schedule - courses · example winter 2002 arthur keller – cs 180 5–12 sells = bars = barinfo =...
TRANSCRIPT
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
1
Sche
dule
•T
oday
: Jan
. 17
(TH
)◆
Rel
atio
nal A
lgeb
ra.
◆R
ead
Cha
pter
5. P
roje
ct P
art 1
due
.
•Ja
n. 2
2 (T
)◆
SQL
Que
ries
.◆
Rea
d Se
ctio
ns 6
.1-6
.2. A
ssig
nmen
t 2 d
ue.
•Ja
n. 2
4 (T
H)
◆Su
bque
ries
, Gro
upin
g an
d A
ggre
gatio
n.◆
Rea
d Se
ctio
ns 6
.3-6
.4. P
roje
ct P
art 2
due
.
•Ja
n. 2
9 (T
)◆
Mod
ific
atio
ns, S
chem
as, V
iew
s.◆
Rea
d Se
ctio
ns 6
.5-6
.7. A
ssig
nmen
t 3 d
ue.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
2
“Cor
e” R
elat
iona
l Alg
ebra
A s
mal
l set
of
oper
ator
s th
at a
llow
us
to m
anip
ulat
ere
latio
ns in
lim
ited
but u
sefu
l way
s. T
he o
pera
tors
are
:
1.U
nion
, int
erse
ctio
n, a
nd d
iffe
renc
e: th
e us
ual s
etop
erat
ors.
◆B
ut th
e re
latio
n sc
hem
as m
ust b
e th
e sa
me.
2.Se
lect
ion:
Pic
king
cer
tain
row
s fr
om a
rel
atio
n.
3.P
roje
ctio
n: P
icki
ng c
erta
in c
olum
ns.
4.P
rodu
cts
and
join
s: C
ompo
sing
rel
atio
ns in
use
ful w
ays.
5.R
enam
ing
of r
elat
ions
and
thei
r at
trib
utes
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
3
Rel
atio
nal A
lgeb
ra
•lim
ited
expr
essi
ve p
ower
(su
bset
of p
ossi
ble
quer
ies)
•go
od o
ptim
izer
pos
sibl
e
•ric
h en
ough
lang
uage
to e
xpre
ss e
noug
h us
eful
thin
gs
Fin
itene
ss
σ S
ELE
CT
π P
RO
JEC
T
X C
AR
TE
SIA
N P
RO
DU
CT
FU
ND
AM
EN
TA
L
U U
NIO
N
BIN
AR
Y
– S
ET
-DIF
FE
RE
NC
E
∩ S
ET
-IN
TE
RS
EC
TIO
N
θ T
HE
TA
-JO
IN
C
AN
BE
DE
FIN
ED
N
AT
UR
AL
JOIN
IN
TE
RM
S O
F
÷ D
IVIS
ION
or
QU
OT
IEN
T
F
UN
DA
ME
NT
AL
OP
S
UN
AR
Y
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
4
Ext
ra E
xam
ple
Rel
atio
ns
DE
POSI
T(b
ranc
h-na
me,
acc
t-no
,cus
t-na
me,
bala
nce)
CU
STO
ME
R(c
ust -
nam
e,st
reet
,cus
t-ci
ty)
BO
RR
OW
(bra
nch-
nam
e,lo
an-n
o,cu
st-n
ame,
amou
nt)
BR
AN
CH
(bra
nch-
nam
e,as
sets
, bra
nch-
city
)
CL
IEN
T(c
ust-
nam
e,em
pl-n
ame)
Bo
rro
w
B-N
L
-#
C-N
A
MT
T1
Mid
tow
n
123
F
red
6
00 T
2
M
idto
wn
23
4
Sal
ly 1
200
T3
Mid
tow
n
235
S
ally
150
0 T
4
D
ow
nto
wn
612
T
om
20
00
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
5
Sele
ctio
nR
1 =
σC(R
2)
whe
re C
is a
con
ditio
n in
volv
ing
the
attr
ibut
es o
f re
latio
n R
2.
Exa
mpl
eR
elat
ion Sells
:
JoeMenu
= σ
bar=Joe's(Sells
)
bar
beer
pric
eJo
e's
Bud
2.50
Joe'
sM
iller
2.75
Sue
'sB
ud2.
50S
ue's
Coo
rs3.
