Discr
e
te Str
uctur
es
Pr
eposition
al Logic
2
Dr
.
Muhammad
Huma
y
oun
Assis
tan
t
Pr
of
essor
C
OM
S
A
T
S
Ins
ti
tut
e
o
f
Co
mput
er
Sci
e
nce
,
Laho
re.
mh
um
a
youn@ci
it
lahor
e.e
du.p
k
h
ttp
s://
si
tes.
g
oogl
e.c
om/
a/
ci
itl
ahor
e.edu.pk/
d
struct
/
So
me
o
f
the
ma
teri
al
i
s t
ak
en f
ro
m
Dr
.
Muh
ammad
A
ti
f’
s
sl
ides
1
R
ecap
•
T
ruth t
able:
A tr
uth
t
abl
e di
spl
a
ys
the
r
el
a
ti
onsh
ip
be
tw
een
the trut
h
val
ues
o
f pr
oposi
ti
ons.
A
tabl
e
has
ʹ
row
s
wher
e
݊
i
s number
of pr
oposi
ti
on
vari
abl
es.
•
Ex
clusi
v
e
or:
ْ
ْ
ݍ
i
s
true
when e
xactl
y
one of
and
is true
and
is
fal
se
otherwi
se.
•
Ex
er
ci
se
:
Dr
a
w a truth
t
abl
e
of
ሺ
ْ
ݍሻ
ש
ሺ
֜
ݍሻ
2
Special
De
finit
ions
՜
In
v
er
se:
՜
Con
v
er
se:
՜
Con
tr
aposi
tiv
e:
՜
3
Ex
ample
P
aki
st
an
i t
eam
win
s
whene
v
er
it is r
ai
nin
g
p: It
is
rai
ni
ng
q:
P
aki
st
ani
t
eam
wi
n
s
q
wh
ene
ve
r
p
ؠ
if
p,
th
en
q
ሺ
՜
ݍሻ
If
it
is
rai
nin
g
, then P
aki
st
an
i
team
win
s.
In
v
er
se:
՜
If
it
isn’
t r
ai
ni
ng
,
then
P
aki
st
ani
t
eam d
oesn’
t
wi
n.
Con
v
er
se
:
՜
If
P
aki
st
ani
t
eam
wi
n
s, then
i
t i
s r
ai
ni
ng.
Con
tr
ap
osi
tiv
e:
՜
If
P
aki
st
ani
t
eam
doesn’
t
wi
n,
then
it
isn’
t
rai
ni
ng.
4
Condi
ti
onal
In
v
er
se
C
on
v
erse
Contr
ap
osi
ti
v
e
ݍ
Ø
Ø
ݍ
®
ݍ
Ø
®
Ø
ݍ
ݍ
Ø
ݍ
®
Ø
ܶ
ܶ
ܨ
ܨ
ܶ
ܶ
ܶ
ܶ
ܶ
ܨ
ܨ
ܶ
ܨ
ܶ
ܶ
ܨ
ܨ
ܶ
ܶ
ܨ
ܶ
ܨ
ܨ
ܶ
ܨ
ܨ
ܶ
ܶ
ܶ
ܶ
ܶ
ܶ
5
•
ؠ
•
՜
ݍ
ؠ
ݍ
՜
•
In
ver
se
ؠ
Con
ver
se
•
՜
ݍ
ؠ
ݍ
՜
6
Bic
onditi
onals
De
finitio
n
6
Le
t
p
and
q
be p
roposi
ti
ons.
The
bi
condi
ti
onal
st
at
emen
t
p
↔
q
is
the pr
oposi
ti
on
“
p
if
and
onl
y i
f
q
.”
The
bi
condi
ti
onal
s
ta
temen
t
p
↔
q
is
true whe
n
p
and
q
ha
v
e the
same
truth
v
alues
,
and
is
f
al
se
otherwi
se.
Bi
condi
ti
onal
s
ta
temen
ts
ar
e
al
so c
al
led
b
i-imp
li
cati
ons
.
