logic introduction to logic gates that perform boolean operations … · 2006-09-26 · intro to...
TRANSCRIPT
CSC
258
Lect
ure
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06Sl
ide
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20
Intr
odu
ctio
n t
o Lo
gic
Gat
es
•U
sing
tra
nsis
tor
tech
nolo
gy, w
e ca
n cr
eate
bas
ic lo
gic
gate
sth
at p
erfo
rm b
oole
anop
erat
ions
on
high
(5V
) an
d lo
w (
0V)
sign
als.
•Ex
ampl
e: N
AND
gat
e
CSC
258
Lect
ure
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06Sl
ide
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Intr
o to
Log
ic G
ates
•D
igita
l log
ic g
ates
for
m t
he b
asis
for
all
impl
emen
ted
com
putin
g m
achi
nes.
•Ba
sic
gate
s:–
AND
: *,
∧–
OR
: +,
∨–
NO
T:
¬, ¯
–N
AND
–N
OR
–XO
R:
⊕–
XNO
R
•Re
latio
nshi
ps b
etw
een
inpu
ts a
nd o
utpu
ts a
re
outli
ned
in t
ruth
tab
les
CSC
258
Lect
ure
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06Sl
ide
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Tru
th T
able
s
•AN
D g
ate
C =
A
B
•O
R ga
te
C =
A+
B
CSC
258
Lect
ure
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06Sl
ide
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Tru
th T
able
s (c
ont’
d)
•Bu
ffer
•N
OT
Gat
e
B =
A
CSC
258
Lect
ure
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ide
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Tru
th T
able
s (c
ont’
d)
•N
AND
Gat
e
•N
OR G
ate
CSC
258
Lect
ure
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06Sl
ide
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Tru
th T
able
s (c
ont’
d)
•XO
R G
ate
C =
A ⊕
B
•XN
OR
Gat
e
CSC
258
Lect
ure
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es ©
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s, 20
06Sl
ide
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20
Logi
c O
pera
tion
s•
Logi
c ga
tes
all f
ollo
w t
he s
ame
rule
s as
logi
c op
erat
ors
in p
rogr
amm
ing
lang
uage
s.
AND
:1
x =
x0
x =
0O
R:
1 +
x =
10
+ x
= x
NO
T:x
+ x
= 1
x x
= 0
XO
R:
x ⊕
y =
y ⊕
x
•O
rder
of
oper
atio
ns a
lso
appl
ies
to t
hese
ope
ratio
ns,
whe
n pu
ttin
g to
geth
er t
he d
igita
l circ
uit.
CSC
258
Lect
ure
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06Sl
ide
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Des
ign
ing
a C
ircu
it
•Ta
sk #
1:G
iven
a lo
gic
expr
essi
on, d
eter
min
e th
e eq
uiva
lent
gat
e re
pres
enta
tion.
–Ex
ampl
e:
f =
x1
x 2+
x1
x 2
•G
roup
ter
ms
acco
rdin
g to
ord
er o
f op
erat
ions
:–
NO
T te
rms
first
–AN
D t
erm
s ne
xt–
OR t
erm
s la
st
•Th
e ex
ampl
e ge
ts r
ewrit
ten
as:
f =
((x
1)
x 2)
+ (
x 1(x
2))
CSC
258
Lect
ure
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06Sl
ide
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20
Cir
cuit
Exa
mpl
e
•By
eva
luat
ing
the
term
s of
the
exp
ress
ion
from
the
in
side
-out
, we
cons
truc
t th
e di
agra
m a
bove
.•
Of
cour
se, t
his
can
also
be
repr
esen
ted
as a
sin
gle
gate
, but
the
und
erly
ing
com
plex
ity is
the
sam
e.