00
bar
beer
pric
eJo
e's
Bud
2.50
Joe'
sM
iller
2.75
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
6
S
ELE
CT
(σσσσ
)
ar
ity(σσσσ
(R))
= a
rity
(R)
0 ≤≤≤≤
car
d(σσσσ
(R))
≤≤≤≤ c
ard
(R)
σσσσ c
(R
)
σσσσ
c (
R) ⊆⊆⊆⊆
(R
)
c is
sel
ecti
on
co
nd
itio
n:
term
s o
f fo
rm:
attr
op
val
ue
att
r o
p a
ttr
op
is o
ne
of
< =
> ≤≤≤≤ ≠≠≠≠ ≥≥≥≥
e
xam
ple
of
term
: b
ran
ch-n
ame
= "M
idto
wn
"
ter
ms
are
con
nec
ted
by
∧∧∧∧ ∨∨∨∨
¬¬¬¬
σσσσ b
ran
ch-n
ame
= "M
idto
wn
" ∧∧∧∧
am
ou
nt
> 10
00 (
Bo
rro
w)
σσσσ c
ust
-nam
e =
em
p-n
ame
(clie
nt)
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
7
Proj
ecti
onR
1 =
π L
(R2)
whe
re L
is a
list
of
attr
ibut
es f
rom
the
sche
ma
of R
2.
Exa
mpl
eπ b
eer,price(Sells
)
•N
otic
e el
imin
atio
n of
dup
licat
e tu
ples
.
beer
pric
eB
ud2.
50M
iller
2.75
Coo
rs3.
00
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
8
Pro
ject
ion
(ππππ
)
0 ≤≤≤≤
car
d (ππππ
A (
R))
≤≤≤≤ c
ard
(R
)
ari
ty (ππππ
A (
R))
= m
≤≤≤≤ a
rity
(R)
= k
ππππ
i 1,..
.,im
(R)
1 ≤≤≤≤
i j ≤≤≤≤
k d
isti
nct
pro
du
ces
set
of
m-t
up
les ⟨⟨⟨⟨ a
1 ,.
..,a
m ⟩⟩⟩⟩
su
ch t
hat
∃∃∃∃ k
-tu
ple
⟨⟨⟨⟨ b
1,...
,bk
⟩⟩⟩⟩ in
R w
her
e a j
= b
i j f
or
j = 1
,...,m
ππππ b
ran
ch-n
ame,
cu
st-n
ame
(
Bo
rro
w)
Mid
tow
n
F
red
Mid
tow
n
S
ally
Do
wn
tow
n T
om
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
9
Prod
uct
R =
R1
× R
2
pair
s ea
ch tu
ple
t 1 o
f R
1 w
ith e
ach
tupl
e t 2
of
R2
and
puts
in R
a tu
ple
t 1t 2
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
10
Car
tesi
an P
rod
uct
(×××× )
a
rity
(R)
= k1
a
rity
(R ××××
S)
= k1
+ k
2
a
rity
(S)
= k2
c
ard
(R ××××
S)
= ca
rd(R
) ××××
card
(S)
R ××××
S is
th
e se
t al
l po
ssib
le (
k1 +
k2)
-tu
ple
s
wh
ose
fir
st k
1 at
trib
ute
s ar
e a
tup
le in
R
last
k2
attr
ibu
tes
are
a tu
ple
in S
R
S
R ××××
S
A B
C
D
D E
F
A
B
C D
D'
E F
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
11
The
ta-J
oin
R =
R1
C R
2is
equ
ival
ent t
o R
= σ
C(R
1 ×
R2)
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
12
Exa
mpl
eSells
=
Bars
=
BarInfo = Sells Sells.Bar=Bars.Name Bars
bar
beer
pric
eJo
e's
Bud
2.50
Joe'
sM
iller
2.75
Sue
'sB
ud2.
50S
ue's
Coo
rs3.
00
nam
ead
drJo
e's
Map
le S
t.S
ue's
Riv
er R
d.
bar
beer
pric
ena
me
addr
Joe'
sB
ud2.
50Jo
e's
Map
le S
t.Jo
e's
Mill
er2.
75Jo
e's
Map
le S
t.S
ue's
Bud
2.50
Sue
'sR
iver
Rd.
Sue
'sC
oors
3.00
Sue
'sR
iver
Rd.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
13
The
ta-J
oin
Rar
ity(R
) =
r
arity
(S)
= s
arity
(R
S)
= r
+ s
0 ≤
card
(R
S
) ≤
card
(R)
× ca
rd(S
)
S
i θ j
σ$i
θ $
(r+
j)
(R
× S
)
R
S
1 . .
