MATH 1473: Mathematics for Critical Thinking
The University of Oklahoma, Dept. of Mathematics
Study Guide and Notes for Chapter 1
Compiled by John Paul Cook
For use in conjunction with the course textbook:
Mathematics: A Practical Odyssey (Sixth Edition)
By David Johnson and Thomas Mowry
The exercises in this packet come from a vaiiety of sources: the course textbook, and supplementary
course materials provided by Ms. Christine Tinsley, University of
Oklahoma; additionally, some of
exercises are originaL.
the
Section 1.1: Deductive versus Inductive Reasoning
Key Terms and Concepts
Deductive reasoning, inductive reasoning, valid argument, invalid argument, Venn Diagrams
ì ",c9u
o1-\vE'
'. s-pec'.-Ç; c. ~ ~e.~1
Practice Problems J¿i UC' t. ve : 3e.~ \ -- Sf0c; f'; C
1) Classify each argument type as inductive or deductive:
My television set did not work two nights ago.
My television set did not work last night. \ V\e&uc t \ Ve
Therefore, my television set is broken.
All electronic devices give their owners grief.
My television set is an electronic device. Ó)tcQuc: ve
Therefore, my television set gives me grief.
VC\ t., ~ = \~ \ ('0( \ \y \lo.dì / Vlb1 ~efYVI~ ~\ti +r "- ti
VQi'~~ euS"'VV~t: Cl Ct\A~+ ì~ lAL~C-~ ~~ edl0lu~~~ 'ie: Uio.vÐJ.(.~1Q
2) Construct a Venn Diagram to determine the validity of the given argument
All roads lead to Rome.
Route 66 is a road.
Therefore, Route 66 leads to Rome.
\JQ l',&
All pesticides are harmful to the environment.
No fertilizer is a pesticide.
Therefore, no fertilizer is harmful to the environment.
-, /' )
'X 1
'---------_._..~
'I:: .tev+~ \; ~
\, Y\ \JOt; \'~&
-. --
x
No professor is a millionaire.
No millionaire is illiterate.
Therefore, no professor is illiterate.
Some women are police officers.
Some police officers ride motorcycles.
Therefore, some women ride motorcycles.
\'
ç~-e iI \AQuS 0.+ \eas1 fWe.
\i+n G'1e t t-
V\~¡Aj \" ke cW3\A~t
D~CfWS
~Jèt- \i 'ô-tr e y~Je () ~~C.ers
1' ~ /\4e~.Q0+ W~€.
. Writing and Understanding
3) Explain the difference between deductive and inductive reasoning.
i""Jiuit\ve ~~¡t.I'i~ -6k~ q -sre"t~-c se-l ~ ~e. ~e.QI'iM
~+) &~Uc.i-\le. 1f~Gv~"j JDP~ ~~ ;reMQ.\ '\ Sf(\:t~c
4) Explain why validity does not imply truth. In other words, why does saying that an
argument is valid not necessarily imply that the conclusion is true?
\/()h9.~ ~s \ft 'l~~ hn, \cQu~e \fot~~;1: 1'3 ~L'j c~ceV\~
~ì~ rte. \fl~;C oti ~€ ~t, 9';\f ~.e f-I'€.;S.Q~~ v\1)tii')
. IS ~Oi\óì Q~'" r\e ~ ~ ~-e ~rewìse~, .
Section 1.1 Homework: 1,4,7,8, 10, 15, 18, 19,22
Section 1.2: Symbolic Logic
Key Terms and Concepts
Statement, compound statement, the word "some", negation, conjunction, disjunction,
conditional, inclusive or; ~e W- ~e ~l . ~ t 1(~ te~
Practice '\O,-.\M~\'\
Problems
S..lNl~c.~
L. ..
' rrlÁe ô/- ~\~
ex ~t€vc-e
'iSziRev
1) Determine which of the following sentences are statements. Explain.
a. Apple manufactures computers. 'I QS
b. Apple manufactures the world's best computers. vy - op\/\ 'i~
c. Did you buy an IBM? V\Q-ei \.QS+~~
tA3U\~.(') ev
d. A $2000 computer purchased at a 250/0 discount will cost $1000. 'i.e ~
~\ Che
e. Make sure to tum the lights off when you leave the room. VìO ,- Cü ~
\r * ~s +rue
\)( ~t~e
f. This sentence is false. V1 'D - ~ Vò c9)Ç
2) Draw the "negation chart", and use it to negate the statements below:
?:j21' ~
A I) If eve q No \? 0Je 9.
