Tuple Relational Calculus

A tuple relational calculus expression is of the form

{t | Q(t)}

Where Q is known as formula.

We also use the notation
• t[a] to indicate the value of tuple t on attribute a.
• t Є r to show that tuple t is in relation r.

1. A tuple variable is said to be a free variable unless it is quantified by Then it is said to be a bound variable.

2. A formula is built of atoms. An atom is one of the following forms:

• δ Є r, where δ is a tuple variable, and r is a relation (¢ is not allowed).
• δ[x] Θ u[y], where δ and u are tuple variables, and x and y are attributes, and Θ is a comparison operator (<, ≤ , =, ≠, >, ≥).
• δ[x] Θ c, where is a constant in the domain of attribute x.

3. Formulae are built up from atoms using the following rules:

An atom is a formula.
• If Q is a formula, then so are Q and (Q).
• If Q1 and Q2 are formulae, then so are Q1 V Q2, Q1 Λ Q2 and Q1 => Q2.
• If Q(δ) is a formula containing a free tuple variable δ, then
δ Є r(Q(δ)) and δ Є r(Q(δ))
are formulae also.

3. Note some equivalences:
• Q1 Λ Q2 = (Q1 V Q2).
• t Є r(Q(t)) = t Є r( Q(t)).
• Q1 => Q2 = Q1 V Q2.