1. Relational Algebra : #I
Based on Chapter 2.4 & 5.1
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2. Relational Languages Query Language : Languages
Define data retrieval operations for relational model
Express easy access to large data sets in high-level language, not complex application programs
Languages
Relational Algebra : procedural semantics based on set or bag theory
Relational Calculus : logic-based language of denoting what is to be retrieved (but not how)
SQL: syntactic sugar for relational calculus.
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3. Basics Query Algebra : Example Query : Closure of Relational Algebra :
nested expression of algebra operators that
accept as input relations and outputs a relation
Example Query :
SELECT [gpa > 3.0] ( UNION (Ugrads,Grads) )
Closure of Relational Algebra :
operators work on relations and returns a relation
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4. Set Operators Union, Intersection, Difference
Defined only for union compatible relations.
Relations are union compatible if
they have same sets of attributes and
the same types (domains) of attributes
Student (sNumber, sName)
Course (cNumber, cName)
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5. Union:  Consider two bags R1 and R2 that are union-compatible.
R1  R2 R1 R2 A B 1 2 3 4 5 6 A B 1 2 3 4 A B 1 2 3 4 5 6
Suppose a tuple t appears in R1 m times, and in R2 n times.
Then in the union, t appears m + n times.
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6. Intersection: ∩ Consider two bags R1 and R2 that are union-compatible.
R1 ∩ R2 A B 1 2 3 4 A B 1 2 3 4 5 6 A B 1 2 3 4
Suppose tuple t appears in R1 m times, and in R2 n times.
Then in intersection, t appears min (m, n) times.
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7. Difference: - Suppose tuple t appears in R1 m times & in R2 n times.
Then in R1 – R2, t appears max (0, m - n) times. R1 R2
R1 – R2 A B 1 2 3 4 A B 1 2 3 4 5 6 A B 1 2
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8. Idempotent property Idempotent property : Example :
Operation applied twice gives same result as when applied once
Example :
Filter-BLUE ( Filter-BLUE ( images ))
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9. Bag vs Set Semantics Union is idempotent for sets:
(R1  R2)  R2 = R1  R2
What about union for bags ?
Union is not idempotent for bags.
What about intersection ?
Intersection is idempotent for sets
Intersection is idempotent for bags
What about difference ?
Difference is idempotent for sets
Difference is not idempotent for bags
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10. Cross Product (Cartesian Product): X
Consider two bags R1 and R2.
Suppose a tuple t1 appears in R1 m times,
and a tuple t2 appears in R2 n times.
Then in R1 X R2, t1t2 appears m*n times.
R1 X R2 R1 R2 A
R1.B
R2.B C 1 2 3 4 5 A B 1 2 B C 2 3 4 5
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11. Basic Relational Operations
Select, Project
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12. Basic Relational Operations
Select: σC (R):
selects subset of tuples of R that satisfies selection condition C.
σ(C ≥ 6) (R) R A B C 1 2 5 3 4 6 7 A B C 3 4 6 1 2 7
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13. Select Select is commutative: Select is idempotent:
σC2 (σC1 (R)) = σC1 (σC2 (R))
Select is idempotent:
σC (σC (R)) = σC (R)
We can combine multiple select conditions into one condition.
σC1 (σC2 (… σCn (R)…)) = σC1 AND C2 AND … Cn (R)
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14. Project: πA1, A2, …, An (R) πA, B (R)
πA1, A2, …, An (R), with A1, A2, …, An  attributes AR
returns tuples in R, but only columns A1, A2, …, An.
πA, B (R) R A B C 1 2 5 3 4 6 7 8 A B 1 2 3 4
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15. Equivalences with Select/Project
σ [sal>100k) (π sal ( Employee )) = πsal ( σ [sal>100k) ( Employee ))
σ [sal>100k) (πsal,name ( Employee )) = πsal,name ( σ [sal>100k) (πsal,name ( Employee ))
σ [sal>100k) (π name ( Employee )) = πname ( σ [sal>100k) ( Employee ))
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16. Summary So Far Key Property: Basic Operators: Logical Rewrite Rules:
Closure of Relational Algebra
Basic Operators:
Set Operators: Union, Intersection, Difference
Cartesian Product (simple form of “Join”)
Select, Project
Logical Rewrite Rules:
Idempotent, commutative, associative.
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