Consider the following instance of a relation District – Which district names…

Concept: Selection σₙₐₘₑ(R) keeps only the tuples of relation R that satisfy the predicate; projection πₙₐₘₑ(R) then extracts just that attribute from the surviving tuples (duplicates removed). For two such projected sets, intersection (∩) keeps tuples common to both, and set difference (A − B) keeps the tuples of A that are absent from B.

1. Left operand: πₙₐₘₑ (σₙₐₘₑ≠"AJ" (District)) — σₙₐₘₑ≠"AJ" keeps every row whose name is not "AJ". From District {JA:10, AJ:50, MM:5}, this is {JA, MM}. πₙₐₘₑ of this is {JA, MM}.

2. First factor of the bracket: πₙₐₘₑ (σₚₒₚᵤₗₐₜᵢₒₙ>5 (District)) — σₚₒₚᵤₗₐₜᵢₒₙ>5 keeps every row with population greater than 5. Both JA (10) and AJ (50) satisfy this; MM (5) does not, since 5 is not strictly greater than 5. So the selection is {JA, AJ}, and πₙₐₘₑ gives {JA, AJ}.

3. Second factor of the bracket: πₙₐₘₑ (σₙₐₘₑ="AJ" (District)) — σₙₐₘₑ="AJ" keeps only the row named "AJ", i.e. {AJ}. πₙₐₘₑ gives {AJ}.

4. Intersect the two projected sets: {JA, AJ} ∩ {AJ} = {AJ}.

5. Subtract from the left operand: {JA, MM} − {AJ}. Since "AJ" is not an element of {JA, MM}, removing it changes nothing, leaving {JA, MM}.

Cross-check: πₙₐₘₑ (σₙₐₘₑ="AJ" (District)) can never contain any name other than "AJ", so the bracket’s intersection is always a subset of {AJ} — at most {AJ}, never more, regardless of any population value. The left operand πₙₐₘₑ (σₙₐₘₑ≠"AJ" (District)) excludes "AJ" by construction. Subtracting a set that is at most {AJ} from a set that never contains "AJ" removes nothing, so the expression evaluates to {JA, MM} as long as AJ exists in District — independent of the exact population figures. Final result: {JA, MM}.