Zhao
0213
Cheng
CS12115
Assignment
Qnl
Cal lil
planar
u
panga but pay
vfrx panel
Efrapna
r
given
v
Curnutte
commutative law
pulping
ra
urn
I
I
I
I
I
I
I
distributive law
CPM ulpaqjulupaq
Palau
Inra
traumas
Cupa
a
distributive lay
distributive tan
negation law
patrue U Gpa
rn
identity law
Pu Gpa
wrk pump x pug
distributive law
urn
n
ra
true x
VA
pug
ur
xp
negation law I
identity law
distributive lay
rag
pug
C
Lii
S
E
lur xp
I
urn
identity law
pug
wftv vulpua
v9
Ll
In Lrv
Ciii l
nor
AE
aub
E
AND I
nor
la
till
fro
varna
Demoga's
double
low
negation
law
a
Cro Laib
nor Gorda
a
nola b
nor
bibl
Chl l The argument is valid when all premises
are true and the conclusion is also true
I I Assuring H and T are the
1 2 If we sub te true at true
into
putt 7 T we round
untrue
getcputne
7 metre
1 3 Cpu true
I
I
true
true
7
false
I
tee
universal hound
laws
negation of true
L
t
1
4
true v
I
false Limpliation law
I
n
I
false u false
I
false
negation
identity law
of
true
However DUH J T E false
which ion radius to 1 The premise
is
false
1 5 This means that the argument is
vacuously valid
2 For an argument to be sound it must
be valid at all premises are true Here it
is unsound statement
p req
2 Ca
i
y
prey
Pree
x
y
Y Z
x
z
ASEN Vx y EL Kaken Cs Y XPregex Y 7 Taken Six
Lil
lee Aiken be
a
Taken La R
i
Taken
a
Taken
x
P
x
Taken
a
a
s
Q
1 The argument is valid when the premise is
true and the
conclusion is also the
1 I Consider That TakenLair x Tarentaise
Ital
1 2 Taken la R
tee
and
Taken as
true
spelialization
Taken Ca R Ete
1 3.1 Aiken has taken R
consider
1 4
Taken cars
true
1 4 1
Taken cars
falsecnegat
1
3
Consider that
1 4.2
Aiken has
not
taken S
1 5 From l Ida and the knowledge base
Aiken has either taken P or Q
both
1G
Aiken has taken both I od
LS
is true
in ie
the
premise is
contusion is
true
true
or
Q
and the
Theargument is
valid
3 Ca
1 Consider elements 1 5,9 where
I EA 5 EB ad GEC
1 71 11 51 4 al ly 21 15 91 4
1 I
1 2 4 is not smaller than 4 ad here
Ix yl is not smaller than ly 21
2
Henie the statement is false
Cbl l Suppose
and
I I
that the statement is
negation is tie
LUXEAG EBU EC
its
false
ext Nycz RCA
I true
I
12
13
EXEAEYEBEZE
x Cy NYC2
NEX
I true
Consider all the elements
from BartC
Hy EB K ZEC Cy 2 I false
EYE BEZEL Cy 2
EX EA Ey EBEZEC
Ex EA Ey E BEZEL
I
a
false
afalse nan
false false
Cariversal bond laws
1 4 This iontraditts 1 2 that
EXEAEYEBEZEC KY NYC2 NEX
are
derived
from t t
I 5 Henie the statement must be true
whish
is
since its
1
l
false
Consider the
first lase where 4 2
I I When X L 2 0 74 5 it H Y 1251 3
and ly 21 1501 5
H y kl y 21
1 2 when x 2 2 10 74 1 S T Nyt 2 11
and
2
negation is
ly 21
11 101 9
H ykly
Consider the second case where x
21 when x 4 2
0
74 7
2
4
5 t 1 71
14 71 3
I
I
2
3
4
S
4
1 4144 21
ad 14 21 17 01 7
I when
4,2 10 74 3 5 t 1 41 14 31 1
