Which of the following tuple relational calculus expression(s) is/are…

Answer: Only the expression ¬∃t ∈ r (¬P(t)) is equivalent to ∀t ∈ r P(t).

Reason:

• Apply the standard quantifier negation equivalence: ∀x φ ↔ ¬∃x ¬φ.

• Using this, ∀t ∈ r P(t) is equivalent to ¬∃t ∈ r ¬P(t), which matches the third expression.

• The expression ¬∃t ∈ r (P(t)) is equivalent to ∀t ∈ r (¬P(t)) and therefore is the negation of the target, not equivalent to it.

• Any expression that quantifies over tuples not in r (for example, ∃t ∉ r (P(t)) or ∃t ∉ r (¬P(t))) talks about tuples outside the relation and is unrelated to the universal claim about tuples inside r.

Therefore, only ¬∃t ∈ r (¬P(t)) is equivalent to ∀t ∈ r P(t).