What the Heck is a Relation? From Tables to Cartesian Products to Logic

relational-algebradatabaseslogicset-theorysqlprolog

Abstraction: Unifying relational databases, logic, and set theory through the relation abstraction

Key points:

A relation has four equivalent definitions: subset of a Cartesian product, a table, a generalization of a function, and a predicate function assigning truth-values to k-tuples
Rows = tuples/individuals/points; columns = dimensions; column types = domains — all the same mathematical objects
Functions are relations but not vice versa: functions impose a "vertical line test" (one output per input); relations have no privileged output dimension (illustrated via Prolog's plus/3 vs functional plus)
Closed World Assumption (SQL/Prolog): absence of a fact means false; Open World Assumption (Semantic Web/RDF triples): absence means unknown — a fundamental design divide
Relational algebra and relational calculus are provably equivalent (Codd 1972); relational calculus is a modified predicate calculus
Apache Hive and Pig implement relational semantics on Hadoop/MapReduce while retaining SQL-like DSL character

Connections: Apache Hive · Hadoop · Relational Algebra · Database Theory · Logic Programming