OpenLLM-France/wikimedia huggingface dataset:
# Equivalence relation/Symmetric accessibility relation/Example

Suppose that we are in a situation, where certain places (or Objects) can be reached from certain other places or not. This accessibility can be determined by the choice of means of transport, or by a more abstract kind of movement. Such an accessibility relation yields often an equivalence relation. A place can be reached by starting at this place, this gives reflexivity. The symmetry of accessibility means that if it is possible to reach $A$ starting from $B$, then it is also possible to reach $B$ starting from $A$. This is a natural condition for accessibility, though it is probably not fulfilled for every kind of accessibility. The transitivity does hold whenever we can do the movements after each other, like going from $A$ to $B$, and then from $B$ to $C$.
If, for example, accessible is given by walking overland, then two places are equivalent if and only if they lie on the same island (or continent). Islands and continents are the equivalence classes. In topology, the concept of path-connectedness plays an important role: two points are path-connected if they can be connected by a continuous path. Or: On the set of integer number, there is a colony of fleas, and every jump of a flea has the length of five unites (in both directions). How many flea populations are there, which fleas can meet each other?
# Equivalence relation/N times N/Jumps (2,0) and (3,3)/Example

We consider the lattice product set $M=\mathbb {N} \times \mathbb {N}$. We fix the jumps (think about a Meadow jumping mouse that can perform these jumps and no other jumps)
    $\pm (2,0)\,\,{\text{ and }}\,\,\pm (3,3).$
We say that two points $P=(a,b),\,Q=(c,d)\in M$ are equivalent if it is possible, starting in $P$, to reach the point $Q$ with a sequence of such jumps. This defines an equivalence relation (the main reason is that for every jump also its negative jump is allowed). Typical questions are: How can we characterize equivalent points, how can we decide whether two points are equivalent or not?
# Equivalence relation/Real projective space/Example

Let $n\in \mathbb {N}$, and set $M=\mathbb {R} ^{n+1}\setminus \{0\}$. $\mathbb {R} ^{n+1}$ is a real vector space, the scalar multiplication of $\lambda \in \mathbb {R}$ and $x\in \mathbb {R} ^{n+1}$ is denoted by $\lambda \cdot x$ Moreover, set
    $R={\left\{(x,y)\in M\times M\mid {\text{ there exists some }}\lambda \in \mathbb {R} \setminus \{0\}{\text{ such that }}\lambda \cdot x=y\right\}}\,.$
Therefore, two points are defined to be equivalent if they can be transformed to each other by scalar multiplication with a scalar $\lambda \neq 0$. We can also say that two points are equivalent if they define the same line through the origin.
This is indeed an equivalence relation. The reflexivity follows from $x=1x$ for every $x\in M$. To prove symmetry, suppose $xRy$, that is, there exists some $\lambda \neq 0$ such that $\lambda x=y$. Then also $y=\lambda ^{-1}x$ holds, as $\lambda$ has an inverse element. To prove transitivity, suppose that $xRy$ and $yRz$ holds; this means that there exist $\lambda ,\delta \neq 0$ such that $\lambda x=y$ and $\delta y=z$. Then $z=\delta y=(\delta \lambda )x$ with $\delta \lambda \neq 0$. The equivalence classes of this equivalence relation are the lines through the origin (but without the origin). The quotient set is called the real-projective space (of real dimension $n$), and is denoted by $[[:Template:Op:Projectivespace]]$.
# Integer numbers/Construction from natural numbers/Example

