Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
versahasanimpactonthemeaningandvalueofthestatement.Thiscanbe
easilyseenbyexaminingthefirstexample.
EXAMPLES
Example1
ReadthefollowingexpressionsandfindtheirBooleanvalues:
∀
x
∈
R
y
∃
∈
R
x
+
y
=
0
,
y
∃
∈
R
∀
x
∈
R
x
+
y
=
0
.
Solution
Atthebeginningofthefirstexpressionthereistheuniversalquantifierwith
variablex,whichshouldread:“foreverynumberxbelongingtothesetofreal
numbers”.Anotherelementoftheexpressionistheexistentialquantifierwith
variabley,soreadit:“thereisanumberythatbelongstothesetofrealnumbers
suchthat”.Finally,thesentencesoundslike“foreverynumberxbelongingto
thesetofrealnumbers,thereisanumberythatbelongstothesetofreal
numberssuchthatxplusyisequalto0”.
Isthistrue?Yes.Why?Becauseforeachxitisenoughtotakey=–x.Then
indeedx+y=x+(–x)=0.
Andwhatwouldhappenifweswitchedthequantifiers?Thesentencewould
readasfollows:“thereexistsarealnumberysuchthatforeachrealnumberx
thesumofxandyis0”.Thisisnottrue–thereisnosuchnumber,whichwould
give0aftercombiningwithanyrealnumber.
Asyoucansee,theorderofthequantifiersreallymatters.
Attheend,payattentiontothefactthatoncewehaveusedtheterm“real
numberx”,andonce“xbelongingtothesetofrealnumbers”.Botharecorrect.
Youcanalternativelyuseanyother,similarlysoundingexpressions,itisonly
importanttopreservethesense(xisthenumberfromaspecificset).
Example2
Writethefollowingsentenceusingquantifiers:“foranytwonaturalnumbersm
andn,theirproductisanonnegativenumber”.
Solution
Theexpression„forany”hasinthiscontextexactlythesamemeaningas„for
each”.Sowehavetousetwouniversalquantifiers–oneformoneforn,orone
universalquantifierformandn.Inthefirstcasetheformulalookslikethis:
17
