Definition:Affirmative Categorical Statement

An '''affirmative categorical statement''' is either one of: === Universal Affirmative === {{:Definition:Universal Affirmative}} === Particular Affirmative === {{:Definition:Particular Affirmative}}


Definition:Negative Categorical Statement

A '''negative categorical statement''' is either one of: === Universal Negative === {{:Definition:Universal Negative}} === Particular Negative === {{:Definition:Particular Negative}} Category:Definitions/Categorical Statements i4tlzj2k8juuoz2pt0zforyz6b8lyw6


Definition:Reduction to First Figure

'''Reduction to the first figure''' is a method for determining the validity of a categorical syllogism, as follows: :$(1): \quad$ Certain "self-evident" patterns are identified in the first figure of the categorical syllogism. :$(2): \quad$ Using various rules of categorical statements, the valid patterns of other figures of the categorical syllogism are deduced. There are two forms this procedure takes: === Direct Reduction === {{:Definition:Reduction to First Figure/Direct}} === Indirect Reduction === {{:Definition:Reduction to First Figure/Indirect}}


Definition:Reduction to First Figure/Indirect

'''Indirect reduction to the first figure''' is a form of the reduction to the first figure method for determining the validity of a categorical syllogism. This form of the method works backward by supposing that a pattern is not valid, and then deriving a contradiction against a known valid pattern in the first figure. Hence by Reductio ad Absurdum the pattern is deduced to be valid.


Definition:Square of Opposition/Categorical Statements/Vacuous Terms

The traditional treatment of the categorical syllogism makes the assumption that no term is vacuous. However, from the point of view of the full predicate logic, this assumption may not be valid. Note that if $S$ is empty, then the square of opposition no longer holds. Although ''All $S$ are $P$'' is vacuously true for such an empty universe, ''Some $S$ are $P$'' is not. Thus ''Some $S$ are $P$'' is no longer subimplicant to ''All $S$ are $P$''. Similarly, as ''Some $S$ are not $P$'' is also false, it follows that ''All $S$ are $P$'' and ''Some $S$ are not $P$'' are no longer subcontrary.


Definition:Categorical Syllogism/Shorthand

In order to specify the pattern of a categorical syllogism completely, it is necessary and sufficient to specify: :$(1) \quad$ The figure ($\text I$ to $\text {IV}$) of the syllogism and :$(2) \quad$ The types of the three categorical statements that compose the syllogism. Hence, for example, the following categorical syllogism, which is of the first figure: :$\begin{array}{r|rl} \text I & & \\ \hline \\ \text{Major Premise}: & \mathbf E & \left({M, P}\right) \\ \text{Minor Premise}: & \mathbf I & \left({S, M}\right) \\ \hline \\ \text{Conclusion}: & \mathbf O & \left({S, P}\right) \\ \end{array}$ is specified completely by: :$\text I: EIO$


Definition:Categorical Statement/Abbreviation

A '''categorical statement''' connecting $S$ and $P$ can be abbreviated as: :$\mathbf{\Phi} \left({S, P}\right)$ where $\Phi$ is one of either $\mathbf{A}$, $\mathbf{E}$, $\mathbf{I}$ or $\mathbf{O}$, signifying Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative respectively. Thus: :$\mathbf{A} \left({S, P}\right)$ denotes ''All $S$ are $P$'' :$\mathbf{E} \left({S, P}\right)$ denotes ''No $S$ are $P$'' :$\mathbf{I} \left({S, P}\right)$ denotes ''Some $S$ are $P$'' :$\mathbf{O} \left({S, P}\right)$ denotes ''Some $S$ are not $P$''.


Definition:Categorical Statement/Subject

The symbol $S$ can be referred to as the '''subject''' of $\map {\mathbf {\Phi} } {S, P}$.


Definition:Categorical Statement/Predicate

The symbol $P$ can be referred to as the '''predicate''' of $\map {\mathbf {\Phi} } {S, P}$.


Definition:Universal Categorical Statement

A '''universal categorical statement''' is either one of: === Universal Affirmative === {{:Definition:Universal Affirmative}} === Universal Negative === {{:Definition:Universal Negative}} Category:Definitions/Categorical Statements kd494xo7vtho2h5zy959svqc60zum76


Definition:Particular Categorical Statement

A '''particular categorical statement''' is either one of: === Particular Affirmative === {{:Definition:Particular Affirmative}} === Particular Negative === {{:Definition:Particular Negative}} Category:Definitions/Categorical Statements 9n19giae1i2ax8o8kalelftnxe1gifw


Definition:Distributed Term of Categorical Syllogism/Subject

Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be a universal categorical statement. Then $S$ is described as being '''distributed'''.


A term in a syllogism is '''distributed''' if the categorical statement in which it occurs refers to the whole of the class designated by the term. === Distributed Subject === {{:Definition:Distributed Term of Categorical Syllogism/Subject}} === Distributed Predicate === {{:Definition:Distributed Term of Categorical Syllogism/Predicate}}


Definition:Distributed Term of Categorical Syllogism/Predicate

Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be a negative categorical statement. Then $S$ is described as being '''distributed'''.


A term in a syllogism is '''undistributed''' if the categorical statement in which it occurs does not refer to the whole of the class designated by the term. === Undistributed Subject === {{:Definition:Undistributed Term of Categorical Syllogism/Subject}} === Undistributed Predicate === {{:Definition:Undistributed Term of Categorical Syllogism/Predicate}}


Definition:Undistributed Term of Categorical Syllogism/Predicate

Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be an affirmative categorical statement. Then $P$ is described as being '''undistributed'''.


Definition:Undistributed Term of Categorical Syllogism/Subject

Let $\mathbf{\Phi} \left({S, P}\right)$ be a categorical statement, expressed in abbreviated form. Let $\mathbf{\Phi}$ be a particular categorical statement. Then $S$ is described as being '''undistributed'''.


=== Universal Affirmative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Universal Affirmative}} === Universal Negative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Universal Negative}} === Particular Affirmative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Particular Affirmative}} === Particular Negative === {{:Definition:Distributed Term of Categorical Syllogism/Examples/Particular Negative}} This can be tabulated as follows: :$\begin{array}{rcl} \mathbf{A} & (S, & P) \\ & d & u \end{array} \qquad \begin{array}{rcl} \mathbf{E} & (S, & P) \\ & d & d \end{array}$ :$\begin{array}{rcl} \mathbf{I} & (S, & P) \\ & u & u \end{array} \qquad \begin{array}{rcl} \mathbf{O} & (S, & P) \\ & u & d \end{array}$ where $d$ denotes a distributed term and $u$ denotes an undistributed term.


Definition:Non-Trivial Subgroup

Let $\struct {G, \circ}$ be a group whose identity is $e$. A '''non-trivial subgroup''' of $\struct {G, \circ}$ is a subgroup $\struct {H, \circ}$ of $G$ such that $H \ne \set e$.


Definition:Non-Trivial Element

Let $\struct {G, \circ}$ be a group whose identity is $e$. A '''non-trivial element''' of $G$ is an element of $G$ distinct from $e$.


Definition:Commutative/Set

Let $\struct {S, \circ}$ be an algebraic structure. Let $X \subseteq S$ be a subset of $S$ such that: :$\forall a, b \in X: a \circ b = b \circ a$ That is, every element of $X$ commutes with every other element. Then $X$ is a '''commuting set of elements''' of $S$.


Definition:Proper Subgroup/Non-Trivial

Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$ such that $\set e \subset H \subset G$, that is: :$H \ne \set e$ :$H \ne G$ Then $\struct {H, \circ}$ is a '''non-trivial proper subgroup of $\struct {G, \circ}$'''.


Definition:Group Type

An equivalence class under group isomorphism is a '''(group) type'''.


Definition:Real Function/Definition by Formula

A function $f: S \to T$ can be considered as a '''formula''' which tells us how to determine what the value of $y \in T$ is when we have selected a value for $x \in S$. === As an Equation === {{:Definition:Real Function/Definition by Equation}}


Definition:Real Function/Definition by Equation

It is often convenient to refer to an equation or formula as though it were a function. What is meant is that the equation ''defines'' the function; that is, it specifies the rule by which we obtain the value of $y$ from a given $x$. === Square Function === For example, let $x, y \in \R$. {{:Definition:Real Square Function|Square Function}} We may express this as $y = x^2$, and use this equation to ''define'' this function. This may be conceived as: :For each $x \in \R$, the number $y \in \R$ assigned to it is that which we get by squaring $x$. Another useful notation is: :$\forall x \in \R: x \mapsto x^2$


Definition:Nu Function

The '''$\nu$ function''' is the function $\nu: \Z_{>0} \to \Z_{>0}$ is defined as: :$\forall n \in \Z_{>0}: \map \nu n = $ the number of types of group of order $n$


Definition:Ring of Sets/Definition 1

A '''ring of sets''' $\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {RS} 1_1         | lc= Non-Empty:         | m = \RR \ne \O }} {{axiom | n = \text {RS} 2_1         | lc= Closure under Intersection:         | q = \forall A, B \in \RR         | m = A \cap B \in \RR }} {{axiom | n = \text {RS} 3_1         | lc= Closure under Symmetric Difference:         | q = \forall A, B \in \RR         | m = A * B \in \RR }} {{end-axiom}}


Definition:Ring of Sets/Definition 2

A '''ring of sets''' $\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {RS} 1_2         | lc= Empty Set:         | m = \O \in \RR }} {{axiom | n = \text {RS} 2_2         | lc= Closure under Set Difference:         | q = \forall A, B \in \RR         | m = A \setminus B \in \RR }} {{axiom | n = \text {RS} 3_2         | lc= Closure under Union:         | q = \forall A, B \in \RR         | m = A \cup B \in \RR }} {{end-axiom}}


Definition:Limit Inferior of Sequence of Sets/Definition 1

Let $\sequence {E_n : n \in \N}$ be a sequence of sets. Then the '''limit inferior''' of $\sequence {E_n : n \in \N}$, denoted $\ds \liminf_{n \mathop \to \infty} E_n$, is defined as: {{begin-eqn}} {{eqn | l = \liminf_{n \mathop \to \infty} E_n       | o = :=       | r = \bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty E_n }} {{eqn | r = \paren {E_0 \cap E_1 \cap E_2 \cap \ldots} \cup \paren {E_1 \cap E_2 \cap E_3 \cap \ldots} \cup \cdots }} {{end-eqn}}


Definition:Limit Superior of Sequence of Sets/Definition 1

Let $\sequence {E_n : n \in \N}$ be a sequence of sets. Then the '''limit superior''' of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as: {{begin-eqn}} {{eqn | l = \limsup_{n \mathop \to \infty} E_n       | o = :=       | r = \bigcap_{i \mathop = 0}^\infty \bigcup_{n \mathop = i}^\infty E_n }} {{eqn | r = \paren {E_0 \cup E_1 \cup E_2 \cup \ldots} \cap \paren {E_1 \cup E_2 \cup E_3 \cup \ldots} \cap \ldots }} {{end-eqn}}


Definition:Limit Superior of Sequence of Sets/Definition 2

Let $\sequence {E_n: n \in \N}$ be a sequence of sets. Then the '''limit superior''' of $\sequence {E_n: n \in \N}$, denoted $\ds \limsup_{n \mathop \to \infty} E_n$, is defined as: :$\ds \limsup_{n \mathop \to \infty} E_n = \set {x : x \in E_i \text { for infinitely many } i}$


Definition:Limit Inferior of Sequence of Sets/Definition 2

Let $\sequence {E_n : n \in \N}$ be a sequence of sets. Then the '''limit inferior''' of $\sequence {E_n : n \in \N}$, denoted $\ds \liminf_{n \mathop \to \infty} E_n$, is defined as: :$\ds \liminf_{n \mathop \to \infty} E_n := \set {x: x \in E_i \text { for all but finitely many } i}$


Definition:Ring of Sets/Definition 3

A '''ring of sets''' $\RR$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {RS} 1_3         | lc= Empty Set:         | m = \varnothing \in \RR }} {{axiom | n = \text {RS} 2_3         | lc= Closure under Set Difference:         | q = \forall A, B \in \RR         | m = A \setminus B \in \RR }} {{axiom | n = \text {RS} 3_3         | lc= Closure under Disjoint Union:         | q = \forall A, B \in \RR         | m = A \cap B = \O \implies A \cup B \in \RR }} {{end-axiom}}


Definition:Sigma-Ring/Definition 1

A '''$\sigma$-ring''' is a ring of sets which is closed under countable unions. That is, a ring of sets $\Sigma$ is a '''$\sigma$-ring''' {{iff}}: :$\ds A_1, A_2, \ldots \in \Sigma \implies \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma$


Definition:Sigma-Ring/Definition 2

A '''$\sigma$-ring''' $\Sigma$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {SR} 1         | lc= Empty Set:         | m = \O \in \Sigma }} {{axiom | n = \text {SR} 2         | lc= Closure under Set Difference:         | q = \forall A, B \in \Sigma         | m = A \setminus B \in \Sigma }} {{axiom | n = \text {SR} 3         | lc= Closure under Countable Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}


Definition:Formal Semantics

A '''formal semantics''' for $\mathcal L$ comprises: * A collection of objects called '''structures'''; * A notion of '''validity''' of $\mathcal L$-WFFs in these structures. Often, a '''formal semantics''' provides these by using a lot of auxiliary definitions.