S~e ? cd il ~ s,()e ? 0J V\.o-t 9.
a. She is not a vegetarian. ,Ç~e \ ~ V\+ () veß e +QJ\Qv ,
~co,.t\',
b. Some elephants are pink. No e \e.ph:wJç QJ ~ \iì j¿.
S:Q.e := eXt \ eo s t 1- h.
&te c. All candy promotes tooth decay. S~e ~ &os; \J ~fOO;-e de.ro,
d. No lunch is free. ~o \.f? ~ ~ ec OJe f íQe ,
3) Write the following conditionals in "if. . . then" form:
a. All mammals are warm-blooded.
rt \+ \~ Q ""a\M~l) R~ \t is 0)o.JI'VV ~l~eQ \
b. No snake is warm blooded.
Tr ~t \~ Q s;V\Qke./ ~ \+ ,s: \i+ tMQ\fW\ \oio~e&.
4) Write the following conditionalsp~q in the form q ifp:
a. If I exercise regularly, then I am healthy.
I ~ ~(A\tL'f ~~ l' ~)(evch~ v~\ÁtCùl~.
b. If I eat junk food and do no exercise, then I am not healthy.
i ~ ",d- "'et \fi \-\ 1: \h 1- J VJ \.( ~~ c: &0 \/ot
~xevt;~~.
5) Using the symbolic representations:
v ov"
Go oV0. R-e S'1\A\oot~ 'I /\ ~~
'" V\~1-~
p: The lyrics are controversiaL.
q: The performance is banned.
l S:l","'tIM0.'\S: QJe ~ V'erreç'a..E'~ ~
~JOCJ "',Ç. -
express the following compound statements in symbolic form.
a. The lyrics are controversial and tht performance is banned. p A 9.
b. If
the lyrics are not controversial, the performance is not banned. V'? ~ ~ Ct
c. It is not the case that the lyrics are controversial or the performance is banned. \.(?"i) \oy c~+ei
d. The lyrics are controversial and the performance is not banned.
?l\~~
6) Using the symbolic representations:
S'(lI \00 L ~ l' -Ç I? .1Levq,
p: The car costs $40,000.
q: The car goes 140mph.
~S P ~i
r: The car is red.
express the following compound statements in symbòlic form.
a. All red cars go i 40mph. if \Á'" ~ \~'''. :r l ',t ;~ Oi 1/"'& co) ~ 0- 3~ \ 'I""¡pl..
i 'i~9.
b. The car is red, goes 140mph, and does not cost $40,000.
rl\ei/\ V'p
c. If the car does not cost $40,000, it does not go 140mph.
\/? -- V\~
d. The car is red and it does not go 140mph or cost $40,000.
'Í ¡\ V\ lq VP)
7) Translate the sentence into symbolic form. Be sure to define each letter you use.
a. All squares are rectangles.
?: 1= + ,,~ 0. s'et u.(ÆV-t? .
'ìt, "1+ 'is. (A 'Vee~~\~.
l~'1
b. I do not sleep soundly if I drink coffee or eat chocolate. V'ewr ~ìe'.
sA-o ~Q\N~\~
't: :t ,s\~ S~lj,
dQ~)I\~ \/\
1:~ :t et~I'k c$ee.
() ~òSi+ìve
~~Iíuto
Q\JO~ cQ
C.~fuS\~
~ '-01 ~le€f S~Uv~l~.
~c. If you àrnk and drive, you are fined or you go to jaiL.
V-'. l' ful- ~/'it)('-ot(À+e.
~', y~ it;"k.
ei 'i YOu c9r\v€,
.,: ~Du D.e ~~r4.
ç~ (OM 301'JCi\1.
1:r" :t &r',/\ \( Û)~~ee oJ
~+ C.~OCo\O\-t~) ~J
(? Act) -- (rvs)
(qví) ~ ?
