User:Ascii/Definitions (by Meaning 1-800)

Metaconcepts

Logic

Definition:Logic
- Logic is the study of the structure of statements and their truth values, divorced from their conceptual content.
Definition:Symbolic Logic
- Symbolic logic is the study of logic in which the logical form of statements is analyzed by using symbols as tools. Instead of explicit statements, logical formulas are investigated, which are symbolic representations of statements, and compound statements in particular. In symbolic logic, the rules of reasoning and logic are investigated by means of formal systems, which form a good foundation for the symbolic manipulations performed in this field.

Definition:Statement
- A statement is a sentence which has objective and logical meaning.
Definition:Proposition
- A proposition is a statement which is offered up for investigation as to its truth or falsehood. Loosely, a proposition is a statement which is about to be proved (or disproved).
Definition:True
- A statement has a truth value of true if and only if what it says matches the way that things are.
Definition:False
- A statement has a truth value of false if and only if what it says does not match the way that things are.
Definition:Truth Value
- In Aristotelian logic, a statement can be either true or false, and there is no undefined, in-between value. Whether it is true or false is called its truth value. Note that a statement's truth value may change depending on circumstances.
Definition:Proof System
- Let $\LL$ be a formal language. A proof system $\mathscr P$ for $\LL$ comprises:
  - Axioms and/or axiom schemata;
  - Rules of inference for deriving theorems.
- It is usual that a proof system does this by declaring certain arguments concerning $\LL$ to be valid. Informally, a proof system amounts to a precise account of what constitutes a (formal) proof.
Definition:Axiom
- In all contexts, the definition of the term axiom is by and large the same. That is, an axiom is a statement which is accepted as being true. A statement that is considered an axiom can be described as being axiomatic.
Definition:Assumption
- An assumption is a statement which is introduced into an argument, whose truth value is (temporarily) accepted as True. In mathematics, the keyword let is often the indicator here that an assumption is going to be introduced. For example: Let $p$ (be true) ... can be interpreted, in natural language, as: Let us assume, for the sake of argument, that $p$ is true ...
Definition:Premise
- A premise is an assumption that is used as a basis from which to start to construct an argument. When the validity or otherwise of a proof is called into question, one may request the arguer to "check your premises".

Definition:Valid Argument
- A valid argument is a logical argument in which the premises provide conclusive reasons for the conclusion.
Definition:Sequent
- A sequent is an expression in the form: $\phi_1, \phi_2, \ldots, \phi_n \vdash \psi$ where $\phi_1, \phi_2, \ldots, \phi_n$ are premises (any number of them), and $\psi$ the conclusion (only one), of an argument.
Definition:Logical Language
- Let $\LL$ be a formal language used in symbolic logic. Then $\LL$ is called a logical language.
Definition:Logical Formula
- Let $\LL$ be a formal language used in the field of symbolic logic. Then the well-formed formulas of $\LL$ are often referred to as logical formulas. They are symbolic representations of statements, and often of compound statements in particular.
Definition:Parenthesis
- Parenthesis is a syntactical technique to disambiguate the meaning of a logical formula. It allows one to specify that a logical formula should (temporarily) be regarded as being a single entity, being on the same level as a statement variable. Such a formula is referred to as being in parenthesis. Typically, a formal language, in defining its formal grammar, ensures by means of parenthesis that all of its well-formed words are uniquely readable. Generally, brackets are used to indicate that certain formulas are in parenthesis. The brackets that are mostly used are round ones, the left (round) bracket $($ and the right (round) bracket $)$.

Definition:Propositional Logic
- Propositional logic is a sub-branch of symbolic logic in which the truth values of propositional formulas are investigated and analysed. The atoms of propositional logic are simple statements. There are various systems of propositional logic for determining the truth values of propositional formulas, for example:
  - Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, each of which themselves are either "self-evident" axioms or themselves derived from other valid sequents.
  - The Method of Truth Tables, which consists of the construction of one or more truth tables which exhaustively list all the possible truth values of all the statement variables with a view to determining the required answer by inspection.
Definition:Language of Propositional Logic
- There are a lot of different formal systems expressing propositional logic. Although they vary wildly in complexity and even disagree (to some extent) on what expressions are valid, generally all of these use a compatible formal language.

Definition:RHS

Set Theory

Definition:Set Theory

Definition:Strictly Precede

Definition:Minimally Inductive Set

Definition:Cardinal

Definition:Ordered Subset

Mappings

Definition:Ordinal

Number Systems

Definition:Quaternion

Definition:Standard Number Field

Definition:Usual Ordering

Number Theory

Definition:Apotome

Definition:Convergent of Continued Fraction

Abstract Algebra

Definition:Abstract Algebra

Topology

Metric Spaces

Definition:Pseudometric
- A pseudometric on a set $A$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions:
  - $(\text M 1) : \forall x \in A : d \left({x, x}\right) = 0$
  - $(\text M 2) : \forall x, y, z \in A : d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$
  - $(\text M 3) : \forall x, y \in A : d \left({x, y}\right) = d \left({y, x}\right)$
Definition:Pseudometric/Pseudometric Space
- A pseudometric space $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \O$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.
Definition:Metric Space
Axiom:Metric Space Axioms
Definition:Metric Space/Distance Function
Definition:Metric Subspace

