Tuple Relational Calculus (TRC) is a declarative query language for the relational model. It expresses queries as set-builder notation {t | condition} that returns tuples t satisfying the condition. The condition is a formula combining atoms with logical operators like AND and OR. TRC allows safe expressions but cannot perform non-query operations like aggregation or modify the database.

1. TUPLE
RELATIONAL
CALCULUS
By RANA MAYUR

2. What is TRC?
• it is a calculus that was introduced by Edgar F.
Codd as part of the relational model, in order
to provide a declarative database-query
language for this data model.

3. BASICS

4. Tuple Relational Calculus
• It is a nonprocedural query language.
• A query is expressed as { t | cond }
• Return all tuples T that satisfy the
condition Cond.
• t is a tuple variable, t[A] denotes the
value of tuple t on attribute A

5. Atoms
• For the construction of the formula we will
assume an infinite set V of tuple variables.
• Examples of atoms are:
• (t.age = s.age) — t has an age attribute
and s has an age attribute with the same
value
• (t.name = “priyank”) — tuple t has a name
attribute and its value is ”Priyank"
• Book(t) — tuple t is present in relation
Book.

6. Formulas
• Formulas are combined using operators ∧
(and), ∨ (or) and ¬ (not), and we can use the
existential quantifier (∃) and the universal
quantifier (∀) .
• Here ∃ and ∀ quantifier are called bound
variable..

7. Predicate Calculus Formula
• Set of comparison operatores:
<, ≤, >, ≥ , = , ≠
• Set of connectives:
and ( ^), or (v), not (¬)
• Implication (=>):
x => y means: if x is true, then y is true.

8. Predicate Calculus Formula
(cont’d)
• ∃ t ∈ r ( Q(t) ) ≡ There exists a tuple t
in relation r such that predicate Q (t)
is true
• ∀ t ∈ r ( Q(t) ) ≡ Q is true for all
tuples t in relation r

9. Formal semantics
• " f1 ∧ f2 " is true if and only if " f1 " is true and
" f2 " is true,
• " f1 ∨ f2 " is true if and only if " f1 " is true or
" f2 " is true or both are true,
• " ¬ f " is true if and only if " f " is not true,

10. Example
• student(rollNo, name, degree, year,
sex, deptNo, advisor)
• department(deptId, name, hod, phone)
Question:
Obtain the rollNo, name of all girl
students in the Maths Dept (deptNo=
2)

11. • Answer
• {s.rollNo,s.name| student(s)^ s.sex=‘F’^
s.deptNo=2}
• Here student(s) is true whenever value of s is
a tuple in the student relation,
• false otherwise

12. Example Relational Schemes
• student(rollNo, name, degree, year, sex,
deptNo, advisor)
• department (deptId, name, hod, phone)
• course (courseId, cname, credits, deptNo)
• enrollment (rollNo, courseId, sem, year, grade)

13. Example Queries
• 1)Determine the departments that
do not have any girl students .

14. • student (rollNo, name, degree, year, sex,
deptNo, advisor)
• department (deptId, name, hod, phone)
• Answer is …
• {d.name|department(d) ^ ¬(∃s)(student(s) ^
s.sex=‘F’^ s.deptNo= d.deptId)

15. • 2)Obtain the names of courses enrolled
by student named Mahesh

16. Answer is …
{c.name | course(c) ^ (∃s) (∃e) ( student(s) ^
enrollment(e)^ s.name = “Mahesh”^ s.rollNo=
e.rollNo^ c.courseId = e.courseId}

17. • 3)Get the names of students who have taken
at least one course taught by their advisor..

18. Output…
• {s.name | student(s) ^ (∃e)(∃t)(enrollment(e)
^ teaching(t) ^ e.courseId= t.courseId
^e.rollNo= s.rollNo^ t.empId= s.advisor}

19. • 4) Display the departments whose HODs are
teaching at least one course in the current
semester.

20. Output
• {d.name | department(d) ^(∃t)(teaching(t)
^t.empid= d.hod^ t.sem= ‘odd’^ t.year=
‘2012’)}

21. Safety of Expressions
• It is possible to write tuple calculus
expressions that generate infinite
relations.
• For example, expression
{ t | ¬ t ∈ r } results in an infinite relation
if the domain of any attribute of relation
r is infinite.

22. To solve this problem
• We restrict the set of allowable
expressions to safe expressions.
• What is safe expression?

23. Definition of Safe Expression
• An expression { t | P(t) } in the tuple
relational calculus is safe if every
component of t appears in one of
the relations, tuples, or constants
that appear in tuple relational
formula P.

24. Example
• { t | ¬ ( t ∈loan) } is NOT safe.
It is possible to have a tuple t not in
loan that contains values that do not
appear in loan.
• All examples in the previous
expressions are safe.

25. Limitations of TRC
• It can not express queries involving
aggregations,closure.
• It can not express non-query operations like
insert,delete,update.
• because those involve modeling the changes
in the state of a database.

26. Summary
• The relational calculus provides an
• alternate way to express queries
• • A formal model based on logical formulae
and set theory
• Equivalence with algebra means can use
• either or both – but only one for formal
• proofs