x 1 x 2
f
x 1 x 2f
CSC
258
Lect
ure
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ide
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Mor
e C
ircu
it D
esig
n•
Task
#2:
Giv
en a
tru
th t
able
th
at s
peci
fies
a lo
gic
circ
uit’s
be
havi
our,
des
ign
the
equi
vale
nt c
ircui
t.–
Exam
ple:
thr
ee-in
put
circ
uit
•Su
m-o
f-pr
oduc
tte
chni
que:
–G
roup
all
row
s w
ith a
n ou
tput
of
f=1
into
a s
ingl
e AN
D t
erm
(pr
oduc
t)–
Com
bine
the
se A
ND
ter
ms
with
a s
ingl
e O
R g
ate
(sum
)
•N
ote:
All t
ruth
tab
les
can
be
conv
erte
d in
to g
ate
form
by
usin
g th
is t
echn
ique
.
fx 3
x 2x 1
01
11
10
11
01
01
00
01
11
10
10
10
11
00
00
00
CSC
258
Lect
ure
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06Sl
ide
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Sum
-of-
Pro
duct
s•
A m
inte
rmis
a p
rodu
ct t
erm
w
here
eac
h of
the
inpu
t va
riabl
es o
ccur
s ex
actly
on
ce.
–Th
e su
m-o
f-pr
oduc
ts t
echn
ique
is
als
o re
ferr
ed t
o as
a
disj
uctio
nof
min
term
s
•G
roup
ing
toge
ther
the
m
inte
rms
for
all r
ows
whe
re
f=1
yiel
ds t
he f
ollo
win
g ex
pres
sion
:
x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3 x 1 x 2 x 3
f
f = x
1x2x
3+
x 1x 2
x 3+
x 1x 2
x 3+
x 1x 2
x 3
CSC
258
Lect
ure
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es ©
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s, 20
06Sl
ide
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Red
uci
ng
Min
term
s•
This
is a
n ug
ly e
xpre
ssio
n.Ca
n w
e fin
d so
me
way
to
min
imiz
eth
e ex
pres
sion
, to
mak
e it
mor
e co
mpa
ct?
•W
e ca
n, b
y em
ploy
ing
a se
t of
bin
ary
logi
c ru
les:
f = x
1x2x
3+
x 1x 2
x 3+
x 1x 2
x 3+
x 1x 2
x 3
xy=
x +
yx
+ y
= x
y
de M
orga
n
xx=
0x
+ x
= 1
Com
plem
ent
x =
xIn
volu
tion
xx=
xx
+ x
= x
Idem
pote
nt
x(y
+ z
) =
xy
+ x
zx
+ y
z=
(x
+ y
)(x
+ z
)D
istr
ibut
ive
(xy)
z =
x(y
z)(x
+ y
) +
z =
x +
(y
+ z
)As
soci
ativ
e
xy=
yx
x +
y =
y +
xCo
mm
utat
ive
Alge
brai
c Id
entit
yRul
e N
ame
CSC
258
Lect
ure
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06Sl
ide
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Sim
plif
icat
ion
Exa
mpl
e•
Redu
ce t
he f
ollo
win
g su
m o
f pr
oduc
ts:
•St
eps:
–D
istr
ibut
ive:
–Co
mpl
emen
t:–
Iden
tity:
–Id
empo
tent
:–
Asso
ciat
ive:
–D
istr
ibut
ive:
–Co
mpl
emen
t:–
Iden
tity:
f = x
1x2x
3+
x 1x 2
x 3+
x 1x 2
x 3+
x 1x 2
x 3+
x 1x 2
x 3
f = x
1x3(x
2+
x 2) +
x1x
2x3
+ x 1
x 2x 3
+ x 1
x 2x 3
f = x
1x3
1+
x 1x 2
x 3+
x 1x 2
x 3+
x 1x 2
x 3
f = x
1x3
+ x 1
x 2x 3
+ x 1
x 2x 3
+ x 1
x 2x 3
f = x
1x3
+ x 1
x 2x 3
+ x 1
x 2x 3
+ (x
1x2x
3+
x 1x 2
x 3)
f = x
1x3
+ (x
1x2x
3+
x 1x 2
x 3)+
(x1x
2x3
+ x 1
x 2x 3
)f =
x1x
3+
x 1x 3
(x2
+ x 2
)+ x
1x2 (x
3+
x 3)
f = x
1x3
+ x 1
x 31
+ x 1
x 21
f = x
1x3
+ x 1
x 3+
x 1x 2
CSC
258
Lect
ure
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es ©
