COPE-01 Digital Logic.indd
20
D
ig
it
al
L
o
g
ic
©
2
0
2
1
U
w
e
R
.
Z
im
m
er
, T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
p
ag
e
2
0
o
f
4
8
1
(
ch
ap
te
r
1
:
“D
ig
it
al
L
o
gi
c”
u
p
t
o
p
ag
e
9
6
)
C
o
m
b
in
at
io
n
al
L
o
gi
c
Fu
n
ct
io
n
s
L
o
g
ic
i
s
re
d
u
ci
b
le
/e
q
u
iv
al
e
n
t
to
p
u
re
f
u
n
ct
io
n
s:
t
h
e
re
a
re
n
o
s
ta
te
s!
IF
t
h
e
f
u
n
ct
io
n
i
s
co
m
b
in
at
io
n
al
t
h
e
n
t
h
e
re
i
s
o
n
ly
o
n
e
o
u
tp
u
t
fo
r
an
y
co
m
b
in
at
io
n
o
f
in
p
u
ts
,
e
.g
. t
h
e
f
u
n
ct
io
n
c
an
b
e
w
ri
tt
e
n
o
u
t
as
a
t
ru
th
t
ab
le
:
a
b
c
O
u
tp
u
t
q
m
in
te
rm
s
S
im
p
li
fi
e
d
m
in
te
rm
s
F
F
F
F
F
F
T
F
F
T
F
T
a
b
c
/
/
b
c
/
F
T
T
F
T
F
F
T
a
b
c
/
/
a
b
/
T
F
T
T
a
b
c
/
/
T
T
F
T
a
b
c
/
/
T
T
T
F
S
u
m
o
f
m
in
te
rm
s:
q
=
a
b
c
a
b
c
a
b
c
a
b
c
/
/
/
/
/
/
/
/
0
0
0
^
^
^
^
h
h
h
h
S
u
m
o
f
si
m
p
li
fi
e
d
m
in
te
rm
s:
q
=
a
b
b
c
/
0
/
^
^
h
h
S
im
p
li
fi
ca
ti
o
n
s
ca
n
b
e
d
o
n
e
b
y
(a
u
to
m
at
e
d
)
al
g
e
b
ra
ic
t
ra
n
sf
o
rm
at
io
n
s,
K
ar
n
au
g
h
m
ap
s
o
r
o
th
e
rs
16
D
ig
it
al
L
o
g
ic
©
2
0
2
1
U
w
e
R
.
Z
im
m
er
, T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
p
ag
e
1
6
o
f
4
8
1
(
ch
ap
te
r
1
:
“D
ig
it
al
L
o
gi
c”
u
p
t
o
p
ag
e
9
6
)
C
o
m
m
o
n
B
o
o
le
an
o
p
er
at
o
rs
C
o
m
m
o
n
ly
u
se
d
o
p
e
ra
to
rs
o
n
e
xp
re
ss
io
n
s
a,
b
t
o
d
e
fi
n
e
b
o
o
le
an
a
lg
e
b
ra
s:
a
b
0
(a
k
a
a
b
+
o
r “
a
O
R
b
”
o
r
S
U
M
)
a
b
/
(a
k
a
a
b
$
o
r “
a
A
N
D
b
”
o
r
P
R
O
D
U
C
T
)
a
(a
k
a
a
J
o
r
al
o
r “
n
o
t
a ”
)
O
th
e
r
h
an
d
y
o
p
e
ra
to
rs
:
a
b
”
=
a
b
0
^
h
(a
k
a
“a
I
M
P
LI
ES
b
”)
a
b
=
^
h
=
a
b
a
b
/
0
/
^
^
h
h
(a
k
a
“a
E
Q
U
A
LS
b
”)
a
b
5
=
a
b
a
b
/
0
/
^
^
h
h
(a
k
a
“a
E
X
C
LU
S
IV
E-
O
R
b
”
o
r
“a
X
O
R
b
”)
a
b
/
=
a
b
0
^
h
(a
k
a
“a
N
O
T-
A
N
D
b
”o
r “
a
N
A
N
D
b
”)
a
b
0
=
a
b
/
^
h
(a
k
a
“a
N
O
T-
O
R
b
”o
r “
a
N
O
R
b
”)
N
A
N
D
a
n
d
N
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R
a
re
t
h
e
o
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ly
s
o
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s
u
ffi
c
ie
n
t
b
o
o
le
an
o
p
e
ra
to
rs
,
i.
e
. y
o
u
c
an
r
e
d
u
ce
a
n
y
b
o
o
le
an
e
xp
re
ss
io
n
t
o
o
n
ly
N
A
N
D
o
r
o
n
ly
N
O
R
o
p
e
ra
to
rs
.
