Question

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

Short answer

The tuple relational calculus representation of the given query is $\mathrm{NOT}(\mathrm{\exists }x)[((\mathrm{EMP}\mathrm{\_}\mathrm{PROJECT}(\mathrm{EMP}\mathrm{\_}\mathrm{SSN}{=}^{\prime }{123456789}^{\prime },\mathrm{PROJECT}\mathrm{\_}\mathrm{ID}=x)\text{ AND NOT }\mathrm{EMP}\mathrm{\_}\mathrm{PROJECT}(\mathrm{EMP}\mathrm{\_}\mathrm{SSN}=e,\mathrm{PROJECT}\mathrm{\_}\mathrm{ID}=x))]$

Step-by-step solution

Determine The Variables

First, it is important to determine the variables provided and what they represent. 'x' is a tuple variable that ranges over the PROJECT relation. The employee with ssN=123456789 working on project x is represented by $P$, and any employee e working on project x is represented by $Q$. Understanding these variables is key to solving this problem.

Construct The Query

Using the given equivalence rules, the query can be constructed. According to the first rule, the FOR ALL statement can be translated to a NOT EXISTS statement. Using the second rule, the IF-THEN statement can be translated to a NOT OR statement. Based on these rules, the query becomes: $\mathrm{NOT}(\mathrm{\exists }x)[(P(x)\text{ AND NOT }Q(x))]$.

Defining The Variables

Now we should replace the variables $P$ and $Q$ with their provided definitions. Doing so, the final form of the query in tuple relational calculus is: $\mathrm{NOT}(\mathrm{\exists }x)[((\mathrm{EMP}\mathrm{\_}\mathrm{PROJECT}(\mathrm{EMP}\mathrm{\_}\mathrm{SSN}{=}^{\prime }{123456789}^{\prime },\mathrm{PROJECT}\mathrm{\_}\mathrm{ID}=x)\text{ AND NOT }\mathrm{EMP}\mathrm{\_}\mathrm{PROJECT}(\mathrm{EMP}\mathrm{\_}\mathrm{SSN}=e,\mathrm{PROJECT}\mathrm{\_}\mathrm{ID}=x))]$

Interpret the Result

The final query correctly expresses the initial task. It retrieves the ssN of employees who work on at least the same projects as the employee with ssN=123456789. If a project exists that the specific employee works on but another does not, it contradicts the assertion and thus, the employee's ssN is not selected.