Multiple choice

Which of the following tuple relational calculus expression(s) is/are equivalent to $\forall t \in r \left(P\left(t\right)\right)$?

I. $\neg \exists t \in r \left(P\left(t\right)\right)$ II. $\exists t \notin r \left(P\left(t\right)\right)$ III. $\neg \exists t \in r \left(\neg P\left(t\right)\right)$ IV. $\exists t \notin r \left(\neg P\left(t\right)\right)$

  1. I only

  2. II only

  3. III only

  4. III and IV only
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using De Morgan's laws for quantifiers: ¬∃t∈r(P(t)) is NOT equivalent to ∀t∈r(P(t)) - it means 'no element satisfies P', while the original means 'all elements satisfy P'. ¬∃t∈r(¬P(t)) simplifies to ∀t∈r(P(t)) - if there's no element that doesn't satisfy P, then all elements must satisfy P. Expression III is correct.

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