Chapter 1
Numbers an d Logi c
1.1. Numbers . Calculu s an d rea l analysi s begi n wit h numbers :
The natural numbers
N = {1, 2 , 3 , . . . } .
The integers
Z = {... , - 3 , - 2 , - 1 , 0 , 1, 2 , 3 , . . . }
(Z stand s fo r th e Germa n wor d Zah l fo r number) .
The rationals
Q {p/q i n lowes t terms : p G Z, g E N }
= {repeatin g o r terminatin g decimals} .
(Q stand s fo r quotients) .
The reals
R = {al l decimals }
with th e understandin g tha t .99 9 = 1, etc . Real s whic h ar e no t rationa
are calle d irrational Thu s th e se t o f irrationals i s the complement o f the se t
of rationals , an d w e writ e
{irrationals} = Q
C
= R - Q .
1.2. Interval s i n R . Fo r a b, define intervals
[a, 6] = { x G R : a x 6} , a # &
(a, 6) = { x G R : a x 6}, a o 0 6
[a, oo) = {x G R : a x} , a # ^
and s o on .
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