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 .