. r
1
. . .
s
θ
θ
θ ca
n be
< >
= ≠
≤ ≥
If eq
ual (
=),
then
it is
an E
QU
IJO
IN
RS
=σ
(R
× S
)c
c
R(A
B C
)
S
(C D
E)
resu
lt ha
s sc
hem
a T
(A B
C C
' D E
)
R.A
<S
.D
ij
R(A
BC
) S
(CD
E)
T(A
BC
C’D
E)
1
3 5
2 1
1
1
3 5
1 2
2
2 4
6
1
2 2
1 3
5 3
3 4
3
5 7
3 3
4
1
3 5
4 4
3
4 6
8
4
4 3
2 4
6 3
3 4
2
4 6
4 4
3
3 5
7 4
4 3
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
14
Nat
ural
Joi
nR
= R
1
R2
calls
for
the
thet
a-jo
in o
f R
1 an
d R
2 w
ith th
e co
nditi
on th
atal
l attr
ibut
es o
f th
e sa
me
nam
e be
equ
ated
. The
n, o
neco
lum
n fo
r ea
ch p
air
of e
quat
ed a
ttrib
utes
is p
roje
cted
out
.
Exa
mpl
eSu
ppos
e th
e at
trib
ute name
in r
elat
ion Bars
was
cha
nged
to bar
, to
mat
ch th
e ba
r na
me
in Sells
.BarInfo = Sells Bars
bar
beer
pric
ead
drJo
e's
Bud
2.50
Map
le S
t.Jo
e's
Mill
er2.
75M
aple
St.
Sue
'sB
ud2.
50R
iver
Rd.
Sue
'sC
oors
3.00
Riv
er R
d.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
15
Ren
amin
gρ S
(A1,
…,A
n) (R
) p
rodu
ces
a re
latio
n id
entic
al to
R b
utna
med
S a
nd w
ith a
ttrib
utes
, in
orde
r, n
amed
A1,
…,A
n.
Exa
mpl
eBars
=
ρ R(bar,addr
) (Bars
) =
•T
he n
ame
of th
e se
cond
rel
atio
n is
R.
nam
ead
drJo
e's
Map
le S
t.S
ue's
Riv
er R
d.
bar
addr
Joe'
sM
aple
St.
Sue
'sR
iver
Rd.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
16
Un
ion
(R
∪∪∪∪ S
) a
rity
(R)
= ar
ity(
S)
= a
rity
(R ∪∪∪∪
S)
m
ax(c
ard
(R),
card
(S))
≤≤≤≤ c
ard
(R ∪∪∪∪
S)
≤≤≤≤ c
ard
(R)
+ ca
rd(S
)
set
of
tup
les
in R
or
S o
r b
oth
R ⊆⊆⊆⊆
R ∪∪∪∪
S
S
⊆⊆⊆⊆ R
∪∪∪∪ S
Fin
d c
ust
om
ers
of
Per
ryri
dg
e B
ran
ch
ππππ Cu
st-N
ame
(σσσσ
Bra
nch
-Nam
e =
"Per
ryri
dg
e" (
BO
RR
OW
∪∪∪∪ D
EP
OS
IT)
)
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
17
Dif
fere
nce
(R −−−−
S)
ari
ty(R
) =
arit
y(S
) =
arit
y(R
– S
)
0
≤≤≤≤ c
ard
(R –
S)
≤≤≤≤ c
ard
(R)
∅∅∅∅
⊆⊆⊆⊆ R
– S
⊆⊆⊆⊆ R
is t
he
tup
les
in R
no
t in
S
Dep
osi
tors
of
Per
ryri
dg
e w
ho
are
n't
bo
rro
wer
s o
f P
erry
rid
ge
ππππC
ust
-Nam
e (σσσσ
Bra
nch
-Nam
e =
"Per
ryri
dg
e" (D
EP
OS
IT –
BO
RR
OW
) )
Dep
osi
t <
Per
ryri
dg
e, 3
6, P
at, 5
00 >
Bo
rro
w
< P
erry
rid
ge,
72,
Pat
, 100
00 >
π C
ust-
Nam
e (σ
Bra
nch-
Nam
e =
"P
erry
ridge
" (
DE
PO
SIT
) )
—π C
ust-
Nam
e (σ
Bra
nch-
Nam
e =
"P
erry
ridge
" (
BO
RR
OW
) )
Doe
s σ
(π
(D) − π
(B)
) w
ork?
