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+40 votes

Which of the following query transformations (i.e., replacing the l.h.s. expression by the r.h.s expression) is incorrect? R_{1} and R_{2} are relations, C_{1} and C_{2 }are selection conditions and A_{1} and A_{2} are attributes of R_{1.}

A. $\sigma_{C_1} \left(\sigma_{C_2}\left(R_1\right)\right) \to \sigma_{C_2} \left(\sigma_{C_1}\left(R_1\right)\right)$

B. $\sigma_{C_1} \left(\pi_{A_1}\left(R_1\right)\right) \to \pi_{A_1} \left(\sigma_{C_1}\left(R_1\right)\right)$

C. $\sigma_{C_1} \left(R_1 \cup R_2\right) \to \sigma_{C_1}\left(R_1\right) \cup \sigma_{C_1} \left(R_2\right)$

D. $\pi_{A_1} \left(\sigma_{C_1}\left(R_1\right)\right) \to \sigma_{C_1} \left(\pi_{A_1}\left(R_1\right)\right)$

0

y B is correct bcz same reasoning valid for B also wch is for D ten ans shd B& D both incoorect kindly explain

+4

a) & c) are correct! only b) & d) we need to look deep :

Actually, we are assuming here that LHS to be specific and trying to transform the same meaning to RHS. So in b) LHS specifically selects attribute A1 then applies condition C1 so interchanging them in RHS will do no harm.

But in d) we are keeping projections options open by applying first C1 then A1 but in RHS we are restricting by projecting only A1 then C1 so it's not safe.

For RHS in d) C1 may have been related to other attributes but since they will not be present then the query might return empty but in LHS we are keeping options and not restricting.

Thus answer d)

Actually, we are assuming here that LHS to be specific and trying to transform the same meaning to RHS. So in b) LHS specifically selects attribute A1 then applies condition C1 so interchanging them in RHS will do no harm.

But in d) we are keeping projections options open by applying first C1 then A1 but in RHS we are restricting by projecting only A1 then C1 so it's not safe.

For RHS in d) C1 may have been related to other attributes but since they will not be present then the query might return empty but in LHS we are keeping options and not restricting.

Thus answer d)

+28 votes

Best answer

+6

In (B), first we select columns and then apply select conditions on them. So, it can be assumed that select conditions are based the selected columns only.

+14

For that LHS of B should be an invalid query. Since the question talks about query transformation and not just query equivalence, we can assume LHS is a valid one.

+11

actually in (b) option for validity of L.H.S condition c1 must be applied on attribute a1 and same condition will be applicable after projection will give same answer

but in option (d) for validity condition may be any thing because selection is being here first suppose if selection is based on any other attribute other than a1 then in RHS after projection how can we apply same condition because attribute other than a1 are not present there

am i right sir

0

I have a questions:

1) for LHS of option B, can C1 be applied on attribute A2 ?

OR

2) for RHS of option D, can C1 be applied on attribute A2 ?

Will the above operations return anything (empty relation) ? or they are invalid operations?

@Arjun Sir

1) for LHS of option B, can C1 be applied on attribute A2 ?

OR

2) for RHS of option D, can C1 be applied on attribute A2 ?

Will the above operations return anything (empty relation) ? or they are invalid operations?

@Arjun Sir

+2

I have tried to explain this way. @Arjun Sir, please check if is right way to do it.

A->B ..if A is true, then B is true to make "A implies B is valid"

Else A is true and B is false, then "A implies B is not valid"

So either of the cases, LHS has to be true. Now we need to check whether RHS is true or false. Here I am not bothered when A is false.

Here true means getting some attribute values or non empty set.

Option B:

____________

LHS: first obtain columns A1, then apply the condition C1. Since LHS is true means some values are there. So we can confirm that C1 is related to A1 and no other attributes, so we got the LHS is true.

Option B RHS: Since it is confirmed from LHS that C1 is related to A1, so RHS will give same number of values.

So option B is valid.

Option D.

__________

If C1 is based on only attribute A1, then both LHS and RHS will give same result.

But in LHS , C1 is not bound to only A1. It may have any attributes from A1, A2,A3,...An.

So let's say C1 is based on attribute A2, then it will select only those tuples where A2 condition satisfy and from those tuples it will select only attribute A1. By doing so we got some attribute values on left side since LHS is true.

Now check the RHS: Select all the attribute A1. On top of that when I apply C1 which is based on A2, then there will be no values. So RHS is empty.

So LHS has some values(true) and RHS is empty set(false). So for option D, LHS->RHS is not valid.

So only option B is true.

A->B ..if A is true, then B is true to make "A implies B is valid"

Else A is true and B is false, then "A implies B is not valid"

So either of the cases, LHS has to be true. Now we need to check whether RHS is true or false. Here I am not bothered when A is false.

Here true means getting some attribute values or non empty set.

Option B:

____________

LHS: first obtain columns A1, then apply the condition C1. Since LHS is true means some values are there. So we can confirm that C1 is related to A1 and no other attributes, so we got the LHS is true.

Option B RHS: Since it is confirmed from LHS that C1 is related to A1, so RHS will give same number of values.

So option B is valid.

Option D.

__________

If C1 is based on only attribute A1, then both LHS and RHS will give same result.

But in LHS , C1 is not bound to only A1. It may have any attributes from A1, A2,A3,...An.

So let's say C1 is based on attribute A2, then it will select only those tuples where A2 condition satisfy and from those tuples it will select only attribute A1. By doing so we got some attribute values on left side since LHS is true.

Now check the RHS: Select all the attribute A1. On top of that when I apply C1 which is based on A2, then there will be no values. So RHS is empty.

So LHS has some values(true) and RHS is empty set(false). So for option D, LHS->RHS is not valid.

So only option B is true.

+10 votes

but in option (d) for validity condition may be any thing because selection is being here first suppose if selection is based on any other attribute other than a1 then in RHS after projection how can we apply same condition because attribute other than a1 are not present there

am i right sir

+2 votes

a) & c) are correct! only b) & d) we need to look deep :

Actually, we are assuming here that LHS to be specific and trying to transform the same meaning to RHS. So in b) LHS specifically selects attribute A1 then applies condition C1 so interchanging them in RHS will do no harm.

But in d) we are keeping projections options open by applying first C1 then A1 but in RHS we are restricting by projecting only A1 then C1 so it's not safe.

For RHS in d) C1 may have been related to other attributes but since they will not be present then the query might return empty but in LHS we are keeping options and not restricting.

Thus answer d

Actually, we are assuming here that LHS to be specific and trying to transform the same meaning to RHS. So in b) LHS specifically selects attribute A1 then applies condition C1 so interchanging them in RHS will do no harm.

But in d) we are keeping projections options open by applying first C1 then A1 but in RHS we are restricting by projecting only A1 then C1 so it's not safe.

For RHS in d) C1 may have been related to other attributes but since they will not be present then the query might return empty but in LHS we are keeping options and not restricting.

Thus answer d

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