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| DATE | 2014-12-18 |
| FROM | Ruben Safir
|
| SUBJECT | Re: [LIU Comp Sci] Need tutoring on Relational Calculus
|
From owner-learn-outgoing-at-mrbrklyn.com Thu Dec 18 06:10:15 2014
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Date: Thu, 18 Dec 2014 06:10:28 -0500
From: Ruben Safir
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To: learn-at-mrbrklyn.com, learn-at-nylxs.com,
Ping-Tsai Chung ,
Ping-Tsai Chung
Subject: Re: [LIU Comp Sci] Need tutoring on Relational Calculus
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Sorry, the last one the mail client did what it wanted, not what I wanted...
In predicate logic, a universal quantification is a type of quantifier,
a logical constant which is interpreted as "given any" or "for all". It
expresses that a propositional function can be satisfied by every member
of a domain of discourse. In other terms, it is the predication of a
property or relation to every member of the domain. It asserts that a
predicate within the scope of a universal quantifier is true of every
value of a predicate variable.
It is usually denoted by the turned A (?) logical operator symbol,
which, when used together with a predicate variable, is called a
universal quantifier ("?x", "?(x)", or sometimes by "(x)" alone).
Universal quantification is distinct from existential quantification
("there exists"), which asserts that the property or relation holds only
for at least one member of the domain.
Quantification in general is covered in the article on quantification
(logic). Symbols are encoded U+2200 ? for all (HTML ∀ · ∀ ·
as a mathematical symbol).
~~~~~~~~~~~~~
In mathematics, a predicate is commonly understood to be a
Boolean-valued function P: X? {true, false}, called the predicate on X.
However, predicates have many different uses and interpretations in
mathematics and logic, and their precise definition, meaning and use
will vary from theory to theory. So, for example, when a theory defines
the concept of a relation, then a predicate is simply the characteristic
function or the indicator function of a relation. However, not all
theories have relations, or are founded on set theory, and so one must
be careful with the proper definition and semantic interpretation of a
predicate.
On 12/18/2014 05:09 AM, Ruben Safir wrote:
>> I'm pouring over the HW assignment and the explanation and notes from
>> the text is not adequate to explain relational calculus. The book is
>> actually in shambles. This is a very advanced mathamtical topic and
>> there is no way to learn this in a single lesson, although you might be
>> able to bluff you way through it.
>>
>> Does anyone understand this? It makes little sense as it is presented,
>> and I'm not in any way certain of the meaning of the syntax.
>>
>> I have no idea what this whole section of 6.6.63 is out of the text.
>>
>>
>> Quote:
>>
>> In addition, two special symbols called quantifiers can appear in
>> formulas; these are
>> the universal quantifier (?) and the existential quantifier (?). Truth
>> values for
>> formulas with quantifiers are described in Rules 3 and 4 below;
>>
>> *first, however, we need to define the concepts of free and bound tuple
>> variables in a formula.***
>> ENDQUOTE
>>
>>
>> _*Here is a question, Why does it matter if Tuple Variables are Free or
>> Bound?****
>>
>> Why do we make them Free or Bound? ****
>>
>> QUOTE
>> Informally, a tuple variable t is bound if it is quantified, meaning
>> that it appears in an (?t) or (?t) clause; otherwise, it is free.
>> ENDQUOTE
>>
>> ??????
>>
>> QUOTE
>> Formally, we define a tuple variable in a formula as free or bound
>> according to the following rules:
>>
>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~`
>>
>> Stop this author doesn't know the material. Either that or he doesn't
>> know how to explain it. You can't LEARN math by memorizing rules inside
>> rule inside rule. You have to lay out the theory and show practical
>> models. You have to understand why rules are used and developed and
>> there is no effort to even attempt an explanation.
>>
>>
>> QUOTE
>> An occurrence of a tuple variable in a formula F that is an atom is free
>> in F.
>>
>> An occurrence of a tuple variable t is free or bound in a formula made up of
>> logical connectives—(F 1 AND F2), (F1 OR F2 ), NOT(F1 ), and NOT(F2 )—
>> depending on whether it is free or bound in F1 or F2 (if it occurs in
>> either).
>>
>> Notice that in a formula of the form F = (F1 AND F2) or F = (F1 OR F2), a
>> tuple variable may be free in F1 and bound in F2, or vice versa; in this
>> case,
>> one occurrence of the tuple variable is bound and the other is free in F.
>>
>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>> Why does it matter?
>>
>> QUOTE:
>>
>> All free occurrences of a tuple variable t in F are bound in a formula
>> F of the
>> form F= (? t)(F) or F = (?t)(F).
>>
>> UNQUOTE
>>
>> I have no idea what he is talking about here. He just said above that
>> any ? t or ?t means that t is BOUND. Now it is free?
>>
>>
>> QUOTE
>> The tuple variable is bound to the quantifier specified in F. For
>> example, consider the following formulas:
>> F1 : d.Dname=‘Research’
>> F2 : (? t)(d.Dnumber=t.Dno)
>> F3 : (?d)(d.Mgr_ssn=‘333445555’)
>> The tuple variable d is free in both F1 and F2, whereas it is bound to
>> the (?) quan-
>> tifier in F3. Variable t is bound to the (?) quantifier in F2.
>> We can now give Rules 3 and 4 for the definition of a formula we started
>> earlier:
>> ?
>> ?
>> Rule 3: If F is a formula, then so is (?t)(F), where t is a tuple
>> variable. The
>> formula (?t)(F) is TRUE if the formula F evaluates to TRUE for some (at
>> least
>> one) tuple assigned to free occurrences of t in F; otherwise, (?t)(F) is
>> FALSE.
>> Rule 4: If F is a formula, then so is (?t)(F), where t is a tuple
>> variable. The
>> formula (?t)(F) is TRUE if the formula F evaluates to TRUE for every tuple
>> (in the universe) assigned to free occurrences of t in F; otherwise,
>> (?t)(F) is
>> FALSE.
>> The (?) quantifier is called an existential quantifier because a formula
>> (?t)(F) is
>> TRUE if there exists some tuple that makes F TRUE. For the universal
>> quantifier,
>>
>> (?t)(F) is TRUE if every possible tuple that can be assigned to free
>> occurrences of t
>> in F is substituted for t, and F is TRUE for every such substitution. It
>> is called the uni-
>> versal or for all quantifier because every tuple in the universe of
>> tuples must make F
>> TRUE to make the quantified formula TRUE.
>>
>> ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>>
>> So then we have this homework and I'm doing it, barerly working through
>> this methodology and then you hit this:
>>
>> e) Retrieve the names of employees who work on every project:
>>
>> This question is insane with SQL and Relational Algebra (where we can
>> use a division) but solving it with relational calculus??
>>
>> Who can explain an answer this?
>>
>>
>> This is an image of the database scheme
>>
>> http://www.nylxs.com/images/database_3.3_company.png
>>
>>
>>
>>
>>
>
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