The document provides a comprehensive overview of relational calculus in the context of database management, detailing its non-procedural nature and main types: tuple relational calculus (TRC) and domain relational calculus (DRC). It explains the structure of queries, the use of tuple and domain variables, and the roles of quantifiers in formulating expressions. Key concepts include safe expressions and the differentiations between tuple and domain relational calculus in defining atomic formulas and outputs.

1. Dr. Amiya Ranjan Panda
Assistant Professor [II]
School of Computer Engineering,
Kalinga Institute of Industrial Technology (KIIT),
Deemed to be University,Odisha
Relational Calculus
KALINGA INSTITUTE OF INDUSTRIAL
TECHNOLOGY
School Of Computer
Engineering
4 Credit Lecture Note 14

2. Chapter Contents
q Relational Calculus
q Tuple Relational Calculus
q Safe Expressions
q Queries
q Domain Relational Calculus
q Queries
2

3. Ø Relational calculus is non-procedural.
Ø In relational calculus, a query is solved by defining a solution relation in a
single step.
Ø Relational calculus is mainly based on the well-known propositional calculus,
which is a method of calculating with sentences or declarations.
Ø Various types of relational calculus are:
ü Tuple Relational Calculus (TRC)
ü Domain Relational Calculus (DRC)
3
Relational Calculus

4. Ø A tuple variable is a variable that takes on tuples of a particular relation schema
as values.
Ø A tuple relational calculus query has the form: {T/ P(T)}
Ø The result of this query is the set of all tuples t for which the formula P(T)
evaluates to TRUE with T = t
Sailors (sid, sname, rating, age)
Query: Find all the sailors with a rating above 4
{S/S ∈ Sailors ∧ S.rating > 4}
4
Tuple Relational Calculus

5. Ø Let Rel → be a relation name, R, S → be the tuple variables, a → an attribute
of R, b → an attribute of S, op → operator in the set {<, ≤, >, ≥, =, ^=}. An
Atomic formula is one of the following:
ü R ∈ Rel
ü R.a op S.b
ü R.a op Constant or Constant op R.a
Ø To represent the join and division of relational algebra by relational calculus,
we need quantifiers such as: existential for join and universal for division.
Ø A quantifier quantifies or indicates the quantity of something.
Ø The existential quantifier (∃) states that at least one instance of a particular
type of thing exist Similarly, the universal quantifier (∀) states that some
condition applies to all or to every row of some type.
Ø A formula is recursively defined by using the following rules:
ü Any atomic formula
ü If p and q are formulae, then ¬ p, p ∧ q, p ∨ q, or p ⇒ q are also
formulae
ü If p is a formula that contains T as a variable, then ∃ T(p) and ∀ T(p) are
also formulae.
5
Tuple Relational Calculus

6. Ø The quantifiers ∃ and ∀ are said to bind the tuple variable R; whereas a
variable is said to be free in a formula if the formula does not contain an
occurrence of a quantifier that binds it.
Ø In most of the queries, the output is shown by using the free variables.
Ø Safe Expressions: Whenever we use universal quantifiers or existential
quantifiers in a calculus expression, we must make sure that the resulting
expression makes sense.
Ø A safe expression in relational calculus is one that is guaranteed to yield a
finite number of tuples as its result; otherwise, the expression is called unsafe.
Ø That means, an expression is said to be safe if all values in its result are from
the domain of the expression.
6
Tuple Relational Calculus

7. 7
Queries

8. 8
Queries…

9. 9
Queries…

10. Ø In tuple relational calculus, the variables range over the tuples whereas in
domain relational calculus, the variables range over the domains.
Ø Let Rel → be a relation name, X, Y → be the domain variables, op → an
operator in the set {<, ≤, >, ≥, =, ^=}. An Atomic formula in domain relational
calculus is one of the following:
ü <x1, x2, ... xn > ∈ Rel
ü X op Y
ü X op Constant or Constant op X
Ø A formula is recursively defined by using the following rules:
üAny atomic formula
üIf p and q are formulae, then ¬p, p ∧ q, p ∨ q, or p ⇒ q are also formulae
üIf p is a formula that contains X as a domain variable, then ∃X(p) and ∀
X(p) are also formulae
Ø The quantifiers ∃ & ∀ are said to bind the domain variable X. Whereas a
variable is said to be free in a formula if the formula does not contain an
occurrence of a quantifier that binds it.
10
Domain Relational Calculus (DRC)

11. 11
Queries

12. 12
Queries…

13. 13