Relational Algebra Operations — Let R = { (1,2), (2,3) }, S = { (3,4) }. Which is true for the R ∘ S composition?

In R ∘ S, we look for b such that (a,b) ∈ R and (b,c) ∈ S. Here (2,3) ∈ R and (3,4) ∈ S → (2,4). For a=1, (1,2) ∈ R, but there is no (2,4) in S. So only {(2,4)}.

Relational Algebra

Databases

Relational algebra is a formal system for handling relations. With it, we can perform various operations on relations, similar to how we can perform union, intersection, or difference on sets. The result of the operations is always another relation, so we stay within the set of relations (closure).

Basic Operations

Union: R ∪ S = { (a,b) | (a,b) ∈ R or (a,b) ∈ S }. Defined only if R and S come from the same universe.
Intersection: R ∩ S = { (a,b) | (a,b) ∈ R and (a,b) ∈ S }.
Difference: R - S = { (a,b) | (a,b) ∈ R and (a,b) ∉ S }.
Complement: R^c = all pairs in the universe not in R.
Composition: R ∘ S = { (a,c) | ∃ b: (a,b) ∈ R and (b,c) ∈ S }.

Example

Let R = { (1,2), (2,3) }, S = { (3,4) }, universe U = {1,2,3,4} × {1,2,3,4}.

R ∪ S = { (1,2), (2,3), (3,4) }.
R ∩ S = ∅.
R - S = { (1,2), (2,3) }.
R ∘ S = { (2,4) } because (2,3) ∈ R and (3,4) ∈ S.

Properties

Operations are closed on the set of relations.
Union and intersection are commutative and associative.
Composition is associative but generally not commutative.
Complement behaves according to De Morgan's laws.

Summary

Relational algebra provides a formal tool for handling relations. Its basic operations allow combining, filtering, and transforming relations, which is of central importance in mathematics and the world of databases.

Practice Exercise

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