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43 votes
43 votes

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

  1. $\neg \exists t \in r \left(P\left(t\right)\right)$
  2. $\exists t \notin r \left(P\left(t\right)\right)$
  3. $\neg \exists t \in r \left(\neg P\left(t\right)\right)$
  4. $\exists t \notin r \left(\neg P\left(t\right)\right)$
    1. I only
    2. II only
    3. III only
    4. III and IV only

5 Answers

Best answer
51 votes
51 votes
Only III is correct.
The given statement means for all tuples from r, P is true. III means there does not exist a tuple in r where P is not true. Both are equivalent.

IV is not correct as it as saying that there exist a tuple, not in r for which P is not true, which is not what the given expression means.
selected by
47 votes
47 votes

∀t∈r(P(t)):  All boys in the class are in a relationship.
 

I) ¬∃t∈r(P(t)) : There is not a single boy in the class who is in a relationship.  (X)
II) ∃t∉r(P(t)) : There is at least one boy in the class who does not belong those boys who are in a relationship. (X)
III) ¬∃t∈r(¬P(t)) : There is not a single boy in the class who is not in a relationship. (Matches the qsn)
IV) ∃t∉r(¬P(t)) : There is at least one boy in the class who does not belong to those boys who are not in a relationship. (X)

Only IV is little bit tricky. Please read again carefully.  We need "all boys" instead of "at least one" to match our original qsn.
So only option III is true.
Hence  C is the answer. 

16 votes
16 votes

(IV) is incorrect as Arjun sir have explained.Above is simplification of given expression to (III).

Note that, ∀x∈t (P(x))=∀x (x∊t-->P(x))

                ∃x∈t (P(x))= ∃x (x∊t^P(x))

15 votes
15 votes

Concept:- A statement is equivalent to its double negation.

i.e. x=~(~x)

Similarly compare below expression with above expression.

$\forall t \in r \left(P\left(t\right)\right)$ = $\neg$ ( $\exists t \in r \left(\neg P\left(t\right)\right)$ )

Hence C is Ans.

Answer:

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