Definition:Sigma-Algebra/Definition 4

Let $X$ be a set. A '''$\sigma$-algebra''' $\Sigma$ over $X$ is an algebra of sets which is closed under countable unions.


Definition:Sigma-Algebra/Definition 1

Let $X$ be a set. A '''$\sigma$-algebra''' $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties: {{begin-axiom}} {{axiom | n = \text {SA} 1         | lc= Unit:         | m = X \in \Sigma }} {{axiom | n = \text {SA} 2         | lc= Closure under Complement:         | q = \forall A \in \Sigma         | m = \relcomp X A \in \Sigma }} {{axiom | n = \text {SA} 3         | lc= Closure under Countable Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}


Definition:Logical Language

Let $\mathcal L$ be a formal language used in symbolic logic. Then $\mathcal L$ is called a '''logical language'''.


Definition:Tautology

A '''tautology''' is a statement which is ''always true'', independently of any relevant circumstances that could theoretically influence its truth value. It is epitomised by the form: :$p \implies p$ that is: :'''if $p$ is true then $p$ is true.''' An example of a "relevant circumstance" here is the truth value of $p$. The archetypal '''tautology''' is symbolised by $\top$, and referred to as Top.


Definition:Formal Semantics/Valid

Part of specifying a formal semantics $\mathscr M$ for $\LL$ is to define a notion of '''validity'''. Concretely, a precise meaning needs to be assigned to the phrase: :"The $\LL$-WFF $\phi$ is '''valid''' in the $\mathscr M$-structure $\MM$." It can be expressed symbolically as: :$\MM \models_{\mathscr M} \phi$


Definition:Formal Semantics/Structure

Part of specifying a formal semantics $\mathscr M$ for $\LL$ is to specify '''structures''' $\MM$ for $\mathscr M$. A '''structure''' can in principle be any object one can think of. However, to get a useful formal semantics, the '''structures''' should support a meaningful definition of validity for the WFFs of $\LL$. It is common that '''structures''' are sets, often endowed with a number of relations or functions.


Definition:Model (Logic)

Let $\mathscr M$ be a formal semantics for a logical language $\LL$. Let $\MM$ be a structure of $\mathscr M$. === Model of Logical Formula === {{:Definition:Model (Logic)/Logical Formula}} === Model of Set of Logical Formulas === {{:Definition:Model (Logic)/Set of Logical Formulas}}


Definition:Sigma-Algebra/Definition 3

A '''$\sigma$-algebra''' $\Sigma$ is a $\sigma$-ring with a unit.


Definition:Boolean Interpretation/Formal Semantics

The boolean interpretations for $\mathcal L_0$ can be interpreted as a formal semantics for $\mathcal L_0$, which we denote by $\mathrm{BI}$. The structures of $\mathrm{BI}$ are the boolean interpretations. A WFF $\phi$ is declared ($\mathrm{BI}$-)valid in a boolean interpretation $v$ iff: :$v \left({\phi}\right) = T$ Symbolically, this can be expressed as: :$v \models_{\mathrm{BI}} \phi$


Definition:Model (Logic)/Logical Formula

Let $\phi$ be a logical formula of $\LL$. Then $\MM$ is a '''model of $\phi$''' {{iff}}: :$\MM \models_{\mathscr M} \phi$ that is, if $\phi$ is valid in $\MM$.


Definition:Model (Logic)/Set of Logical Formulas

Let $\FF$ be a set of logical formulas of $\mathcal L$. Then $\MM$ is a '''model of $\FF$''' {{iff}}: :$\MM \models_{\mathscr M} \phi$ for every $\phi \in \FF$ that is, if it is a model of every logical formula $\phi \in \FF$.


Definition:Sound Proof System

Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$, and let $\mathscr M$ be a formal semantics for $\LL$. Then $\mathscr P$ is said to be '''sound for $\mathscr M$''' {{iff}}: :Every $\mathscr P$-theorem is an $\mathscr M$-tautology. Symbolically, this can be expressed as the statement that, for every logical formula $\phi$ of $\LL$: :$\vdash_{\mathscr P} \phi$ implies $\models_{\mathscr M} \phi$ === Strongly Sound Proof System === {{:Definition:Sound Proof System/Strongly Sound}}


Definition:Complete Proof System

Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$, and let $\mathscr M$ be a formal semantics for $\LL$. Then $\mathscr P$ is said to be '''complete for $\mathscr M$''' {{iff}}: :Every $\mathscr M$-tautology is a $\mathscr P$-theorem. Symbolically, this can be expressed as the statement that, for every logical formula $\phi$ of $\LL$: :$\models_{\mathscr M} \phi$ implies $\vdash_{\mathscr P} \phi$ === Strongly Complete Proof System === {{:Definition:Complete Proof System/Strongly Complete}}


Definition:Unsatisfiable/Formula

A logical formula $\phi$ of $\LL$ is '''unsatisfiable for $\mathscr M$''' {{iff}}: :$\phi$ is valid in none of the structures of $\mathscr M$ That is, for all structures $\MM$ of $\mathscr M$: :$\MM \not\models_{\mathscr M} \phi$


Definition:Sigma-Ring/Definition 3

A '''$\sigma$-ring''' $\Sigma$ is a system of sets with the following properties: {{begin-axiom}} {{axiom | n = \text {SR} 1'         | lc= Empty Set:         | m = \O \in \Sigma }} {{axiom | n = \text {SR} 2'         | lc= Closure under Set Difference:         | q = \forall A, B \in \Sigma         | m = A \setminus B \in \Sigma }} {{axiom | n = \text {SR} 3'         | lc= Closure under Countable Disjoint Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}


Definition:Sigma-Algebra/Definition 2

Let $X$ be a set. A '''$\sigma$-algebra''' $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties: {{begin-axiom}} {{axiom | n = \text {SA} 1'         | lc= Unit:         | m = X \in \Sigma }} {{axiom | n = \text {SA} 2'         | lc= Closure under Complement:         | q = \forall A \in \Sigma         | m = \relcomp X A \in \Sigma }} {{axiom | n = \text {SA} 3'         | lc= Closure under Countable Disjoint Unions:         | q = \forall A_n \in \Sigma: n = 1, 2, \ldots         | m = \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma }} {{end-axiom}}


Definition:Satisfiable/Formula

A logical formula $\phi$ of $\mathcal L$ is '''satisfiable for $\mathscr M$''' iff: :$\phi$ is valid in some structure $\mathcal M$ of $\mathscr M$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \models_{\mathscr M} \phi$


Definition:Falsifiable/Formula

A logical formula $\phi$ of $\mathcal L$ is '''falsifiable for $\mathscr M$''' iff: :$\phi$ is not valid in some structure $\mathcal M$ of $\mathscr M$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \not\models_{\mathscr M} \phi$


Definition:Satisfiable

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. === Satisfiable Formula === {{:Definition:Satisfiable/Formula}} === Satisfiable Set of Formulas === {{:Definition:Satisfiable/Set of Formulas}} === Satisfiable for Boolean Interpretations === {{:Definition:Satisfiable/Boolean Interpretations}}


Definition:Satisfiable/Set of Formulas

A collection $\mathcal F$ of logical formulas of $\mathcal L$ is '''satisfiable for $\mathscr M$''' iff: :There is some $\mathscr M$-model $\mathcal M$ of $\mathcal F$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \models_{\mathscr M} \mathcal F$


Definition:Falsifiable

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. === Falsifiable Formula === {{:Definition:Falsifiable/Formula}} === Falsifiable Set of Formulas === {{:Definition:Falsifiable/Set of Formulas}} === Falsifiable for Boolean Interpretations === {{:Definition:Falsifiable/Boolean Interpretations}}


Definition:Falsifiable/Set of Formulas

A collection $\mathcal F$ of logical formulas of $\mathcal L$ is '''falsifiable for $\mathscr M$''' iff: :Some $\mathscr M$-structure $\mathcal M$ is not a model of $\mathcal F$ That is, there exists some structure $\mathcal M$ of $\mathscr M$ such that: :$\mathcal M \not\models_{\mathscr M} \mathcal F$


Definition:Unsatisfiable/Set of Formulas

A collection $\FF$ of logical formulas of $\LL$ is '''unsatisfiable for $\mathscr M$''' {{iff}}: :There is no $\mathscr M$-model $\MM$ for $\FF$ That is, for all structures $\MM$ of $\mathscr M$: :$\MM \not \models_{\mathscr M} \FF$


Definition:Rooted Tree/Child Node/Grandchild Node

A child of a child node of a node $t$ can be referred to as a '''grandchild node''' of $t$. In terms of the parent mapping $\pi$ of $T$, a '''grandchild node''' of $t$ is a node $s$ such that: :$\pi \left({\pi \left({s}\right)}\right) = t$


Definition:Propositional Tableau/Graphical Representation

The conditions by which a propositional tableau may be identified, respectively constructed, can be graphically represented like so: ::{| style="text-align: center" |- | $\begin{xy}\xymatrix @R=1em @C=3em{  \ar@{.}[d] & \ar@{.}[d] & \ar@{.}[d] & \ar@{.}[d] \\  *++{\neg \neg \mathbf A} \ar@{.}[d] &  *++{\mathbf A \land \mathbf B} \ar@{.}[d] &  *++{\neg \left({\mathbf A \lor \mathbf B}\right)} \ar@{.}[d] &  *++{\neg \left({\mathbf A \implies \mathbf B}\right)} \ar@{.}[d] \\  *++{t} \ar@{-}[d] &  *++{t} \ar@{-}[d] &  *++{t} \ar@{-}[d] &  *++{t} \ar@{-}[d] \\  \mathbf A &  \mathbf A \ar@{=}[d] &  \neg \mathbf A \ar@{=}[d] &  \mathbf A \ar@{=}[d] \\   & \mathbf B & \neg \mathbf B & \neg \mathbf B \\  \boxed{\neg\neg} & \boxed \land & \boxed{\neg\lor} & \boxed{\neg\implies} }\end{xy}$ |- | $\begin{xy}\xymatrix @R=1em @C=1em{  & \ar@{.}[d] & & & & & & \ar@{.}[d] & & & & & & \ar@{.}[d] \\  & *++{\mathbf A \lor \mathbf B} \ar@{.}[d] &  & & & &  & *++{\neg \left({\mathbf A \land \mathbf B}\right)} \ar@{.}[d] & & & & &  & *++{\mathbf A \implies \mathbf B} \ar@{.}[d] \\  & *++{t} \ar@{-}[dl] \ar@{-}[dr] & & & & &  & *++{t} \ar@{-}[dl] \ar@{-}[dr] & & & & &  & *++{t} \ar@{-}[dl] \ar@{-}[dr] \\  \mathbf A & & \mathbf B & & & &  \neg \mathbf A & & \neg \mathbf B & & & &  \neg \mathbf A & & \mathbf B \\  & \boxed{\lor} & & & & & & \boxed{\neg \land} & & & & & & \boxed \implies  }\end{xy}$ |- | $\begin{xy}\xymatrix @R=1em @C=0.7em{  & \ar@{.}[d] & & & & & & \ar@{.}[d] \\  & *++{\mathbf A \iff \mathbf B} \ar@{.}[d] &  & & & &  & *++{\neg \left({\mathbf A \iff \mathbf B}\right)} \ar@{.}[d] \\  & *++{t} \ar@{-}[dl] \ar@{-}[dr] & & & & &  & *++{t} \ar@{-}[dl] \ar@{-}[dr] \\  *++{\mathbf A \land \mathbf B} & & *++{\neg \mathbf A \land \neg \mathbf B} & & & &  *++{\mathbf A \land \neg \mathbf B} & & *++{\neg \mathbf A \land \mathbf B} \\  & \boxed{\iff} & & & & & & \boxed{\neg \iff} }\end{xy}$ |}


Definition:Projective Geometry

'''Projective Geometry''' is the field of geometry concerned with properties which are invariant under projective transformations.


Definition:Analytic Geometry/Plane

'''Plane analytic geometry''' is the branch of analytic geometry that makes an algebraic study of the real number plane $\R^2$.


