Plan of the lecture
• Some properties of relational algebra operators
G53RDB:
Theory of Relational Databases
• Relational algebra on multisets
Lecture 3
Natasha Alechina
School of Computer Science & IT
nza@cs.nott.ac.uk
Lecture 3
Some properties of ∪
Relational algebra operators
• We defined operations on relations which take relations as
operands and return a relation.
• Commutativity: R ∪ S = S ∪ R
• By relational algebra we will mean the following set of
operators: (×, −, ∪, π, σ, ρ). Sometimes we also consider
∩ and ><, but they are definable using the rest.
• Next we look at the algebraic properties of our operations
(we need them because we are going to use them in query
rewriting).
• If R ⊆ S then R ∪ S = S
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• Associativity: R ∪ (P ∪ S) = (R ∪ P ) ∪ S
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Some properties of ∩
Some properties of ∪ and ∩
• Commutativity: R ∩ S = S ∩ R
• Distributivity: R ∩ (P ∪ S) = (R ∩ P) ∪ (R ∩ S)
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R ∪ (P ∩ S) = (R ∪ P) ∩ (R ∪ S)
• Associativity: R ∩ (P ∩ S) = (R ∩ P ) ∩ S
• If R ⊆ S then R ∩ S = R
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Some properties of difference
Some properties of ×
• R − (P ∪ S) = (R − P) ∩ (R − S)
• Associativity: R × (P × S) = (R × P) × S)
• R − (P ∩ S) = (R − P) ∪ (R − S)
• Distributivity: R × (P ∪ S) = (R × P) ∪ (R × S)
• R−R=∅
R × (P ∩ S) = (R × P) ∩ (R × S)
• R− ∅=R
• R ×∅= ∅
• ∅× R= ∅
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Some properties of σ
Some properties of π
• σα (R) ∩ σβ (R) = σα ∧β (R)
• If all attributes of R and S involved in the join condition
are in X, then
• σα (R) ∪ σβ (R) = σα ∨β (R)
π X(R><S) = π X (R) >< π X (S)
• If R and S are union compatible,
• If condition α only involves attributes of R, then
σα (R×S) = σα (R) × S
π X(R ∪ S) = π X (R) ∪ π X (S)
• If condition α only involves attributes of R, then
σα (R><S) = σα (R) >< S
• Give a counterexample to:
• If condition α only involves attributes from set X, then
π X(R ∩ S) = π X (R) ∩ π X (S)
for union-compatible R and S.
• πX σα (R) = σα πX (R)
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Relational algebra and duplicates
Multisets
• Relational algebra assumes that operations are defined on
relations which are sets of tuples.
• A multiset is a collection of elements, where each element
has a specified number of occurrences.
• Tables in a database are not exactly sets of rows: a table
may contain duplicate rows.
• For example, A = {x,x,y,z,z,z} is a multiset; x occurs 2
times, y once, z occurs 3 times.
• Solution: define operations of relational algebra on bags or
multisets (sets where each element may occur a number of
times) instead of sets.
• Multisets are unordered, so {x,x,y,z,z,z} is the same as
{x,y,z,x,z,z}.
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• The number of occurrences matters, so {x,x,y,z,z,z} is
different multiset from {x,y,z}.
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∪, ∩, −, × for multisets
Set theory for multisets
• For two multisets A and B:
– A = B if they have the same elements and each
elements x occurs in A the same number of times as it
occurs in B.
– A ⊆ B if each element of A occurs in B at least as many
times as it occurs in A. A ⊂ B if A ⊆ B and A ≠ B.
– For example,
{x,x,x,y} ⊆ {x,x,x,y,y,z}
but not
{x,x,x,y} ⊄{x,y,z}
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• Let A, B be multisets.
• If there are i copies of x in A and j copies of x in B, then
there are i+j copies of x in A ∪ B
• If there are i copies of x in A and j copies of x in B, then
there are min(i,j) copies of x in A ∩ B
• If there are i copies of x in A and j copies of x in B, then
there are max(i-j, 0) copies of x in A − B
• If there are i copies of x in A and j copies of y in B, then
there are ij copies of (x,y) in A × B
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Example
Projection and selection for multisets
• Let A = {x,y,z,z} and B = {x,y,z,w}. Then
• If a relation R is a multiset, and there are i tuples in R with
the same value for X, then πX R contains i copies of those
values. (For normal sets, πX R would have just one tuple
with those values).
• A ∪ B = {x,x,y,y,z,z,z,w}
• A ∩ B = {x,y,z}
• A − B = {z}
• σα works the same way as for normal relations: if there
were i duplicate tuples in R satisfying property α, then
σαR contains all those i tuples.
• A × B = { (x,x), (x,y), (x,z), (x,w),
(y,x), (y,y), (y,z), (y,w),
(z,x), (z,y), (z,z), (z,w),
(z,x), (z,y), (z,z), (z,w) }
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Example: projection for multisets
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Example: selection for multisets
πName R
R
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σAge>20R
R
Name
Age
Name
Name
Age
Name
Age
John
Dave
Tom
John
Dave
21
30
20
22
30
John
Dave
Tom
John
Dave
John
Dave
Tom
John
Dave
21
30
20
22
30
John
Dave
John
Dave
21
30
22
30
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Relational algebra for multisets
Exercise
• All operations of relational algebra can be reformulated for
multisets.
• Exercise: show that for multisets A, B and C
A × (B ∪ C) = (A × B) ∪ (A × C)
• We know (from set theory) that the two multisets contain
the same elements, we need to check if they have the same
number of occurrences for each element.
• Suppose x occurs i times in A, y occurs j times in B and k
times in C. How many copies of (x,y) in A × (B ∪ C) and
in (A × B) ∪ (A × C)?
• y occurs j+k times in B ∪ C, so i(j+k) times in A× (B ∪ C)
• (x,y) occurs ij times in A × B, ik times in A × C, so ij+ik =
i(j+k) times in (A × B) ∪ (A × C).
• Most equivalences (rewriting rules) still hold and
algorithms for evaluating relational algebra expressions
can be easily modified for multisets.
• Note: some don’t. For example, A ∪ A ≠ A.
• Exercise: show that for multisets A, B and C
A × (B ∪ C) = (A × B) ∪ (A × C)
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Lecture 3
Exercises
Recommended reading
• Prove distributivity of Cartesian product over union:
R × (P ∪ S) = (R × P) ∪ (R × S) using definitions of
union and product.
• Ullman, Widom: Chapter 5.1 – 5.3
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• Does selection satisfy the following properties:
– σα (σα (R)) = σα ( R)
– σβ (σα (R)) = σα&β ( R)
– σβ (σα (R)) = σα (σβ ( R))
• Does projection satisfy the following properties:
– π X ( π X (R)) = π X (R)
– π X ( π Y (R)) = π Y ( π X (R))
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4