12
D
ig
it
al
L
o
g
ic
©
2
0
2
1
U
w
e
R
.
Z
im
m
er
, T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
p
ag
e
1
2
o
f
4
8
1
(
ch
ap
te
r
1
:
“D
ig
it
al
L
o
gi
c”
u
p
t
o
p
ag
e
9
6
)
A
xi
o
m
at
ic
B
o
o
le
an
A
lg
eb
ra
(
W
h
it
eh
ea
d
1
8
9
8
)
0
-L
aw
s
/
-L
aw
s
a
a
0
=
a
a
a
/
=
a
(r
e
d
u
n
d
an
t)
a
b
0
=
b
a
0
a
b
/
=
b
a
/
(c
o
m
m
u
ta
ti
ve
)
a
b
c
0
0
^
h
=
a
b
c
0
0
^
h
a
b
c
/
/
^
h
=
a
b
c
/
/
^
h
(a
ss
o
ci
at
iv
e
)
a
a
b
0
/
^
h
=
a
(a
b
so
rp
ti
o
n
)
a
b
c
/
0
^
h
=
a
b
a
c
/
0
/
^
^
h
h
(d
is
tr
ib
u
ti
o
n
)
T
ru
e
a
/
=
a
(i
d
e
n
ti
ty
)
(c
o
n
st
an
t)
a
a
0
=
T
ru
e
a
a
/
=
F
al
se
(i
n
ve
rs
e
)
D
e
M
o
rg
an
(d
o
u
b
le
n
o
t)
N
ti
l
U
i
it
A
lg
e
b
ra
s
al
lo
w
f
o
r
e
as
ie
r
re
as
o
n
in
g
t
h
an
t
ru
th
t
ab
le
s.
8
1
D
ig
it
al
L
o
gi
c
U
w
e
R
.
Z
im
m
er
–
T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
C
o
m
p
u
te
r
O
rg
an
is
at
io
n
&
P
ro
g
ra
m
E
xe
cu
ti
o
n
2
02
1
21
D
ig
it
al
L
o
g
ic
©
2
0
2
1
U
w
e
R
.
Z
im
m
er
, T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
p
ag
e
2
1
o
f
4
8
1
(
ch
ap
te
r
1
:
“D
ig
it
al
L
o
gi
c”
u
p
t
o
p
ag
e
9
6
)
C
o
m
b
in
at
io
n
al
L
o
gi
c
Fu
n
ct
io
n
s
L
o
g
ic
i
s
re
d
u
ci
b
le
/e
q
u
iv
al
e
n
t
to
p
u
re
f
u
n
ct
io
n
s:
t
h
e
re
a
re
n
o
s
ta
te
s!
IF
t
h
e
f
u
n
ct
io
n
i
s
co
m
b
in
at
io
n
al
t
h
e
n
t
h
e
re
i
s
o
n
ly
o
n
e
o
u
tp
u
t
fo
r
an
y
co
m
b
in
at
io
n
o
f
in
p
u
ts
,
e
.g
. t
h
e
f
u
n
ct
io
n
c
an
b
e
w
ri
tt
e
n
o
u
t
as
a
t
ru
th
t
ab
le
:
a
b
c
O
u
tp
u
t
q
m
in
te
rm
s
S
im
p
li
fi
e
d
m
in
te
rm
s
F
F
F
F
F
F
T
F
F
T
F
T
a
b
c
/
/
b
c
/
F
T
T
F
T
F
F
T
a
b
c
/
/
a
b
/
T
F
T
T
a
b
c
/
/
T
T
F
T
a
b
c
/
/
T
T
T
F
S
u
m
o
f
m
in
te
rm
s:
q
=
a
b
c
a
b
c
a
b
c
a
b
c
/
/
/
/
/
/
/
/
0
0
0
^
^
^
^
h
h
h
h
S
u
m
o
f
si
m
p
li
fi
e
d
m
in
te
rm
s:
q
=
a
b
b
c
/
0
/
^
^
h
h
S
im
p
li
fi
ca
ti
o
n
s
ca
n
b
e
d
o
n
e
b
y
(a
u
to
m
at
e
d
)
al
g
e
b
ra
ic
t
ra
n
sf
o
rm
at
io
n
s,
K
ar
n
au
g
h
m
ap
s
o
r
o
th
e
rs
Ev
e
ry
c
o
m
b
in
at
io
n
al
f
u
n
ct
io
n
c
an
b
e
w
ri
tt
e
n
a
s
a
su
m
o
f
p
ro
d
u
ct
s!
17
D
ig
it
al
L
o
g
ic
©
2
0
2
1
U
w
e
R
.