Definition:Analytic Geometry/Solid

'''Solid analytic geometry''' is the branch of analytic geometry that makes an algebraic study of the  real vector space $\R^3$.


Definition:Collinear Points

Three or more points which belong to the same line are '''collinear'''.


Definition:Concurrent Lines

A set of $3$ or more (straight) lines through a single point are '''concurrent'''.


Definition:Coplanar Points

Four or more points which belong to the same plane are '''coplanar'''.


Definition:Collinear Planes

A set of planes which share the same (straight) line are '''collinear'''.


Definition:Contradictory Branch

Let $T$ be a labeled tree for propositional logic. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is a '''contradictory branch''' iff, for some WFF of propositional logic $\mathbf A$, both $\mathbf A$ and $\neg \mathbf A$ occur along $\Gamma$.


Definition:Tableau Confutation

Let $\mathbf H$ be a set of WFFs of propositional logic. A '''tableau confutation''' of $\mathbf H$ is a propositional tableau $T$ with root $\mathbf H$ such that every branch of $T$ is contradictory.


Let $\mathbf H$ be a set of WFFs of propositional logic. Let $\mathbf A$ be a WFF. A '''tableau proof''' of $\mathbf A$ from $\mathbf H$ is a tableau confutation of $\mathbf H \cup \set {\neg \mathbf A}$. This definition also applies when $\mathbf H = \varnothing$. Then a '''tableau proof''' of $\mathbf A$ is a tableau confutation of $\set {\neg \mathbf A}$. If there exists a '''tableau proof''' of $\mathbf A$ from $\mathbf H$, one can write: :$\mathbf H \vdash_{\mathrm{PT} } \mathbf A$ Specifically, the notation: :$\vdash_{\mathrm{PT} } \mathbf A$ means that there exists a '''tableau proof''' of $\mathbf A$. === Proof System === {{:Definition:Tableau Proof (Propositional Tableaus)/Proof System}}


Definition:Consistent (Logic)

Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$. === Proof System === {{:Definition:Consistent (Logic)/Proof System}} === Set of Formulas === {{:Definition:Consistent (Logic)/Set of Formulas}}


Definition:Provable Consequence

Let $\mathscr P$ be a proof system for a formal language $\mathcal L$. Let $\mathcal F$ be a collection of WFFs of $\mathcal L$. Denote with $\mathscr P \left({\mathcal F}\right)$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\mathcal F$ as axioms. Let $\phi$ be a theorem of $\mathscr P \left({\mathcal F}\right)$. Then $\phi$ is called a '''provable consequence''' of $\mathcal F$, and this is denoted as: :$\mathcal F \vdash_{\mathscr P} \phi$ Note in particular that for $\mathcal F = \varnothing$, this notation agrees with the notation for a $\mathscr P$-theorem: :$\vdash_{\mathscr P} \phi$


Definition:Monotone Sequence of Sets

Let $X$ be a set. Let $\mathcal S \subseteq \powerset X$ be a collection of subsets of $X$. A '''monotone sequence of sets (in $\mathcal S$)''' is a sequence $\sequence {A_n}_{n \mathop \in \N}$ in $\mathcal S$, such that either: :$\forall n \in \N: A_n \subseteq A_{n + 1}$ or: :$\forall n \in \N: A_n \supseteq A_{n + 1}$ That is, such that $\sequence  {A_n}_{n \mathop \in \N}$ is either increasing or decreasing


Definition:Inconsistent (Logic)

Let $\LL$ be a logical language. Let $\mathscr P$ be a proof system for $\LL$. A collection $\FF$ of logical formulas is '''inconsistent for $\mathscr P$''' {{iff}}: :For every logical formula $\phi$, $\FF \vdash_{\mathscr P} \phi$. That is, ''every'' logical formula $\phi$ is a provable consequence of $\FF$.


Definition:Proof System/Formal Proof

Let $\phi$ be a WFF of $\mathcal L$. A '''formal proof of $\phi$''' in $\mathscr P$ is a collection of axioms and rules of inference of $\mathscr P$ that leads to the conclusion that $\phi$ is a theorem of $\mathscr P$. The term '''formal proof''' is also used to refer to specific presentations of such collections. For example, the term applies to tableau proofs in natural deduction.


Definition:Tarski's Geometry

Tarski's geometry is an axiomatic treatment of geometry.  Unless specified otherwise, {{ProofWiki}} will use the term Tarski's geometry to be a formal systematic treatment of geometry containing only: :$(1):\quad$ The language and axioms of first-order logic, and the disciplines preceding it :$(2):\quad$ The undefined terms of Tarski's Geometry :$(3):\quad$ Tarski's Axioms of Geometry.


Definition:Boolean Lattice/Definition 3

A '''Boolean lattice''' is a bounded lattice $\left({S, \vee, \wedge, \preceq, \bot, \top}\right)$ together with a unary operation $\neg$ called '''complementation''', subject to: $(1): \quad$ For all $a, b \in S$, $a \preceq \neg b$ {{iff}} $a \wedge b = \bot$ $(2): \quad$ For all $a \in S$, $\neg \neg a = a$.


Definition:Symmetric Relation/Definition 1

$\RR$ is '''symmetric''' {{iff}}: :$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$


Definition:Symmetric Relation/Definition 2

$\RR$ is '''symmetric''' {{iff}} it equals its inverse: :$\RR^{-1} = \RR$


Definition:Reflexive Relation/Definition 1

$\RR$ is '''reflexive''' {{iff}}: :$\forall x \in S: \tuple {x, x} \in \RR$


Definition:Reflexive Relation/Definition 2

$\RR$ is '''reflexive''' {{iff}} it is a superset of the diagonal relation:  :$\Delta_S \subseteq \RR$


Definition:Transitive Relation/Definition 1

$\RR$ is '''transitive''' {{iff}}: :$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$ that is: :$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$


Definition:Transitive Relation/Definition 2

$\RR$ is '''transitive''' {{iff}}: :$\RR \circ \RR \subseteq \RR$ where $\circ$ denotes composite relation.


Definition:Field of Relation

Let $S$ and $T$ be sets. Let $\RR \subseteq S \times T$ be a relation. The '''field''' of $\RR$ is defined as: :$\map {\operatorname {Field} } \RR := \set {x \in S: \exists t \in T: \tuple {x, t} \in \RR} \cup \set {x \in T: \exists s \in S: \tuple {s, x} \in \RR}$ That is, it is the union of the preimage of $\RR$ with its image.


Definition:Coreflexive Relation/Definition 1

$\mathcal R$ is '''coreflexive''' {{iff}}: :$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$


Definition:Coreflexive Relation/Definition 2

$\mathcal R$ is '''coreflexive''' {{iff}}: :$\mathcal R \subseteq \Delta_S$ where $\Delta_S$ is the diagonal relation.


Definition:Finitely Branching

Let $T$ be a tree. Then $T$ is '''finitely branching''' {{iff}} $T$ is locally finite.


Definition:Preordering/Definition 1

$\RR$ is a '''preordering''' on $S$ {{iff}}: {{begin-axiom}} {{axiom | n = 1         | lc= $\RR$ is reflexive         | q = \forall a \in S         | m = a \mathrel \RR a }} {{axiom | n = 2         | lc= $\RR$ is transitive         | q = \forall a, b, c \in S         | m = a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c }} {{end-axiom}}


Definition:Preordering/Notation

Symbols used to denote a general preordering relation are usually variants on $\lesssim$, $\precsim$ or $\precapprox$. A symbol for a preordering can be reversed, and the sense is likewise inverted: :$a \precsim b \iff b \succsim a$ The notation $a \sim b$ is defined as: :$a \sim b$ {{iff}} $a \precsim b$ and $b \precsim a$ The notation $a \prec b$ is defined as: :$a \prec b$ {{iff}} $a \precsim b$ and $a \not \sim b$


Definition:Preordering/Definition 2

$\mathcal R$ is a '''preordering''' on $S$ {{iff}}: :$(1): \quad \mathcal R \circ \mathcal R = \mathcal R$ :$(2): \quad \Delta_S \subseteq \mathcal R$ where: : $\circ$ denotes relation composition : $\Delta_S$ denotes the diagonal relation on $S$.


Definition:Total Ordering/Definition 1

$\RR$ is a '''total ordering''' on $S$ {{iff}}: :$(1): \quad \RR$ is an ordering on $S$ :$(2): \quad \RR$ is connected That is, $\RR$ is an ordering with no non-comparable pairs: :$\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$


Definition:Total Ordering/Definition 2

$\RR$ is a '''total ordering''' on $S$ {{iff}}: :$(1): \quad \RR \circ \RR = \RR$ :$(2): \quad \RR \cap \RR^{-1} = \Delta_S$ :$(3): \quad \RR \cup \RR^{-1} = S \times S$


Definition:Ordering/Notation

Symbols used to denote a general ordering relation are usually variants on $\preceq$, $\le$ and so on. On {{ProofWiki}}, to denote a general ordering relation it is recommended to use $\preceq$ and its variants: :$\preccurlyeq$ :$\curlyeqprec$ To denote the conventional ordering relation in the context of numbers, the symbol $\le$ is to be used, or its variants: : $\leqslant$ : $\leqq$ : $\eqslantless$ The symbol $\subseteq$ is universally reserved for the subset relation. :$a \preceq b$ can be read as: :'''$a$ precedes, or is the same as, $b$'''. Similarly: : $a \preceq b$ can be read as: : '''$b$ succeeds, or is the same as, $a$'''. If, for two elements $a, b \in S$, it is not the case that $a \preceq b$, then the symbols $a \npreceq b$ and $b \nsucceq a$ can be used.


Definition:Ordering/Definition 1

An '''ordering on $S$''' is a relation $\RR$ on $S$ such that: {{begin-axiom}} {{axiom | n = 1         | lc= $\RR$ is reflexive         | q = \forall a \in S         | m = a \mathrel \RR a }} {{axiom | n = 2         | lc= $\RR$ is transitive         | q = \forall a, b, c \in S         | m = a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c }} {{axiom | n = 3         | lc= $\RR$ is antisymmetric         | q = \forall a \in S         | m = a \mathrel \RR b \land b \mathrel \RR a \implies a = b }} {{end-axiom}}


Definition:Ordering/Definition 2

An '''ordering on $S$''' is a relation $\RR$ on $S$ such that: :$(1): \quad \RR \circ \RR = \RR$ :$(2): \quad \RR \cap \RR^{-1} = \Delta_S$ where: :$\circ$ denotes relation composition :$\RR^{-1}$ denotes the inverse of $\RR$ :$\Delta_S$ denotes the diagonal relation on $S$.


Definition:Operation/Binary Operation/Postfix Notation

Let $\circ: S \times T \to U$ be a binary operation. The convention that places the symbol for the operation after the two operands: :$z = \left ({x, y}\right) \circ$ is called '''postfix notation'''.


Definition:Operation/Binary Operation/Prefix Notation

Let $\circ: S \times T \to U$ be a binary operation. The convention that places the symbol for the operation before the two operands: :$z = \circ \left ({x, y}\right)$ is called '''prefix notation'''.


Definition:Surjection/Definition 1

$f: S \to T$ is a '''surjection''' {{iff}}: :$\forall y \in T: \exists x \in \Dom f: \map f x = y$ That is, {{iff}} $f$ is right-total.


Definition:Surjection/Definition 2

$f: S \to T$ is a '''surjection''' {{iff}}: :$f \sqbrk S = T$ or, in the language and notation of direct image mappings: :$\map {f^\to} S = T$ That is, $f$ is a '''surjection''' {{iff}} its image equals its codomain: :$\Img f = \Cdm f$


Definition:Ordered Tuple/Ordered Couples and Ordered Pairs

Notice the difference between ordered pairs and ordered couples. By definition, an ordered couple $\tuple {a, b}$ is in fact the set $\set {\tuple {1, a}, \tuple {2, b} }$, where each of $\tuple {1, a}$ and $\tuple {2, b}$ are ordered pairs. It is not possible to use the definition of ordered couple as the definition of ordered pair, as the latter is used to define a mapping, which is then used to define an ordered couple. However, in view of the Equality of Ordered Tuples, it is generally accepted that it is valid to use the notation $\tuple {a, b}$ to mean both an ordered couple and an ordered pair. It is worth bearing this in mind, as there are times when it is important not to confuse them.


Definition:Singleton Relation

Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Then $\mathcal R$ is a singleton relation iff: :$\exists_1 \left({x, y}\right) \in S \times T: \left({x, y}\right) \in \mathcal R$ That is, iff $\mathcal R$ has exactly one element.