8) Using the symbolic representations:
p: The car costs $40,000.
q: The car goes 140mph.
express the following in words:
a. p /\ q l1e CD Cos,ts, -$.hD ~ 3()Q~ 14- 1M¡:.
b. p--q ir- ~e \D æs.ts -$/00 i"-B \,t 3()E'S ('+""~.
c. - q --- P l t' Í1e a. &e~ V\'Dt 30 NO 1N~1 nev tl~ (!o, &e~ "totC-Os t- ~ YQ;Ð&O"
d. qv ~ p
iie (f 3oeS, 1l\""~ Oí n-e íl &o~ lI'òt CEls-t
9) Using the symbolic representations:
:tYD)OO~
p: I am an environmentalist.
q: I recycle my aluminum cans.
r: I recycle my newspapers.
express the following in words:
a. (q v r) -- P:t-Ç i V"0 ye. t e \/ ~ ext lJ ) JHM.AA ~ 'l Ð. Vvy
V\~S;~~UçJ ~~ -: ~ OJ ~V~f'OV\\Me,tet~.S-t,
b. - p --- (q v r)"l.Ç "" Q. V\cl Q. ei u\.r VV.eA/ìk \ h") R-€ :: &v i
"eJ:-~ cle \/'f C\t\.~I\\. ('WAS I!í VQ(\'ic\~ Vvy VlQ.~f(X
C. (q/\)v- p l' vec'I,1e \)'( QlUw\f\~ cr~ u, ~'f vi,,~~~
~~ ~ \
c. :t Ct V\ot Cu ~v\roV' ~~~\;~" \
d. (r /\ - q) --- P 'I vel) 'Ie t e VI't V\0.S- ~le. S () \/.c 1M 'f Q \ ~ 1M .
CQN~) ~~ ~ ~ V\Dl- Q. ~V~fOt~~iQl~~t~
Writing and Understanding
10) Explain why the negation of
lQs~ /\3 VeV\V'
the statement "Some pare q" is not "Some p are not q".
L)~Ctôr~~ :
~
we ~ S'~Q ~C\+ Re2.""J s~fe\/eA+ \s. ~oI~\:d. ..ro ~e
11) Explain why the negation of the statement "All pare q" is not "No pare q". ~ Jl .
¡
Aii 'i+ ~'S vie~~ "\ ~1e ow 1\ uiU 11 S~1e1MBtt
ì~ \ C'l.we)(U\le/ c~tJdki\\)\ \Ál.\~ ~s lÁl.'1
tL" VO~J& t\StJetl \~ ~~.
12) Why is it helpful when breaking statements down to define the statements without
negations? For instance, we will often do this:
We can write "This car is not fast" as ~p, where we let p be the statement "This car is
fast."
Why would we not just let the original statement be p?
T-t ~keC; ì"\ ~'S;~ +0 \~l'\~ "'es01ted 8'\1~e.tS
ì", s'1""bl\c ~. D~~w\ç'€/ it w-el~ \c vev~
~~'S~ ~ UJdQCIl.
Section 1.2 Homework: 1,4,8,9,12, 15, 18,20,21,28,30
Section 1.3: Truth Tables
Key Terms and Concepts
truth table, conjunction, disjunction, conditional, equivalent expressions, DeMorgan's Laws
?';INO-:i ì~ËQ ~\~ ~ti ~Yes'. .t~ ~t Q)(ød~ ú)l-~a.
Practice Problems
~~EM-+ ;s ~e / ~l~ ~ u.\"' Q. --\ole
~ ~)(Ct\M\~e oJ\ ~Sg\'6l¡t~~
the following compound statements
1) Write the truth table for each of
a. Negation: ~p
T t=
r: T
TIV\t
b. Conjunction: p /\ q
~ 1.
?I\~
Tt
F
i- T T
t\
f: F
c. Disjunction: p v q
t=
r:
PV9
T
L
T
ç:
:1
?~ci
T
P
~
T
T
w"'V ? ~eC.CUse \ ~ 't ~ ~ n,,€
iO 1 +,,~ tw i tk.e CÐ&:'\ '~a
~~ v~otû\+~.. ~J Cve ~O\td. Conditional: p -- q
~-tJ1'~ w\k QU .r: &~ \li(T
vjo~O\+e \t \cC'lU~ \+ &~s~4
\l~~~ ~ e lf~'i~e.