7
Truth T
abl
e
•
p ↔ q
has
e
xactl
y
the
same trut
h
val
ue as
(
p →
q)
ר
(q →
p
)
8
Common
w
a
y
s t
o e
xpr
ess
p
↔
q
•
“
p
is
necessary and
suf
fi
ci
en
t
for
q
”
•
“i
f
p
the
n
q
,
and c
on
v
er
sel
y
”
•
“
p
if
f
q
”
Ex
ampl
e
p:
“
You c
an t
ak
e the
fl
igh
t”
q:
“
You bu
y a
ti
ck
e
t”
p
↔
q
:
You
can t
ak
e the fl
igh
t
if
and
onl
y i
f y
ou buy a
ti
ck
e
t
You
can t
ak
e the fl
igh
t
if
f
you buy
a
ti
ck
e
t
The f
act t
ha
t
you
can t
ak
e th
e fl
igh
t
is
necess
ar
y
and
suf
fi
ci
en
t
for buyi
ng
a
ti
ck
e
t
1
0
p:
You c
an t
ak
e
fl
igh
t
q: Y
ou buy a
ti
ck
e
t
՞
ݍ
Y
ou
can
tak
e fl
igh
t i
f and on
ly
if y
ou
buy
a
tick
e
t
Wh
a
t i
s
the
truth
val
ue
w
he
n:
•
you
buy
a
ti
ck
e
t
and
you
can
tak
e the
fl
igh
t
??
•
ܶ
՞
ܶ
ؠ
ܶ
•
y
ou don’
t buy
a ti
ck
e
t
and
y
ou c
an’t
tak
e the
fl
igh
t ?
?
•
ܨ
՞
ܨ
ؠ
ܶ
•
you
buy
a
ti
ck
e
t
but y
ou
can’t
tak
e the
fl
igh
t
??
•
ܶ
՞
ܨ
ؠ
ܨ
•
y
ou c
an’t buy
a ti
ck
e
t
but
can
tak
e the
fl
igh
t
?
?
•
ܨ
՞
ܶ
ؠ
ܨ
1
1
Pr
ec
edence of
Logic
al
Oper
a
tor
s
12
ሺ
ْ
ݍሻ
ש
ሺ
֜
ݍሻ
Can be
wri
tt
en as
ሺ
ْ
ݍሻ
ש
֜
ݍ
(T
/F)
?
ܽ
ר
ܾ
ܽ
ש
ܾ
֞
ܾ
ש
ܽ
ܽ
ר
ܾ
ש
ܿ
ܽ
ר
ܾ
ר
ܿ
ܽ
ש
ܾ
ש
ܿ
Ex
er
ci
se:
For whi
ch
v
al
ues
o
f
a, b
and
c
one
g
e
ts
0
in th
e trut
h
tabl
e of
ܽ
ר
ܾ
֜
ܿ
֜
ሺ
ܾ
֜
ܽ
ר
ܿሻ
1
3
Logi
c and
Bit Op
er
a
tio
ns
•
Bool
ean v
al
ues
c
an be r
epr
esen
ted
as 1 (true)
and
0 (f
al
se)
•
A bi
t
stri
ng
is
a seri
es
of
Bool
ean v
al
ues
.
Leng
th
of
the s
tri
ng
is
the number
of bi
ts
.
–
1011
0100
is
ei
gh
t Bool
ean v
al
u
es
in
one
stri
ng
•
W
e
can then do
oper
a
ti
ons on thes
e Boo
lean
stri
ngs
–
E
ac
h c
ol
u
mn
is i
ts
o
wn
bool
ean
oper
a
ti
on
1
4
1.2
Applic
a
tions
of P
ropositio
nal
Logic
•
Tr
ansl
a
ti
ng
Engl
ish
sen
tences (
Formal
iz
a
ti
on
)
•
S
ys
tem Speci
fi
ca
ti
ons
•
Bool
ean Sear
ches
•
Logi
c ci
rcui
ts
•
…
1
5
T
ransla
ting
English
Sen
tenc
es
•
You c
an access
th
e In
terne
t fr
om c
ampus
onl
y i
f
you ar
e
a c
ompu
ter sci
en
ce
major or
y
ou ar
e
not
a fr
eshm
an.
