Database Management Systems by Raghu Ramakrishnan, Johannes Gehrke ~ 2nd edition Q4.3.5

Question

Given schema:

$Suppliers$ ($sid:$$ integer$, $sname:$ $string$, $address:$ $string$)
$Parts$ ($pid:$ $integer$, $pname:$ $string$, $color:$ $string$)
$Catalog$ ( $sid:$ $integer,$ $pid:$ $integer,$ $cost:$ $real$)

Find the $sids$ of suppliers who supply every part.

Is My solution correct?: TRC and DRC:

$\{ R$ $| Ǝc ∈$ $Catalog ( ∀p$ ∈ $Parts(c.pid=p.pid$ ^ $c.sid = R.sid))\}$

$\{<A> $$|Ǝ(A,D,G)∈ Catalog$ ^ $∀(D,E,F)∈ Parts\}$

Comments

{R{R |Ǝc∈|Ǝc∈ Catalog(∀pCatalog(∀p ∈ Parts(c.pid=p.pidParts(c.pid=p.pid ^ c.sid=R.sid))}

Why Ǝc∈ Catalog existential quantifier here ?

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Correction 1: use different tuple variable for parts and catalog.

Ǝc∈ Catalog here as we need to refer to some of the catalog tuples, not all. Some of the catalog tuples will only fulfill our conditions not all ie. some catalog tuples will be used in the query.

However each and every parts tuples will be used in our query to find final "R" Query so, ∀p ∈ Parts.

Pls comment if errors or more explaination needed.

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Bro im confident about TRC but half confident about DRC

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No need to refer to Supplier schema for the TRC part for first question... It is redundant. But it is correct.

For DRC "For All" quantifier has to be used.

Thanks for the answers, but my query(not of DBMS) was if my queries are wrong , why are they incorrect as solutions do not have this and I am finding no problem in my solution for the TRC.

Why cannot we use same domain variable for "sid" and "pid" for the 3 schema respectively? Is it becaue of the possibility that tuples from supplier or parts tables may not be related in the catalog table??