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ve E
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s, 20
06Sl
ide
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Sim
plif
icat
ion
Exa
mpl
e
•St
eps
(con
t’d):
–(f
rom
pre
viou
s sl
ide)
:–
Dis
trib
utiv
e:–
Com
plem
ent:
–Id
entit
y:
•So
lutio
n is
:
f = x
1x3
+ x 1
x 3+
x 1x 2
f = x
3(x1
+ x 1
) + x
1x2
f = x
31
+ x 1
x 2f =
x3
+ x 1
x 2
f = x
3+
x 1x 2
f
x 1 x 2 x 3
CSC
258
Lect
ure
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es ©
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s, 20
06Sl
ide
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Mor
e de
Mor
gan
•Im
plic
atio
ns o
f de
Mor
gan’
s La
w:
x 1 x 2x 1
+ x 2
x 1 x 2x 1
x 2
x +
y =
x y
x 1 x 2x 1
x 2x 1 x 2
x 1+
x 2
x y
= x
+ y
CSC
258
Lect
ure
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es ©
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ve E
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s, 20
06Sl
ide
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20
Kar
nau
ghM
aps
•A
Karn
augh
map
is a
noth
er w
ay o
f re
pres
entin
g a
circ
uit’s
tru
th t
able
–Pr
esen
ted
as a
tw
o-di
men
sion
al g
rid o
f 2n
squa
res.
–Ea
ch a
xis
repr
esen
ts t
he d
iffer
ent
inpu
t va
lues
to
the
circ
uit,
an
d th
e co
nten
ts o
f ea
ch s
quar
e re
pres
ents
the
out
put
valu
e fo
r th
e in
ters
ectin
g in
put
valu
es.
•H
oriz
onta
lly a
nd v
ertic
ally
adj
acen
t sq
uare
s on
ly d
iffer
by
a si
ngle
inpu
t va
riabl
e•
Num
ber
of r
ows
and
colu
mns
mus
t be
a p
ower
of
2•
Not
e:ro
w a
nd c
olum
n la
bels
mus
t al
so d
iffer
by
a si
ngle
dig
it.
–Si
mpl
ified
circ
uit
is f
ound
by
circ
ling
grou
ps o
f ad
jace
nt 1
’s
on t
he g
rid.
–Res
ult:
The
min
imal
exp
ress
ion
for
the
circ
uit.
CSC
258
Lect
ure
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ide
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Kar
nau
ghM
aps
•Ex
ampl
e:f =
x1x
2x3
+ x 1
x 2x 3
+ x 1
x 2x 3
+ x 1
x 2x 3
+ x 1
x 2x 3
11
10
01
10
x 1x 2
x 300
0111
10
0 1
CSC
258
Lect
ure
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ide
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Kar
nau
ghM
aps
•N
ext
step
: ci
rcle
blo
cks
of 1
’s–
Cann
ot c
onta
in a
ny 0
’s in
the
blo
ck–
Hei
ght
and
wid
th o
f bl
ock
mus
t be
a p
ower
of
2–
Bloc
ks a
re a
llow
ed t
o ov
erla
p
•Ci
rcle
d bl
ocks
cor
resp
ond
to x
3an
d x 1
x 2
11
10
01
10
x 1x 2
x 300
0111
10
0 1
CSC
258
Lect
ure
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06Sl
ide
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Kar
nau
ghEx
ampl
e
11
11
1
00
11
1
11
01
1
10
01
1
11
10
1
00
10
1
11
00
1
10
00
1
11
11
0
10
11
0
11
01
0
00
01
0
11
10
0
00
10
0
11
00
0
00
00
0
XD
CB
A
Out
put
Inpu
ts•
Task
:G
iven
the
tru
th t
able
on
the
right
, det
erm
ine
the
sim
ples
t eq
uiva
lent
gat
e ar
rang
emen
t.