Let $\mathbb {N}$ be the set of the natural numbers, and let $M=\mathbb {N} \times \mathbb {N}$ denote the product set with the componentwise addition. We define on $M$ a relation by
    $(a,b)\sim (c,d),{\text{ if }}a+d=b+c.$
This holds, in case $a\leq c$, if and only if there exists an $e\in \mathbb {N}$ (namely $e=c-a$) such that
    $(c,d)=(a,b)+(e,e)\,.$
This means that the pairs differ by an diagonal element, that is, a pair, where both components coincide. This relation is an equivalence relation on $M$, see exercise. If we consider $\mathbb {N} \times \mathbb {N}$ as a square lattice (a "discrete coordinate system“), then the equivalence classes are given by the points on a "discrete line“ parallel to the diagonal. The points $(a,b)$ with $a\geq b$ are equivalent with $(a-b,0)$; they have a representative where the second component is $0$. The points $(a,b)$ with $a\leq b$ are equivalent to $(0,b-a)$; they have a representative where the first component is $0$. The points $(a,a)$ are equivalent to $(0,0)$. We call the representative of an equivalence class where at least one component is $0$, the standard representative of this equivalence class. The standard representatives are the discrete points of the confining lines of the quadrant; for a point, we find its standard representative by going, parallel to the diagonal, in direction of these lines. Two points are equivalent if and only if they have the same standard representative.
We call now the quotient set, that is, the set of the equivalence classes under this equivalence relation, the set of the integer numbers, and denote it with $\mathbb {Z}$. Every integer number has exactly one standard representative of the form $n:=(n,0)$ with $n\in \mathbb {N} _{+}$, or of the form $0:=(0,0)$, or of the form $-n:=(0,n)$ with $n\in \mathbb {N} _{+}$. We consider a natural number $n$ as the integer number $(n,0)$.
We want to "add“ two integer numbers, that is, two such equivalence classes $[(a,b)]$ and $[(c,d)]$, in order to get a binary operation $\oplus$ on $\mathbb {Z}$. The first idea that comes to mind is to define this via
    $[(a,b)]\oplus [(c,d)]:=[(a+c,b+d)]\,.$
However, here we encounter the problem whether this is well-defined, since this operation is defined using certain representatives; it is not clear at all whether different representatives give the same result. Therefore, if $(a,b)\sim (a',b')$ and $(c,d)\sim (c',d')$ holds, then we have to check that also
    $(a+c,b+d)\sim (a'+c',b'+d')\,$
holds, which means $[(a+c,b+d)]=[(a'+c',b'+d')]$. This is the case, see exercise. Moreover, one can show that this operation on $\mathbb {Z}$ is associative and commutative, that $[(0,0)]$ is the neutral element of this operation, and that for every element $[(a,b)]$ there exists an inverse element, namely $[(b,a)]$.
We define now a multiplication on $\mathbb {Z}$ via
    $[(a,b)]\cdot [(c,d)]:=[(ac+bd,ad+bc)]\,.$
Again, this is well-defined, and one can show that the multiplication is associative and commutative, that $1=[(1,0)]$ is the neutral element, and that the distributive law holds.
# Equivalence relation/Quotient set/Universal property/Fact

Let $M$ be a set, and let $\sim$ denote an equivalence relation on $M$, with the quotient set $M/\sim$. Let $\varphi \colon M\rightarrow W$ be a mapping with $\varphi (x)=\varphi (y)$ for all $x,y\in M$ with $x\sim y$.
${\overline {\varphi }}\colon M/\sim \,\,\rightarrow W$ fulfilling
$\varphi ={\overline {\varphi }}\circ q$
# Equivalence relation/Quotient set/Universal property/Fact/Proof

Let $[x]\in M/\sim$ be given. The only possibility for ${\overline {\varphi }}$ is to set ${\overline {\varphi }}([x]):=\varphi (x)$. We have to show that this mapping is well-defined, that is, independent of the choice of the representatives. For this, let $[x]=[y]$, that is, $x\sim y$. By the condition on $\varphi$, we have $\varphi (x)=\varphi (y)$.
# Research methods in psychology

A critical understanding of research methods in psychology examines how different types of scientific research methods are used to investigate psychological phenomena, highlightint their benefits and limitations.

## Overview

Psychology is a science. Psychological research uses the scientific method to systematically investigate and understand human experience and behaviour (Willig, 2019). The scientific method is a systematic refinement of everyday thinking (Einstein, 1936).
A useful distinction is made between research which is:
* quantitative (using numerical data) and
* qualitative (using non-numerical data) research.
The approaches represent two sides of the same coin. Or, as Karl Marx wrote:
Though predating the term "qualitative research", early pioneers such as Freud, Jung, and Piaget utilised such methods in their early work (e.g., case studies) and, by doing so, provided the foundation for modern psychology (del Rio Carral & Tseliou, 2019).
Since the first experimental psychology laboratory was established in 1879 by Wundt, psychological research become highly quantitative during the 20th century. During the 21st century, the methods and foci of psychological research have expanded and diversified (Nelson, 2015).
Each research method has strengths and weaknesses, so the selection of which method(s) to use should depend on the aims and nature of the research question.

