Show how to express the relational operators of projection, selection, Cartesian product, natural join, union, and set-di?erence using Datalog.

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Projection on the ?rst and third arguments of relation Rel with four attributes:

Projection(?A,?C) :- Rel(?A,?B,?C,?D).

Selection on the second attribute of the form ?B=1 and ?B > 12:

Selection1(?A,1,?C,?D) :- Rel(?A,1,?C,?D).

Selection2(?A,?B,?C,?D) :- Rel(?A,?B,?C,?D), ?B > 12.

Cartesian product and natural join (for join, we assume that the second attribute of Rel1 has the same name as the ?rst attribute of Rel2:

Product(?A,?B,?X,?Y,?Z) :- Rel1(?A,?B), Rel2(?X,?Y,?Z).

Join(?A,?B,?Y,?Z) :- Rel1(?A,?B), Rel2(?B,?Y,?Z).

Union and set-di?erence:

Union(?A,?B) :- Rel1(?A,?B) ; Rel2(?A,?B).

Difference(?A,?B) :- Rel1(?A,?B), not Rel2(?A,?B).