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Asyouwillnotice,theselawsareanalogoustothetautologiesfromtheprevious
chapter(Table1.3).Indeed,inordertoprovethemwewilluserespective
tautologies.
Wearemostlyinterestedinthe
setsofnumbers
(ornumbersets),i.e.thesets
havingonlythemembersbeingnumbers.Anumberofsuchsetsyoucanfind
alreadyinthepreviouschapter.
Particularlyinterestingsetsofnumbersarethefollowinginfinitesets:
•
thesetofnaturalnumbers:N={0,1,2,3,ł};
•
thesetofintegers:Z={ł,–3,–2,–1,0,1,2,3,ł};
•
thesetofrationalnumbers:Q={x:∃p∈Z∃q∈Z\{0}:x=p/q},i.e.
thesetoffractionshavingintegernumeratorsanddenominators,for
example1/2,–3/17,43/3;
•
thesetofrealnumbers:R;formaldefinitionofthissetisrathertricky
(andcompletelynotusefulforourpurposes),soletususesomeintuition
anddescribeitas“thesetofallthenumbersthatcanbewrittenusing
thedecimalrepresentation,finiteorinfinite”;therationalnumbershave
alsofiniteorinfinitedecimalrepresentations,butinthecaseofinfinite
representationithasaperiod(itmeansthatstartingfromsomepointthe
samesequenceofdigitsrepeatsinfinitelymanytimes);therearealso
realnumberswhicharenotrational–theirdecimalrepresentationsare
infiniteanddonothaveperiods,examplesofsuchnumbersare:
−
2
=
−
1
.
41421
...
,
e
=
2
.
71818
...
and
π
=
3
.
14159
...
Thefollowinginclusionrelationsaresatisfiedforthesetslistedabove:
N
⊂
Z
⊂
Q
⊂
R
.Inotherwords,allthenaturalnumbersareintegers,allthe
integers(soalsonaturalnumbers)arerationalnumbers,allrationalnumbers
(includingnaturalnumbersandintegers)arereal.
Usingthegivennotationtogetherwithsymbol“+”or“–”,wecanrestrict
ourselvestopositiveornegativenumbers.Forexample“R
–
”denotesthesetof
positiverealnumbers,and“Z
+
∪{0}”isthesetofpositiveintegerstogether
withthenumber0,soinotherwords–thesetofnaturalnumbers.
ThesubsetsofN,Z,QandRmaybedescribedbyenumerationorusingthe
formulae(justasinthebeginningofthissection).ThesubsetsofRmaybealso
describedas
intervals
.Theintervalscanbe:
•
open
;forinstance(1,2.2)isthesetofalltherealnumbersgreaterthan1
andlowerthan2.2;(3.27,Ż)inturnisthesetofalltherealnumbers
greaterthan3.27(thesymbol“Ż”denotesinfinity);
•
closed
(fromonesideorfrombothsides);forexampleleft-closed
interval[2.11,Ż)isthesetofalltherealnumbersgreaterthanorequal
to2.11;closedinterval[4,7]isinturnthesetofalltherealnumbers
greaterthanorequalto4andnotgreaterthan7.
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