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
18
Com
bini
ng O
pera
tions
Alg
ebra
=1.
Bas
is a
rgum
ents
+2.
Way
s of
con
stru
ctin
g ex
pres
sion
s.Fo
r re
latio
nal a
lgeb
ra:
1.A
rgum
ents
= v
aria
bles
sta
ndin
g fo
rre
latio
ns +
fin
ite, c
onst
ant r
elat
ions
.2.
Exp
ress
ions
con
stru
cted
by
appl
ying
one
of th
e op
erat
ors
+ p
aren
thes
es.
•Q
uery
= e
xpre
ssio
n of
rel
atio
nal a
lgeb
ra.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
19
ππππ Cu
st-N
ame,
Cu
st-C
ity
(σσσσC
LIE
NT
.Ban
ker-
Nam
e =
"Jo
hn
son
"
(C
LIE
NT
×××× C
US
TO
ME
R)
)
=
ππππ C
ust
-Nam
e,C
ust
-Cit
y (
CU
ST
OM
ER
)
•Is
th
is a
lway
s tr
ue?
ππππC
LIE
NT
.Cu
st-N
ame,
CU
ST
OM
ER
.Cu
st-C
ity
(σσσσ
CL
IEN
T.B
anke
r-N
ame
= "J
oh
nso
n"
∧∧∧∧ C
LIE
NT
.Cu
st-N
ame
= C
US
TO
ME
R.C
ust
-Nam
e
(C
LIE
NT
×××× C
US
TO
ME
R)
)
ππππC
LIE
NT
.Cu
st-N
ame,
CU
ST
OM
ER
.... Cu
st-C
ity
(σσσσ
CL
IEN
T.C
ust
-Nam
e =C
US
TO
ME
R.C
ust
-Nam
e
(CU
ST
OM
ER
×××× ππππ
Cu
st-N
ame
(((( σσσσ C
LIE
NT
.Ban
ker-
Nam
e="J
oh
nso
n"
(C
LIE
NT
) )
) )
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
20
SE
T IN
TE
RS
EC
TIO
N
ari
ty(R
) =
arit
y(S
) =
arit
y (R
∩∩∩∩ S
)
(R
∩∩∩∩ S
)
0 ≤≤≤≤
car
d (
R ∩∩∩∩
S) ≤≤≤≤
min
(ca
rd(R
), c
ard
(S))
tu
ple
s b
oth
in R
an
d in
S
R −−−−
(R −−−−
S)
= R
∩∩∩∩ S
SR
∅ ⊆
R ∩
S ⊆
R
∅ ⊆
R ∩
S ⊆
S
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
21
Ope
rato
r Pr
eced
ence
The
nor
mal
way
to g
roup
ope
rato
rs is
:
1.U
nary
ope
rato
rs σ
, π, a
nd ρ
hav
e hi
ghes
t pre
cede
nce.
2.N
ext h
ighe
st a
re th
e “m
ultip
licat
ive”
ope
rato
rs,
,
C ,
and
×.
3.L
owes
t are
the
“add
itive
” op
erat
ors,
∪, ∩
, and
—.
•B
ut th
ere
is n
o un
iver
sal a
gree
men
t, so
we
alw
ays
put
pare
nthe
ses
arou
nd th
e ar
gum
ent o
f a
unar
y op
erat
or, a
nd it
is a
good
idea
to g
roup
all
bina
ry o
pera
tors
with
par
enth
eses
encl
osin
g th
eir
argu
men
ts.
Exa
mpl
eG
roup
R ∪
σS
T
as
R ∪
(σ
(S )
T ).
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
22
Eac
h E
xpre
ssio
n N
eeds
a S
chem
a
•If
∪, ∩
, — a
pplie
d, s
chem
as a
re th
e sa
me,
so
use
this
sche
ma.
•Pr
ojec
tion:
use
the
attr
ibut
es li
sted
in th
e pr
ojec
tion.
•Se
lect
ion:
no
chan
ge in
sch
ema.
•Pr
oduc
t R ×
S: u
se a
ttrib
utes
of
R a
nd S
.◆
But
if th
ey s
hare
an
attr
ibut
e A
, pre
fix
it w
ith th
e re
latio
nna
me,
as
R.A
, S.A
.
