# Relational Calculus

## Tuple Relational Calculus

* It is a non-procedural query language
* This means that it will mention the required information without specifying the procedure to obtain it
* An expression in tuple relational calculus is denoted as follows: **{t | P(t)}**
* In other words, it is the set of all tuples t such that predicate P is true for t
* t is a tuple variable and t\[A] denotes the value of tuple t for attribute A
* t $$\in$$ r denotes that tuple t is in relation r
* The predicate can contain:
  * A set of attributes and constants
  * A set of comparison operators: (<, ≤, =, ≠, >, ≥)
  * A set of connectives: and (∧), or (v)‚ not (¬)
  * An implication (⇒): x ⇒ y, i.e. if x if true, then y is true

    x ⇒ y ≡ ¬x v y
  * A set of quantifiers:

    $$\exists$$ t$$\in$$r (Q (t)) ≡ "there exists" a tuple t in relation r such that predicate Q(t) is true

    $$\forall$$ t$$\in$$r (Q (t)) ≡ Q(t) is true "for all" tuples t in relation r
* **Ex**. Finding ID, name, dept\_name and salary of all instructors who have salary greater than 80000:\
  {t | t ∈ instructor ∧ t \[salary] > 80000}

## Domain Relational Calculus

* It is a non-procedural language as well
* It is as powerful as tuple relational calculus
* An expression in domain relational calculus is denoted as follows: {<$$x\_1, x\_2, ... x\_n$$>|P($$x\_1, x\_2, ..., x\_n$$)}

  where $$x\_1, x\_2, ..., x\_n$$ are domain variables and P is the predicate
* **Ex**. Finding ID, name, dept\_name and salary of all instructors who have salary greater than 80000:

  {\<i, n, d, s> | \<i, n, d, s> ∈ instructor ∧ s > 80000}