ࢇǣ
Y
ou c
an acc
ess
the In
terne
t fr
om
campus
ࢉǣ
You
ar
e
a c
omput
er sci
ence
major
ࢌǣ
y
ou
ar
e
a
fr
eshm
an
ࢇ
՜
ሺ
ࢉ
ש
ࢌሻ
1
6
•
You c
annot ri
de the r
ol
ler
coas
ter
if
y
ou ar
e
und
er 4 f
ee
t
tal
l
unl
ess
y
ou ar
e
ol
der
t
han
16
year
s ol
d.
ݎǣ
y
ou c
an ri
de r
ol
ler
coas
ter
݂
y
ou a
re
under 4 f
ee
t
you ar
e
ol
der than
16 y
ear
s ol
d
݂
ר
՜
ݎ
1
7
S
y
st
em
Specific
a
tio
ns
•
The
aut
oma
ted
repl
y
cannot be sen
t wh
en the
fi
le s
ys
tem i
s
ful
l
p:
The a
ut
oma
ted r
epl
y
can
be
sen
t
q:
The s
ys
tem
is ful
l
ݍ
ื
1
Consis
tenc
y
•
S
ys
tem s
peci
fi
ca
ti
ons
shoul
d
be
consis
ten
t
,
–
The
y
shou
ld
not c
on
tai
n
con
fl
ict
ing
requi
remen
ts
tha
t
coul
d be us
ed t
o
deri
ve a
con
tr
adi
ct
ion
•
When
speci
fi
ca
ti
ons
ar
e
not
consi
st
en
t,
ther
e
w
oul
d be no w
a
y t
o
de
vel
op a s
ys
tem tha
t
sa
ti
sfi
es al
l
spe
ci
fi
ca
ti
ons
1
9
De
termi
ne whet
her
thes
e s
ys
tem s
peci
fi
ca
ti
ons
ar
e
consis
ten
t
:
1
.
The
di
agnos
ti
c
mess
ag
e i
s s
tor
ed i
n
the buf
fer
or
it
i
s r
e
tr
ansm
it
ted
.
2
.
The
di
agnos
ti
c
mess
ag
e i
s not
st
or
ed i
n the
buf
fer
.
3
.
If
the
di
agnos
ti
c
mes
sag
e i
s s
tor
ed i
n
the
buf
fer
,
then
it
i
s r
e
tr
ansmi
tt
ed
.
2
0
De
termi
ne whet
her
thes
e s
ys
tem s
peci
fi
ca
ti
ons
ar
e
consis
ten
t
:
1
.
The
di
agnos
ti
c
mess
ag
e i
s s
tor
ed i
n
the buf
fer
or
it
i
s r
e
tr
ansm
it
ted
.
2
.
The
di
agnos
ti
c
mess
ag
e i
s not
st
or
ed i
n the
buf
fer
.
3
.
If
the
di
agnos
ti
c
mes
sag
e i
s s
tor
ed i
n
the
buf
fer
,
then
it
i
s r
e
tr
ansmi
tt
ed
.
p
:
The di
agnos
ti
c
mess
ag
e i
s s
tor
ed i
n
the buf
fer
q
:
The di
agnos
ti
c
mess
ag
e i
s
re
tr
ansmi
tt
ed
1
.
ש
2.
3.
՜
2
1
1
.
ש
2
.
3
.
՜
R
easoning
•
An assi
gnmen
t
of t
rut
h
val
ues
tha
t
mak
es al
l
thr
ee
spe
ci
fi
ca
ti
ons
tru
e m
us
t
ha
ve
p
fal
se t
o mak
e
㻀
tru
e.
•
Bec
ause
w
e
w
an
t
ש
ݍ
to
be true bu
t
mus
t
be
fal
se,
q
mus
t
be true
.
