Description
MATH 2305
Exam exam
renew
Review
practice problems
5) prove the argument valid
6 (apvqr
©
(~t)
e lapan) → (25)
o decide whether or not there
a number whoresquare equals 4 times the number itself. Explain: 3 xed [x2= 4x]
• let X4 (4) 2 = 4 (4)
16 = 16
nypotheses are assumed true
0~put 2 ~p
de Morgan's M.T. O and © M.P. © and © M.P. 0,0 and cut and
We also discuss several other topics like Who killed her family for jason?
ova
(2) Explain why the sentence We also discuss several other topics like What are some of the reasons we first use assessment tools when first meeting a patient?
that you are currently reading is false." is not a proposition.
proposition: a sentence with
a truth valne a the sentence doesn't have
a truth value (ie, if the sentence is true, it's falser,
and vice versa) We also discuss several other topics like What are the strongest words in the english language?
(6) what is the difference between
"Propositions and predicate"?
• Propositions are just sentences
with otvИТИ yajи
• predicates have a variable,
become propositions when variables are replaced with
avaine from the domain
(3) Statt de Morgan's laws: Don't forget about the age old question of Why cellphones must be put away at all times in lab?
(AB) = (CA) V (B)
• (A/B) = (~A) A (B) 14) Decide if the argument
(7) negate the logical expression: WAXED, 3 y&D, [P(X,Y) * Q(x)])
AXED, YyED (v [~P(x,y) VQ(x)]) EX ED, VY ED Ev (v p(x,y)^(~Q] We also discuss several other topics like What physically defines an element?
2 XED, YyEp [P(x,y) ^ ^Q(x)] (6) what is universal instantiation? We also discuss several other topics like What is the ph of the egg yolk?
every sub. Of an element ED WIU be true in a universal argument
pf: VxED[PC] (M. I.?
Hopea Pq P*99 p/ Pvq
.
copval
(9) compare prop. and Pred. logic (11) thm: if kis odd MATH 2305
wing the concepts of Ttables, and mis even, Exam / proof methods, tautologies,
Review predicater, and propositions:
pfi 0.1e7 Ranodd number → Prop. logici
2 = 2at 1
• Proposentence, letters, TIF
let m = an even number
• can show proof through
m=2b tables (if there aren't too
1 k 2 + m 7 = (29+12+(26) 2 many letters), annotated
=4492749+1) +(463) listings or proof sequences
= 2 (20229) + 1] + 2 (262) valid argumente: M.A.M.T. cut, generalization,
= (2n+1+2v Specialization, 1 transitivity
 22+2v+1 = 2(n+v)41 tantology shown in tables → pred logici
pred: sentence, variable is
generic H2+ņa ir odd incinded, becomes a prop
©:, Theorem istrue when variables are
replaced by values (12) prove or disprove: ttables won't work for
toimi if n is odd, Disodd Universal statements
• V=all values
pf: if n=5, nirodd 3 at least one valne
but 1517  4 = 2 even
• truth Of Y can be shown
thronan U.I. and M. p.
•] only needs one true (13) repeating decimal 1.2323 val nie; can be found
as a rational number thronan guessing, testing all options when finite)
112 32 32 line up
I repeating vare tanto logies
100 X = 12312323) decimal theorem:
9 9 X = 1221 (10) prove that the square of an
odd number 1. odd:
X=122 >pfi
Olet =anodd number ~ 9 = 29+1 def. Of odd)
X = (20 + 1)2 = 492 +4+1 = 2(29+20)+1
(292+29) = b = 26+1 3 generic ý zis odd! es theorem is true
MATH230S exam Review
(14) thm: sum of any 3
I consecutive integers is divisible by 3
(17) Matching
terms may be used more than once, or not at all
Pf:
let â = an integer
ât câ+1)+(6+2) = 3 + 3 = 3(@+D (= 36 ) :
36 generic â makes theorem
TERMS:
B. variable C a proposition E cutlelimination FAB
de Morgan's laws ( modus ponens
NO Term appears
hence, theorem is true
(15) thm sum of any two rational
number is a rational
DEFINITIONS:
1. a formula which is logically
eanivalent to AVB
E. AB
of 9 let
a
and C be rational
2. ~(AMB) = ~AVNB and
~ (AVB) = ~ANNB are called
I. de Morgan's laws
c. ad bc
=m
3. a name used for an unknown val
variable
bd to because bad to
4. place where variable values
may be found
NO Term Appears ► domain
theorem works for generic I values (a, b, c, d ) ☺ hence, theorem is true
5. meinod to conclude "Bmust be
true when "A" and "AB" are true
modusponens
(16) what is wrong with this "Proof"
ihm for any integer ko
K2+2K+L is a composite [Pf 22 +222 +1=9 and 9 = 3*31
6. Valid proof that can be used to
implement M.P. and M.1. © Cut/elimination
thm. says "any integer, meaning it must be
universally quontified (1), not just one example. They should have ured a
generic value for ķ
7. A → A is this type of formula
statement form
No term appears  implication
pencil = proof
green commentary red: justifications
(5) prove that the following argument is valid hypotheses (@lup v
q r are labeled o sv (9)
. this is This is
our @@,50 L can easily
theorem refer back in proof
Le (Par) (us)
. (9)
1 @ pt
Proofi
Justification' @ these are our
hypotheses from above.) all hypotheses
• I'm just too lazy to } are assumed e rewrite them *
true o uput
Togically equivalent to a (Not demorgan's. I wrote
@ the wrong thing before )
@pt = ~puto 2 ~P
modus tollens using 0 and ©
orovt
@rt 3 :. ~P
wpvq)
I talked to Dr. Thomas and he
said my Original Mełnod needed the explicit addition of these steps
generalization using ☺
3 up ®:.,(npVq) Modus Ponens using 3 and 4
@ (~PV q ) ►r
3 (~Pvq) Or specialization using @ and ☺
I
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them before)
VPNO
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VOUSOU
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@ (upar) ~ 6 (~Par)
anos pris
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math 2305 10.4.16
alternative solutions (9) compare prop and pred
logici (Prop. pred.
