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e)Thereisanaturalnumbersuchthatthereisnorealnumberthat
wouldbegreaterthanit.
f)Eachtimewhenarealnumberisnegative,thenalsoitssquareis
negative.
g)Ifthereisanaturalnumbergreaterthan0,thenthereexistsa
realnumberlowerthan0.
h)Onlythesquaresofnaturalnumbersarenaturalnumbers.
i)
Squaresofsomerealnumbers,notbeingnaturalnumbers,are
naturalnumbers.
j)
Foreachnumber
ε
greaterthan0thereexistsanumberMsuch
thatforeachnaturalnumbermthefollowingistrue:ifmis
greaterthanM,thentheabsolutevalueofthedifference
betweenthevalueoffunctionfatpointmandthenumbergis
lowerthan
ε
.
3.Writethenegationsoftheexpressionsfromexercise1,usingtheDe
Morgan’sformulae.
SOLUTIONS
1.Forexample:
a)Foreachrealnumberxitssquareisgreaterthan0.False.
b)Foreachrealnumberxitssquareisgreaterthanorequalto0.
True.
c)Thereisarealnumberxsuchthatitssquareisgreaterthan0.
True.
d)Thereexistsarealnumberxsuchthatitssquareislowerthan0.
False.
e)Foreachpairofrealnumbersxandytheexpressionx
2
+y
2
+1
ispositive.True.
f)Thereisapairofrealnumbersxandysuchthattheirsumis
lowerthannegative5.True.
g)Foreachnaturalnumbermthereisarealnumberxsuchthatm
tothepowerxisgreaterthan9932.False
2
.
h)Thereisapairofnaturalnumbersminsuchthatforeachreal
numberxtheproductofmandnequalstotheproductofm
andx.True.
i)
Foreveryrealnumberxthereisanaturalnumbermsuchthat
thesquareofthedifferencebetweenxandmisnegative.False.
j)
Foreach
ε
greaterthan0thereisa
δ
greaterthan0suchthatfor
everyrealnumberxthefollowingistrue:iftheabsolutevalue
2
Inthisbookweassumethat0isanaturalnumber.Theexpression0
x
equalsto0,when
x>0andisnotdefinedforx≤0.
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