•
Bec
ause
՜
ݍ
is
true when
is
f
al
se and
ݍ
is
tru
e
•
w
e
concl
u
de t
ha
t
the
se speci
fi
ca
ti
ons
ar
e
consis
ten
t
•
Le
t u
s
do i
t
wi
th
truth
table
no
w
2
2
•
Is
i
t
remai
n
consi
st
en
t
if
the speci
fi
ca
ti
on
“
The
diagnos
tic
messag
e
is not r
e
tr
ansmit
ted”
is
added
?
p
:
The
di
agn
os
ti
c
mes
sag
e i
s s
tor
ed i
n
the
buf
fer
q
:
The di
agnos
ti
c
mess
ag
e i
s
re
tr
ansmi
tt
ed
1.
ש
2.
3.
՜
2
3
•
Is
i
t
remai
n
consi
st
en
t
if
the speci
fi
ca
ti
on
“
The
diagnos
tic
messag
e
is not r
e
tr
ansmit
ted”
is
added
?
p
:
The
di
agn
os
ti
c
mes
sag
e i
s s
tor
ed i
n
the
buf
fer
q
:
The di
agnos
ti
c
mess
ag
e i
s
re
tr
ansmi
tt
ed
1.
ש
2.
3.
՜
4
.
Inc
onsis
ten
t
2
4
Boo
lean
Sear
ches
•
Logi
cal
c
onnecti
ves ar
e
used e
xt
ensi
vel
y i
n
sear
ches
of l
ar
g
e c
ol
lec
ti
ons
of i
n
forma
ti
on,
such as
inde
xes of
W
eb pag
es.
•
Bec
ause
thes
e sear
ches
empl
o
y t
echni
q
ues
fr
om pr
oposi
ti
onal
l
ogi
c, the
y
ar
e
cal
led
Bool
ean
sear
ches
.
2
5
•
Fi
ndi
ng
W
eb
pag
es about
un
iv
er
sit
ie
s
in
Ne
w
Me
xico
:
•
Ne
w
AN
D Me
xi
co
AN
D Uni
ver
si
ti
es
–
‘N
e
w
Me
xi
co’
Uni
v
er
si
ti
es
–
Ne
w
Uni
ver
si
ti
es
i
n
Me
xi
co
•
“Ne
w
Me
xi
co” A
ND
Uni
ver
si
ti
es
•
(Ne
w
AN
D Me
xi
co
OR Ar
iz
ona)
AN
D
Uni
ver
si
ti
es
–
‘N
e
w
Me
xi
co’
Uni
ver
si
ti
es
–
Ari
zona
Uni
ver
si
ti
es
•
(Me
xi
co
AN
D
Uni
ver
si
ti
es
)
NO
T
Ne
w
2
6
Quiz
•
Le
t x = “
ك!"
”
Then x +
“
ا
”
=
$%!"
W
ri
te B
ool
ean
sear
ch c
ap
tur
ing
thi
s
pa
tt
ern
2
Logi
c Puzzles
•
An
is
land
has
tw
o ki
nds
of
inhab
it
an
ts
,
–
K
ni
gh
ts
,
who al
w
a
ys t
el
l
the
tru
th
–
K
na
ves
,
who
al
w
a
ys l
ie.
•
You e
nc
oun
ter t
w
o
peopl
e
A
and
B
.
•
Wha
t
ar
e
A
and
B
if
–
A
sa
y
s “
B
is a
knig
h
t”
–
B
sa
y
s “
The t
w
o
of us
ar
e
oppo
sit
e
types?
2
8
–
A
sa
y
s “
B
is a
kni
gh
t”
–
B
sa
y
s
“
The tw
o of us
ar
e
opp
osi
te
ty
pes
?
p:
A
is
a kn
igh
t
:
A
is
a
kna
ve
q:
B
is
a
kni
gh
t
ݍ
:
B
is
a kna
ve
2
9
–
A
sa
y
s “
B
is a
kni
gh
t”
–
B
sa
y
s
“
The tw
o of us
ar
e
opp
osi
te
ty
pes
?
p:
A
is
a kn
igh
t
:
A
is
a
kna
ve
q:
B
is
a
kni
gh
t
ݍ
:
B
is
a kna
ve
Fi
rs
t
possi
bi
li
ty:
A
is
a
kni
gh
t;
tha
t
is
p
is
true
.