01
11
01
11
11
10
01
10
CD
AB
0001
1110
00 01 11 10
X =
AC
+ D
+ A
BC
CSC
258
Lect
ure
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es ©
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ve E
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s, 20
06Sl
ide
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20
“Don
’t C
are”
Con
diti
ons
•So
met
imes
cer
tain
out
puts
are
not
def
ined
–Ex
ampl
e:A
thre
e-in
put
circ
uit
whe
re t
he o
utpu
t is
1 if
any
of
the
inpu
ts is
1, a
nd 0
if a
ll of
the
inpu
ts a
re 0
.–
The
outp
ut c
an b
e an
ythi
ng in
the
cas
e w
here
tw
o or
mor
e in
puts
ar
e 1,
sin
ce w
e do
n’t
cons
ider
tho
se c
ases
.–
In t
hose
cas
es, l
abel
the
out
put
as “
don’
t ca
re”
(or
“X”)
.
XX
X1
1X
10
x 1x 2
x 300
0111
10
0 1
•“D
on’t
care
” co
nditi
ons
are
usef
ul b
ecau
se t
hey
can
be t
reat
ed a
s ei
ther
1
or 0
, dep
endi
ng o
n w
hich
mak
es t
hing
s ea
sier
.–
In r
eal l
ife, y
ou u
sual
ly
have
to
set
it to
som
e se
nsib
le v
alue
.
CSC
258
Lect
ure
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06Sl
ide
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“Don
’t C
are”
Exa
mpl
e
•Ad
ding
milk
to
beve
rage
s:–
If a
pat
ron
orde
rs c
offe
e (C
), a
dd m
ilk. I
f th
e pa
tron
ord
ers
tea
(T),
onl
y ad
d m
ilk if
lem
on (
L) h
asn’
t al
read
y be
en a
dded
.
–Th
e “d
on’t
care
” co
nditi
ons
are
som
etim
es w
ritte
n as
“d”
(as
in
the
tex
tboo
k, f
or e
xam
ple)
. In
indu
stry
tho
ugh,
“X”
is m
ore
com
mon
ly u
sed,
as
in t
he p
hras
e “X
-pro
paga
tion”
.
1X
0X
1X
1X
LC
T00
0111
10
0 1
CSC
258
Lect
ure
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06Sl
ide
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The
Impo
rtan
ce o
f N
AN
D
•N
AND
gat
es a
re c
onsi
dere
d to
be
the
“uni
vers
al”
gate
, bec
ause
any
oth
er g
ate
can
be s
ynth
esiz
ed
usin
g N
AND
.•
In f
act,
mos
t ga
tes
are
impl
emen
ted
in s
olid
-sta
te
TTL
chip
s (T
rans
isto
r-Tr
ansi
stor
Log
ic)
–e.
g. 7
4LS0
0 in
tegr
ated
circ
uit
(IC)
CSC
258
Lect
ure
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ide
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Get
tin
g to
NA
ND
•H
ow d
o w
e cr
eate
a N
AND
-bas
ed c
ircui
t, if
our
ap
proa
ch s
o fa
r ha
s be
en t
o fin
d su
m-o
f-pr
oduc
ts?
–An
swer
:By
usi
ng d
e M
orga
n’s
rule
!
•Ac
cord
ing
to d
e M
orga
n (s
ee s
lide
15),
•Th
eref
ore,
the
fol
low
ing
are
equi
vale
nt a
s w
ell:
equi
vale
nt to
f
x 1 x 2 x 3f
x 1 x 2 x 3
CSC
258
Lect
ure
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es ©
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06Sl
ide
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Nei
ther
…N
OR
•N
AND
and
NO
R g
ates
are
the
che
apes
t to
mak
e,
and
the
mos
t co
mm
only
fou
nd in
IC
chip
s•
Typi
cal p
roce
ss f
or c
reat
ing
a N
AND
-bas
ed c
ircui
t is
:1.
repr
esen
t tr
uth
tabl
e in
Kar
naug
h m
ap2.
isol
ate
smal
lest
ter
ms
that
pro
duce
hig
h ou
tput
3.pr
oduc
e su
m-o
f-pr
oduc
t m
odel
with
AN
D &
OR g
ates
4.co
nver
t m
odel
to
NAN
D r
epre
sent
atio
n
•W
hat
abou
t cr
eatin
g an
equ
ival
ent
circ
uit
out
of
NO
R g
ates
? Is
tha
t po
ssib
le?