## List of psychological research methods

Many different research methods are used in psychological research, including:
* Action research
* Autoenthnography
* Case study in psychology
* Content analysis
* Correlational research
* Delphi method
* Discourse analysis
* Ethnography
* Experiment
* Exploratory research
* Grounded theory
* Indigenous research
* Meta-analysis
* Observation
* Participatory research
* Qualitative research
* Quantitative research
* Quasi-experiment
* Randomised experiment
* Secondary research (Wikipedia)
* Survey research - Survey research and design in psychology
# Brezhoneg Tri/Lesson 2

Brezhoneg Tri/Lesson 2
You will learn
How to ask where something is
How to tell someone where something is
How to tell someone something isn't there
Position
You use these to describe where an object is:
Position
BretonEnglish
e-barzh
inside
er-maezh
outside
dirak
in front
a-dreñv
behind
e-kichen
beside
etre
between
e-kreiz
in the center
aze
here
amañ
there
a-us
above
en-traoñ
in the bottom
en-nec'h
in the top
dindan
below
war
upon
diwar
(coming) from
gant
with
digant
without
uhel
high
izel
low
a-gleiz
to the left
a-zehoù
to the right
ouzh
at
diouzh
from
a-led
horizontal
a-sonn
vertical
eus
from
da
to
a-hed
war-eeun
straight ahead
Position : Questions & answers
This is the basic format for a question:
Ez eus _____? - Is there ____?
Emañ _____? - Is there ____?
Eo _____? - Is there ____?
Eg.
Ez eus ur c'harr er karrdi?
Ez eus ur gador dirag an daol?
N'emañ ket ur c'hi en ti?
This is the basic format for a positive answer:
Ez eus, emañ _____. - There is!(=Yes), there is ____.
Geo, emañ _____. - There is!(=Yes), there is ____.
Examples :
Ez eus, emañ ur c'harr er karrdi.
Ez eus ur gador dirag an daol.
Geo, emañ ur c'hi en ti.
A negative question N'eus ket ... ? Isn't it? is answered as follows:
the positive answer is Geo, ...
the negative answer is N'eus ket ....
A negative question N'eo ket ... ? Isn't it? is answered as follows:
the positive answer is Geo, ...
the negative answer is N'eo ket ....
In the case of asking of a position. You give the position afterwards.
Eus ar c'har dirag an ti?
N'eus ket, n'eo ket ar c'har dirag an ti. Emañ ar c'harr er karrdi.
Is the car in front of the house? No, the car isn't in front of the house. The car is in the garage.
Note:
When giving answers about yourself (eg. Laouenn out? Are you happy?). The positive answer is the appropriate personal form of the same verb. The negative is preceded by the negative particle Ne, eg.: Laouenn out? = Eo, Laouenn on. or N'on ket, trist on.
Geriadur Vocabulary
Emañ - there is
N'eus ket - there isn't
Eus! - There is! (Yes)(a)
en - in the
N'eo ket ...?
Isn't there...?
Ez eus ...? (used with an indefinite noun)
Is there a ...?
Muioc'h skoueroù/More Examples
Eg. 1.
Ez eus ur karr er karrdi?
Eus. Emañ ur karr er karrdi.
Is there a car in the garage? Yes/there is. There is a car in the garage.
Eg. 2.
Emañ ar c'haz dre an tan?
N'eo ket. N'emañ ket ar c'haz dre an tan. Emañ ar c'haz er kegin.
Is a cat by the fire? No. The cat isn't by the fire. The cat is in the kitchen.
Eg. 3.
Ha n'eus ket un daol en ty ker?
Ez eus. Emañ un daol en ty ker!
Isn't there a table in the town hall?
Yes. There is a table in the town hall.
==Poelladennoù/Exercises==
====Poelladenn unan====
Using vocabulary you have learnt from the previous courses, make at least 10 sentences. Form a negative and positive option for each sentence.
Eg.:
Ez eus ur den er stal?
Eus. Emañ ur den er stal.
Nann. N'eus ket ur den er stal. :
| previous lesson | previous page | next page | lesson intro | next lesson | Department | Division |
# Linear form/Introduction/Section
## Definition

Let $K$ be a field and let $V$ be a $K$-vector space. A linear mapping
    $V\longrightarrow K$
$V$

## Example

A linear form on $K^{n}$ is of the form
    $K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}x_{i},$
for a tuple $\left(a_{1},\,\ldots ,\,a_{n}\right)$. The projections
    $p_{j}\colon K^{n}\longrightarrow K,\left(x_{1},\,\ldots ,\,x_{n}\right)\longmapsto x_{j},$
are the easiest linear forms.
The zero mapping to $K$ is also a linear form, called the zero form.

