C) = (An B) u (An C), Au (B n C) = (Au B) n (Au C)

7. Ac AU B, A :J An B


8. 01 = U, U' = 0


9. (A')' = A


10. A U A' = U, A n A' = 0

11. (AUB)' =A' nB', (AnB)' =A' UB'

12. If A c B, then A' :J B'

Some of the properties above are similar to well-known properties of

addition and multiplication in a number field. For instance, in ( 4) we have

the associative laws and in (5) the commutative laws. The first of (6) is

similar to the distributive law of multiplication with respect to addition.

However, the second equality in (6), called correlative of the distributive