Definition:Symmetric Relation/Definition 3

$\mathcal R$ is '''symmetric''' {{iff}} it is a subset of its inverse: :$\mathcal R \subseteq \mathcal R^{-1}$


Definition:Root of Unity Modulo m

Let $n \in \Z_{>0}$ be a positive integer. Let $m \in \Z_{>1}$ be a positive integer greater than one. Then $a$ is an '''$n^{th}$ root of unity modulo $m$''' {{iff}}: :$a^n \equiv 1 \pmod m$ Solving the following equation over the smallest integers modulo $m$: :$\displaystyle \left({a^n - 1}\right) = \left({a - 1}\right) \left({\sum_{k \mathop = 0}^{n-1} a^k}\right) \equiv 0 \pmod m$ will produce the roots. Any root found will also have $a + k m$ as a solution, where $k \in \Z$ is any integer. Euler's Function $\phi \left({m}\right)$ root of unity modulo $m$ is the set of all positive integers less than $m$. {{explain|The above statement needs tightening: what has been described "$\phi \left({m}\right)$ root of unity modulo $m$" is not clearly defined and (while it possibly may be used to define a set by predicate, is does not appear itself actually to be a set. Difficult to tell.}}


Definition:Subtree (Graph Theory)

Let $T = \left({V, E}\right)$ be a tree. A '''subtree''' of $T$ is a subgraph of $T$ that is also a tree.


Definition:Rooted Subtree

Let $\left({T, r_T}\right)$ be a rooted tree. A '''rooted subtree''' of $T$ is a rooted tree $\left({S, r_S}\right)$ such that: :$S$ is a subtree of $T$ :$r_S = r_T$ Note that the second condition implies that $r_T \in S$.


Definition:Extension of Rooted Tree

Let $\left({T, r_T}\right)$ and $\left({S, r_S}\right)$ be rooted trees. Then $\left({S, r_S}\right)$ is an '''extension of $T$''' {{iff}}: :$T$ is a subtree of $S$; :$r_S = r_T$. That is, {{iff}} $T$ is a rooted subtree of $S$. {{expand|Define using parent mapping}}


Definition:Extension of Branch of Rooted Tree

Let $T$ be a rooted tree, and let $\Gamma$ be a branch of $T$. Let $S$ be an extension of $T$, and let $\Gamma'$ be a branch of $S$. Then $\Gamma'$ is an '''extension of $\Gamma$''' {{iff}} $\Gamma \subseteq \Gamma'$. Informally, a branch may be '''extended''' by successively adding children to its leaf node, one at a time.


Let $\left({T, \mathbf H, \Phi}\right)$ be a propositional tableau. === Definition 1 === {{:Definition:Extension of Propositional Tableau/Definition 1}} === Definition 2 === {{:Definition:Extension of Propositional Tableau/Definition 2}}


=== Definition 1 === {{:Definition:Convergent Series/Normed Vector Space/Definition 1}} === Definition 2 === {{:Definition:Convergent Series/Normed Vector Space/Definition 2}}


Definition:Convergent Series/Number Field

Let $S$ be one of the standard number fields $\Q, \R, \C$. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$. Let $\sequence {s_N}$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$. It follows that $\sequence {s_N}$ can be treated as a sequence in the metric space $S$. If $s_N \to s$ as $N \to \infty$, the series '''converges to the sum $s$''', and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$. A series is said to be '''convergent''' {{iff}} it converges to some $s$.


Definition:Extension of Branch of Propositional Tableau

Let $T$ be a propositional tableau, and let $\Gamma$ be a branch of $T$. Let $S$ be an extension of $T$, and let $\Gamma'$ be a branch of $S$. Then $\Gamma'$ is an '''extension of $\Gamma$''' iff $\Gamma \subseteq \Gamma'$. Informally, a branch may be '''extended''' by successively adding children (with WFFs attached to them) to its leaf node, one at a time.


Definition:Geometric Sequence/Common Ratio

The parameter: :$r \in \R: r \ne 0$ is called the '''common ratio''' of $\sequence {x_n}$.


Definition:Geometric Series

Let $\sequence {x_n}$ be a geometric sequence in $\R$: :$x_n = a r^n$ for $n = 0, 1, 2, \ldots$ Then the series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty x_n = a + a r + a r^2 + \cdots + a r^n + \cdots$ is a '''geometric series'''.


Definition:Isolated Point (Metric Space)/Subset

Let $S \subseteq A$ be a subset of $A$. $a \in S$ is an '''isolated point''' of $S$ {{iff}} there exists an open $\epsilon$-ball of $x$ in $M$ containing no points of $S$ other than $a$: :$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \cap S = \set a$ That is: :$\exists \epsilon \in \R_{>0}: \set {x \in S: \map d {x, a} < \epsilon} = \set a$


Definition:Isolated Point (Metric Space)/Space

$a \in A$ is an '''isolated point''' of $M$ {{iff}} there exists an open $\epsilon$-ball of $x$ containing no points other than $a$: :$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a = \set a$ That is: :$\exists \epsilon \in \R_{>0}: \set {x \in A: \map d {x, a} < \epsilon} = \set a$


{{:Definition:Natural Logarithm/Positive Real/Definition 1}}


Definition:Natural Logarithm/Complex

{{:Definition:Natural Logarithm/Complex/Definition 1}}


Definition:General Logarithm/Common

Logarithms base $10$ are often referred to as common logarithms.


Definition:Series/Sequence of Partial Sums

The sequence $\sequence {s_N}$ defined as the indexed summation: :$\displaystyle s_N: = \sum_{n \mathop = 1}^N a_n = a_1 + a_2 + a_3 + \cdots + a_N$ is the '''sequence of partial sums''' of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.


Definition:Series/General/Sequence of Partial Products

The sequence $\left \langle {s_N} \right \rangle$ defined as the indexed iterated operation: :$\displaystyle s_N = \sum_{n \mathop = 1}^N a_n = a_1 \circ a_2 \circ \cdots \circ a_N$ is the '''sequence of partial products''' of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.


Definition:Positive Series

Let $\displaystyle s = \sum_{n \mathop = 1}^\infty a_n$ be a series in the real numbers $\R$. The series $s$ is a '''positive series''' {{iff}} either: :$\forall n \in \N: a_n > 0$ or: :$\forall n \in \N: a_n < 0$ That is, if all terms of $\sequence {a_n}$ are either all (strictly) positive or (strictly) negative.


Definition:Absolutely Convergent Series/General

Let $V$ be a normed vector space with norm $\norm {\, \cdot \,}$. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ in $V$ is '''absolutely convergent''' {{iff}} $\displaystyle \sum_{n \mathop = 1}^\infty \norm {a_n}$ is a convergent series in $\R$.


Definition:Absolutely Convergent Series/Real Numbers

Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in the real number field $\R$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is '''absolutely convergent''' {{iff}}: :$\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}$ is convergent where $\size {a_n}$ denotes the absolute value of $a_n$.


Definition:Alternating Series

Let $\displaystyle s = \sum_{n \mathop = 1}^\infty a_n$ be a series in the real numbers $\R$. The series $s$ is an '''alternating series''' {{iff}} the terms of $\sequence {a_n}$ alternate between positive and negative.


Definition:Absolutely Convergent Series/Complex Numbers

Let $S = \displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in the complex number field $\C$. Then $S$ is '''absolutely convergent''' {{iff}}: :$\displaystyle \sum_{n \mathop = 1}^\infty \cmod {a_n}$ is convergent where $\cmod {a_n}$ denotes the complex modulus of $a_n$.


Definition:Harmonic Series

The series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n = 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \cdots$ is known as '''the harmonic series'''.


Definition:Mercator's Constant

'''Mercator's constant''' is the real number: {{begin-eqn}} {{eqn | l = \ln 2       | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n       | c =  }} {{eqn | r = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb       | c =  }} {{eqn | r = 0 \cdotp 69314 \, 71805 \, 59945 \, 30941 \, 72321 \, 21458 \, 17656 \, 80755 \, 00134 \, 360 \ldots \ldots       | c =  }} {{end-eqn}}


Definition:Cauchy Product

Let $A := \displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $B := \displaystyle \sum_{n \mathop = 0}^\infty b_n$ be two  series. The '''Cauchy product''' $C$ of $A$ and $B$ is defined as: :$\displaystyle C := \sum_{n \mathop = 0}^\infty c_n = \sum_{n \mathop = 0}^\infty a_n \cdot \sum_{n \mathop = 0}^\infty b_n$ where: :$\displaystyle \forall n \in \N: c_n = \sum_{k \mathop = 0}^n a_k b_{n - k} = \sum_{k \mathop = 0}^n a_{n - k} b_k$ {{NamedforDef|Augustin Louis Cauchy|cat=Cauchy}}


Definition:Power Series/Coefficient

The terms: :$a_0, a_1, \ldots, a_n, \ldots$ are the '''coefficients''' of $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$.


Definition:Maclaurin Series

Let $f$ be a real function which is smooth on the open interval $\openint a b$. Then the '''Maclaurin series expansion of $f$''' is: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$ It is ''not'' necessarily the case that this power series is convergent with sum $\map f x$. {{NamedforDef|Colin Maclaurin|cat = Maclaurin}}


Definition:Determinant/Matrix/Order 1

:$\begin {vmatrix} a_{1 1} \end {vmatrix} = a_{1 1}$ Thus the determinant of an order $1$ matrix is that element itself.


Definition:Determinant/Matrix/Order 2

{{begin-eqn}} {{eqn | l = \begin {vmatrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end{vmatrix}       | r = \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1}       | c =  }} {{eqn | r = a_{1 1} a_{2 2} - a_{1 2} a_{2 1}       | c =  }} {{end-eqn}}


Definition:Determinant/Matrix/Order 3

:$\map \det {\mathbf A} = \begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} \\ a_{2 1} & a_{2 2} & a_{2 3} \\ a_{3 1} & a_{3 2} & a_{3 3} \end {vmatrix}$ Then: {{begin-eqn}} {{eqn | l = \map \det {\mathbf A}       | r = a_{1 1} \begin {vmatrix} a_{2 2} & a_{2 3} \\ a_{3 2} & a_{3 3} \end {vmatrix} - a_{1 2} \begin {vmatrix} a_{2 1} & a_{2 3} \\ a_{3 1} & a_{3 3} \end {vmatrix} + a_{1 3} \begin {vmatrix} a_{2 1} & a_{2 2} \\ a_{3 1} & a_{3 2} \end{vmatrix}       | c =  }} {{eqn | r = \map \sgn {1, 2, 3} a_{1 1} a_{2 2}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {1, 3, 2} a_{1 1} a_{2 3}  a_{3 2}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {2, 1, 3} a_{1 2} a_{2 1}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {2, 3, 1} a_{1 2} a_{2 3}  a_{3 1}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {3, 1, 2} a_{1 3} a_{2 1}  a_{3 2}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {3, 2, 1} a_{1 3} a_{2 2}  a_{3 1}       | c =  }} {{eqn | r = a_{1 1} a_{2 2}  a_{3 3}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 1} a_{2 3}  a_{3 2}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 2} a_{2 1}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = a_{1 2} a_{2 3}  a_{3 1}       | c =  }} {{eqn | o =        | ro= +       | r = a_{1 3} a_{2 1}  a_{3 2}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 3} a_{2 2}  a_{3 1}       | c =  }} {{end-eqn}} and thence in a single expression as: :$\ds \map \det {\mathbf A} = \frac 1 6 \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^3 \sum_{k \mathop = 1}^3 \sum_{r \mathop = 1}^3 \sum_{s \mathop = 1}^3 \sum_{t \mathop = 1}^3 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$ where $\map \sgn {i, j, k}$ is the sign of the permutation $\tuple {i, j, k}$ of the set $\set {1, 2, 3}$. The values of the various instances of $\map \sgn {\lambda_1, \lambda_2, \lambda_3}$ are obtained by applications of Parity of K-Cycle.


Definition:Extension of Propositional Tableau/Definition 2

An '''extension of $T$''' is a propositional tableau $\left({S, \mathbf H', \Phi'}\right)$ such that: :$S$ is an extension of $T$; :$\mathbf H = \mathbf H'$; :$\Phi'$ is an extension of $\Phi$.


Definition:Extension of Propositional Tableau/Definition 1

A tableau $T'$ is an '''extension of $T$''' if $T'$ can be obtained from $T$ by repeatedly adding nodes to the leaf nodes of $T$ (by means of the tableau extension rules).


Definition:Ordered Statistic

Let $X_1, X_2, X_3, \ldots, X_N$ be a sequence of $N$ independent random variables chosen from an identical probability space. Let $X_{(1)},X_{(2)},X_{(3)},\ldots,X_{(N)}$ be these measures re-arranged in increasing order. Then $X_{(i)}$ is said to be the '''$i$th ordered statistic'''. Category:Definitions/Probability Theory rzaj5o9lujxt7js071sccdi9t649l84


Definition:Exterior (Topology)/Notation

The exterior of $H$ can be denoted: :$\map {\mathrm {Ext} } H$ :$H^e$ The first is regarded by some as cumbersome, but has the advantage of being clear. $H^e$ is neat and compact, but has the disadvantage of being relatively obscure. On {{ProofWiki}}, the notation of choice is $H^e$.


