1549901369-Elements_of_Real_Analysis__Denlinger_

B.l Sets and the Algebra of Sets 617

Definition B.1.8 (Operations on Collections of Sets)

Let C = {A>. : .A E A} be a collection of sets, "indexed" by some set A of

"indices" .A. Then

(a) nC = n A>.= {x: x EA>. for every .A EA}.

>.EA

(b) UC = LJ A>. = { x : x E A>. for at least one .A E A}. 0

>.EA

Examples B.1.9 (a) n { (-~, 1 + ~) : n EN}= n (-~, 1 + ~) = [O, l].
nEJ\I

(b) LJ { ( -~, 1 + ~) : n E N} = LJ ( -~, 1 + ~) = ( -1, 2).

nEJ\I

1 1 +-A-

Figure B.2

Theorem B.1.10 (Algebra of Collections of Sets) Let C = {A>. : .A EA}
be a collection of sets and let B be any set. Then

(a) ( n A>.)c = LJ A1. (de Morgan's law)

>.EA >.EA

(b) ( LJ A>.)c = n A1. (de Morgan's law)

>.EA >.EA

(d) Bu ( n A>.) = n (Bu A>.). (distributive law)

>.EA >.EA

(d) B n ( LJ A>.) = LJ (B n A>.)-(distributive law)

>.EA >.EA

(e) B - ( n A>.) = LJ (B - A>.)- (de Morgan's law)

>.EA >.EA

(f) B - ( LJ A>.) = n (B - A>.)-(de Morgan's law)

>.EA >.EA

Proof of ( e): Let C = {A>. : .A E A} be a collection of sets and let B be
any set.