self doubt

Question

While writing out tuple relational calculas queries as nested parenthesis , can we change the order of parenthesis ?
What does the new expression (after changing order of parenthesis) mean in terms of relational algebra query or even in normal language ?

Comments

  yes , In order to express a relation using relational calculus scheme , we don't have to give details of all the steps or sequence of steps by which we obtained the relation. That is why Relational Calculus is Non-Procedural.

Example :- Suppose we have 2 relations :-

1) User (b_name,b_addr,card_no.)

2) Borrow(acc_no , card_no, date_of_issue)

Now, suppose Query is :-

Find out all the borrowers (B_name) and their addresses for those who have issued a book on "05/08/2018"

Now we can write it as :-

1) {t | $\exists$ u $\in$ User (t[b_name] = u[b_name] $\wedge$ t[b_addr] = u[b_addr] $\wedge$ $\exists$ w $\in$ Borrow(u[card_no]=w[card_no] $\wedge$ w[date_of_issue] = "05/08/2018") )}

                                                   (or)

2) {t | $\exists$ w  $\in$ Borrow (w[date_of_issue]= "05/08/2018" $\wedge$ $\exists$ u $\in$ User(w[card_no]=u[card_no] $\wedge$ t[b_name] = u[b_name] $\wedge $t[b_addr] = u[b_addr]) )}

Now , equivalent Relational Algebra Expression would be :-

$\prod _{b_name , b_addr}$ ($\sigma$_{date_of_issue = "05/08/2018"} (Borrow $\Join$ User))

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Thanks for you reply.
In the example you gave it is quite right that both queries make same sense.
how do we convince ourselves thsi is always going to be the case for any kind of query where nesting occurs.

I was studying from lectures of PPC- IITKGP
https://www.youtube.com/playlist?list=PLyvBGMFYV3auVdxQ1-88ivNFpmUEy-U3M

He also gives a similar example and a few more .
But he also mentions in passing that this is not always the case.
I am unable to reason about this ; how to think of it mentally that what changing order of parenthesis is doing to our query. 

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@Rameesh , Relational Calculus is based on Formal Mathematical Logic. So, order of quantifiers matters here. if we use either existential quantifier or universal quantifier in the whole query then we can change the order of sequence in the query but if we use the combination of both quantifier then we can't change the order because then meaning will be changed. In both cases we can write equivalent relational algebra expression if the relational calculus query expression is safe.   So, Prof, Chakraborty said correct that it is not always the case , we can change the order.

I know that you know the order of quantifiers matters  in formal logic but still I am attaching the screenshot from Rosen. Please find the attachments and please correct me if I have said anything wrong...

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@ankitgupta.1729
yes got it , thanks for your detailed answer
I was just looking for something of this sort. This is a good and very important connection to have - Relational Calculas and Formal logic.