Definition:Exterior (Topology)/Definition 1

The '''exterior''' of $H$ is the complement of the closure of $H$ in $T$.


Definition:Exterior (Topology)/Definition 2

The '''exterior''' of $H$ is the interior of the complement of $H$ in $T$.


Definition:Enumeration/Finite

Let $X$ be a finite set of cardinality $n \in \N$. An '''enumeration''' of $X$ is a bijection $x: \N_n \to X$, where $\N_n = \set {1, \ldots, n}$.


Definition:Enumeration/Countably Infinite

Let $X$ be a countably infinite set. An '''enumeration''' of $X$ is a bijection $x: \N \to X$.


Definition:Euclidean Plus Metric

Let $\R$ be the set of real numbers. Let $\set {r_i}$ be an enumeration of the rational numbers $\Q$. The '''Euclidean plus metric''' $d: \R \times \R \to \R$ is the metric defined as: :$\map d {x, y} := \set {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Thus $d$ adds to the Euclidean metric between $x$ and $y$ a contribution which measures the relative distances of $x$ and $y$ from $\Q$.


Definition:Tautology/Formal Semantics

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. A logical formula $\phi$ of $\mathcal L$ is a '''tautology for $\mathscr M$''' {{iff}}: :$\phi$ is valid in every structure $\mathcal M$ of $\mathscr M$ That $\phi$ is a '''tautology for $\mathscr M$''' can be denoted as: :$\models_{\mathscr M} \phi$


Definition:Tautology/Formal Semantics/Boolean Interpretations

Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is called a '''tautology (for boolean interpretations)''' {{iff}}: :$\map v {\mathbf A} = T$ for every boolean interpretation $v$ of $\mathbf A$. That $\mathbf A$ is a '''tautology''' may be denoted as: :$\models_{\mathrm {BI} } \mathbf A$


Definition:Contradiction

A '''contradiction''' is a statement which is ''always false'', independently of any relevant circumstances that could theoretically influence its truth value. This has the form: :$p \land \neg p$ or, equivalently: :$\neg p \land p$ that is: :'''$p$ is true and, at the same time, $p$ is not true.'''


Definition:Rational Number Space

Let $\Q$ be the set of rational numbers. Let $d: \Q \times \Q \to \R$ be the Euclidean metric on $\Q$. Let $\tau_d$ be the topology on $\Q$ induced by $d$. Then $\struct {\Q, \tau_d}$ is the '''rational number space'''.


Definition:Irrational Number Space

Let $\mathbb I := \R \setminus \Q$ be the set of irrational numbers. Let $d: \mathbb I \times \mathbb I \to \R$ be the Euclidean metric on $\mathbb I$. Let $\tau_d$ be the topology on $\mathbb I$ induced by $d$. Then $\struct {\mathbb I, \tau_d}$ is the '''irrational number space'''.


Definition:Unsatisfiable/Boolean Interpretations

Let $\mathbf A$ be a WFF of propositional logic. $\mathbf A$ is called '''unsatisfiable (for boolean interpretations)''' {{iff}}: :$\map v {\mathbf A} = F$ for every boolean interpretation $v$ for $\mathbf A$. In terms of validity, this can be rendered: :$v \not \models_{\mathrm {BI} } \mathbf A$ that is, $\mathbf A$ is invalid in every boolean interpretation of $\mathbf A$.


Definition:Boolean Interpretation/Formal Semantics/Invalid

$\phi$ is declared '''($\mathrm{BI}$-)invalid''' in a boolean interpretation $v$ iff: :$v \left({\phi}\right) = F$ Symbolically, this can be expressed as: :$v \not\models_{\mathrm{BI}} \phi$


Definition:Scattered Space/Definition 1

A topological space $T = \struct {S, \tau}$ is '''scattered''' {{iff}} it contains no non-empty subset which is dense-in-itself. That is, $T = \struct {S, \tau}$ is '''scattered''' {{iff}} every non-empty subset $H$ of $S$ contains at least one point which is isolated in $H$.


Definition:Scattered Space/Also defined as

According to {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition}}, a topological space $T$ is defined as '''scattered''': : ''... if it contains no non-empty dense-in-itself subsets; ...'' and it is immaterial whether those subsets are closed or not. On the other hand, [http://planetmath.org/encyclopedia/ScatteredSpace.html PlanetMath's definition] specifically requires that in order for a space to be classified as '''scattered''', only its closed subsets are required to contain one or more isolated points. There are few other reliable definitions to be found (the concept can be found neither on Wikipedia nor even MathWorld), and when the concept is used at all, the definitions go either way. However, it is apparent that the two definitions are equivalent, and so ultimately it does not matter which definition is used.


Definition:Scattered Space/Definition 2

A topological space $T = \left({S, \tau}\right)$ is '''scattered''' {{iff}} it contains no non-empty closed set which is dense-in-itself. That is, $T = \left({S, \tau}\right)$ is '''scattered''' {{iff}} every non-empty closed set $H$ of $S$ contains at least one point which is isolated in $H$.


{{:Definition:Isolated Point (Topology)/Subset/Definition 1}}


Definition:Isolated Point (Topology)/Space

$x \in S$ is an '''isolated point of $T$''' {{iff}}: :$\exists U \in \tau: U = \set x$ That is, {{iff}} there exists an open set of $T$ containing no points of $S$ other than $x$.


Definition:Integer Reciprocal Space

Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$. Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ That is: :$A := \set {1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \ldots}$ Then $\struct {A, \tau_d}$ is the '''integer reciprocal space'''.


Definition:Exponential Distribution

Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $X$ has the '''exponential distribution with parameter $\beta$''' {{iff}}: :$\map X \Omega = \R_{\ge 0}$ :$\map \Pr {X < x} = 1 - e^{-\frac x \beta}$ where $0 < \beta$.


Definition:Expectation/Continuous

Let $X$ be a continuous random variable over the probability space $\struct {\Omega, \Sigma, \Pr}$. Let $F = \map \Pr {X < x}$ be the cumulative probability function of $X$. The '''expectation of $X$''' is written $\expect X$, and is defined over the probability measure as: :$\expect X := \displaystyle \int_{x \mathop \in \Omega} x \rd F$ whenever the integral is absolutely convergent, i.e. when: :$\displaystyle \int_{x \mathop \in \Omega} \size x \rd F < \infty$


Definition:Logical Implication

In a valid argument, the premises '''logically imply''' the conclusion. If the truth of one statement $p$ can be shown in an argument directly to cause the meaning of another statement $q$ to be true, then $q$ follows from $p$ by '''logical implication'''. We may say: :'''$p$, therefore $q$''' and write $p \vdash q$. :'''$q$, because $p$''' and write $q \dashv p$. In symbolic logic, the concept of '''logical consequence''' occurs in the form of semantic consequence and provable consequence. In the context of proofs of a conventional mathematical nature on {{ProofWiki}}, the notation: :$p \leadsto q$ is preferred, where $\leadsto$ can be read as '''leads to'''. === Semantic Consequence === {{:Definition:Semantic Consequence}} === Provable Consequence === {{:Definition:Provable Consequence}}


<section notitle="true" name="tc"> Let $\mathcal F$ be a collection of WFFs of propositional logic. Then a WFF $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ iff: :$v \models_{\mathrm{BI}} \mathcal F$ implies $v \models_{\mathrm{BI}} \mathbf A$ where $\models_{\mathrm{BI}}$ is the models relation. </section> {{transclude:Definition:Semantic Consequence/Boolean Interpretations/Single Formula   |section = tc   |title = Semantic Consequence of Single Formula   |header = 3   |link = true   |increase = 1 }} === Notation === That $\mathbf A$ is a '''semantic consequence''' of $\mathcal F$ is denoted as: :$\mathcal F \models_{\mathbf{BI}} \mathbf A$


Definition:Union of Adjacent Open Intervals

Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$. Let $a, b, c \in \R$ where $a < b < c$. Let $A$ be the union of the two open intervals: :$A := \openint a b \cup \openint b c$ Then $\struct {A, \tau_d}$ is the '''union of adjacent open intervals'''.


Definition:Logical Equivalence

If two statements $p$ and $q$ are such that: :$p \vdash q$, that is: $p$ therefore $q$ :$q \vdash p$, that is: $q$ therefore $p$ then $p$ and $q$ are said to be '''(logically) equivalent'''. That is: :$p \dashv \vdash q$ means: :$p \vdash q$ and $q \vdash p$. Note that because the conclusion of an argument is a single statement, there can be only one statement on either side of the $\dashv \vdash$ sign. In symbolic logic, the notion of '''logical equivalence''' occurs in the form of provable equivalence and semantic equivalence. === Provable Equivalence === {{:Definition:Provable Equivalence}} === Semantic Equivalence === {{:Definition:Semantic Equivalence}}


Definition:Provable Equivalence

Let $\mathcal P$ be a proof system for a formal language $\mathcal L$. Let $\phi, \psi$ be $\mathcal L$-WFFs. Then $\phi$ and $\psi$ are '''$\mathscr P$-provably equivalent''' {{iff}}: :$\phi \vdash_{\mathscr P} \psi$ and $\psi \vdash_{\mathscr P} \phi$ that is, {{iff}} they are $\mathscr P$-provable consequences of one another. The '''provable equivalence''' of $\phi$ and $\psi$ can be denoted by: :$\phi \dashv \vdash_{\mathscr P} \psi$


Definition:Semantic Equivalence

Let $\mathscr M$ be a formal semantics for a formal language $\mathcal L$. Let $\phi, \psi$ be $\mathcal L$-WFFs. Then $\phi$ and $\psi$ are $\mathscr M$-'''semantically equivalent''' {{iff}}: :$\phi \models_{\mathscr M} \psi$ and $\psi \models_{\mathscr M} \phi$ that is, iff they are $\mathscr M$-semantic consequences of one another.


Let $\mathbf A, \mathbf B$ be WFFs of propositional logic. === Definition 1 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 1}} === Definition 2 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 2}} === Definition 3 === {{:Definition:Semantic Equivalence/Boolean Interpretations/Definition 3}}


<section notitle="true" name="tc"> Let $\mathbf A, \mathbf B$ be WFFs of propositional logic. === Definition 1 === {{:Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 1}} === Definition 2 === {{:Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 2}} </section> === Notation === That $\mathbf A$ is a '''consequence of $\mathbf B$ for boolean interpretations''' can be denoted as: :$\mathbf B \models_{\mathrm{BI}} \mathbf A$


Definition:Semantic Equivalence/Boolean Interpretations/Definition 1

Then $\mathbf A$ and $\mathbf B$ are '''equivalent for boolean interpretations''' {{iff}}: :$\mathbf A \models_{\mathrm{BI}} \mathbf B$ and $\mathbf B \models_{\mathrm{BI}} \mathbf A$ that is, each is a semantic consequence of the other. That is to say, $\mathbf A$ and $\mathbf B$ are '''equivalent''' {{iff}}: :$v \left({\mathbf A}\right) = T$ {{iff}} $v \left({\mathbf B}\right) = T$ for all boolean interpretations $v$.


Definition:Semantic Equivalence/Boolean Interpretations/Definition 2

Then $\mathbf A$ and $\mathbf B$ are '''equivalent for boolean interpretations''' {{iff}}: :$\map v {\mathbf A} = \map v {\mathbf B}$ for all boolean interpretations $v$.


Definition:Semantic Equivalence/Boolean Interpretations/Definition 3

Then $\mathbf A$ and $\mathbf B$ are '''equivalent for boolean interpretations''' {{iff}}: :$\mathbf A \iff \mathbf B$ is a tautology where $\iff$ is the biconditional connective.


Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 1

Then $\mathbf A$ is a '''semantic consequence''' of $\mathbf B$ iff: :$v \models_{\mathrm{BI}} \mathbf B$ implies $v \models_{\mathrm{BI}} \mathbf A$ for all boolean interpretations $v$. Here, $\models_{\mathrm{BI}}$ is the models relation.


Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 2

Then $\mathbf A$ is a '''semantic consequence''' of $\mathbf B$ {{iff}}: :$\mathbf A \implies \mathbf B$ is a tautology where $\implies$ is the conditional connective.