~V"
L.t '1~" j \J€ ""e -$ \ù) 1: tJ, \ \
E 'l CUl \~ '.
'à ~\.e 0 'lt).iL ne +"c.l.(e-t.
2) Complete the following truth table:
p
T
T
F
F
q/\ ~ P
~P
q
T
F
T
F
c
F-
i
F
T
pv(q/\~p)
T
T
F-
T
T
T
T
"\
l
~ Sin1~~1 \ike
h'is wL\c-~ ~s U\ lt'ß
1-u.e ,~ CD.\t~ ex
TCl.L\to VX_
A (\n"()',~~ 1-0 l)
3) How many rows/are required in a truth table?
t oJ? st-k..",e...;tJ l~ttc; ~
4) Complete the following truth table f?r the symbolic expression (q /\ p) ~ (~ r v p):
p
q
r
q /\ P
T
T
T
T
F
F
F
T
T
T
T
F
F
T
T
T
T
FP-
F
F
F
F
T
t:
F
F-
F
~r
t
T
F
T
i-
F
T:
T
I¡
T
~
T
~rv P
IT
T
T
f'
1-'
F
t.
(q/\p)~(~rv p)
T
\T
T
IT
T
T
5) Construct a truth table for the following expressions to determine when each statement is
tre or false:
a. No square is a triangle.
iæ-r'i-le', "L~ ~t ~s. U\ S:C1lAQ\ft:j tiev ;ì- \s, V\O+ 0 t'Î'i~te,
'P ,i l-t-- \ S Q s.q \.Q ~ (? .
l', ì+-. '\~ O\t-V'\~E',
S'f \N~tìCD.tt~\. ? ~ L/l
? L
T T
\At
?~J\~
f:
r:
T \= T
r. T
F F-
ç.
T
T
T
T
I?~ YOu kl.t 0. &.\ve.c: \:C'~~ 't: Y\W \nQve Q. c,v~'. CQIf. r: Vauif ,
,s
you have a driver's license or a credit card. C~Q.~
Û(c~aQ
b. Your check is accepted if
\Íe IJf \ ~e', 1: ri \.,~ "'QV.e Q ~\\le.'C; \ ~ (ev~.e ö" 0- C vei9 ~ + en ~ J
Th~ \ltlv c",eek ;~ QCX~'~,
P
T
'\
,i
f:
F
F
1
T
T
F:
P-
T
T~
y
?V't
\T
( l\l't)
t:
T
\
"\
r:
L
t:
T
F
T
,~
~v
iP
"\
\
T
Sy IN \ohcO\ll'::
(pV~)~ V
ITF
Fi:
ç: \
t6) Determine if the given statements are equivalent using truth tables:
T wo s~teW\~ì-s Olve e.lA~ \la.\~t:\- ~.e~r ~th vG\luP~ \I~+rL\ ttp E:X'qdl~
If the spotted owl is on the endangered species list, then lumber jobs are lost. ~ ~l.
If lumber jobs are not lost, then the spotted owl is not on the endangered species list. ~ ~ V" P
?:lle ~tott~ OC/\ ìs: ~ l1e ev~ev~ ~~Q"es I\~,
'1: LlA ~ J D\oS' ON:e lo~_
V'C( ~?
?
T
r. F
T T:
\
t:. r T T T T/
/ ti~ c.e E'q I. ~ \/Q\.evJ
\ T-
ç:
J
7) Using truth tables, show that the expression Yp v q is equivalent to the conditional
p~q:
?
ei
T T
T f:
ç:
p
?--l
\A ~
\Apvq
~
1=
T
i-
r:
T
\i
J ~~ CW ~tA~VQ\et+
\)e Mov3cvs L(0~ ON€ '."'~ ~/ ~-e~Ül~\Aj s;'k\~ '\S lù:~
8) State DeMorgan's Laws CUcQ tJ-r Ov \n n~
'- ( ? \I t: \:: \/ ? /\ \A ~
"" (?I\"L) ~ V'~VV'~
9) Using truth tables, verify the given part of
DeMorgan's Laws (the other is for
homework): ~ (p 1\ q) =~ pv ~ q
P L
T T
T r:
F T
~f
? 1\ et
\rp ~l
T
t: t:
r-
\' r
F
J ~ei evfè
t: T
~ll~\J(Xt~ 1-
T T
F-
a. p -- q (for this part, use the result from question 7)
? ~OC :: ~~V9. ~ ~ (p~(t) =: ~ (v-rrV1 ) -: ?A \A1
b. p/\ ~ (q v r)
v' ( ? A. v' C ,,Vii ì ):: \A P V ("t V V)
i 1) Write the statement in symbolic form, and then use DeMorgan's laws to find the
negation. Then rewrite the negated staJement in words. _ .