usually 1020 matching
(definitions)
Truin tables
prop. logic only not in pred. logic
 X2 4X=0
X (X4)=0 X=0 X4=0
X=0 or X = 4
stable
(2) can't establish a
truth value
Tantologie
always true prop.: checked through
truth tables for all Possible truth value assignments, the
formula is true
•Pred for all possible.
pred. definitions n all possible domains
FX ED [P(X) V ~P(x)]
(3) ~YXED[ ] = 7XED ~[ ]
waXE DE 1= AXED [ ] (can include these universal
oner if you want)
(4)

T p q lettersdon't need to be T q pf D true for the line flio, pval to be true
predicates;
ca senience, contains
variables, becomes a es proposition when variables are replaced by values Pname (x, V, Z)
A() = a prop. in pred. logic proposition
a sentence that has exactly one truth Value A inse a letter to signiful
(51 0 up
Mite © and 2 ~pva) generalization and
M.P. @ and a 9 (up 1r) specialization 0 and
us
Mip. and
cut and b  work: opet ☺ (prqder 3 (upvq) ar
vt vp
fupva) up (vPvgi ir P
p^n + SV69) heart
1 ~q
10.4.16
alternative solutions conta... (1) Olet ke an odd number (14) ► problems like this won't be
land me an even number on the exam. 2 k = 2atl. atz I m = 26. bez
(15) Pf: let rur EQ crational)
r. = 9/6 anbez, bto = 492 +49 +1 +4b2
 P ă cadez, d to 2.[222 +29 +262)+1
3 a C  [ac]  m mnez = C, CEZ
b d [bo] n nto t = 2C+1 Caef. Of oad)
Ifine producte 2 generic k24 m2 isoad
numbers is o, one hence inlorem is true
or both must be o
*on previous review solved for (12) prove or disprove
sum of 2 rationais, problem
15 says Product! if trueshow
 Itaire,
OOPS Proof
give a
Counterexample (16) cant substitute ("proof by example") expression (thm if n is oad, then
for infinitely many possible  values
universal
Ipf:
Counter example
(17) 7. A A is a tantology
_ ®NO Term appears
n = 5
151  4,2 leven)
2
21
ithm is false
"the first time it's La thick, the rest
. Of the time it's at ChiИИС"
MATH 2305 exami Review
setbuilder notation
S = x.suon that Pex is true}
S = EXP(x) abbreviated
CHAPTER 1: Speaking mathematically
1. variables
· used for unknown values ex: is there a number such
that doubling it and adding 3 gives the same
result as sanaring it?
2x+ 3 = x2
X3 or X=1 used for arbitrary generic values ("all" values) ex' if a number is >2,
its square is >4 > 107 X= "a numbet."
if X2, X274 #used in proofs
• universal statements: "for
all/every x ino"
• conditional Statements:
if O, then 4"
AS B if yea, then XeB YEB, but YEA Russein's paradox: problem
wim Sctbuilder notation
{XIXX3=B if BEB→ B&B
BOB >> BEB → repair by assigning a
universe (U) B = { x} X3 B + 6 because
640 (B must not
be in the universe) cartesian Products
ordered pairs
(a, b) = (0, d) iff a: c^=d.
