[Www.pkbulk.blogspot.com]dbms04

Relational Algebra
1
Fall 2001 Database Systems 1
Relational Algebra
• A relation is a set of tuples. Each relational
algebra operation takes as input a list of
relations and produces a single relation.
• General form:
OperatorArguments (List of Relations)

New Relation
• After each operation, the remaining attributes
are carried to the new relation. The attributes
may be renamed, but their domain remains the
same.
Set Theoretic Operations
• Regular set operations on relations
• A set operation requires two participating relations R and
S to be compatible
– R and S should have the same attributes
• R(Name:D1, Email:D2)
• S(Name:D1, Email:D2, Address:D3)
• T(Name:D1, Email:D4)
• V(Name:D1, Email:D2)
Which relations above are union (set operation)
compatible?
– Union compatibility may require type conversion
(casting).

Relational Algebra
2
Set Operations
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Relational Algebra
3
Assignment and Alias
• Given a relation R, head(R)=A1…An is the list of attributes
in R.
• For each attribute, dom(A) gives the domain of A,
name(A) gives the name of the attribute
• Suppose B1…Bn is a list of attributes such that for each i,
dom(Bi) = dom(Ai)
Then assignment (:=) changes the names of some or all of
the attributes in R and results in a new relation S,
S(B1…Bn) := R(A1…An)
and dom(B1) = dom(A1) … dom(Bn) = dom(An).
Set Operations
• Find all people who are registered as buyers or owners
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$65 798@3¤ACB Johns@geocities.com
DE% F¦GIHCP Gray@microsoft.com
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BUYERS-2(oid,name,email) :=
BUYERS(buyid,name,email)
RESULT := BUYERS-2 ∪ OWNERS

Relational Algebra
4
Set Operations
• Find all people who are owners but not buyers
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RESULT:= OWNERS - BUYERS-2
Set Operations
• Find all people who are both owners and buyers
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RESULT := OWNERS ∩ BUYERS-2

Relational Algebra
5
Projection
• Select all tuples from relation R, restrict each tuple to the
input attributes
– written as R[A1…An] or
for attributes A1…An from R
– meaning
nAA ...1
π
for each tuple in R
construct a new tuple containing
only attributes A1…An
place the new tuple in the result
end-for
remove duplicate tuples to preserve
the set property
Projection
• Find the names and email addresses of people who are
both buyers and owners
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RESULT:= OWNERS[name,email] ∩ BUYERS[name,email]

Relational Algebra
6
Projection
• Find all the items on which at least one bid was placed
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RESULT := BIDS[iid]
Selection
• Select certain tuples from a relation R based on a boolean
condition C
– written as R where C or
– meaning:
RCσ
for each tuple t in R
if t makes C true then
return t in the resulting relation
end-for

Relational Algebra
7
Selection
• The selection condition C is built of
– atomic conditions of the form Ai ∝ Aj or Ai ∝ c where
• Ai, Aj are attributes and c is a constant of the same domain of
values, and
• ∝ is a comparison predicate of the form: =, , , = , = , etc.
• If C1, C2 are selection conditions, then so are
• C1 AND C2
• C1 OR C2
• NOT C1
Selection
• Find the bid identifiers of bids for item number “I1”:
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I1-BIDS := BIDS WHERE iid = “I1”
RESULT := I1-BIDS[bid]

