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GO Classes 2024 | IIITH Mock Test 5 | Question: 1

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Let $\mathrm{R}(a, b, c, d)$ be a RDBMS relation having the following set of functional dependencies (FD set) :
FD Set $$\mathrm{F}=\{b \rightarrow c ; c \rightarrow d ; ab \rightarrow d ; bc \rightarrow cd\}$$

We apply some queries on a relational instance $r$ of $\text{R}$. Which of the following queries are guaranteed to return the same number of tuples as in relation instance $r$?

  1. $\prod_a(r)$
  2. $\prod_{a, c}(r)$
  3. $\prod_{a, b}(r)$
  4. $\prod_{a, b, d}(r)$
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From the given FD set we can find candidate key which is “$ab$” because these attributes do not appear on the RHS of any FD.
The number of tuples in a relation resulting from a PROJECT operation is always less than or equal to the number of tuples in $\text{R}$.
If the projection list is a superkey of $\text{R}$—that is, it includes some key of $\text{R}$— the resulting relation has the same number of tuples as $\text{R}.$
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