Definition:Trace (Linear Algebra)/Matrix

Let $A = \sqbrk a_n$ be a square matrix of order $n$. The '''trace''' of $A$ is: :$\ds \map \tr A = \sum_{i \mathop = 1}^n a_{ii}$


Definition:Trace (Linear Algebra)/Linear Operator

Let $V$ be a vector space. Let $A: V \to V$ be a linear operator of $V$. The '''trace''' of $A$ is the trace of the matrix of $A$ with respect to some basis.


Definition:P-Element

Let $\left({G, \circ}\right)$ be a group. Let $p$ be a prime number. Let $x \in G$ be an element of $G$. Then $x$ is a '''$p$-element of $G$''' {{iff}} its order is a power of $p$.


Definition:P-Group

A '''$p$-group''' is a group whose elements are all $p$-elements.


Definition:Elementary Abelian P-Group

Let $G$ be an abelian group. Then $G$ is an '''elementary abelian $p$-group''' {{iff}} all its non-trivial elements are of order $p$.


Definition:Abelian Group/Definition 1

An '''abelian group''' is a group $G$ where: : $\forall a, b \in G: a b = b a$ That is, every element in $G$ commutes with every other element in $G$.


Definition:Abelian Group/Definition 2

An '''abelian group''' is a group $G$ {{iff}}: : $G = \map Z G$ where $\map Z G$ is the center of $G$.


Definition:Normal Subset/Definition 6

:$\map {N_G} S = G$ where $\map {N_G} S$ denotes the normalizer of $S$ in $G$.


Definition:Normal Subset/Definition 7

:$\forall g \in G: g \circ S \subseteq S \circ g$ or: :$\forall g \in G: S \circ g \subseteq g \circ S$


Definition:Normal Subgroup/Notation

The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \lhd G$. A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal). To use the notation introduced in the definition of the conjugate: :$N \lhd G \iff \forall g \in G: N^g = N$


Definition:Refinement of Normal Series/Proper Refinement

A '''proper refinement''' of a normal series is a refinement which is not equal to the original normal series. That is, it contains extra (normal) subgroups which are not present in the original normal series.


Definition:Characteristic Subset

Let $G$ be a group. Let $S \subseteq G$ be a subset of $G$ such that: :$\forall \phi \in \Aut G: \map \phi S = S$ where $\Aut G$ is the automorphism group of $G$. Then $S$ is  '''characteristic (in $G$)''', or '''a characteristic subset of $G$'''.


Definition:Minimal Subgroup

Let $G$ be a group. Let $M \le G$ be a non-trivial subgroup of $G$. Then $M$ is a '''minimal subgroup of $G$''' {{iff}}: :For every subgroup $H$ of $G$, $H \subseteq M$ means $H = M$ or $H = \set e$. That is, if there is no subgroup of $M$, except $M$ and $\set e$ itself, which is a subset of $M$.


Definition:Asymptotics

Suppose the task to evaluate a certain number is sufficiently unwieldy that it becomes impractical to calculate it directly. In such cases it is frequently useful to have a completely different method to calculate an approximation to that number. Let the accuracy of this approximation improve as the number of operations involved in its calculation increases. The field of mathematics concerned with studying such techniques is called '''asymptotics'''.


Definition:Primitive Triple

Let $x, y, z \in \Z_{>0}$ be (strictly) positive integers. Let $A, B, C \in \Z_{>0}$ be (strictly) positive integers that satisfy: :$A^x + B^y = C^z$ Let $\left({A, B, C}\right)$ be pairwise coprime. Then $\left({A, B, C}\right)$ is a '''primitive triple'''.


Definition:Pythagorean Triple/Primitive/Canonical Form

Let $\tuple {x, y, z}$ be a primitive Pythagorean triple. The convention for representing $\tuple {x, y, z}$ as a (primitive) Pythagorean triple is that $x$ is the even element, while $y$ and $z$ are both odd. This is the '''canonical form''' of a Pythagorean triple.


Definition:Pythagorean Triple/Primitive

Let $\tuple {x, y, z}$ be a Pythagorean triple such that $x \perp y$ (that is, $x$ and $y$ are coprime). Then $\tuple {x, y, z}$ is a '''primitive Pythagorean triple'''.


Definition:Perfect Set/Definition 1

A '''perfect set''' of a topological space $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that: : $H = H'$ where $H'$ is the derived set of $H$. That is, where: : every point of $H$ is a limit point of $H$ and : every limit point of $H$ is a point of $H$.


Definition:Perfect Set/Definition 2

A '''perfect set''' of a topological space $T = \left({S, \tau}\right)$ is a subset $H \subseteq S$ such that: : $H$ is a closed set of $T$ : $H$ has no isolated points.


Definition:Perfect Set/Definition 3

A '''perfect set''' of a topological space $T = \struct {S, \tau}$ is a subset $H \subseteq S$ such that: :$H$ is dense-in-itself. :$H$ contains all its limit points.


Definition:Conjugate Symmetric Mapping

Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ Let $\innerprod \cdot \cdot: V \times V \to \mathbb F$ be a mapping. Then $\innerprod \cdot \cdot: V \times V \to \mathbb F$ is '''conjugate symmetric''' {{iff}}: :$\forall x, y \in V: \quad \innerprod x y = \overline {\innerprod y x}$ where $\overline {\innerprod y x}$ denotes the complex conjugate of $\innerprod x y$.


Definition:Symmetric Mapping (Linear Algebra)

Let $\R$ be the field of real numbers. Let $\F$ be a subfield of $\R$. Let $V$ be a vector space over $\F$ Let $\innerprod \cdot \cdot: V \times V \to \mathbb F$ be a mapping. Then $\innerprod \cdot \cdot: V \times V \to \mathbb F$ is '''symmetric''' {{iff}}: :$\forall x, y \in V: \innerprod x y = \innerprod y x$


Definition:Non-Negative Definite Mapping

Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping. Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is '''non-negative definite''' {{iff}}: :$\forall x \in V: \quad \left \langle {x, x} \right \rangle \in \R_{\geq 0}$ That is, the image of $\left \langle {x, x} \right \rangle$ is always a non-negative real number.


Definition:Positiveness

Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ Let $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ be a mapping. Then $\left \langle {\cdot, \cdot} \right \rangle : V \times V \to \mathbb F$ is '''positive''' iff: :$\forall x \in V: \quad \left \langle {x, x} \right \rangle = 0 \implies x = \mathbf 0_V$ where $\mathbf 0_V$ denotes the zero vector of $V$.


Definition:Prime Number/Definition 4

A '''prime number''' $p$ is an integer greater than $1$ that has no positive integer divisors other than $1$ and $p$.


Definition:Partial Subtraction

'''Partial subtraction''' is the operation $\dot - : \N \times \N \to \N$ defined on the set of natural numbers $\N$ as follows: :$\forall a, b \in \N: a \mathop {\dot -} b = \begin{cases} 0 & : a < b \\ a - b & : a \ge b \end{cases}$ where $a - b$ denotes natural number subtraction.


Definition:Algebraically Closed Field/Definition 1

:The only algebraic field extension of $K$ is $K$ itself.


Definition:Algebraically Closed Field/Definition 2

:Every irreducible polynomial $f$ over $K$ has degree $1$.


Definition:Algebraically Closed Field/Definition 3

:Every polynomial $f$ over $K$ of strictly positive degree has a root in $K$.


Definition:Hilbert 23

The '''Hilbert 23''' is a list of $23$ at-the-time unsolved problems in mathematics published by {{AuthorRef|David Hilbert}} during $1900$. They are as follows: === 1: The Continuum Hypothesis === {{:Definition:Hilbert 23/1}} === 2: Consistency of Axioms of Mathematics === {{:Definition:Hilbert 23/2}} === 3: Finite Dissection of Polyhedra === {{:Definition:Hilbert 23/3}} === 4: Construction of all Metrics where Lines are Geodesics === {{:Definition:Hilbert 23/4}} === 5: Whether Continuous Groups are Differential Groups === {{:Definition:Hilbert 23/5}} === 6: Axiomatize all of Physics === {{:Definition:Hilbert 23/6}} === 7: The Gelfond-Schneider Theorem === {{:Definition:Hilbert 23/7}} === 8a: The Riemann Hypothesis === {{:Definition:Hilbert 23/8a}} === 8b: The Goldbach Conjecture === {{:Definition:Hilbert 23/8b}} === 8c: The Twin Prime Conjecture === {{:Definition:Hilbert 23/8c}} === 9: General Reciprocity Theorem in Algebraic Number Field === {{:Definition:Hilbert 23/9}} === 10: Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution === {{:Definition:Hilbert 23/10}} === 11: Quadratic Forms with Algebraic Numerical Coefficients === {{:Definition:Hilbert 23/11}} === 12: Extension of Kronecker-Weber Theorem to any base Number Field === {{:Definition:Hilbert 23/12}} === 13: Solution of 7th Degree Equations using Two Parameter Functions === {{:Definition:Hilbert 23/13}} === 14: Proof of Finiteness of certain Complete Systems of Functions === {{:Definition:Hilbert 23/14}} === 15: Rigorous foundation of Schubert's Enumerative Calculus === {{:Definition:Hilbert 23/15}} === 16: Topology of Algebraic Curves and Surfaces === {{:Definition:Hilbert 23/16}} === 17: Definite Rational Function as Quotient of Sums of Squares === {{:Definition:Hilbert 23/17}} === 18a: Existence of Non-Regular Space-Filling Polyhedron === {{:Definition:Hilbert 23/18a}} === 18b: Kepler's Conjecture (Densest Sphere Packing) === {{:Definition:Hilbert 23/18b}} === 19: Solutions of Lagrangian are Analytic === {{:Definition:Hilbert 23/19}} === 20: Existence of Solutions of Variational Problems with certain Boundary Conditions === {{:Definition:Hilbert 23/20}} === 21: Existence of Linear Differential Equation with prescribed Monodromic Group === {{:Definition:Hilbert 23/21}} === 22: Uniformization of Analytic Relations by means of Automorphic Functions === {{:Definition:Hilbert 23/22}} === 23: Further Development of the Calculus of Variations === {{:Definition:Hilbert 23/23}}


Definition:Millennium Problems

The '''Millennium problems''' are a collection of seven mathematical problems stated by the {{WP|Clay_Mathematics_Institute|Clay Mathematics Institute}} on $24$ May $2000$. Each carries a prize of $\$1 \, 000 \, 000$ (US dollars). Six of them remain unsolved. They are as follows: # P versus NP # The Hodge Conjecture # The Poincaré Conjecture (the only one of the seven to be solved) # The Riemann Hypothesis # Yang-Mills Existence and Mass Gap # Navier-Stokes Existence and Smoothness # The Birch and Swinnerton-Dyer Conjecture


Definition:Almost Perfect Number

Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Almost Perfect Number/Definition 1}} === Definition 2 === {{:Definition:Almost Perfect Number/Definition 2}} === Definition 3 === {{:Definition:Almost Perfect Number/Definition 3}} Thus an '''almost perfect number''' can be described as "deficient, but only just".


Definition:Quasiperfect Number

Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Quasiperfect Number/Definition 1}} === Definition 2 === {{:Definition:Quasiperfect Number/Definition 2}} === Definition 3 === {{:Definition:Quasiperfect Number/Definition 3}}


Definition:Abundant Number/Definition 1

Let $A \left({n}\right)$ denote the abundance of $n$. $n$ is '''abundant''' {{iff}} $A \left({n}\right) > 0$.


Definition:Abundant Number/Definition 2

Let $\sigma \left({n}\right)$ be the sigma function of $n$. $n$ is '''abundant''' {{iff}} $\dfrac {\sigma \left({n}\right)} n > 2$.


Definition:Deficient Number/Definition 1

Let $A \left({n}\right)$ denote the abundance of $n$. $n$ is '''deficient''' {{iff}} $A \left({n}\right) < 0$.


Definition:Deficient Number/Definition 2

Let $\sigma \left({n}\right)$ be the sigma function of $n$. $n$ is '''deficient''' {{iff}}: :$\dfrac {\sigma \left({n}\right)} n < 2$


Definition:Antisymmetric Matrix

Let $R$ be a ring. Let $\mathbf A$ be a square matrix over $R$. $\mathbf A$ is '''antisymmetric''' {{iff}}: :$\mathbf A = - \mathbf A^\intercal$ where $\mathbf A^\intercal$ is the transpose of $\mathbf A$.


Definition:Orthogonal Matrix

Let $R$ be a ring with unity. Let $\mathbf Q$ be an invertible square matrix over $R$. Then $\mathbf Q$ is '''orthogonal''' {{iff}}: :$\mathbf Q^{-1} = \mathbf Q^\intercal$ where: :$\mathbf Q^{-1}$ is the inverse of $\mathbf Q$ :$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$


Definition:Proper Orthogonal Matrix

Let $\mathbf Q$ be an orthogonal matrix. Then $\mathbf Q$ is a '''proper orthogonal matrix''' {{iff}}: :$\map \det {\mathbf Q} = 1$ where $\map \det {\mathbf Q}$ is the determinant of $\mathbf Q$.