and ly 21 13 101 7
H y Cly 21
Consider the lase where x 6
2 I when x G 2 0 74 7 s t H if 1071 1
and ly 21 17 01 7 1 41 14 21
3 2 when x 6 2 10 7 1 3 1 t 1 41 11 31 3
1 71 14 21
and ly 21 13 101 7
Consider the lase where x 8
4 I when x 8 2 0 77 7 it 1 41 1871 1
and ly 4 17 01 7 1 41 74 21
4 2 when x 8 2 10 77 1 s t 1 71 18 11 7
and ly 21 11 101 9 1 7144 21
Henie by insider all loses of X ad 2
suit that
there always exists a
y
is
true
21
The statement is true
lx Y Cly
a
it 4 0
True for
Ci it
any non empty A
Ciii
A 2133
I BY
Liv
A
e
g A
1,2
b i 1 Sine
sets
A and B are
two non
empty sets
PLA and RB are also two non
sets
empty
a set
also
that
PLA PLA is
I I This implies
ad it can either be empty whish
on cars
when
whish is
A B
or
non
empty
At B
1 2
By theorem 6.2.4 0 is a
subset of every set invading both
2
empty set and
O E PLA PLB
i
non
l let A
ii
1
when
non
empty sets it
I PLA
PCB
Ai Az
O
A
0 B
empty
is true
a
dB
for all
Any ad B
A Az
Az
Bi 1323
123
sets
Bi
Bi
in
Hi Hi An
9131,132
Bn
since both A and B are non empty lets
1 2 PCH P B Will be a set containing all
elements that are in A ad not in
B fromit
B Sine OG PLA and
LPCA PCB
04 Pat PLRt
0
2
c
The
non
restriction
OEI
is
true
for all
empty sets A al B
c nd d EZ
is old
f
denotes all integers
greater than 3
let a
and let Pa refers to n being a primenumber
In EZ 3 sit Pcn x Pinta x I n 41
l Suppose that the statement is fake and its
negation is
Un Ets Cupen v
true
of exfatian
be either odd or
negation
I I
n
can
Piney u
n
I
Pent41
true
e en
but not
both
axiom
1 2 consider when n is even
even no
1 2 1 7 KEI s t ne 2K def
of
since n 3 21h is tree for all
n
7 V15 Eat n ers a Kren a Ksan
1 2 2
where either r or s is 2 Henceby
numbers
the definition
composite
of
n is
not prime
7 2
3
For all
n when his e er
Un62,3 pen union 2 UnPinta
13 consider when n is odd
1.3.1 IKEA s t nakti def odd
of
number
1.3.2 In order
nd n 4
to be prime 34nA 34614 3 6 4
for
the result
n
of a mod 3
equal to 1
2 where
or
t or n 4
1 3.2.1 Consider n mod 3
n
i
n
ht 2
2 od
n
be
must
a
at a 2mod
243k where
KEZ
mod 3
1.3.22 Consider nmods andCnt4mod3
Nt
n
4 and 473k where KEL
n mod 3
n 4
1 3 2.3 Consider Mt 2 mod 3 and
Cnt 4 mod 3
nth
n
anti
where
na
2
n 121
KEZ
mod 3
nt 4
1 3.3 This shows that h
Sine there are only 3
and 243k
h
nt4 mod3
possible elements in
the set
containing the result ofmodulo 3
hit are o I ad 2
N mod 3 0 or
n 2 mod 3 0
or
n mod 4
0
I
31 n u 3km2 V 316 47
ad it contradicts 1.3.2 whish
means that there must be
at least one number in n
that
n 2 and nt4 sash
EET CK
r s
k is
where
in
n
n
least one
2
since
ad nt4
of
the number
Henie
at
the number in n nd
is not
htt
ACKraKINKSSK
at least one
t2
of
rs
a
for every
prime number
even
ad
ad odd
the statement is
the negation
of the statement
is
true
the negation
n
true
is
for
false
all
n
of
Henie
the statement