The relational algebra and calculus
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The relational algebra and calculus. … SQL. The big 3:. Selection and projection are unary ops Join is binary Selection is based on a formula and returns a table that contains all tuples from a given table where the formula is valid

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The relational algebra and calculus

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The relational algebra and calculus

The relational algebra and calculus

… SQL


The big 3

The big 3:

  • Selection and projection are unary ops

  • Join is binary

  • Selection is based on a formula and returns a table that contains all tuples from a given table where the formula is valid

  • Projection returns a table consisting of a subset of attributes from a given table, with dupes removed

  • Join creates tuples with attributes from two given tables, where a specific attribute in one matches a specific attribute in another (often a PK, FK pair)


Algebraic closure

Algebraic closure

  • Any relational algebra operation returns a legal derived table

  • The set operators are also part of the algebra

  • From a formal perspective, the join operator is not a minimal operator, and is therefore represented as a cross product followed by a selection (where the PK equals the FK)

  • Note that joins are symetric


Joins can be generalized

Joins can be generalized

  • Complex join conditions

  • Non-equi joins

  • A “natural” join is based on matching all attributes with equal names in both tables

  • “Outer” join creates null-packed tuples when tuples on the left do not match any on the right; there is also a right outer join


The calculus

The calculus

  • It is a tuple calculus, not a domain calculus

  • SQL is equivalent

  • Select From Where

  • The part after the Where is declarative

  • A tuple calculus (SQL)

  • Notice that the variables are indeed tuples

  • Note that set operators often act on tables that are being created in the query


The relational algebra and calculus

More

  • IN operator is “element of”

  • EXISTS

  • Nesting

  • FOR ALL

  • FOR SOME

  • Putting computations in the SELECT clause

  • COUNT, SUM, AVG, MAX, MIN operators


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