Definition:Einstein Summation Convention

The '''Einstein summation convention''' is a notational device used in the manipulation of matrices and vectors, in particular square matrices in the context of physics and applied mathematics. If the same index occurs twice in a given expression involving matrices, then summation over that index is automatically assumed. Thus the summation sign can be omitted, and expressions can be written more compactly.


Definition:Trace (Linear Algebra)/Matrix/Einstein Summation Convention

The trace of $A$, using the Einstein summation convention, is: :$\map \tr A = a_{ii}$


Definition:Determinant/Matrix/Order 3/Einstein Summation Convention

The determinant of a square matrix of order $3$ $\mathbf A$ can be expressed using the Einstein summation convention as: :$\map \det {\mathbf A} = \dfrac 1 6 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$ Note that there are $6$ indices which appear twice, and so $6$ summations are assumed.


Definition:Matrix/Square Matrix/Order

Let $\mathbf A$ be an $n \times n$ square matrix. That is, let $\mathbf A$ have $n$ rows (and by definition $n$ columns). Then the '''order''' of $\mathbf A$ is defined as being $n$.


Definition:Matrix Product (Conventional)/Einstein Summation Convention

The matrix product of $\mathbf A$ and $\mathbf B$ can be expressed using the Einstein summation convention as: Then: :$c_{i j} := a_{i k} \circ b_{k j}$ The index which appears twice in the expressions on the {{RHS}} is the entry $k$, which is the one summated over.


Definition:Dot Product/Definition 1

The '''dot product''' of $\mathbf a$ and $\mathbf b$ is defined as: :$\ds \mathbf a \cdot \mathbf b = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = \sum_{i \mathop = 1}^n a_i b_i$ If the vectors are represented as column matrices: :$\mathbf a = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix} , \mathbf b = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$ we can express the dot product as: :$\mathbf a \cdot \mathbf b = \mathbf a^\intercal \mathbf b$ where: :$\mathbf a^\intercal = \begin{bmatrix} a_1 & a_2 & \cdots & a_n \end{bmatrix}$ is the transpose of $\mathbf a$ :the operation between the matrices is the matrix product.


Definition:Dot Product/Definition 2

The '''dot product''' of $\mathbf a$ and $\mathbf b$ is defined as: :$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \angle \mathbf a, \mathbf b$ where: :$\norm {\mathbf a}$ denotes the length of $\mathbf a$ :$\angle \mathbf a, \mathbf b$ is the angle between $\mathbf a$ and $\mathbf b$, taken to be between $0$ and $\pi$.


Definition:Left Module Axioms

{{begin-axiom}} {{axiom | n = \text M 1         | lc= Scalar Multiplication (Left) Distributes over Module Addition         | q = \forall \lambda \in R: \forall x, y \in G         | ml= \lambda \circ \paren {x +_G y}         | mo= =         | mr= \paren {\lambda \circ x} +_G \paren {\lambda \circ y} }} {{axiom | n = \text M 2         | lc= Scalar Multiplication (Right) Distributes over Scalar Addition         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= \paren {\lambda +_R \mu} \circ x         | mo= =         | mr= \paren {\lambda \circ x} +_G \paren {\mu \circ x} }} {{axiom | n = \text M 3         | lc= Associativity of Scalar Multiplication         | q = \forall \lambda, \mu \in R: \forall x \in G         | ml= \paren {\lambda \times_R \mu} \circ x         | mo= =         | mr= \lambda \circ \paren {\mu \circ x} }} {{end-axiom}}


Definition:Unitary Module Axioms

{{begin-axiom}} {{axiom | n = \text {UM} 1         | q = \forall \lambda \in R: \forall x, y \in G         | m = \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y} }} {{axiom | n = \text {UM} 2         | q = \forall \lambda, \mu \in R: \forall x \in G         | m = \paren {\lambda +_R \mu} \circ x = \paren {\lambda \circ x} +_G \paren {\mu \circ x} }} {{axiom | n = \text {UM} 3         | q = \forall \lambda, \mu \in R: \forall x \in G         | m = \paren {\lambda \times_R \mu} \circ x = \lambda \circ \paren {\mu \circ x} }} {{axiom | n = \text {UM} 4         | q = \forall x \in G         | m = 1_R \circ x = x }} {{end-axiom}}


Definition:Simultaneous Equations/Solution

An ordered $n$-tuple $\tuple {x_1, x_2, \ldots, x_n}$ which satisfies each of the equations in a system of $m$ simultaneous equations in $n$ variables is called '''a solution''' of the system.


Definition:Simultaneous Equations/Consistency

A system of simultaneous equations: :$\forall i \in \set {1, 2, \ldots m} : \map {f_i} {x_1, x_2, \ldots x_n} = \beta_i$ that has at least one solution is '''consistent'''. If a system has no solutions, it is '''inconsistent'''.


{{:Definition:Echelon Matrix/Echelon Form/Non-Unity Variant/Definition 1}}


Definition:Echelon Matrix/Reduced Echelon Form

The matrix $\mathbf A$ is in '''reduced echelon form''' {{iff}}, in addition to being in echelon form, the leading $1$ in any non-zero row is the only non-zero element in the column in which that $1$ occurs. Such a matrix is called a '''reduced echelon matrix'''.


Definition:Generalized Sum/Net Convergence

Let $\left({g_n}\right)_{n \in \N}$ be a sequence in $G$. The series $\displaystyle \sum_{n \mathop = 1}^\infty g_n$ '''converges as a net''' or '''has net convergence''' {{iff}} the generalized sum $\displaystyle \sum \left\{{g_n: n \in \N}\right\}$ converges.


Definition:Generalized Sum/Absolute Net Convergence

Let $V$ be a Banach space. Let $\family {v_i}_{i \mathop \in I}$ be an indexed subset of $V$. Then $\displaystyle \sum \set {v_i: i \in I}$ '''converges absolutely''' {{iff}} $\displaystyle \sum \set {\norm {v_i}: i \mathop \in I}$ converges. This nomenclature is appropriate as we have Absolutely Convergent Generalized Sum Converges.


Definition:Finite Cyclic Group

Let $\struct {G, \circ}$ be a cyclic group. Then $\struct {G, \circ}$ is a '''finite cyclic group''' {{iff}} it is a finite group.


Definition:Sequential Continuity/Point

Let $x \in X$. Then $f$ is '''sequentially continuous at $x$''' iff: :For every sequence $\left\langle{x_n}\right\rangle_{n \ge 1}$ in $X$ which converges to $x$, the sequence $\left\langle{f \left({x_n}\right)}\right\rangle_{n \ge 1}$ in $Y$ converges to $f \left({x}\right)$.


Definition:Complex Number/Definition 1

A '''complex number''' is a number in the form $a + b i$ or $a + i b$ where: :$a$ and $b$ are real numbers :$i$ is a square root of $-1$, that is, $i = \sqrt {-1}$.


Definition:Complex Number/Definition 2

A '''complex number''' is an ordered pair $\tuple {x, y}$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows: === Complex Addition === {{:Definition:Complex Number/Definition 2/Addition}} === Complex Multiplication === {{:Definition:Complex Number/Definition 2/Multiplication}} === Scalar Product === {{:Definition:Complex Number/Definition 2/Scalar Product}}


Definition:Complex Number/Imaginary Unit

The entity $i := 0 + 1 i$ is known as the '''imaginary unit'''.


Definition:Complex Number/Wholly Real

A complex number $z = a + i b$ is '''wholly real''' {{iff}} $b = 0$.


Definition:Complex Number/Wholly Imaginary

A complex number $z = a + i b$ is '''wholly imaginary''' {{Iff}} $a = 0$.


Definition:Complex Number/Complex Plane

Because a complex number can be expressed as an ordered pair, we can plot the number $x + i y$ on the real number plane $\R^2$: :400px This representation is known as the '''complex plane'''. === Real Axis === {{:Definition:Complex Number/Complex Plane/Real Axis}} === Imaginary Axis === {{:Definition:Complex Number/Complex Plane/Imaginary Axis}}


Definition:Complex Number/Polar Form

For any complex number $z = x + i y \ne 0$, let: {{begin-eqn}} {{eqn | l = r       | r = \cmod z = \sqrt {x^2 + y^2}       | c = the modulus of $z$, and }} {{eqn | l = \theta       | r = \arg z       | c = the argument of $z$ (the angle which $z$ yields with the real line) }} {{end-eqn}} where $x, y \in \R$. From the definition of $\arg z$: :$(1): \quad \dfrac x r = \cos \theta$ :$(2): \quad \dfrac y r = \sin \theta$ which implies that: :$x = r \cos \theta$ :$y = r \sin \theta$ which in turn means that any number $z = x + i y \ne 0$ can be written as: :$z = x + i y = r \paren {\cos \theta + i \sin \theta}$ The pair $\polar {r, \theta}$ is called the '''polar form''' of the complex number $z \ne 0$.  The number $z = 0 + 0 i$ is defined as $\polar {0, 0}$.


Definition:Complex Number/Complex Plane/Real Axis

Complex numbers of the form $\tuple {x, 0}$, being wholly real, appear as points on the $x$-axis.


Definition:Complex Number/Complex Plane/Imaginary Axis

Complex numbers of the form $\tuple {0, y}$, being wholly imaginary, appear as points on the points on the $y$-axis. This line is known as the '''imaginary axis'''.


Definition:Complex Number/Definition 2/Addition

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers. Then $\tuple {x_1, y_1} + \tuple {x_2, y_2}$ is defined as: :$\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$


Definition:Complex Number/Definition 2/Multiplication

Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers. Then $\tuple {x_1, y_1} \tuple {x_2, y_2}$ is defined as: :$\tuple {x_1, y_1} \tuple {x_2, y_2} := \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$


Definition:Argument of Complex Number/Principal Argument

Let $R$ be the principal range of the complex numbers $\C$. The unique value of $\theta$ in $R$ is known as the '''principal value of the argument''', or just '''principal argument''', of $z$. This is denoted $\Arg z$. Note the capital $A$. The standard practice is for $R$ to be $\hointl {-\pi} \pi$. This ensures that the '''principal argument''' is continuous on the real axis for positive numbers. Thus, if $z$ is represented in the complex plane, the '''principal argument''' $\Arg z$ is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.


Definition:Pairwise Disjoint Events

Let $\family {A_i}_{i \mathop \in I}$ be an indexed family of events in a probability space. Then $\family {A_i}$ is '''pairwise disjoint''' {{iff}}: :$\forall i, j \in I, i \ne j: A_i \cap A_j = \O$ That is, a set of events is '''pairwise disjoint''' {{iff}} every pair of them is disjoint.


Definition:Complex Number/Polar Form/Exponential Form

From Euler's Formula: :$e^{i \theta} = \cos \theta + i \sin \theta$ so $z$ can also be written in the form: :$z = r e^{i \theta}$


Definition:Hyperbolic Cotangent

{{:Definition:Hyperbolic Cotangent/Definition 1}}


Definition:Hyperbolic Secant

{{:Definition:Hyperbolic Secant/Definition 1}}


Definition:Hyperbolic Cosecant

{{:Definition:Hyperbolic Cosecant/Definition 1}}


Definition:Hyperbolic Tangent/Definition 1

The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\tanh: X \to \C$: :$\forall z \in X: \tanh z := \dfrac {e^z - e^{-z} } {e^z + e^{-z} }$ where: : $X = \set {z : z \in \C, \ e^z + e^{-z} \ne 0}$


Definition:Hyperbolic Tangent/Definition 2

The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\tanh: X \to \C$: :$\forall z \in X: \tanh z := \dfrac {\sinh z} {\cosh z}$ where: :$\sinh$ is the hyperbolic sine :$\cosh$ is the hyperbolic cosine :$X = \set {z : z \in \C, \ \cosh z \ne 0}$


Definition:Hyperbolic Tangent/Definition 3

The '''hyperbolic tangent''' function is defined on the complex numbers as: :$\tanh: X \to \C$: :$\forall z \in X: \tanh z := \dfrac {e^{2 z} - 1} {e^{2 z} + 1}$ where: : $X = \set {z: z \in \C, \ e^{2 z} + 1 \ne 0}$


Definition:Hyperbolic Cotangent/Definition 1

The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\coth: X \to \C$: :$\forall z \in X: \coth z := \dfrac {e^z + e^{-z} } {e^z - e^{-z}}$ where: : $X = \set {z : z \in \C, \ e^z - e^{-z} \ne 0}$


