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ofthem:universalquantifiercorrespondingtothephrase“forall”and
existentialquantifiercorrespondingtotheexpression“exists”.Theyaregiven
intheTable1.13:
Table10130Quantifiers
Type
Universal
Existential
Howwereadit?
“Foreachł”
“Existsł”
Symbol
*
∀
∃
*
Sometimesyoucanalsomeetthefollowingnotation:“
∧
”fortheuniversalquantifierand“
∨
”
fortheexistentialquantifier.
SymbolsshowninTable1.13areusedinsuchawaythattherangeofavariable
isgivenbelowthesymbol,andthepredicateiswrittenontheright,orboththe
scopeofthevariable,aswellasthepredicatearewrittenontheright,separated
byacolon.Forexamplethesentence“Foreachrealnumberx,x
2
≥0”wecan
writeatleastintwoways–thisway:
∀
x
∈
R
:
x
2
≥
0
,
orthisway:
∀
x
∈
R
x
2
≥
0
.
Sentenceswrittenbylogicquantifiersaresubjecttocertaindependencies.In
particular,ifP(x)denotesanypredicatewithvariablex,thenthetwofollowing
properties,calledDeMorgan’slawsforquantifiers,aretrue:
§
¨
©
∀
x
¬
P
(
x
)
·
¸
¹
⇔
¬
§
¨
©
∃
x
P
(
x
)
·
¸
¹
,
§
¨
©
∃
x
¬
P
(
x
)
·
¸
¹
⇔
¬
§
¨
©
∀
x
P
(
x
)
·
¸
¹
.
Asyoucansee,DeMorgan'slawssaythataftermovingthenegationinsidethe
predicate,wehavetochangeitstype(ifyouseetheanalogytotheDeMorgan’s
lawslogicalstatements?).
Quantifiersofthesametypecanbestoredinanyorder.Itdoesnotmatterin
whatorderwewritesomeuniversalquantifiersortheexistentialones.Itisin
turnimportanttokeeptheorderwhilewritingthequantifiersofdifferenttypes.
Whetheritappearsfirstexistentialquantifier,thenanuniversalone,orvice
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