Topology

Definition:Interior (Topology)
Definition:Basis (Topology)
Definition:Basis (Topology)/Analytic Basis
Definition:Local Basis, open neighbourhood

countable basis

Definition:First-Countable Space
Definition:Second-Countable Space, countable basis

Definition:Neighborhood (Topology)/Point
Definition:Open Neighborhood, neighbourhood, open set
Definition:Limit Point/Topology/Set, open neighbourhood
Definition:Closure (Topology), derived set (/closed set)
Definition:Boundary (Topology), interior, closure
Definition:Everywhere Dense, closure
Definition:Separable Space, everywhere dense
Definition:Connected (Topology)/Topological Space, separation
Definition:Connected Set (Topology), separated sets
Definition:Component (Topology)
Definition:Dense-in-itself

Definition:Product Space (Topology)

Definition:Topological Group

Definition:Borel Sigma-Algebra

Analysis

Definition:Extended Real Number Line

Definition:Cauchy Sequence

Definition:Gamma Function

Definition:Laplace Transform

Definition:Hilbert Space

To be Categorised

Geometry

Definition:Geometry

Definition:Dimension (Geometry)

Graph Theory

Category Theory

Definition:Metacategory

Probability

Definition:Sigma-Algebra
- Let $X$ be a set. A $\sigma$-algebra $\RR$ over $X$ is a system of subsets of $X$ with the following properties:
  - $(SA 1)$ Unit: $X \in \RR$
  - $(SA 2)$ Closure under Complement: $\forall A \in \RR : \complement_X \left({A}\right) \in \RR$
  - $(SA 3)$ Closure under Countable Unions: $\forall A_n \in \RR: n = 1, 2, \ldots : \bigcup_{n \mathop = 1}^\infty A_n \in \RR$
Definition:Sigma-Algebra Generated by Collection of Subsets
- Let $X$ be a set. Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$. The $\sigma$-algebra generated by $\GG$, $\map \sigma \GG$, is the smallest $\sigma$-algebra on $X$ that contains $\GG$.

Definition:Measurable Space
- Let $\Sigma$ be a $\sigma$-algebra on a set $X$. Then the pair $\left({X, \Sigma}\right)$ is called a measurable space.
Definition:Measure (Measure Theory)
- Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\mu: \Sigma \to \overline{\R}$ be a mapping, where $\overline{\R}$ denotes the set of extended real numbers. Then $\mu$ is called a measure on $\Sigma$ iff it has the following properties:
  - $(1): \quad$ For every $E \in \Sigma$: $\mu \left({E}\right) \ge 0$
  - $(2): \quad$ For every sequence of pairwise disjoint sets $\set {S_n} \subseteq \Sigma$: $\ds \map \mu {\bigcup_{n \mathop = 1}^\infty S_n} = \sum_{n \mathop = 1}^\infty \map \mu {S_n}$ (that is, $\mu$ is a countably additive function).
  - $(3): \quad$ There exists at least one $E \in \Sigma$ such that $\mu \left({E}\right)$ is finite.
Definition:Measurable Set
- Let $\left({X, \Sigma}\right)$ be a measurable space. A subset $S \subseteq X$ is said to be ($\Sigma$-)measurable if and only if $S \in \Sigma$.
Definition:Measurable Function
- Let $\left({X, \Sigma}\right)$ be a measurable space. Let $E \in \Sigma$. Then a function $f: E \to \R$ is said to be $\Sigma$-measurable on $E$ iff: $\forall \alpha \in \R: \left\{{x \in E : f \left({x}\right) \le \alpha}\right\} \in \Sigma$
Definition:Measure Space
- A measure space is a triple $\left({X, \Sigma, \mu}\right)$ where:
  - $X$ is a set
  - $\Sigma$ is a $\sigma$-algebra on $X$
  - $\mu$ is a measure on $\Sigma$.

Definition:Probability Space
- A probability space is a measure space $\left({\Omega, \Sigma, \Pr}\right)$ in which $\Pr \left({\Omega}\right) = 1$. A probability space is used to define the parameters determining the outcome of an experiment $\EE$. In this context, the elements of a probability space are generally referred to as follows:
  - $\Omega$ is called the sample space of $\EE$
  - $\Sigma$ is called the event space of $\EE$
  - $\Pr$ is called the probability measure on $\EE$.
Definition:Experiment
Definition:Event Space
Definition:Event
Definition:Probability

User:Ascii/Definitions (by Meaning 1-800)

Contents

Metaconcepts

Logic

Set Theory

Mappings

Number Systems

Number Theory

Abstract Algebra

Topology

Metric Spaces

Topology

Analysis

To be Categorised

Geometry

Graph Theory

Category Theory

Probability

Physics

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User:Ascii/Definitions (by Meaning 1-800)

Metaconcepts

Logic

Set Theory

Mappings

Number Systems

Number Theory

Abstract Algebra

Topology

Metric Spaces

Topology

Analysis

To be Categorised

Geometry

Graph Theory

Category Theory

Probability

Physics

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