Distribution, commutation and association in database queries

An image showing the commutativity of addition
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Another reminder note.

The RDBMS query engine relies on these laws to build efficient queries:

Commutation:

Numbers may travel (commute)

a + b = b + a or in relational algebra A \Join B = B \Join A

Association:

Numbers may freely associate

(a + b) + c = a + (b + c) or in relational algebra (A \Join B) \Join C = A \Join (B \Join C)

Distribution:

Operators can be distributed

3 \times (4 + 5) = 3 \times 4 + 3 \times 5 or, in relational algebra \sigma(A \Join B) = \sigma(A) \Join \sigma(B)

A more general statement of these laws:

Commutative law: f(a, b) = f(b, a)
Associative law: f(a, f(b, c)) = f(f(a, b), c)
Distributive law: f(g(a,b)) = g(f(a,b))

Date has an excellent discussion of all this on pp 124 – 127 of SQL and Relational Theory: How to Write Accurate SQL Code

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