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16
(HAPTER1 logic and Proofs
(h)
(i)
).
The fish bite only when the moon is full,
A time of 3 minutes, 48 secondsor less is necessaryto qualify for the
Olympic team.
Write the converseandcontrapositiveof eachconditionalsentencein Exercise1.
3. What can be said about the truth value of Q when
(a ) P i sfal seandP * Q i strue? (b) P i strueandP * Qi st r ue?
(c ) P i strueandP + Q i sfal se? (d) P i sfal seandP + ) Qi st r ue?
(e) P is true and P €f Q is false?
4. Identify the antecedentand consequentfor each conditional sentencein the
following statementsfrom this book.
(a) Theorem1.3.1(a)
(b) Exercise3 of Section1.6
(c) Theorem2.1.4
(d) ThePMI, Section2.4
(e) Theorem2.6.4
(0 Theorem3.4.2
(g) Theorem4.2.2
(h) Theorem5.1.7(a)
*
Which of the following conditional sentencesare true?
(a) Iftriangles have three sides,then squareshave four sides.
(b) If a hexagon has six sides,then the moon is made of cheese.
(c ) It7 + 6:14,then5 * 5:10.
(d ) If 5 < 2,then 10 < 7.
(e) If one interior angle of a right triangle is 92", then the other interior
angleis 88o.
(f) If Euclid's birtLday was April 2, then rectangleshave four sides.
(g) 5 is prime if 42 is not irrational.
(h ) 1 + 1:2i ssuffi ci entfor3 > 6.
6. Which of the following are true?
* (a) Triangles have three sidesiff squareshave four sides.
(b )
(c)
(d)
(e )
(f)
(g)
(h)
(i)
0)
(k )
",7-i
;
7 + 5:12i ff
I1-l :2.
b is even iff b + I is odd. (Assume that b is some fixed integer.)
n is odd iff m2 is odd. (Assume that m is some fixed integer.)
5* 6:6+ 5i ff7* l :10.
A parallelogramhas three sidesiff 27 is prime.
The Eiffel Tower is in Paris if and only if the chemical symbol for
helium is H.
.
"/ro+ Jn Ji + Jnn JB Jn.. Jn - ho.
xz > O iff -r > 0. (Assume that x is a fixed real number.)
*2 - y': 0 iff (x - y)(x * y) : 0. (Assume that .r and y are fixed real
numbers.)
(x-ty)2:50. (A ssumethat x and y are fixed r eal
xz+ y2:50i ff
numbers.)
Make truth tablesfor thesepropositional forms.
(a) P+(QnP).
(c) -Q+ (Q e P).
(e) (PnQ)v(Q^R)=+PvR.
* (b) (-P + e)v (e <+p).
- ( d) ( PvQ) + e^Q) .
(f) l( Q+ s) ( 0 + R) l+ te v Q)+ ( s v R) 1.
^
andBiconditionals
1.2 Gonditionals
17
truth tables for each equivalence'
8. Prove Theorem I.2.2by constructing
g, Determine whether each statementqualifies as a definition'
(a) y'y : f (x)is a linearfunctionwhenits qranhfsa totgl:llT
an x- term'
: (x) is a quadratic function when it contarns
'm f
integer n'
is apetfect'iquare when m , n2 fyr some
two of its interior angles
of
sum
the
when
A triangle it iigttt ttl"ngle
"
is 90'.
the samenumber'
(e) Two lines are parallel when their slopesare
time'
measuring
(fi
A sundial is an instrument for
logical connectives.Assume
each of the following sentencesusing
,rfi,
^'"' R"rrite
fixed object'
';
symbol/, xo, n, x,S' B representssome
,rt"i
"""rt
* (a )Ifl h a s u ," l u ti u " m i n i mumatxsandi f/i sdi fferenti abl eat' x0' then
iU
i"t
(d)
//(x6) : 0'
if n is prime, then n :2 or zris odd'
and not rational wheneverx is irrational'
A numberu is
'eul
1,
then l x | : 1'
If x : 1 o t x :
xs\ff f '(xs): 0 or/'(xs) doesnot exist'
/has a critical point at
S is compact iff S is closed and bounded'
condition for det B + O'
B is invertible is a necessaryand sufficient
4or n> I0'
i rtl 6 > -n - 3 o n l Yif n>
convergent'
is
x
(i) r is Cauchy implies
whenever
x6
f (x) : f (x$'
0) /is continuous at
,l$
at rs, then/'(.ro) > 0.