3
0
–
A
sa
y
s
“
B
is
a
kni
gh
t”
–
B
sa
y
s
“
The
tw
o o
f
us
ar
e o
pposi
te
types?
p:
A
is a kni
gh
t
:
A
is
a
kna
ve
q:
B
is
a
kni
gh
t
ݍ
:
B
is
a kna
ve
Fi
rs
t possi
bi
li
ty:
A
is
a
kni
gh
t;
tha
t
is
p
is
true
.
•
If
A
is
a
kni
gh
t,
then
he
i
s t
el
li
ng t
he
truth
when
he
sa
ys tha
t
B
is
a kni
gh
t
,
so
tha
t
q
is
true
, and
A
and
B
ar
e
the
same
type
(both
kni
gh
t).
•
But,
if
B
is a kni
gh
t,
then
B
’s
st
a
teme
n
t
tha
t
A
and
B
ar
e
of
opposi
te type
s
(p
ר
㻀
q)
ש
(
㻀
p
ר
q
)
,
ha
ve
to
be
true
. But
it
is not; bec
ause
A
and
B
ar
e
both
kni
gh
ts
.
Not
co
nsis
ten
t.
•
Concl
u
si
on:
A
is not a
kni
gh
t
(
p
is
fal
se).
3
1
–
A
sa
y
s “
B
is a
kni
gh
t”
–
B
sa
y
s
“
The tw
o of us
ar
e
opp
osi
te
ty
pes
?
p:
A
is
a kn
igh
t
:
A
is
a
kna
ve
q:
B
is
a
kni
gh
t
ݍ
:
B
is
a kna
ve
Sec
ond poss
ibi
li
ty:
A
is
a
kna
ve; tha
t
is
p
is
f
al
se
.
•
If
A
is
a
kn
a
ve
,
the
n
he i
s
tel
li
ng
li
e
when
he s
a
ys
tha
t
B
is
a
kni
gh
t
.
So B
i
s kna
ve (
q
is
fals
e)
.
•
Al
so when
B
sa
ys t
ha
t
A
and
B
ar
e
of
opposi
te
types
(p
ר
㻀
q)
ש
(
㻀
p
ר
q
)
,
he ag
ai
n l
ies
.
•
Concl
us
ion:
A
and
B
ar
e
both
kna
ves
.
3
2
Logic
Cir
cuits
•
Pr
oposi
ti
onal
l
ogi
c
can
b
e
appl
ied
t
o
the
desi
gn
o
f
comput
er
har
dw
ar
e
•
A
logic
cir
cu
it
(or
digit
al
cir
cu
it
)
re
cei
ves
i
nput
si
gna
ls
ଵ ǡଶ
ǡǤ
ǤǤ
ǡ
,
e
a
ch
a
bi
t
[ei
ther
0
(of
f)
o
r
1
(o
n)]
,
an
d
pr
od
u
ces
output
si
gnal
s
ݏଵ ǡݏଶ
ǡǤ
ǤǤ
ǡݏ
,
e
a
ch
a
bi
t.
3
3
Qui
z:
Dr
a
w
ר
ש
࢘
3
4
Qui
z:
Dr
a
w
ר
ש
࢘
3
5
1.3
Pr
opo
sit
iona
l
E
quiv
alence
•
An
i
mport
an
t
typ
e of s
tep used
i
n
a m
a
the
ma
ti
cal
ar
gumen
t
is
the r
epl
acemen
t
of a
st
a
temen
t wi
th
another s
ta
temen
t wi
th
the same
tru
th
val
ue
•
Pr
oposi
ti
onal
E
qui
val
ence
is
e
xt
ensi
vel
y used i
n
the c
ons
tru
cti
on
of m
a
them
a
ti
cal
ar
gumen
ts
.
3
Taut
olo
gy and Co
n
tr
adicti
on
•
A c
ompound
pr
oposi
ti
on
whi
ch
is
al
w
a
ys
true
,
is
c
al
led
taut
olog
y
.
