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 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. 

Answer 2

both a) and c) should be true....can someone plz explain the correct answer

Comments

A) A is not unsafe, because we have intersection with R1, so at max we can get all tuples from R1. There is no such restriction in C. So A is not answer
!

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Thanks.