(k) If/is difl.erentiablear xe and/is strictty increasing
** (b)
(c)
* (d )
* i"j
(f)
(g)
ll.Dictionariesindicatethattheconditionalmeaningofunlessispreferred,but
thereareotherinterpretationsaSaconverseorabiconditional.Discussthe
translation of each sentence'
(a) I will go to the store unlessit is raining'
* (b )T h e D o l p h i n s w i l l n otmakethepl ayoffsunl esstheB earsw i nal l therest
of their games'
your homework first'
(c) You cannot go to the game unlessyou do
unlessyou buy a ticket'
idl You won't win the lottery
are equivalent'
Show that the following pairs of statements
fD
\-/*-(a)
(P v 0) + R and -R J (-P r'-Q)'
* inl e rQ)+Ran d (P n -R)+ -Q '
P + (Qn R) and(-Qv -R) + -P'
-- i.)
(P
(d ) P+ (Qv R )a n d
^-R )+ Q'
e + Q)i R a n d (P n-Q)vR '
P )'
v Q)^eQv
P e Qa n d (-P
sentencefor which
conditional
true
a
13. Give, if possible,an example
^ of
is false'
converse
(b)
the
* (a) the converseis true'
is true'
contrapositive
(d)
the
* (c) the contrafositive is false'
for
false conditional sentence which
14. Give, if possible, an example of a
(b) the converseis.false'
(a) the converseis true'
(d) the contrapositiveis true'
false'
(c) the contrapositiveis
G)
(f)
l8
CHAPTER
I Logicand proofs
15' Give the converseand contrapositive
ofeach sentenceofExercises 10(a),(b),
(c), and (d). Tell whether each
converse and contrapositive is true
or f-alse.
\ 16' Determine whethereachof the following
is a tautology, a confradiction, or neither.
*
(a ) [ (P = + e )+ 4 a p .
F
(b) P + + pn(P vQ).
(c )
* (d)
(e)
(f )
(g)
(h )
(i)
P + e e p n -e .
P =+[p =+ (p + e)1.
P n (Qv -e) ++ p.
t Q ^ e + e )l= + p .
(P +) g ++
-(-p v e)v (-p q.
^ p )1 .
t P+ (Q v R)l = +le + ilv (R+
p n (p e e )n _ e .
CI ) e v e ) + e + p .
(k ) t P + (e ^ R)l+
t R= +(p + Dl
(l) t P + (e ^ R)l + R+ (p
*A'.'
17. The inverse, or opposite, of the
conditionalsentencep =+
eis _p + _e.
(a) Show that p +
e and its inverse are not equivalent forms.
(b) For what values ofthe propositions p
and e arep ) e andits inverse
both true?
(c) which is equivalent to the
converseof a conditional sentence,
the con_
trapositiveofits inverse,or the inverse
ofits contrapositiufi
1 .3
Quantifiers
unless therehasbeena prior agreement
about the value of x, the statement.? >
3,,is
neithertrue nor false.A sentencethat
containsvanablesis cailed an open
sentenceor
predicate' andbecomesa-proposition
only whenits variablesareassigned
specificval_
ues.For example,"x > 3" is true when
, i, gl"* the value 7 and false when
x:2,
when P is an open sentencewith a
variable x,.the sentenceis symbolized
by
P("r). Likewise,if p hasvariablesxI,
x2,x3,. . .-,*r, m"sentencemay be
denotedby
P (x1, x2, x3,. . . , xn). For example,for
the sentence,,x * ! : 32,,we-wite p (x,
y, z),
andwe seethatP(4,5,3) is truebecause
4 + 5 :3(3), while p(1,2,4) is false.
The collectionof objecrsrharmay
be substituredro make ;;";
sentencea
true proposition is cailed the truth
set of the sentence.Before a truth
set can be
determined,we must be given or must
decidewhat objectsare availablefbr
consid_
eration;that is, we must have specifiea
u u.rro".re of discourse.In many cases
the
universewill be understoodfrom the
context. Fo. u ,"nr"n"e such as ._rlikes
choco_
late"' the universeis presumablythe
set of all people. we will often use the
number
systemsN, Z, Q, R, and C as our universes.
(Seethe preface to the Student.)
Exanrpfe' The truth set ofthe open
sentence"x2 < 5,, dependsupon the
corection
of objectswe choosefor the universe
of discourse.with the universespecified
asthe
set N, the truth set is { 1, 2}. For the
uniuJrr"-2, the truth set is _2, _1,
0,
{
1,2J.
When the universeis R, the truth set
is rhe openint".uul f_ E,"Bi.'