What is the suitable file structure for database if queries are select (relational algebra) operations only?

Searches related to A relation R(A, B, C, D) has to be accessed under the query σB=10(R). Out of the following possible file structures, which one should be chosen and why? i. R is a heap file. ii. R has a clustered hash index on B. iii. R has an unclustered B+ tree index on (A, B).

how to get rows with more than 1 appearance of a specific column in relational algebra

I have a table like this:

PostId   |    Body    |    AuthorId    2             b               F   2             b               E   4             d               A   4             d               E   8             h               F  

So what I want is to get all the rows that have more that 1 appearance of PostId. Here the result would be 2 and 4 because they appear more than 1 time. I want this in relational algebra.
I have already a relation that works just fine but in this relation I use aggregation methods and I don’t really what to use count in my relation. I am wondering if there is a way to this with subtraction or division operators?
What I do for now is π PostId (σ c ≥ 2 ( γ Body; COUNT(PostId)->c R2)) to get the row with more than 1 appearance of PostId.
Thanks in advance for help

Testing whether polynomial is in algebra of other polynomials

A collection $ \Sigma$ of polynomials is an algebra if: (1) $ \lambda f + \eta g \in \Sigma$ for any $ f,g \in \Sigma, \lambda,\eta \in \mathbb{R}$ and (2) $ f,g \in \Sigma$ implies $ fg \in \Sigma$ . We say that $ P$ is in the algebra of $ \{P_1,\dots,P_n\}$ if $ P$ is in the smallest algebra containing $ P_1,\dots,P_n$ .

I was wondering if there was a way, on any computer math software, to check whether a given $ P$ as in the algebra of a given collection $ P_1,\dots,P_n$ .

Example: take $ n \ge 1$ and let $ P_1 = x_1+\dots+x_n, P_2 = x_1^2+\dots+x_n^2,\dots P_n = x_1^n+\dots+x_n^n$ . Then all $ n$ of the following symmetric functions are in the algebra generated by $ P_1,\dots,P_n$ : $ $ x_1+\dots+x_n$ $ $ $ x_1x_2+\dots+x_{n-1}x_n$ $ $ $ x_1x_2x_3+\dots+x_{n-2}x_{n-1}x_n$ $ $ $ \dots$ $ $ $ x_1\dots x_n$ $

I’d appreciate any help.

Is relational algebra a procedural, imperative, and/or declarative language?

In Database System Concepts 6ed,

6.2 The Tuple Relational Calculus

When we write a relational-algebra expression, we provide a sequence of procedures that generates the answer to our query.

The tuple relational calculus, by contrast, is a nonprocedural query language. It describes the desired information without giving a specific procedure for obtaining that information. A query in the tuple relational calculus is expressed as: {t | P(t)}. That is, it is the set of all tuples t such that predicate P is true for t.

Does the above mean that relational algebra is a procedural language?

Is relational algebra a declarative language?

Is the tuple relational calculus a declarative language?

Is the tuple relational calculus more declarative than relational algebra is?

Is a procedural language an imperative language? (This is always what I heard, but I also heard that SQL is a declarative language (so is relational algebra) so is not imperative.)

What is the correct or most reasonable or most accepted definition of procedural languages, imperative languages, and declarative languages?

Thanks.

Using relational algebra to express all with a condition

I encountered this question while revising for my finals exam on database theory.

The following database contains information about car repair workshops. The following tables are used:

$ $ Workshop(\underline{rname})\ Car(\underline{cname},make,model)\ Repairs(\underline{rname,cname},price)$ $

Given two workshops $ W_1$ and $ W_2$ , $ W_1$ is more expensive than $ W_2$ if for every car $ C$ that is repaired by both $ W_1$ and $ W_2$ , the repair price for $ W_1$ is higher than the repair price for $ W_2$ for $ C$ . Write a relational algebra query to find all workshops $ (W_1, W_2)$ where $ W_1$ is more expensive than $ W_2$ . Exclude workshops that do not repair any common cars.

I attempted to use a join naively:

  1. $ \rho_{(w1,car,c1)}(Repairs) \bowtie_{car} \rho{(w2,car,c2)} \rightarrow A$
  2. $ \sigma_{c1 > c2}(A) \rightarrow B$

And then simply project out the $ rname$ on a join with $ Workshop$ . However, I realised that there would exist some Repairs where $ c1 < c2$ and $ c1 > c2$ are both present, and it would still indicate that $ W_1$ is more expensive than $ W_2$ .

I am thinking of using the division $ \text{\}$ operator, but I am not sure how to proceed.

Looking for good book similar Stability/Conditioning in Numerical Linear Algebra,

I am currently reading Numerical Linear Algebra by Trefethen and Bau and I am finding it quite difficult to read. In particular, I have been trying to read the sections on Floating Point Arithmetic, Stability, and Conditioning, and they are pretty confusing to me; it has taken me 3-4 hours to go through the four pages on floating point arithmetic, and there are still a couple things I am unclear on.

I am looking for a book which goes over similar material as Trefefthen and Bau, but which fleshes things out a bit more.

My background is: I am a student of pure mathematics. I am a complete layman when it comes to computers.

Thanks!

Addition, multiplication, and apostrophe used to represent boolean algebra expressions?

I’m looking at a worksheet that expresses boolean logic expressions using multiplication, addition, and apostrophes; something I’ve never seen before.

I can make a guess that the apostrophe is equivalent to ¬ (except it’s suffixed instead of prefixed). But I’m not sure what the addition and multiplication of the variables/propositional atoms would mean. Furthermore, I don’t know how a boolean logic formula can “output” something other than just a truth value…

I can’t seem to piece together with certainty the meaning of this representation. Could anyone take a look at the below example and maybe make a guess as to a translation of this representation to the more traditional ^, v, and ¬ symbols?

This arose in the context of digital logic in terms of logic gates and such on a CPU, if that makes a difference.

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An alleged truth table of the above two expressions (the first row is filled in as an example):

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Consulta Sobre Algebra Relacional

Me dan las siguientes tablas:

Producto<nombreProducto, tipo, precio, esParaDiabetico, esDietetico> Persona<DNI, nombre, edad, estadoCivil, profesion> Fabrica<nombreProducto, nombreFabrica, produccionEnPesos, ciudad> Regalos<DNIRegalador, DNiReceptor, nombreProducto, fecha> 

Como puedo saber todos los productos que han sido regalados a todas la personas existentes en la base de datos?

Gracias de antemano.