a. I have a college degr~ and I am not employed. (\
~ OC ì \' 0J ~'t~D'-~D' l
S'i \I \0 \ ~ Ql~.J \f . '2 1\ V\ ei .
\A ( ? 1\ '"9. ') = V' ~V q
w~r&~; l' &0 \lt) \re Oi C'ò( !~e
J)~ve~) ""tr T: Cf ~iDl fJVc: .
b. If the legislation is approved, the public is uninformed. . l
It 9:" TL-e ~u.\o\;c ì s, ; /\ ~~e£
S'f 1M \00 \',C 0. t \1" l ~ cA 9. - v' ~ V \/'1
== '" l p ~ '-~ \ ~ V' (CJ ~V ~q) ~ ? A'(
wl'&s: n~ \~~'S\c.'\¡Q. ;~ C'~~fO~1 ~ÓL ~e tiu\ol:c ì'S
\V\~I lIe& ,
Writing and Understanding
1) Explain why truth tables are used and studied.
T ý'\Å~ ~'-Ie~ OJe llÇ'~ to f'i~ ~t ((')0.0+1:. if~eu a.
s\cx-1elN.et ¡ So Ì"r \Á e 't r cÇtX \ Se ht lA\ "'3 Q io \a \e i-
~')CtÅJ\/\E' 0.\\ ~ùSS\\; I ~.,\ e~ ,
2) Explain why the only time the truth values of a conditional are false is if the hypothesis is
true and the conclusion is false (i.e. if p is true and q is false).
I ~ P ì ~ iìe O-v.1~ +'i""Q tie CD~i-~()vv:~J h \l\'D~~~ '11' ~t .S:I'D.~s. lA"~ C' r:) VVO~\"3 \ S, vì fi!Oci-eb
,
3) Since there is no set number for the columns in the truth table (unlike the rows), explain a
useful method for how to label the columns of a truth table.
I + I'; ~\?~J ~ Ia.\o\ ~e CObAMIA~ w\'T st",i~ev'';
Th",+ \ofe~\: ~ ke ".ß'Aci\ \I ~MeWìev+ \V\% QCS;;QN)
\/Ðfe \MQvQ~~~le \t~eces,
4) Can you thin of a way to write "Some pare q" symbolically (so that statements of
this
form may be used in truth tables)? Hint: it's a kind of compound statement that we have
already learned about. Think about a specifc example to help.
SOMe ~ OJ€, ~ '3 \? A ei. A Vei'",' \)aJf0J W\a.h~
TlA'ts e(l~\E: ~\- ~~e';
ci V\ot P
. 1.3
P..
1f'ò+9
"l32, J
Section
Homework:
2, 5, 8, 11, PtL&
16, 20, 22, 29,
33, 38, 44, 48, 51, 56
Section 1.4: More on Conditionals
Key Terms and Concepts
Conditional, converse, inverse, contrapositive, biconditionam
Practice Problems
1) Give the following, in both symbols and words, for the following statements:
p: She is a police officer.
q: She carries a gun.
a. Conditional (p -+ q) J: r s"'e ¡~ 0. ?ß t'; cf' JlÇ'te-) ~ s-\.e CO"" ~fS c. 3 LW .
b. Converse Gi -+ ----YI' t sle C1~es. 0:3 U-) 1iQ. s!f? "IJ ex?D \ \~t~ ('
-T CÀ 3\.
c. In~erse (V'O ~~) I~ S~ ~~ t.t Q ?Ð~~ee 'O~Ç\ç\\ C~) ~Q. s;~e &ùe~ \A()~ ~u/ll
d. Contrapositive (~ ~ ~)
l: S:~Q cQes V\-T C'úrj (A ~)~ ~~e ~S \Ar(t a ~d~
ë)~~tGV,
2) Make a truth table for the conditional and its three variations listed above:
\\
~'\ '~C(i
tiveJ.e \/\VEW'i.( C~;''f()~~t~''t
P 9
V\?
