When should you use the existential and universal quantifiers for Relational Calculus? Could someone explain to me WHEN do you use existential and universal quantifiers for Relational Calculus? 

Given this schema:

Hotel(hotelNo, hotelName, city)


The expression below gets all hotel names based in London (expressed in Domain Relational Calculus)

$$\mathrm{HotelName|(\exists hNo,cty)(Hotel(hNo,hotelName,cty)\wedge cty=\,\,"\!\!London\!")}$$

Why does the existential quantifier required for this example? Wouldn't this be the equivalent {hotelName | (Hotel(hNo, hotelName, cty) AND cty='London')}

And why is only hNoand cty selected for the existential quantifier? What about the hotelName? 

Source: Herts University - Advanced Database Course
This question is related to the very basics of database theory, finite model theory and logics. I would strongly suggest Abiteboul's book on Foundations of Databases, or Libkin's book on Finite Model Theory.

Very roughly stated, a database is a collection of facts, and a query is a logical formula, which is used to specify certain patterns to be matched against the database. The most common database query language is unions of conjunctive queries, which is simply a disjunction of conjunctive queries. These are existentially quantified queries and there is NO universal quantification at all. The query in the question is indeed the simple conjunctive query

$\exists \ x \ {Hotel(x, y, london)} $

where $x$ and $y$ are logical variables and $london$ is a constant. Intuitively, the $x$ variables are to be matched to hotel numbers, and $y$ variables to hotel names. Now, in this formula, $y$ is a free variable, i.e. it is not bound to any quantifier. It is wrong to assume that it is bound to a universal quantifier. Such variables are also called the answer variables as these are the variables for which you want to retrieve answers. Note that $x$ is not an answer variable; so, you are not interested in the hotel numbers. All you want to say with this query is: Give me all the names of the hotels in London! Differently, consider this query 

${Hotel(x, y, london)} $

where $x$ is also a free variable. It asks for all names + numbers of the hotels in London. A Boolean query is a special case of a conjunctive query that does not contain any free variables. For instance, the query

$\exists \ x,y \ {Hotel(x, y, london)} $

has no free variables and asks a yes/no question: Is there a hotel in London (with some hotel number and name)? Overall, please have a look at the reference books, and simply learn the query semantics.