–Req
uire
s ob
tain
ing
max
term
sof
the
tru
th t
able
, and
pr
oduc
ing
a pr
oduc
t-of
-sum
sm
odel
CSC
258
Lect
ure
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06Sl
ide
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20
Max
term
s•
A m
axte
rmis
a s
um o
f in
put
valu
es w
here
eve
ry in
put
valu
e oc
curs
exa
ctly
onc
e.•
Whe
n ex
pres
sing
a t
ruth
tab
le,
each
max
term
rep
rese
nts
a ro
w
whe
re t
he o
utpu
t is
0, s
uch
that
th
e m
axte
rm w
ill g
ive
a tr
ue
valu
e in
all
case
s ex
cept
for
tha
t ro
w.
–M
axte
rms
for
exam
ple
tabl
e:•
(x1
+ x
2+
x3)
•(x
1+
x2
+ x
3)•
(x1
+ x
2+
x3)
•(x
1+
x2
+ x
3)
fx 3
x 2x 1
01
11
10
11
01
01
00
01
11
10
10
10
11
00
00
00
CSC
258
Lect
ure
Slid
es ©
Ste
ve E
ngel
s, 20
06Sl
ide
26of
20
PO
S (P
rodu
ct o
f Su
ms)
•In
stea
d of
cre
atin
g a
disj
unct
ion
of c
ases
whe
re t
he
outp
ut o
f th
e ci
rcui
t is
1, t
he p
rodu
ct-o
f-su
ms
tech
niqu
e cr
eate
s a
conj
unct
ion
of t
he c
ases
whe
re
the
outp
ut is
0.
–Fr
om p
revi
ous
exam
ple:
(x1
+ x
2+
x3)
(x
1+
x2
+ x
3)
(x1
+ x
2+
x3)
(x
1+
x2
+ x
3)
•Th
ese
equa
tions
can
be
redu
ced
usin
g th
e sa
me
tech
niqu
es u
sed
on m
inte
rms.
–bo
olea
n lo
gic
rule
s–
Karn
augh
map
s
CSC
258
Lect
ure
Slid
es ©
Ste
ve E
ngel
s, 20
06Sl
ide
27of
20
NO
R C
ircu
it•
A pr
oduc
t-of
-sum
circ
uit
will
res
ult
in s
ever
al O
R
gate
s, u
nite
d by
a s
ingl
e AN
D g
ate.
•Th
roug
h de
Mor
gan’
s ru
le, t
his
conv
erts
to
a ci
rcui
t m
ade
up e
ntire
ly o
f N
OR g
ates
.•
Des
ign
deci
sion
:of
the
max
term
and
min
term
re
pres
enta
tions
, whi
ch h
as t
he f
ewes
t re
duce
d te
rms?
10
00
11
10
x 1x 2
x 300
0111
10
0 1x 1
f
x 3x 2 x 3
CSC
258
Lect
ure
Slid
es ©
Ste
ve E
ngel
s, 20
06Sl
ide
28of
20
Oth
er T
TL L
ogic
•In
add
ition
to
NAN
D c
ircui
ts, i
nver
ter
circ
uits
are
co
mm
only
use
d as
wel
l (e.
g. 7
4LS0
4).
•W
hen
usin
g th
ese
chip
s, m
ake
sure
you
hav
e th
e pi
ns a
nd t
he o
rient
atio
n co
rrec
t. F
or e
xam
ple,
ap
plyi
ng t
he v
olta
ge s
ourc
e to
the
gro
und
pin
can
have
som
e ve
ry u
nple
asan
t re
sults
.