Definition:Hyperbolic Cotangent/Definition 2

The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\coth: X \to \C$: :$\forall z \in X: \coth z := \dfrac {\cosh z} {\sinh z}$ where: :$\sinh$ is the hyperbolic sine :$\cosh$ is the hyperbolic cosine :$X = \set {z : z \in \C, \ \sinh z \ne 0}$


Definition:Hyperbolic Cotangent/Definition 3

The '''hyperbolic cotangent''' function is defined on the complex numbers as: :$\coth: X \to \C$: :$\forall z \in X: \coth z := \dfrac {e^{2 z} + 1} {e^{2 z} - 1}$ where: :$X = \set {z : z \in \C, \ e^{2 z} - 1 \ne 0}$


Definition:Hyperbolic Secant/Definition 1

The '''hyperbolic secant''' function is defined on the complex numbers as: :$\sech: X \to \C$: :$\forall z \in X: \sech z := \dfrac 2 {e^z + e^{-z} }$ where: : $X = \set {z: z \in \C, \ e^z + e^{-z} \ne 0}$


Definition:Hyperbolic Secant/Definition 2

The '''hyperbolic secant''' function is defined on the complex numbers as: :$\sech: X \to \C$: :$\forall z \in X: \sech z := \dfrac 1 {\cosh z}$ where: :$\cosh$ is the hyperbolic cosine :$X = \set {z: z \in \C, \ \cosh z \ne 0}$


Definition:Hyperbolic Cosecant/Definition 1

The '''hyperbolic cosecant''' function is defined on the complex numbers as: :$\csch: X \to \C$: :$\forall z \in X: \csch z := \dfrac 2 {e^z - e^{-z} }$ where: : $X = \set {z: z \in \C, \ e^z - e^{-z} \ne 0}$


Definition:Hyperbolic Cosecant/Definition 2

The '''hyperbolic cosecant''' function is defined on the complex numbers as: :$\csch: X \to \C$: :$\forall z \in X: \csch z := \dfrac 1 {\sinh z}$ where: :$\sinh$ is the hyperbolic sine :$X = \set {z: z \in \C, \ \sinh z \ne 0}$


Definition:Inverse Hyperbolic Function

Let $f: \C \to \C$ be one of the hyperbolic functions on the set of complex numbers. The '''inverse hyperbolic function''' $f^{-1} \subseteq \C \times \C$ is actually a multifunction, as in general for a given $y \in \C$ there is more than one $x \in \C$ such that $y = f \left({x}\right)$. As with the inverse trigonometric functions, it is usual to restrict the codomain of the multifunction so as to allow $f^{-1}$ to be single-valued.


Definition:Inverse Hyperbolic Sine/Real

=== Definition 1 === {{:Definition:Inverse Hyperbolic Sine/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Sine/Real/Definition 2}}


Definition:Inverse Hyperbolic Sine/Complex/Definition 1

The '''inverse hyperbolic sine''' is a multifunction defined as: :$\forall z \in \C: \sinh^{-1} \left({z}\right) := \left\{{w \in \C: z = \sinh \left({w}\right)}\right\}$ where $\sinh \left({w}\right)$ is the hyperbolic sine function.


Definition:Inverse Hyperbolic Sine/Complex/Definition 2

The '''inverse hyperbolic sine''' is a multifunction defined as: :$\forall z \in \C: \sinh^{-1} \left({z}\right) := \left\{{\ln \left({z + \sqrt{\left|{z^2 + 1}\right|} e^{\left({i /  2}\right) \arg \left({z^2 + 1}\right)} }\right) + 2 k \pi i: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 + 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 + 1$ : $\arg \left({z^2 + 1}\right)$ denotes the argument of $z^2 + 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.


=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosine/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosine/Complex/Definition 2}}


Definition:Inverse Hyperbolic Tangent/Complex

Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{-1 + 0 i, 1 + 0 i}\right\}$ === Definition 1 === {{:Definition:Inverse Hyperbolic Tangent/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Tangent/Definition 2}}


Let $S$ be the subset of the complex plane: :$S = \C \setminus \left\{{-1 + 0 i, 1 + 0 i}\right\}$ === Definition 1 === {{:Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2}}


Definition:Inverse Hyperbolic Secant/Complex

=== Definition 1 === {{:Definition:Inverse Hyperbolic Secant/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Secant/Definition 2}}


Definition:Inverse Hyperbolic Cosecant/Complex

=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosecant/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosecant/Definition 2}}


Definition:Inverse Hyperbolic Cosine/Complex/Definition 1

The '''inverse hyperbolic cosine''' is a multifunction defined as: :$\forall z \in \C: \cosh^{-1} \left({z}\right) := \left\{{w \in \C: z = \cosh \left({w}\right)}\right\}$ where $\cosh \left({w}\right)$ is the hyperbolic cosine function.


Definition:Inverse Hyperbolic Tangent/Complex/Definition 1

The '''inverse hyperbolic tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tanh^{-1} \paren z := \set {w \in \C: z = \tanh \paren w}$ where $\tanh \paren w$ is the hyperbolic tangent function.


Definition:Inverse Hyperbolic Cotangent/Complex/Definition 1

The '''inverse hyperbolic cotangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \coth^{-1} \left({z}\right) := \left\{{w \in \C: z = \coth \left({w}\right)}\right\}$ where $\coth \left({w}\right)$ is the hyperbolic cotangent function.


Definition:Inverse Hyperbolic Secant/Complex/Definition 1

The '''inverse hyperbolic secant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \operatorname{sech}^{-1} \left({z}\right) := \left\{{w \in \C: z = \operatorname{sech} \left({w}\right)}\right\}$ where $\operatorname{sech} \left({w}\right)$ is the hyperbolic secant function.


Definition:Inverse Hyperbolic Cosecant/Complex/Definition 1

The '''inverse hyperbolic cosecant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {w \in \C: z = \map \csch w}$ where $\map \csch w$ is the hyperbolic cosecant function.


Definition:Inverse Hyperbolic Cosine/Complex/Definition 2

The '''inverse hyperbolic cosine''' is a multifunction defined as: :$\forall z \in \C: \cosh^{-1} \left({z}\right) := \left\{{\ln \left({z + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)} }\right) + 2 k \pi i: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.


Definition:Inverse Hyperbolic Tangent/Complex/Definition 2

The '''inverse hyperbolic tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tanh^{-1} \left({z}\right) := \left\{{\dfrac 1 2 \ln \left({\dfrac {1 + z} {1 - z} }\right) + k \pi i: k \in \Z}\right\}$ where $\ln$ denotes the complex natural logarithm considered as a  multifunction.


Definition:Inverse Hyperbolic Cotangent/Complex/Definition 2

The '''inverse hyperbolic cotangent''' is a multifunction defined  on $S$ as: :$\forall z \in S: \coth^{-1} \left({z}\right) := \left\{{\dfrac 1 2 \ln \left({\dfrac {z + 1} {z - 1} }\right) + k \pi i: k \in \Z}\right\}$ where $\ln$ denotes the complex natural logarithm considered as a  multifunction.


Definition:Inverse Hyperbolic Secant/Complex/Definition 2

The '''inverse hyperbolic secant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$ where: :$\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ :$\map \arg {1 - z^2}$ denotes the argument of $1 - z^2$ :$\ln$ denotes the complex natural logarithm as a multifunction.


Definition:Inverse Hyperbolic Cosecant/Complex/Definition 2

The '''inverse hyperbolic cosecant''' is a multifunction defined as: :$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \, \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$ where: :$\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$ :$\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$ :$\ln$ denotes the complex natural logarithm considered as a multifunction.


Definition:Periodic Function/Period

The '''period''' of $f$ is the smallest value $\cmod L \in \R_{\ne 0}$ such that: : $\forall x \in X: \map f x = \map f {x + L}$ where $\cmod L$ is the modulus of $L$.


Definition:Periodic Function/Real

Let $f: \R \to \R$ be a real function. Then $f$ is '''periodic''' {{iff}}: :$\exists L \in \R_{\ne 0}: \forall x \in \R: \map f x = \map f {x + L}$


Definition:Periodic Function/Complex

Let $f: \C \to \C$ be a complex function. Then $f$ is '''periodic''' {{iff}}: :$\exists L \in \C_{\ne 0}: \forall x \in \C: f \left({x}\right) = f \left({x + L}\right)$


Definition:Inverse Sine

=== Real Numbers === {{:Definition:Inverse Sine/Real}} === Complex Plane === {{:Definition:Inverse Sine/Complex}}


Definition:Inverse Cosine

=== Real Numbers === {{:Definition:Inverse Cosine/Real}} === Complex Plane === {{:Definition:Inverse Cosine/Complex}}


Definition:Inverse Tangent

=== Real Numbers === {{:Definition:Inverse Tangent/Real}} === Complex Plane === {{:Definition:Inverse Tangent/Complex}}


Definition:Inverse Cosecant

=== Real Numbers === {{:Definition:Inverse Cosecant/Real}} === Complex Plane === {{:Definition:Inverse Cosecant/Complex}}


Definition:Inverse Secant

=== Real Numbers === {{:Definition:Inverse Secant/Real}} === Complex Plane === {{:Definition:Inverse Secant/Complex}}


Definition:Inverse Cotangent

=== Real Numbers === {{:Definition:Inverse Cotangent/Real}} === Complex Plane === {{:Definition:Inverse Cotangent/Complex}}


Definition:Quadratic Equation

A '''quadratic equation''' is a polynomial equation of the form: :$a x^2 + b x + c = 0$ such that $a \ne 0$.


Definition:Cubic Equation/Resolvent

Let: : $y = x + \dfrac b {3 a}$ : $Q = \dfrac {3 a c - b^2} {9 a^2}$ : $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$ Let $y = u + v$ where $u v = -Q$. The '''resolvent equation''' of the cubic is given by: : $u^6 - 2 R u^3 - Q^3$


Definition:Division/Notation

The operation of division can be denoted as: :$a / b$, which is probably the most common in the general informal context :$\dfrac a b$, which is the preferred style on {{ProofWiki}} :$a \div b$, which is rarely seen outside grade school.


Definition:Division/Rational Numbers

Let $\struct {\Q, +, \times}$ be the field of rational numbers. The operation of '''division''' is defined on $\Q$ as: :$\forall a, b \in \Q \setminus \set 0: a / b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\Q$.


Definition:Division/Real Numbers

Let $\struct {\R, +, \times}$ be the field of real numbers. The operation of '''division''' is defined on $\R$ as: :$\forall a, b \in \R \setminus \set 0: a / b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\R$.


Definition:Division/Complex Numbers

Let $\struct {\C, +, \times}$ be the field of complex numbers. The operation of '''division''' is defined on $\C$ as: :$\forall a, b \in \C \setminus \set 0: \dfrac a b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$ in $\C$.


Definition:Division/Divisor

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $b$ is the '''divisor''' of $a$.


Definition:Division/Dividend

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $a$ is the '''dividend''' of $b$.


Definition:Division/Quotient

Let $c = a / b$ denote the division operation on two elements $a$ and $b$ of a field. The element $c$ is the '''quotient of $a$ (divided) by $b$'''.


Definition:Division

Let $\struct {F, +, \times}$ be a field. Let the zero of $F$ be $0_F$. The operation of '''division''' is defined as: :$\forall a, b \in F \setminus \set {0_F}: a / b := a \times b^{-1}$ where $b^{-1}$ is the multiplicative inverse of $b$.


Definition:Irrational Number/Approximation

From its definition, it is not possible to express an irrational number precisely in terms of a fraction. From Decimal Expansion of Irrational Number neither Terminates nor Recurs, it is not possible to express it precisely by a decimal expansion either. However, it is possible to express it to an arbitrary level of precision. Let $x$ be an irrational number whose decimal expansion is $\sqbrk {n.d_1 d_2 d_3 \ldots}_{10}$. Then: :$\displaystyle n + \sum_{j \mathop = 1}^k \frac {d_j} {10^j} \le x < n + \sum_{j \mathop = 1}^k \frac {d_j} {10^j} + \frac 1 {10^k}$ for all $k \in \Z: k \ge 1$. Then all one needs to do is state that $x$ is expressed as '''accurate to $k$ decimal places'''.


Definition:Rational Number/Fraction

By definition, a rational number is a number which can be expressed in the form: :$\dfrac a b$ where $a$ and $b$ are integers. A '''fraction''' is a rational number such that, when expressed in canonical form $\dfrac a b$ (that is, such that $a$ and $b$ are coprime), the denominator $b$ is not $1$.


Definition:Subtraction/Complex Numbers

Let $\struct {\C, +, \times}$ be the field of complex numbers. The operation of '''subtraction''' is defined on $\C$ as: :$\forall a, b \in \C: a - b := a + \paren {-b}$ where $-b$ is the negative of $b$ in $\C$.