AXBga,baeA, beB3 1.3: language of reiationen functions
. dof: reation between sets
used to describe nonfunctions ex: a circle
*9) yvalues are On not unique
1.2 Language of sets
Notations E "in"/"isa member of "
ex: XED "not in"/"not a member of
ex: x € 0 subset ex: A={1,2,33 B={1,2,3,4}
ASB
• useful sets
Z inteqers 1,0,1,2...3
relation
z+ Positive integers {1,2,3...3 set roster notation
S={1,2,33
Lohisiory of members Zt={1,2,3... B infinitely
" def: function
given Xfex is unique every XED must be used
CHAPTER 2: 1091c Of Compound
statements 2.1: logical form and logical
equivalence
def: proposition
a sentence
lear truth value (TED wysymbolized by a letter
logical operators ^ "and" / "conjunction" V orld is junction" ~ "not" negation truth tables shows fruin
values of prons lex: ALB AVB AAB
MATH 2305 exami
Review . de Morgan's laws
MAVB(MAA(UB)
N (AMB) = A (B) 2.2.conditional statements
deft implicanon ()
 if A is true, then B is true  if A is false, the
implication is true I by defain it
ABAB
10gical equivalences * AB & BA
NOT THE SAME! * ABS (VA)VB
AB AB (VA)VB.B TA
TE FT * Syntax and semantics
binary needs two Props (ANB). binary unilateral (one prop
(A) tautology: for all possible truth asian ments, the argument is true ex AMA AVA)
contradicion for all possible
truth assignments, the arqument il faise lex A LA CAMA
negating conditional
statements N (A B ) og. equiv
M (AVBV becomes 1 ( VA)^(^B) ariveinn
An (B) double
negation
• converse: interchange
hup and conc.
ex: A B becomes
logical equivalences BAA) (AAB) (AUB) = (BVA)
"If it rains, then they
cancel school"
becomes Mif they cancel senool,
then it rains
> prove by Cheding the
truin table
 argument fom cont'dono
A B nypothues
A
MATH 230S exam! Review
inverse: negate H and C 16PQ) becomes
ploQ)
if it rains, then they cancer school"
becomes "If it doesn't rain, then
they don't cancel school" Contra positive interchange
the inverse (PQ) becomes
B6 B conclusion
def: vaid arguments when the hup. are all true, the conclusion is always true testing for validity
A make a ttable and
I check it annotated listing of
argument © ure a proof sequence * VALID ARGUMENTS:
MODUS ponens
→ Modus Tollens
Mifit rains, then they
cancel school
becomes Wif they do not cancer school,
then it didn't rain"
• def: biconditional (*)
boin props must have the same truth valve
(if and only if /"iff")
A3 A**B
VB
> cuti elimination
AVB NA
→ Generalization
O, A NE → Specialization
• Order of performing operations
negation 2) AV coni./disi. 3) imp. 'bi cond.
[GA)  (Bņ(yg)]
franshivity
A TMB
> proof bul cases
IPVQ
2.3. Valid and invalid arguments
def: Grayment forma
a list of 109109l expressions in symbolic 10916
. Proof sequence example:
sep
TA
hyp. are assumed
true Mat. using @ and a
oup
@~5
MT. using and
MATH2305
exam,
Review
• def: universallu quantified
statement (A) VXED [P(x)] "Ifor all x valves in D,
P(x) i frue" def: existentially quantified
Statement (3) IXED [PCD] for at least one x ind
P(x) is true" formal v. informal
language → informal human lang,
formal: symbolic lang, FORMAL → INFORMAL
is easy INFORMAL > FORMAL
is hard
• implicit quantification "integers, are rational #s"
La universally quantified
even though we didn't explicitly say
all integersen 3.2. Negating Quantitied
Startements Theorem 3.2.1:
to (VXED [Q(x)]
EXED ~ Q(x37 Theorem 3.2.2 MEXDCQ(x)])
or
cut using
Irus
and
Crot
M.p. using
and ©
* weaknesses of prop. logic
→ no universala statements;
the closest is listing
all Objects (n) →no "existential" statements
without finding an
explicit object
CHAPTER 3: Logid Of Quantified
statements 3.1: Intro. to quantifiers and pred.
• def: predicate
sentence becomes a prop. When variables arereplaced by valueSED
Negating universal
conditional statements
(Vxeb [P(x) = Q(x)]) = 3XED ~ [~P(XVÕx2] LEXED LP(x) V ( QO))]
~ (EXED [P(x) + Q(x)]) = VXED ~ EvP(X) VQ (x)] = WED [POV (Q(x)]
negations of multiplyquantified statements ~ VXED CEYED [P(x) 12 E ]XED IYED [PX) = 3 XEDLEYED INP (x, y)]] Order Of Operators → doesn't matter when
all quantifiers are & ex: VXED VY ED [P(x,y??
