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Relational Algebra Laws

Commutative and Associative Laws:
  • R ⋈ S  ↔  S ⋈ R,    (R ⋈ S) ⋈ T  ↔  R ⋈ (S ⋈ T)    (natural join)
  • R ∪ S  ↔  S ∪ R,    (R ∪ S) ∪ T  ↔  R ∪ (S ∪ T)
  • R ⋈Cond S  ↔  S ⋈Cond R    (theta join)
  • σc ( σd (R))   ↔   σd ( σc (R))
Selection splitting (where c and d are conditions):
  • σc∧d(R)   ↔   σc ( σd (R))
  • σc∨d(R)   ↔   σc(R)  ∪  σd(R)   (but only if R is a set)