GATE CSE 2001 | Question: 2.24

Question

Which of the following relational calculus expression is not safe?

• A. $\left\{t \mid \exists u \in R_1\left(t[A] = u[A]\right) \land \neg \exists s \in R_2 \left(t[A] = s[A]\right)\right\}$
• B. $\left\{ t \mid \forall u \in R_1\left(u[A]="x" \Rightarrow \exists s \in R_2\left(t[A] = s[A] \land s[A] = u[A]\right)\right) \right\} $
• C. $\left\{t \mid \neg (t \in R_1)\right\} $
• D. $\left\{t \mid \exists u \in R_1\left(t[A]=u[A]\right) \land \exists s \in R_2 \left(t[A] = s[A]\right)\right\}$

Comments

Should the variable t not bounded to a relation as well ? If it is not bounded to a relation then shouldn't t yield infinite no. of tupples in A,B and D. From what I have read the notation followed is { t | R(t) ^ <set of conditions> } .

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Someone please explain how to read Queries 1, 2 and 3. Are they from TRC or DRC and how to interpret them in some English language!

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Does not help at all !!!

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Any expression whose result uses “constants / values”that do not appear in the instances of any of the database relations.     

or in Simple words

the expression which results in finite no of output is considered as Safe expression and which expression can give infinite output is unsafe.

Answer 1

Answer: C.

It returns tuples not belonging to R1 (which is infinitely many). So, it is not safe.

Reference: https://people.cs.pitt.edu/~chang/156/10calculus.html

Comments

"A" also has $\displaystyle \neg s$. It shouldn't be safe as well.

I mean that too contains too many tuples.

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Can you please post source again. Its not working anymore

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`JEET

"A" also has ¬s. It shouldn't be safe as well.

I mean that too contains too many tuples.

I think as it contains “There exists”  so even just one out of domain tuple will work. 

and hence option C is more stronger.

Answer 2

Well it’s obvious that the answer is (C).

But can anyone verify my analysis of other Options?

Option (A) : Prints values which are in column A of R1,but not in Column A of R2.

Option (B) : It will print ‘x’ every time it is in column A of R1 and in atleast one time in column A of R2. Oh wait ,bcoz it is a Set ,’x’ will be printed only once!

Option (D) : Prints values if which are in column A
of both relations R1 & R2

Answer 3

{t∣¬(t∈R1)}

If we run this query the tuples we get does not belong to the instances of relation R1, so according to definition this expression is unsafe.

ANSWER: C

 

Answer 4

Answer will be $B,C.$

TRC Queries in both Option B, C are Unsafe Queries. 

Definition of “Unsafe TRC Expression”: Any expression whose result uses “constants / values” that do not appear in the instances of any of the database relations.  

Option B:

NOTE that “$x$” is some constant value, in the domain of variable $A.$

Case 1:

If the instance of $R_1$ has at least one tuple where $R_1.A = $ ”$x$” & the instance of $R_2$ also has at least one tuple where $R_2.A = $ ”$x$” then the result of the TRC query will be a table with single column named $A$ and only one row with value “$x$”.

Case 2:

If the instance of $R_1$ has at least one tuple where $R_1.A = $ ”$x$” BUT the instance of $R_2$ also has NO tuple where $R_2.A = $ ”$x$” then the result of the TRC query will be an Empty table with single column named $A$ i.e. Output is Empty.

Case 3:

If the instance of $R_1$ has NO tuple where $R_1.A = $ ”$x$” then every value in the domain of attribute $A$ will satisfy the condition given in the TRC query & hence, will appear in the output. Hence, in this case, we get an infinite number of tuples in the output.

Hence, Query in Option B is Unsafe. 

Comments

Can we have infinte number of tuples in any relation? if  not then option B can not be unsafe query, because although for the case when there is no tuples for which R1.A="x",then condition will be true every time but as u belongs to R1, and if R1 and R2 can not be infinte then how the result will be infinte?? @Deepak Poonia sir