•T
heta
-joi
n: s
ame
as p
rodu
ct.
•N
atur
al jo
in: u
se a
ttrib
utes
fro
m e
ach
rela
tion;
com
mon
attr
ibut
es a
re m
erge
d an
yway
.
•R
enam
ing:
wha
teve
r it
says
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
23
Exa
mpl
e•
Find
the
bars
that
are
eith
er o
n M
aple
Str
eet
or s
ell B
ud f
or le
ss th
an $
3.Sells(bar, beer, price)
Bars(name, addr)
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
24
Exa
mpl
eFi
nd th
e ba
rs th
at s
ell t
wo
diff
eren
t bee
rs a
t the
sam
e pr
ice.
Sells(bar, beer, price)
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
25
Lin
ear
Not
atio
n fo
r E
xpre
ssio
ns•
Inve
nt n
ew n
ames
for
inte
rmed
iate
rel
atio
ns, a
nd a
ssig
nth
em v
alue
s th
at a
re a
lgeb
raic
exp
ress
ions
.
•R
enam
ing
of a
ttrib
utes
impl
icit
in s
chem
a of
new
rel
atio
n.
Exa
mpl
eFi
nd th
e ba
rs th
at a
re e
ither
on
Map
le S
tree
t or
sell
Bud
for
less
than
$3.
Sells(bar, beer, price)
Bars(name, addr)
R1(name) := π n
ame(σ
addr = Maple St.(Bars))
R2(name) := π b
ar(σ
beer=Bud AND price<$3(Sells))
R3(name) := R1 ∪ R2
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
26
Why
Dec
ompo
sitio
n “W
orks
”?
Wha
t doe
s it
mea
n to
“w
ork”
? W
hy c
an’t
we
just
tear
set
sof
attr
ibut
es a
part
as
we
like?
•A
nsw
er: t
he d
ecom
pose
d re
latio
ns n
eed
to r
epre
sent
the
sam
e in
form
atio
n as
the
orig
inal
.◆
We
mus
t be
able
to r
econ
stru
ct th
e or
igin
al f
rom
the
deco
mpo
sed
rela
tions
.
Proj
ectio
n an
d Jo
in C
onne
ct th
eO
rigi
nal a
nd D
ecom
pose
d R
elat
ions
•Su
ppos
e R
is d
ecom
pose
d in
to S
and
T.
We
proj
ect R
ont
o S
and
onto
T.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
27
Exa
mpl
e
R =
•R
ecal
l we
deco
mpo
sed
this
rel
atio
n as
:
nam
ead
drbe
ersL
iked
man
ffa
vori
teB
eer
Jane
way
Voy
ager
Bud
A.B
.W
icke
dAle
Jane
way
Voy
ager
Wic
kedA
leP
ete'
sW
icke
dAle
Spoc
kE
nter
pris
eB
udA
.B.
Bud
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
28
Proj
ect o
nto Drinkers1(name,
addr,
favoriteBeer
):
Proj
ect o
nto Drinkers3(beersLiked, manf):
Proj
ect o
nto Drinkers4(name, beersLiked):
beer
sLik
edm
anf
Bud
A.B
.W
icke
dAle
Pet
e's
Bud
A.B
.
nam
ead
drbe
ersL
iked
Jane
way
Voy
ager
Bud
Jane
way
Voy
ager
Wic
kedA
leSp
ock
Ent
erpr
ise
Bud
nam
ead
drfa
vori
teB
eer
Jane
way
Voy
ager
Wic
kedA
leSp
ock
Ent
erpr
ise
Bud
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
29
Rec
onst
ruct
ion
of O
rigi
nal
Can
we
figu
re o
ut th
e or
igin
al r
elat
ion
from
the
deco
mpo
sed
rela
tions
?
•So
met
imes
, if
we
natu
ral j
oin
the
rela
tions
.
Exa
mpl
eDrinkers3 Drinkers4
=
•Jo
in o
f ab
ove
with
Drinkers1
= o
rigi
nal R
.
nam
ebe
ersL
iked
man
fJa
new
ayB
udA
.B.
Jane
way
Wic
kedA
leP
ete'
sSp
ock
Bud
A.B
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
30
The
orem
Supp
ose
we
deco
mpo
se a
rel
atio
n w
ith s
chem
a X
YZ
into
XY
and
XZ
and
pro
ject
the
rela
tion
for
XYZ
ont
o X
Y a
nd X
Z.