For e
xampl
e,
ש
,
ܽ
֜
ܽ
,
ܽ
֜
ሺܾ
֜
ܽሻ
•
A c
ompound
pr
oposi
ti
on
whi
ch i
s
al
w
a
ys
false
, i
s c
al
led
con
tr
adi
ction
.
For e
xampl
e,
ר
,
ሺܽ
֜
ܽሻ
,
ܽ
ר
ܾ
ר
ܽ
3
7
Ex
ampl
e on no
teboo
k:
ܽ
֜
ሺܾ
֜
ܽሻ
ܽ
֜
ܽ
3
8
Logi
cal
E
quiv
alences
•
Compound
pr
oposi
ti
ons
tha
t
ha
ve th
e same tr
uth
val
ues
i
n al
l
poss
ibl
e
cases
ar
e
cal
led
log
ic
ally
equ
iv
alen
t
.
•
The
compoun
d
pr
oposi
ti
ons
p
and
q
ar
e
cal
led
logi
cal
ly
equi
va
len
t
if
p
↔
q
is
a t
aut
ol
ogy
.
•
The
not
a
ti
on
p
≡
q
denot
es tha
t
p
and
q
ar
e
logi
cal
ly equi
val
en
t.
3
9
•
Show t
ha
t
ש
ݍ
ؠ
ר
ݍ
4
0
St
andar
d
equiv
alences
Iden
tit
y
•
ר
ࢀ
ؠ
•
ש
ࡲ
ؠ
Domina
tion
•
ש
ࢀ
ؠ
ࢀ
•
ר
ࡲ
ؠ
ࡲ
4
1
St
andar
d e
quiv
alences
Idem
pot
ence
•
ר
ؠ
•
ש
ؠ
Double
Neg
a
tion
•
ؠ
4
2
St
andar
d E
quiv
alences
Commut
a
tiv
e
la
w:
•
ר
ݍ
ؠ
ݍ
ר
•
ש
ݍ
ؠ
ݍ
ש
•
֞
ݍ
ؠ
ݍ
֞
4
3
St
andar
d
equiv
alences
Associa
tivity
•
ר
ݍ
ר
ݎ
ؠ
ר
ݍ
ר
ݎ
•
ש
ݍ
ש
ݎ
ؠ
ש
ݍ
ש
ݎ
•
֞
ݍ
֞
ݎ
ؠ
֞
ሺݍ
֞
ݎሻ
4
4
St
andar
d e
quiv
alences
•
In
ver
si
on
ܶ
ؠ
ܨ
ܨ
ؠ
ܶ
•
Neg
a
ti
on
ؠ
ሺ
֜
ܨሻ
•
Con
tr
adi
ct
ion
ר
ؠ
ܨ
4
Dis
tri
butiv
e
La
w
•
ר
ݍ
ש
ݎ
ؠ
ר
ݍ
ש
ר
ݎ
•
ש
ݍ
ר
ݎ
ؠ
ש
ݍ
ר
ש
ݎ
4
6
4
7
De Mor
g
an’
s La
w
•
ר
ݍ
ؠ
ש
ݍ
•
ሺଵ
ר
ଶ
ר
ר
ሻ
ؠ
ሺ
ଵ
ש
ଶ
ש
ש
ሻ
•
ש
ݍ
ؠ
ר
ݍ
•
ሺଵ
ש
ଶ
ש
ש
ሻ
ؠ
ሺ
ଵ
ר
ଶ
ר
ר
ሻ
4
8
Gener
ali
za
tion
•ٿ
ܿܽ݊
ܾ
݁
ݑݏ݁
݀
݂ݎ
ଵ
ר
ଶ
ר
ڮ
ר
ୀଵ
•ڀ
ୀଵ
ܿܽ݊
ܾ
݁
ݑݏ݁
݀
݂ݎ
ଵ
ש
ଶ
ש