TT
T F
F
~
~ '1
?-4 9 (-4p 'v-~--~9 V'9~~P
T
T F
F T T
~
F T
F:
T
T
T
T T Ff=.
F Ti
..
T T T
T
F
\
T
T
-
the conditional are equivalent, based on the truth table from problem
2? Write these equivalencies in both symbols and words.
3) Which variations of
""--V\l
n~ e('w9~t'ì~~ T ~~?O'3;-\~ve OJe Qqv.ìv.) ì,e. ?4 =
ne Y'v~e 1- GJVWSe We Ô1u.V) LE'. ei~p-¿\.P~4
4) Which out of
the following are equivalent? Explain.
p~~q, ~q~p, ~p~q, q~~p
Q) ~ @) ~
~,~'iC\1 ~v"~~e \1\\J/se a;:iA\ro~t:,i\ve
CD + @) ev e ec LL~ V '\ let-t S~/\c: t"ey OA e tle C0c9)\~1 + C& 'k ~S .
G) + (Ð Cu equ:Vo.JBA+ .s~",æ ~ o. ~e QlVQ~e+ ¡",vei:se
5) Translate the two sentences into symbolic form, and use truth tables to determine whether
the statements are equivalent:
~: i \rve s:ur~j~ q \ l ~QJe ~&~~
~ I cannot have surgery if I do not have health insurance. \A~ ~ '-. P
" "",Su.v"'Qce .
~fI can have surgery, then I do have health insurance. \? ~9.
T \f~ 1-\oles V\+ I/~~) .s\/\Qe CD h ~e ~1'1Q~()S,1\"e
ö.- @)/ T-e ~t9~'t~~\.. ~€ ~tL ~Ue lD~l~ \oe
ì~i-~\ +0 ~e \I~pe~~ve CÐ~LA1S j" \tff\ole.2
~ ~Q \(j'eJ\~~ ~g;,
6) Write equivalent variations of the given conditionals:
a. If it is not raining, I will walk to work.
~"ro ~OS~'\'ive'. ir. 1: & V\+ L0ik +D lÁ)rÐj'~) Tk~ 'it ìç 1/ì:i..\/\\"1'
b. You are a criminal only if
you do not obey the law. vev"~~e: ~-? '-lÐu OJQ Q 0J\'VV\\ f\C\t j
oJ ,- I' () \r\.
. ~O~
+i \_li ~
'tI f:
&. V\*
~"ro~s.A-'i_ve~
-- 'T ,1il
l~.J
~ IaN.
D~e. f1~
YÐ. ù.e \Aot Ol tV'; V\ \E'\ '
7) Determine which of the following pairs of statements are equivalent:
the Giants win, then I am happy. CD,~ CD ' ~
a. If
If
I am happy, then the Giants win. C!1 ~ ci,(( equ~\JoJ~ t-
If
the Giants lose, then I am unhappy(èJ ~ ~~ OJe QqU\ V'()J~+-
If
I am unhappy, then the Giants 10sß
iÍf! r\ fer.
b(!I am a rebel if
I do not have a cause. 1:~ ~ ~ v; ~Q\le 0. CC~ ~~ l' ~ Q ~\.'
f, ~~.J
_fi
I do not have a cause. i~ ~ Q. ~ v-\&~J
:t (X.A-\
~ Q. r
ea~e..
Ø)I am a rebel only if
~ I am not a rebel if I have a cause. J:.t 1: \.ve Q ~ctSQ) tl~ :t ~ \AÐ\ (). v-\r\,
(g If I am not a rebel, I have a cause. ~ S:~e
CD +@) OJQ ~~\ ~\Qv-t
~ ~@ oWe ~LL~t~i
8) Write the following in "if.... then.. .." form: .
a. p only if q 'r\' ~ tÌÒI q .I \..Q, '? ~'1
b. I eat raw fish only if I am in a Japanese restaurant.
i~ l' ~t ~ A2~S~) ~ ~ Cu ~n 0. TQ~Cu~-e
~u'ifCl t .