The
n X
Y
X
Z is
gua
rant
eed
to r
econ
stru
ct X
YZ
if a
nd o
nly
if X
→→
Y (
or e
quiv
alen
tly, X
→→
Z).
•U
sual
ly, t
he M
VD
is r
eally
a F
D, X
→ Y
or
X →
Z.
•B
CN
F: W
hen
we
deco
mpo
se X
YZ in
to X
Y a
nd X
Z, i
t is
beca
use
ther
e is
a F
D X
→ Y
or
X →
Z th
at v
iola
tes
BC
NF.
◆T
hus,
we
can
alw
ays
reco
nstr
uct X
YZ
fro
m it
s pr
ojec
tions
ont
o X
Yan
d X
Z.
•4N
F: w
hen
we
deco
mpo
se X
YZ in
to X
Y a
nd X
Z, i
t is
beca
use
ther
e is
an
MV
D X
→→
Y o
r X
→→
Z th
at v
iola
tes
4NF.
◆A
gain
, we
can
reco
nstr
uct X
YZ f
rom
its
proj
ectio
ns o
nto
XY
and
XZ
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
31
Bag
Sem
antic
s
A r
elat
ion
(in
SQL
, at l
east
) is
rea
lly a
bag
or
mul
tise
t.•
It m
ay c
onta
in th
e sa
me
tupl
e m
ore
than
once
, alth
ough
ther
e is
no
spec
ifie
d or
der
(unl
ike
a lis
t).
•E
xam
ple:
{1,
2,1,
3} is
a b
ag a
nd n
ot a
set
.•
Sele
ct, p
roje
ct, a
nd jo
in w
ork
for
bags
as
wel
l as
sets
.◆
Just
wor
k on
a tu
ple-
by-t
uple
bas
is, a
nd d
on't
elim
inat
e du
plic
ates
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
32
Bag
Uni
on
Sum
the
times
an
elem
ent a
ppea
rs in
the
two
bags
.•
Exa
mpl
e: {
1,2,
1} ∪
{1,
2,3,
3} =
{1,
1,1,
2,2,
3,3}
.
Bag
Int
erse
ctio
nT
ake
the
min
imum
of
the
num
ber
of o
ccur
renc
es in
eac
hba
g.•
Exa
mpl
e: {
1,2,
1} ∩
{1,
2,3,
3} =
{1,
2}.
Bag
Dif
fere
nce
Prop
er-s
ubtr
act t
he n
umbe
r of
occ
urre
nces
in th
e tw
o ba
gs.
•E
xam
ple:
{1,
2,1}
– {
1,2,
3,3}
= {
1}.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
33
Law
s fo
r B
ags
Dif
fer
From
Law
s fo
r Se
ts
•So
me
fam
iliar
law
s co
ntin
ue to
hol
d fo
r ba
gs.
◆E
xam
ples
: uni
on a
nd in
ters
ectio
n ar
e st
ill c
omm
utat
ive
and
asso
ciat
ive.
•B
ut o
ther
law
s th
at h
old
for
sets
do
not h
old
for
bags
.
Exa
mpl
eR
∩ (
S ∪
T) ≡
(R ∩
S) ∪
(R
∩ T
) ho
lds
for
sets
.•
Let
R, S
, and
T e
ach
be th
e ba
g {1
}.•
Lef
t sid
e: S
∪ T
= {
1,1}
; R ∩
(S ∪
T)
= {
1}.
•R
ight
sid
e: R
∩ S
= R
∩ T
= {
1};
(R ∩
S) ∪
(R
∩ T
) =
{1}
∪ {
1} =
{1,
1} ≠
{1}
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
34
Ext
ende
d (“
Non
clas
sica
l”)
Rel
atio
nal A
lgeb
raA
dds
feat
ures
nee
ded
for
SQL
, bag
s.
1.D
uplic
ate-
elim
inat
ion
oper
ator
δ.
2.E
xten
ded
proj
ectio
n.
3.So
rtin
g op
erat
or τ
.
4.G
roup
ing-
and-
aggr
egat
ion
oper
ator
γ.
5.O
uter
join
ope
rato
r o
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
35
Dup
licat
e E
limin
atio
nδ(
R)
= r
elat
ion
with
one
cop
y of
eac
h tu
ple
that
app
ears
one
or m
ore
times
in R
.