= VYEDVXED[P(x,y)]
MATH2305 exami Review
• relationship to de Morgan's when domain is finite v is a generalization
of and statements ] is a generalization
of or statements
• variants Of conditional
Statements (ex : 2.2.5) ~ "if sarg lives in Athens
then she lives in Greece" = 'sara lives in Athens, and
she does not live in Greece she could live in Athens,
Ohio or Athens, GA) 3.3: Statements with multiple
quantifiers ex: "Somebody (3) Supervises
every (v) detail"
3XED VYED [sup.(x, y)] person détail people: * *
3.4: Arguments with Quantified
statements o def: universal instantiation
(YX EDLA(x)  BOX] MA(8), XED
68 B (82
→any substitute of an element in Dis true
by definition нх ухер Грих) C & PC2, XED
proving validity of arguments wiquantified Statements
betails * * * * * *VY Hevery detail has somebody sup."
VYED [IXED[Sup(x, y) detail person people: * * * *
[details:
* ORDER IS IMPORTANT! informal to formal logic "There is a smallest
Positive integer," → one quantifier ]
2X€ 7+ smallest XI > multiple quantifiers
ZX67+WVE 7+ (x=y) it's hard to move from ambiguous informai io concrete forma
E
negations of multiplyquantified statements ~(VXEP (ayed [P(x)]).
XED []ye D[PX EXED VYED (up (y)]] order of operators" → doesn't matter when
all quantifiers are t ex: EXED VYED [P(x, y]
= VyEDVXED[Play]
dy Soup
Studysoup
tour
MATH 7305 exami Review
• relationship to de Morgan's when domain is finite
V is a generalization I of and statements.
I is a generalization
of or statements
• variants of conditional
Statements (ex : 2.2.5) ~"ifsara lives in Athens,
then she lives in Greece" = sara lives in Athens, and
She does not live in Greece Che could live in Athens,
Ohio on Athens, GA) 3.3: Statements with multiple
quantifiers
• ex: "Somebody (7) Supervises
every (v) detail".
]XED[VVED [sup.(x, y)]] person détail people: * *
13.1: Arguments with Quantified
I statements
•def universal instantiation
wVX ED LACX)  BCX)] HEA(R), RED
→ any substitute of an
element in Dis true  by definition H?VXED [P(x) C{ P(8) XED
proving validity of arguments wiquantified statements: use universal instantiation to solve for generic valves → universal modus ponens (UMP)
universal modus tollens (UMIZ
TXED [PCX) + Qx)] ware). RED
bletails * & ** **vy Fevery detail has somebody sup."
VyED [IXED Sup(x, y) detail person people * * * *
Isoup
details: *
* ORDER IS IMPORTANT! informal to formal logic "there is a smallest,
CHAPTER 4: Elementary Number Theory
and Methods of Proof
1.1 direct proofs and counterexample
def even integer : x 2a, atz
loddínteger x+26+1, bez
Prime in tegec: p2,
Pa*b, anbez, avb=! composite integeri CE Q*b, bue anbez, and to
positive integer." → one quantifier
Xe 7 challet (0) multiple quantifiers
IXEZ+VYE 7+ (X=Y) it's hard to move from ambiguou informal to concrete formal
Study So
) studySoup
Soup
getting proofs
MATH2305 Started:
exam = look for familiar Review
language in the
theorem lexithm every complete
bipartie graph, is
connected
Stugu
pf:
let G be a "complete bipartie graphia
a ou
. proving existential statements → find a value, by any method where it is
tre guessing.computing > or snow that is
impossible disproving universal Matements by counterexample: if we can find even one instance where thm. isnt
true, it is false proving universal statements I use an arbitrary value. and the definitions of lodd, eren, etc. to prove memod of direct proof
→]: find one suitable item → V, finite D method of
exhaustion (test every
Possible value in o) → V infinite D: show thm.
true with an arbitrary value (x) to prove suggestions for writing
proofs : o copy thm accurately
"thm: 3 mark beginning of proof
© Hence, Gis
connected 14.2: Pational Numbers
defi rational number
REP, PAQEZ, Q = 0
irrationall number
irt e, Page 7, qt0
theorem 4.2.1 sum of rationals is rational! ef: 9 letr. and he be rational
r = al anbet, bto
I r =
codez, dzo
☺ define all symbols carefully o use complete sentences 6 justify statements
© use therefore, hence, etc.
• common errors:
proof by example in an linfinite domain 2 using the symbol for
2 + ideas 3 conclusions with out
support 4 assuming what is to
be proved
3 ritra= 9.c  adtbc
be at bd = fE Z e enfet, fto
blcbndto © sum of generic numbers
is rational ☺ nence, theorem is true
• misc properties of rationais:
→ sum of rationals is
rational 7 product of rationals
is rational
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