91Ó°ÊÓ

Define the following terms with respect to the tuple calculus: tuple variable, range relation, atom, formula, and expression.

Define the following terms with respect to the domain calculus: domain variable, range relation, atom, formula, and expression.

What is meant by a safe expression in relational calculus?

When is a query language called relationally complete?

In a tuple relational calculus query with \(n\) tuple variables, what would be the typical minimum number of join conditions? Why? What is the effect of having a smaller number of join conditions?

Consider this query: Retrieve the ssns of employees who work on at least those projects on which the employee with \(\operatorname{ss} N=123456789\) works. This may be stated \(\operatorname{as}(\text { FORALL } x)(\text { IF } P \text { THEN } Q),\) where \(\bullet\) \(x\) is a tuple variable that ranges over the PROJECT relation. \(\bullet\) \(P \equiv\) employee with \(\operatorname{ssN}=123456789\) works on project \(x\) \(\bullet\) \(Q \equiv\) employee e works on project \(x\) Express the query in tuple relational calculus, using the rules \(\bullet\) \((\forall x)(P(x)) \equiv \operatorname{NOT}(\exists x)(\operatorname{NOT}(P(x)))\) \(\bullet\) (IF \(P \text { THEN } Q) \equiv(\mathrm{NOT}(P) \text { OR } Q)\)

Show how you may specify the following relational algebra operations in both tuple and domain relational calculus. a. \( \sigma_{A=C}(R(A, B, C))\) b. \(\pi_{}(R(A, B, C))\) c. \(R(A, B, C) * S(C, D, E)\) d. \(R(A, B, C) \cup S(A, B, C)\) e. \(R(A, B, C) \cap S(A, B, C)\) \(f(A, B, C)-S(A, B, C)\) g. \(R(A, B, C) \times S(D, E, F)\) h. \(R(A, B) \div S(A)\)