## Example

Many important examples of linear forms on some vector spaces of infinite dimension arise in analysis. For a real interval $[a,b]$, the set of functions $\operatorname {Map} \,{\left([a,b],\mathbb {R} \right)}$, or the set of continuous functions $C([a,b],\mathbb {R} )$, or the set of continuously differentiable functions $C^{1}([a,b],\mathbb {R} )$ form real vector spaces. For a point $P\in [a,b]$, the evaluation $f\mapsto f(P)$ is a linear form (because addition and scalar multiplication is defined pointwisely on these spaces). Also, the evaluation of the derivative at $P$,
    $C^{1}([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto f'(P),$
is a linear form. For $C([a,b],\mathbb {R} )$, the integral, that is, the mapping
    $C([a,b],\mathbb {R} )\longrightarrow \mathbb {R} ,f\longmapsto \int _{a}^{b}f(t)dt,$
is a linear form. This rests on the linearity of the integral.
The kernel of the zero form is the total space; for any other linear form $f\in \operatorname {Hom} _{K}{\left(V,K\right)}$ with $f\neq 0$, the dimension is $\dim _{K}{\left(V\right)}-1$. This follows from the dimension formula. With the exception of the zero form, a linear form is always surjective.
# Linear algebra (Osnabrück 2024-2025)/Hyperplane/Kernel of a linear form/Fact/Factreferencenumber

Lemma 14.6
# Linear algebra (Osnabrück 2024-2025)/Basis/Dual basis/Tautological lemma/Fact/Factreferencenumber

Lemma 14.13
# Purchase and price tuple/Duality/Motivation/Example

In a shop, there are $n$ different products. A purchase is described by an $n$-tuple (purchase tuple), where the $i$-th entry says how much of the $i$-th product is bought (with respect to a certain unit). The set of all purchases (including refunds) forms an $n$-dimensional vector space. A price list for the products is also described by an $n$-tuple (price tuple), where the $i$-th entry tells the price of the $i$-th product (with respect to the same unit). The set of all price tuples forms also an $n$-dimensional vector space (think about comparing prices, inflation, tax, etc.). It is obviously nonsense to consider a purchase tuple and a price tuple in the same vector space and to add them. To the contrary, the right way to process a purchase tuple $\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)$ and a price tuple $\left(p_{1},\,p_{2},\,\ldots ,\,p_{n}\right)$ is to compute the total price $\sum _{i=1}^{n}p_{i}x_{i}$, which belongs to $\mathbb {R}$. Purchase tuple and price tuple are dual to each other, the vector space of price tuples is dual to the vector space of purchase tuples.
# Linear subspace in dual space/Orthogonal space/Definition

Let $V$ be a $K$-vector space, and $F\subseteq {V}^{*}$ be a linear subspace of the dual space ${V}^{*}$ of $V$. Then
    $F^{\perp }={\left\{v\in V\mid f(v)=0{\text{ for all }}f\in F\right\}}\subseteq V\,$
is called the orthogonal space of $F$.
# Primary Mathematics/Quick Reference/Operator Precedence

This table shows the precedence of primary mathematics operators.
Caption text :
| Operator | Name |
| | Parenthesis |
| Superscript, nth power, ^, ** | Exponentiation |
| ×, / | Multiplication, Division |
| +, - | Addition, Subtraction |
As a simple example, 1+2×3 = 7 not 9 because the multiplication is performed before the addition, superseding the left to right convention. Also, 3 + 5$^2$ = 28 and 3 × 5$^2$ = 75 because of the precedence of exponentiation over multiplication.
The mnemonic PEMDAS can help to remember this. This can be expanded to “Please Excuse My Dear Aunt Sally.”
For a complete discussion, please see the Wikipedia article on order of operations.
# 11-cell

          See also: Wikipedia:11-cell
# LaTeX

LaTeX is a markup language (as is MediaWiki!) for producing mathematical texts of the highest quality. Its use is widespread in the mathematics world. It is built on plain TeX developed by Donald Knuth. You can embed LaTeX markups in MediaWiki by the </math> tags, e.g. <math>e = m c^2</math> is rendered as $e=mc^{2}$.