ڮ
ש
De Mor
g
an’
s La
w
s
•
ڀ
ୀ
ଵ
ؠ
ٿ
ୀ
ଵ
•
ሺٿ
ሻ
ୀ
ଵ
ؠ
ڀ
ୀ
ଵ
4
9
5
0
Ab
sor
p
tio
n
la
w
s
•
ש
ሺ
ר
ݍ
ሻ
ؠ
•
ר
ש
ݍ
ؠ
5
1
Neg
a
ti
on l
a
w
s
•
ש
㻀
ؠ
ࢀ
•
ר
㻀
ؠ
ࡲ
5
2
Implic
a
ti
on
•
֜
ݍ
ؠ
ש
ݍ
•
ש
ݍ
ؠ
֜
ݍ
5
3
Mor
e Implic
a
tio
n
La
w
s
•
՜
ݍ
ؠ
㻀
ݍ
՜
㻀
•
ר
ݍ
ؠ
㻀
ሺ
՜
㻀
ݍሻ
•
ሺ
՜
ݍ
ሻ
ؠ
ר
㻀
ݍ
•
ሺ
՜
ݍ
ሻ
ר
ሺ
՜
ݎ
ሻ
ؠ
՜
ሺ
ݍ
ר
ݎ
ሻ
•
ሺ
՜
ݎ
ሻ
ר
ሺ
ݍ
՜
ݎ
ሻ
ؠ
ሺ
ש
ݍ
ሻ
՜
ݎ
•
ሺ
՜
ݍ
ሻ
ש
ሺ
՜
ݎ
ሻ
ؠ
՜
ሺ
ݍ
ש
ݎ
ሻ
•
ሺ
՜
ݎ
ሻ
ש
ሺ
ݍ
՜
ݎ
ሻ
ؠ
ሺ
ר
ݍ
ሻ
՜
ݎ
5B
i-impli
ca
tions
•
՞
ݍ
ؠ
ሺ
՜
ݍ
ሻ
ר
ሺ
ݍ
՜
ሻ
•
՞
ݍ
ؠ
㻀
՞
㻀
ݍ
•
՞
ݍ
ؠ
ሺ
ר
ݍ
ሻ
ש
ሺ
㻀
ר
㻀
ݍሻ
•
㻀
ሺ
՞
ݍ
ሻ
ؠ
՞
㻀
ݍ
5
5
Using
Logi
cal
E
quiv
alence
•
Show
tha
t
㻀
ሺ
՜
ݍ
ሻ
and
ר
㻀
ݍ
ar
e
logi
cal
ly
equi
val
en
t.
•
Show
tha
t
㻀
ሺ
ש
ሺ
㻀
ר
ݍ
ሻሻ
and
㻀
ר
㻀
ݍ
ar
e
logi
cal
ly equi
val
en
t
b
y dev
el
opi
n
g
a seri
es
of
logi
cal
equi
val
ences
.
•
Pr
ov
e
tha
t
ሺ
ר
ݍሻ
֜
ሺ
ש
ݍሻ
is
a
taut
ol
ogy
.
5
6
Using
Logic
al
E
quiv
alence
Ex:
Pr
o
ve
tha
t
ר
ݍ
֜
ሺ
ש
ݍሻ
i
s
a t
aut
ol
ogy
.
To sho
w t
ha
t
thi
s
st
a
teme
n
t
is
a t
aut
ol
ogy
, w
e
wi
ll
use
logi
cal
eq
ui
val
e
nce
s t
o
de
mons
tr
a
te
tha
t
it
is l
ogi
cal
ly
eq
ui
val
e
n
t t
o
T
ר
ݍ
՜
ש
ݍ
ؠ
ר
ݍ
ש
ש
ݍ
Impl
ic
a
ti
on
eq
ui
val
e
nce
ؠ
ש
ݍ
ש
ש
ݍ
1
st De Mor
g
an l
a
w
ؠ
ש
ሺ
ݍ
ש
ש
ݍ
ሻ
Asso
ci
ati
ve
l
aw
ؠ
ש
ሺ
ש
ש
ݍ
ሻ
Commutat
ive
l
aw
ؠ
ሺ
ש
ሻ
ש
ש
ݍ
Assoc
iati
v
e
l
aw
ؠ
ܶ
ש
ܶ
Taut
ol
ogi
e
s
ؠ
ܶ
Idemp
ot
en
ce
5
7
Do Ex
er
ci
ses
5
