Tuple Calculus

A relation r(A,B) in a relational database has 1200 tuples. The attribute A has integer values ranging from 6 to 20, and the attribute B has integer values ranging from 1 to 20. Assume that the attributes A and B are independently distributed.

The estimated number of tuples in the output of \sigma _{(A > 10)\vee(B=18)}(r) is ____________

Consider a database that has the relation schemas EMP(EmpId, EmpName, DepId) and DEPT(DeptName, DeptId). Note that the DeptId can be permitted to be NULL in the relation EMP. Consider the following queries on the database expressed in tuple relational calculus.

Which of the above queries are safe?

Let R and S be relational schemes such that R={a,b,c} and S={c}. Now consider the following queries on the database:
I.\pi _{R-S}(r)-\pi_{R-S}(\pi_{R-S}(r) \times S -\pi_{R-S,S}(r))
II.\{t|t\in \pi _{R-S}(r)\wedge \forall u \in s (\exists v \in r(u=v[s]\wedge t=v[R-S]))\}
III.\{t|t\in \pi _{R-S}(r)\wedge \forall v\in r(\exists u\in s(u=v[s]\wedge t=v[R-S]))\}
IV. Select R.a, R.b From R,S Where R.c=S.c

Which of the above queries are equivalent?

Which of the following tuple relational calculus expression(s) is/are equivalent to \forall t \in r(P(t))?
I. \neg \exists t \in r(P(t))
II. \neg t \notin r(P(t))
III. \neg \exists t \in r(\neg P(t))
IV. \exists t \in r(\neg P(t))

Consider the relation employee(name, sex, supervisorName) with name as the key. supervisorName gives the name of the supervisor of the employee under consideration. What does the following Tuple Relational Calculus query produce?