Exa
mpl
eR
=A
B1
23
41
2
δ(R
) =
AB
12
34
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
36
Sort
ing
•τ L
(R)
= li
st o
f tu
ples
of
R, o
rder
ed a
ccor
ding
toat
trib
utes
on
list L
.•
Not
e th
at r
esul
t typ
e is
out
side
the
norm
al ty
pes
(set
or
bag)
for
rel
atio
nal a
lgeb
ra.
◆C
onse
quen
ce: τ
can
not b
e fo
llow
ed b
y ot
her
rela
tiona
lop
erat
ors.
Exa
mpl
eR
=A
B1
33
45
2τ B
(R)
= [
(5,2
), (
1,3)
, (3,
4)].
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
37
Ext
ende
d Pr
ojec
tion
Allo
w th
e co
lum
ns in
the
proj
ectio
n to
be
func
tions
of o
ne o
r m
ore
colu
mns
in th
e ar
gum
ent r
elat
ion.
Exa
mpl
eR
=A
B1
23
4π A
+B,A
,A(R
) =
A+B
A1
A2
31
17
33
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
38
Agg
rega
tion
Ope
rato
rs
•T
hese
are
not
rel
atio
nal o
pera
tors
; rat
her
they
sum
mar
ize
a co
lum
n in
som
e w
ay.
•Fi
ve s
tand
ard
oper
ator
s: S
um, A
vera
ge,
Cou
nt, M
in, a
nd M
ax.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
39
Gro
upin
g O
pera
tor
γ L(R
), w
here
L is
a li
st o
f el
emen
ts th
at a
re e
ither
a)In
divi
dual
(gr
oupi
ng)
attr
ibut
es o
rb)
Of
the
form
θ(A
), w
here
θ is
an
aggr
egat
ion
oper
ator
and
A th
e at
trib
ute
to w
hich
it is
app
lied,
is c
ompu
ted
by:
1.G
roup
R a
ccor
ding
to a
ll th
e gr
oupi
ng a
ttrib
utes
on
list L
.2.
With
in e
ach
grou
p, c
ompu
te θ
(A),
for
eac
h el
emen
t θ(A
)on
list
L.
3.R
esul
t is
the
rela
tion
who
se c
olum
ns c
onsi
st o
f on
e tu
ple
for
each
gro
up. T
he c
ompo
nent
s of
that
tupl
e ar
e th
eva
lues
ass
ocia
ted
with
eac
h el
emen
t of
L f
or th
at g
roup
.
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
40
Exa
mpl
eL
et R
=ba
rbe
erpr
ice
Joe'
sB
ud2.
00Jo
e's
Mill
er2.
75Su
e's
Bud
2.50
Sue'
sC
oors
3.00
Mel
'sM
iller
3.25
Com
pute
γbe
er,A
VG
(pri
ce)(R
).
1.G
roup
by
the
grou
ping
attr
ibut
e(s)
, beer
in th
is c
ase:
bar
beer
pric
eJo
e's
Bud
2.00
Sue'
sB
ud2.
50Jo
e's
Mill
er2.
75M
el's
Mill
er3.
25Su
e's
Coo
rs3.
00
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
41
2.C
ompu
te a
vera
ge o
f price
with
in g
roup
s:
beer
AV
G(p
rice
)
Bud
2.25
Mill
er3.
00
Coo
rs3.
00
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
42
Out
erjo
in
The
nor
mal
join
can
“lo
se”
info
rmat
ion,
beca
use
a tu
ple
that
doe
sn’t
join
with
any
from
the
othe
r re
latio
n (d
angl
es)
has
nove
stag
e in
the
join
res
ult.
•T
he n
ull v
alue
⊥ c
an b
e us
ed to
“pa
d”da
nglin
g tu
ples
so
they
app
ear
in th
e jo
in.
•G
ives
us
the
oute
rjoi
n op
erat
or
o .
•V
aria
tions
: the
ta-o
uter
join
, lef
t- a
nd r
ight
-ou
terj
oin
(pad
onl
y da
nglin
g tu
ples
fro
m th
ele
ft (
resp
ectiv
ely,
rig
ht).
Win
ter
2002
Art
hur
Kel
ler
– C
S 18
05–
43
Exa
mpl
e
R =
AB
12
34
S =
BC
45
67
R
o S
=A
BC
34
5pa
rt o
f nat
ural
join
12
⊥pa
rt o
f rig
ht-o
uter
join
⊥6
7pa
rt o
f lef
t-ou
terj
oin