General Notatio n
Here w e se t fort h som e o f th e genera l notatio n tha t i s consistentl y use d i n
the entir e book . Thi s i s standard mathematica l notation , an d w e may refe r
to i t withou t furthe r mention .
Logic an d Se t Theory . Throughout , U and D respectively denot e unio n
and intersection, an d C the subset relation. Occasionall y we may write " a : =
6," wher e a an d b could b e sets , numbers , logica l expressions, functions , etc .
Depending on the context, this may mean either "defin e a to be 6," o r "defin e
b to b e a. " W e will not mak e a distinctio n betwee n th e two .
If A,B C X , the n A
c
denote s th e complemen t o f A [i n X] . Th e de -
pendence o n X i s usuall y suppresse d a s i t i s clea r fro m th e context . Le t
A \ B : = A H
5C
, an d A A B : = {A \ B) U (B \ A). Th e latte r i s calle d th e
set difference o f A an d B.
A se t i s denumerable i f i t i s either countabl e o r finite.
We frequentl y writ e "iff " a s short-han d fo r "i f an d onl y if. "
Finally, "V " an d "3 " respectively stan d fo r "fo r all " an d "ther e exists. "
Euclidean Spaces . Throughout , R = (—o o , oo) denote s th e rea l line , Z =
{0, ± 1 , ±2 ,...} th e integers , N = { 1 , 2 , . . . } th e natura l numbers , an d Q
denotes the rationals. I f X designate s any one of these, then X + denote s th e
non-negative element s o f X , an d X _ denote s th e non-positiv e elements . I f
k G N the n X ^ denote s th e collectio n o f al l fc-tuples {x\ , . . . , Xk) suc h tha t
# i , . . . , Xk are i n X . Fo r instance , R^ _ denote s th e collectio n o f al l k- vectors
that ar e non-negativ e coordinatewise . Th e comple x plan e i s denoted b y C .
xv
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