## Readings

* Wikipedia: LaTeX
* Wikibooks: LaTeX

## Installation

* If you are using Linux, you probably have it already! Just type "latex".
* If you are using Windows, the simplest thing to do is to open Cygwin (the linux emulator) and type "latex". If you haven't installed Cygwin, do it! You will often find it useful. The installation may take less than an hour but it is usually straightforward.
** Note that it is now fairly simple to install a linux dual boot, for example, with wubi. All you need is a number of Gigabytes in a refactored hard-drive. You can delete it like a regular windows directory whenever you don't want it any more.

## Step by step guide for beginners

* Create a file with .tex suffix, e.g. helloworld.tex (see /helloworld.tex).
* Go to your linux/cygwin/ whatever shell, go to the directory of your file helloworld.tex and type latex helloworld.
* A .dvi file will be created - you may view it directly if you have the right tools.
* If not, type dvips helloworld or type dvipdf helloworld to convert it to a postscript or PDF file.
* Use Ghostview to view the .ps file, or the Adobe Acrobat Reader to view the .pdf file.

## Choosing an editor

A good editing environment helps! A useful and free option is the LaTeX editor (LEd) (see their home page).
A LaTeX extension is also available for Open Office.
Or for a quick and easy solution, try Roger's Online Equation Editor.

## Lessons

* helloworld.tex
* Including graphics
* Inserting diagrams
* Two-lined-subscripts
* Serial letter

## Use Latex Source in Wikiversity

You can use latex sources also in Wikiversity. In general you will create a PDF file from your latex source. pdflatex converts the a tex-file hello.tex in hello.pdf. You may need sometimes a different output format. If you want to convert the tex-file into a Wiki markdown output for Wikiversity you can use PanDoc.

### Pandoc

This is useful if you want to use parts from source text in LaTeX also in a Wikiversity learning resource (e.g. for a mathematical text for a lecture). Copy the source in PanDoc and
* select "Latex" as input format and
* select "MediaWiki" as an output format.
When press "convert" you can use the Wiki source code in your learning resource.

### Expand Newcommands

Sometimes it is necessary to expand latex macros because the macros in your source are not available in Wikiversity.
* With the BracketHandler closingbracket.js a newcommand replacement into the source text is possible.
* Use Newcommand Expander as an AppLSAC for replacing the newcommand (GitHub Repository closingbracket.js).

## External Links

* tex by topic - an indepth study of TeX
* closingbracket.js BracketHandler - GitHub demo for newcommand replacement with GitHub-repository 'closingbracket'
# AP Psychology

Advanced Placement examinations (AP exams) are exams offered in United States by the College Board and are taken each May by students. The tests are the culmination of year-long Advanced Placement (AP) courses. AP exams (with few exceptions) have a multiple-choice section and a free-response section.
The AP Psychology exam explores the ideas, theories, and methods of the scientific study of behavior and mental processes.

## Outline

* Introduction
* Scientific Foundations of Psychology
* Biological Bases of Behavior
* Sensation and Perception
* Learning
* Cognitive Psychology
* Developmental Psychology
* Motivation, Emotion, and Personality
* Clinical Psychology
* Social Psychology

## See Also

* Wikipedia: Advanced Placement exams
# Research

Research is defined as human activity based on intellectual application in the investigation of matter. The primary purpose for applied research is discovering, interpreting, and the development of methods and systems for the advancement of human knowledge on a wide variety of scientific matters of our world and the universe. Research can use the scientific method, but need not do so.

## Research resources

* Wikiversity: Research ethics
* Presentations - how to present your research results to others
* High risk research - High risk research can be discussed at that location.
* Business research
* Qualitative research
# Historical introduction to philosophy

Welcome to Historical Introduction to Philosophy.

## Subpages

These are actually subpages of Historical_Introduction_to_Philosophy. See PrefixIndex

### Introduction: What is Philosophy?

* General Introduction
* Philosophical Method
* Presocratics and Socrates

### Philosophy of Religion

* Faith and Reason
* Arguments for God
* The Problem of Evil

### Epistemology

* Epistemology
* Truth, Objectivity, and Relativism
* The Challenge of Skepticism

### The Philosophy of Mind

* The Mind-Body Problem
* Theories of Mind
* Personal Identity

### Free-will and Determinism

* Determinism and the Problem of Free-Will
* Libertarianism
* Compatibilism

### Ethics

* Ethics
* Formalism and Deontological Ethics
* Consequentialism

### Metaphysics

What exists, what are the most general properties of things, is there a god, is life an illusion, what can be said to be real?

### Logic

* Logic

### Other subpages

See List

## Related pages

* Introduction to Philosophy
* Descartes
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