9) The following problems involve the biconditional p ~ q , i.e. p if and only if q, p iff q
a. The symbol p ~ q is given by: p ~ q = (? ~~J ;A (O(~ 4~)
b. Write the biconditional for the statements p and q in problem 1:
S;~e CQ 1(( i ~ 0. 3 WI \\' ~ ~ ~ \ -\ s"'e \~ (A lÐ~; Cf
~cw.
c. Construct the truth table for the biconditional:
\
? ci ?~9. ei~l
T
T T
T r: t:
T
~ i-
f- \ F
, \
T
r:
(?~ct)l\( ei-4p)
T
F-
P
T
d. When is a biconditional statement true?
Ov\~ vù~ ~ aw or ~I'e-ej ).e, ~O-ve ~e
S~€ ~~ \fCÀ~lAe l
two conditionals (use problem 9a):
A triangle is equilateral if and only if it has three equal sides
10) Express the given biconditional as a conjunction of
:tt A '\r~~\e ìs. eq\.-,Jù\~QJCl\ ~~ ;-r ~Ot\i ~ ~Q~\ ~;~e~
~ ì -P ; -\ ~()s. TlJe-' 0tÙDJ s";&ES TtBv ~e tv'iCLI/4e is ~\ b~eJ(
Writing and Understanding
11) Why is it helpful to know equivalencies of symbolic expressions?
i+ h "'tp~\ ~a.ç'.e 'it ""drs II '2 Soft h~31A ~Q
~,1\N\oDls.; \¿V\~\/\q .Q'tLt~ \ttXl~c;es . c~ ~~~Ç)l:~ Q .?Î'D\c~.
12) Give two examples of a biconä'itional statement. Explain why you are abielo wnte them
in this manner.
C0 stÐe~f~' \" Th e- \? "3 -: ; ~t ~ Q. l~ ~ ~ ì ~ClS ~.
(;~ ~e ~)(~Ie)
Section 1.4 Homework: 2, 3, 6, 9, 12ab, 15ab, 17,22,25,28,30,33,37
Section 1.5: Analyzing Arguments
Key Terms and Concepts
Argument, valid argument, tautology
Practice Problems VQ t; d. cxl!lUQvi ~ O- 0Æ9u.evt i,"''b~..
c.~i-~~a\ ~~ ,ç ex ~t- lêiY.
1) Write down the i 3 J ;lie representation of an argument with hypotheses hi ,..., hn and
conclusion c: eo ;î--'ì~ \
~I 1\ ~L A_ ~ ,A h~ ~ c
2) Recall from section 1 that a valid argument is an argument in which the conclusion is
unavoidable. How can we reformulate this definition in terms of truth tables?
~is '\~ equw~~~i "f S;~~"3 ~1- ~e ~c9Ii-~\
\f~\fe~~ ~t~ e. o.bu-e ~s, ot ~td~lJ) Ì-Q, 'r lr~ 0\\\
n ~/\ \~~ 1''IÅ~ ~\O\e Co\~V\
3) Write down the steps for using truth tables to determine if an argument is valid:
CD GO r : 1-Q n e Cu lÁ ~ i " '" .s 't V\ \m \ ~ c ~./
~ wr~~e CX~ c. ~&~1~\
ID u.s.e ""ru. TL to \ole s to See ¡ t' ~ t ',So Ot b ~l ~'J
4) Determine if
~p~q
pvr
:. ~ pvr
the foIl owing argumen is vard
i or mvard
i:
,'\
?
,
T
t=
ci
V
\
t:
r:.
~
\'
r'
T
P-
~
~
~
p
t:
pT
~
T
~
T
T
t:
T
T
T
r
\
\
~
T.
\
T
T
T
æ
t:
cp
T-
T
T
\'
T
\'
T
F
T
T
i
,
r
3 (1,(
@)
Q)I\@) V\'tvr -(
,
\, \
T
v-?~ ci tV-f
\./\~
T T
CD
T
T
P-
T
T
r-
fi
f
T
~
T ~
T
T
T T
5) Using truth tables, determine whether the following argument is valid:
If he is illiterate, he cannot fill out the application.
He can fill out the application.