Z
im
m
er
, T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
p
ag
e
1
7
o
f
4
8
1
(
ch
ap
te
r
1
:
“D
ig
it
al
L
o
gi
c”
u
p
t
o
p
ag
e
9
6
)
A
ll
b
in
ar
y
B
o
o
le
an
o
p
er
at
o
rs
In
p
u
ts
a
, b
Fu
n
ct
io
n
N
am
e
S
u
m
o
f
p
ro
d
u
ct
s
N
A
N
D
D
o
n
’t
c
ar
e
s
a
F
F
T
T
b
u
il
d
,
,
x
/
0
b
F
T
F
T
q
F
F
F
F
Fa
ls
e
C
o
n
st
an
t
FA
LS
E
a,
b
F
F
F
T
a
b
/
A
N
D
a
b
a
b
/
/
/
F
F
T
F
a
b
”
N
O
T-
IM
P
LI
C
A
T
IO
N
a
b
/
^
h
F
F
T
T
a
ID
EN
T
IT
Y
a
b
F
T
F
F
b
a
”
N
O
T-
IM
P
LI
C
A
T
IO
N
a
b
/
^
h
F
T
F
T
b
ID
EN
T
IT
Y
b
a
F
T
T
F
a
b
5
EX
C
LU
S
IV
E-
O
R
, X
O
R
a
b
a
b
/
0
/
^
^
h
h
F
T
T
T
a
b
0
O
R
a
a
b
b
/
/
/
T
F
F
F
a
b
0
N
O
T-
O
R
, N
O
R
a
b
/
^
h
T
F
F
T
a
b
=
EQ
U
A
LI
T
Y,
E
Q
a
a
b
b
0
/
/
^
^
h
h
T
F
T
F
b
IN
V
ER
S
E
b
a
T
F
T
T
b
a
”
IM
P
LI
C
A
T
IO
N
b
a
0
T
T
F
F
a
IN
V
ER
S
E
a
a
a
/
b
T
T
F
T
a
b
”
IM
P
LI
C
A
T
IO
N
a
b
0
T
T
T
F
a
b
/
N
O
T-
A
N
D
, N
A
N
D
a
b
0
T
T
T
T
Tr
u
e
C
o
n
st
an
t
Tr
u
e
a,
b
O
u
tp
u
t
q
13
D
ig
it
al
L
o
g
ic
©
2
0
2
1
U
w
e
R
.
Z
im
m
er
, T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
p
ag
e
1
3
o
f
4
8
1
(
ch
ap
te
r
1
:
“D
ig
it
al
L
o
gi
c”
u
p
t
o
p
ag
e
9
6
)
A
xi
o
m
at
ic
B
o
o
le
an
A
lg
eb
ra
(
H
u
n
ti
n
gt
o
n
1
9
0
4
)
0
-L
aw
s
/
-L
aw
s
(r
e
d
u
n
d
an
t)
a
b
0
=
b
a
0
a
b
/
=
b
a
/
(c
o
m
m
u
ta
ti
ve
)
(a
ss
o
ci
at
iv
e
)
(a
b
so
rp
ti
o
n
)
a
b
c
0
/
^
h
=
a
b
a
c
0
/
0
^
^
h
h
a
b
c
/
0
^
h
=
a
b
a
c
/
0
/
^
^
h
h
(d
is
tr
ib
u
ti
o
n
)
a
Fa
ls
e
0
=
a
T
ru
e
a
/
=
a
(i
d
e
n
ti
ty
)
(c
o
n
st
an
t)
a
a
0
=
T
ru
e
a
a
/
=
F
al
se
(i
n
ve
rs
e
)
D
e
M
o
rg
an
(d
o
u
b
le
n
o
t)
9
D
ig
it
al
L
o
g
ic
©
2
0
2
1
U
w
e
R
.
Z
im
m
er
, T
h
e
A
u
st
ra
li
an
N
at
io
n
al
U
n
iv
er
si
ty
p
ag
e
9
o
f
4
8
1
(
ch
ap
te
r
1
:
“D
ig
it
al
L
o
gi
c”
u
p
t
o
p
ag
e
9
6
)
R
ef
er
en
ce
s
fo
r
th
is
c
h
ap
te
r
[P
at
te
rs
o
n
17
]
D
av
id
A
. P
at
te
rs
o
n
&
J
o
h
n
L
. H
e
n
n
e
ss
y
C
o
m
p
u
te
r
O
rg
an
iz
at
io
n
a
n
d
D
e
si
g
n
–
T
h
e
H
ar
d
w
ar
e
/S
o
ft
w
ar
e
I
n
te
rf
ac
e
A
p
p
e
n
d
ix
A
“
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38
D
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