Relational Algebra
8
Selection
• Find the bid identifiers of bids for item number “I1” that are
strictly more than $15.
I1-BIDS := BIDS WHERE
iid = “I1” AND amount 15
RESULT := I1-BIDS[bid]
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!
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Selection
• Find the dates in which a bid for item number “I1” or
“I5” was placed
I1orI5-BIDS := BIDS WHERE
iid = “I1” OR iid = “i5”
RESULT := I1orI5-BIDS[date]
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Relational Algebra
9
Cartesian Product
• Multiply the relations R and S, producing |R|x|S| many
tuples with attributes head(R)+head(S)
• Cartesian product combines two relations into one “big”
relation
• meaning
for each tuple r in R do
for each tuple s in S do
compute a new tuple containing
all attributes in r and s
end-for
end-for
R := Bids WHERE Amount 20 T := Items WHERE oid=“O2”
S := T[iid,name]
P := R X S
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Relational Algebra
10
Relational Algebra - Join
• For all pairs of tuples r and s from relations R and S where
Head(R)=A1…AnB1…Bk and Head(S)=B1…BkC1…Cm
such that r[B1…Bk] = s[B1…Bk],
include a tuple in R
¢¡ S with
attributes A1…AnB1…BkC1…Cm
• The attributes in common between the two relations (B1…Bk) are
called the “join attributes”
• The join is a cartesian product followed by a selection on the join
attributes, and a projection to remove the duplication of the join
attributes [formally: you have to rename the join attributes in one
relation so that you can express the selection/projection
operations]
Also called equijoin since
the main criteria is the
equality of join attributes
• Find all people with the same name and email address from the
“buyers” and “owners” relations
– join attributes are “Name” and “Email” from the buyer and
owners relations
– the joined relation will have attributes “Buyid, Name, Email, Oid”
– there are 5 × 4 pairs of tuples to compare on the attributes
“Name” and “Email”
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Relational Algebra
11
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Example below shows four of the twenty pairs of tuples
considered for the join
• Find the names of items with bids over $20
• Find the names of the owners of items with bids over $20
and are located in Boston
Temp1 := Bids WHERE Amount 20
Temp2 := Items jlk Temp1
Result := Temp2[Name]
Temp1 := Bids WHERE Amount 20
Temp3 := Items WHERE Location = “Boston”
Temp2 := Temp3 jlk Temp1
Temp4 := Temp2 jlk Owners
Result := Temp4[Owners.Name]

Relational Algebra
12
Relational Algebra - Division
• Let head(R)=A1…AnB1…Bk and head(S)= B1…Bk
• R ÷ S is the “maximal” set of tuples from R[A1…An] such that
R contains all tuples in (R ÷ S) x S.
• Division is often
used for “for-all”
queries, e.g. “Find
tuples in R that are
in a relation with
‘all’ tuples in S.”
• All the attributes in the dividing relation S must exist in the
divided relation R!
for all tuples r in R[A1…An]
if for all tuples s in S, r+s is in R
place r in R ÷ S
end-if
end-for
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Relational Algebra
13
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Find:
• R ÷ S
• R ÷ V
• R ÷ Y
• Find identifiers of buyers who placed a bid on “all” the items
• Find the identifiers of items that have a bid from “all” the buyers
Temp := Items[Iid]
Temp1:= Bids[Buyid,Iid]
Result := Temp1 ÷ Temp
A := Buyers[BuyId]
B := Bids[Iid, BuyId]
Result := B ÷ A

Relational Algebra
14
Relational Algebra Operations
– Union R ∪ S
– Intersection R ∩ S
– Difference R - S
– Assignment S(B1…Bn) := R(A1…An)
– Selection R where C
– Projection R[A1…An]
– Cartesian Product R x S
– Join R
¢¡ S
– Division R ÷ S
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$¥3 465718@9 Johns@geocities.com
$A BC@5EDF8 Brown@netmail.com
$ G H(CPIRQ Gray@microsoft.com
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Auction Database

Relational Algebra
15
Problem 1
Find the name and email of owners of items located in
“Boston”
A := Items WHERE Location = “Boston”
B := A
¢¡ Owners
Result := B[Owners.Name, Owners.Email]
Problem 2
Find the identifiers and amount of bids placed by
buyer “Roberts”
A := Buyers WHERE Name = “Roberts”
B := A
¢¡ Bids
Result := B[Bid, Amount]

Relational Algebra
16
Problem 3
Find the names of buyers who placed a bid on an item
owned by “Brown”
A := Owners WHERE Name = “Brown”
B := (A
¢¡ Items) [Iid]
C := B
¢¡ Bids
Result := (C
¢¡ Buyers) [Buyers.Name]
Problem 4
Find the identifier of items with more than one bid
A := Bids[Bid, Iid]
B := A x Bids[Bid,Iid]
C := B WHERE (A.Bid Bids.Bid)
AND (A.Iid = Bids.Iid)
Result := C[Iid]

Relational Algebra
17
Problem 5
Find the names of items all buyers placed a bid on
A := (Bids[Iid, BuyId]) ÷ (Buyers[Buyid])
B := A
¢¡ Items
Result := B[Name]

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