? ~ V\'t
Therefore, he is not illiterate.
:1
rp', l1e )s, 'i 11\1wÜ\~,
~ 'i He ~ ~ t\ êLit
ite Q~?\kUl1-~,
~~
Cif &'ii ~~i .~~ '\,\i'~
(? -' \Aq) A'1 -" V\ ~
l 4
?
\T
to\
~~
P ~ V\~ 1 (p ~ \A't)A~
P
'P
P
t
T f:
F
T
\'
,i \
T
t:
(? -- "'O¡ ') A 'l ~ '" P
T
T
T
\
~
:- ~ " ._------¡ \,
l,
r-
~
T
F-
!
L
¡
T
~
6) Using truth tables, determine whether the £0 1I0wing argument is valid: \ 't..' \ 'S Q. L ÌD I ~:i i
If the defendant is in.riocent, the defendant does not go to jaiL.
The defendant does not go to jaiL.
Therefore, the defendant is innocent.
\ S-o ~e CU~:rJ.JN~t
L~~""'~""-~.""""----._--..:
',ç \Võl
ep~Re Ö)Ç)~-eNî~t \~ \/'V\OC~11
9.'i The &0~&w+ S\)~ to JOe\\,
?~ \A~
mv&;t~\ I/~V'Q~ '\'\'ì\M'
V\~
(p~v-ci) /\ "'t ~ \?
p
?
~
T T
T- i:
~~
~
\'
~.
; \~
\
T
p~~q
\T
~
T
--
( p ~ Vl't '; A "'OC (p ~ "''1 ') j\ '" 't ~ ti
t=
T
F-
T
\"
T
T (Vt+ (A
CÐ-S~~I
boA ùa,\ JJ 1
7) Using truth tables, determine whether the following argument is valid:
?~ He 'i~ C\ ?~e~Qf~
No professor is a millionaire.
No millionaire is illiterate.
Therefore, no professor is illiterate.
Cf', H-e 'i S. () Vv \ \\ ~~CA \i.e.
r', \i-e ~ ~ \\\ ~~evOI1- .
So,! \M \ot ¡ La. \ lyl.
~~~~~Q' V"e¡re~~~+i~:
?-4 V\C'
(?~~~) 1\ (ci~,-.) ~ (t-7oAr)
c: -7~y
?~v-..
~\A~ ~~\Q
l t
v-
\. q
,
v.
V
\ \
I
T T T ~. F
T T F r:
T P T T ~
T f-. F- T T
tt2 \"
2
(1
(?
~ ~ \/9. O(~",r
\'
r:
F-
P-
P-
T
r ~ T T F:
F ~ P T T
F
F
T
T
\T
P
T
T
T
f:
T
T
,
@)
~~ "'v
,
F-
P
\T
T
T
T
,
(bA@)
(14)~
F
p
T
T
P-
è\
T
ì
\
\ T
T
T
,
~QC~~-e ~e ~&'i +';~\ V'velS~~'D~ \ì-Ç ~.e
~ ~e.t '" \f:;\- ex ~+O\~~) ne ~LLW\-Q+ ìs
\/\ \/0- \ ~c9 ,
\
Writing and Understanding
8) Explain how the two definitions we have for valid argument are equivalent.
Av 4u.e.t 1" lA"'\~ '\e ~tLt~'ì~ 'Is, l.QVq)I~(lU~
ìs eq~ì~tÐA+ 1" SQ~\-: ~~ The ~~ti~\
~~~~~ÛV ì~ C\~tù~ ~L(e/ \.R, Q ~~+oto~y,
Section L.SHomework: 10,11,14, IS, 18, 19,22
Chapter 1 Review
The next few pages contain review materials to help you review for the Chapter 1 Exam. Please
note that the practice exams provide only a means to review for the exam - the actual exam will
not be written directly from the practice exams.
Additionally, a review assignment from the Chapter 1 Review in the textbook (pg. 58) has been
listed below, and may be assigned as homework.
The sources of review that you have for the exam include:
** assigned homework / quizzes
** notes / problems worked in class (i.e. from this packet)
* * practice exams
Chapter 1 Review Homework: 1,5, 9, 11, 13, 15, 17,21,25,29,33, 36,37,40,41,44,45,47