Inversion Mapping is Permutation

Let $\struct {G, \circ}$ be a group. Let $\iota: G \to G$ be the inversion mapping on $G$. Then $\iota$ is a permutation on $G$.


Inversion Mapping is Mapping

Let $\left({G, \circ}\right)$ be a group. Let $\iota: G \to G$ be the inversion mapping on $G$. Then $\iota$ is indeed a mapping.


Cartesian Product of Group Actions

Let $\struct {G, \circ}$ be a group. Let $S$ and $T$ be sets. Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions. Then the operation $*: G \times \paren {S \times T} \to S \times T$ defined as: :$\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {g *_S s, g *_T t}$ is a group action.


Index in Subgroup

Let $G$ be a group. Let $H, K$ be subgroups of finite index of $G$. Then: :$\index H {H \cap K} \le \index G K$ where $\index G K$ denotes the index of $K$ in $G$. Equality happens {{iff}} $G = H K$.


Finite Cyclic Group has Euler Phi Generators

Let $C_n$ be a (finite) cyclic group of order $n$. Then $C_n$ has $\map \phi n$ generators, where $\map \phi n$ denotes the Euler $\phi$ function.


Extendability Theorem for Derivatives Continuous on Open Intervals

Let $f$ be a continuous real function defined on an interval $\closedint a b$ where $a < b$. Then $f$ is continuously differentiable on $\closedint a b$ {{iff}}: :$f$ is continuously differentiable on $\openint a b$ and: :$\displaystyle \lim_{x \mathop \to a^+} \map {f'} x$ and $\displaystyle \lim_{x \mathop \to b^-} \map {f'} x$ exist.


Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous

Consider the categorical statements: :$\mathbf A: \quad$ The universal affirmative: $\forall x: \map S x \implies \map P x$ :$\mathbf E: \quad$ The universal negative: $\forall x: \map S x \implies \neg \map P x$ Then: :$\mathbf A$ and $\mathbf E$ are contrary {{iff}}: :$\exists x: \map S x$ Using the symbology of predicate logic: :$\exists x: \map S x \iff \neg \paren {\paren {\forall x: \map S x \implies \map P x} \land \paren {\forall x: \map S x \implies \neg \map P x} }$


Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous

Consider the categorical statements: :$\mathbf I: \quad$ The particular affirmative: $\exists x: \map S x \land \map P x$ :$\mathbf O: \quad$ The particular negative: $\exists x: \map S x \land \neg \map P x$ Then: :$\mathbf I$ and $\mathbf O$ are subcontrary {{iff}}: :$\exists x: \map S x$ Using the symbology of predicate logic: :$\exists x: \map S x \iff \paren {\paren {\exists x: \map S x \land \map P x} \lor \paren {\exists x: \map S x \land \neg \map P x} }$


Universal Affirmative and Particular Negative are Contradictory

Consider the categorical statements: :$\mathbf A: \quad$ The universal affirmative: $\forall x: \map S x \implies \map P x$ :$\mathbf O: \quad$ The particular negative: $\exists x: \map S x \land \neg \map P x$ Then $\mathbf A$ and $\mathbf O$ are contradictory. Using the symbology of predicate logic: :$\neg \paren {\paren {\forall x: \map S x \implies \map P x} \iff \paren {\exists x: \map S x \land \neg \map P x} }$


Particular Affirmative and Universal Negative are Contradictory

Consider the categorical statements: :$\mathbf I: \quad$ The particular affirmative: $\exists x: \map S x \land \map P x$ :$\mathbf E: \quad$ The universal negative: $\forall x: \map S x \implies \neg \map P x$ Then $\mathbf I$ and $\mathbf E$ are contradictory. Using the symbology of predicate logic: :$\neg \paren {\paren {\exists x: \map S x \land \map P x} \iff \paren {\forall x: \map S x \implies \neg \map P x} }$


Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous

Consider the categorical statements: :$\map {\mathbf A} {S, P}: \quad$ The universal affirmative: $\forall x: \map S x \implies \map P x$ :$\map {\mathbf I} {S, P}: \quad$ The particular affirmative: $\exists x: \map S x \land \map P x$ Then: :$\map {\mathbf A} {S, P} \implies \map {\mathbf I} {S, P}$ {{iff}}: :$\exists x: \map S x$ Using the symbology of predicate logic: :$\exists x: \map S x \iff \paren {\paren {\forall x: \map S x \implies \map P x} \implies \paren {\exists x: \map S x \land \map P x} }$


Socrates is Mortal

:$(1): \quad$ ''All humans are mortal.'' :$(2): \quad$ ''{{AuthorRef|Socrates}} is human.'' :$(3): \quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.''


Right-Hand Differentiable Function is Right-Continuous

Let $f$ be a real function defined on an interval $I$. Let $a$ be a point in $I$ where $f$ is right-hand differentiable. Then $f$ is right-continuous at $a$.


Left-Hand Differentiable Function is Left-Continuous

Let $f$ be a real function defined on an interval $I$. Let $a$ be a point in $I$ where $f$ is left-hand differentiable. Then $f$ is left-continuous at $a$.


Left-Hand and Right-Hand Differentiable Function is Continuous

Let $f$ be a real function defined on an interval $I$. Let $a$ be a point in $I$ where $f$ is left- and right-hand differentiable. Then $f$ is continuous at $a$.


Universal Affirmative and Negative are both False iff Particular Affirmative and Negative are both True

Consider the categorical statements: {{begin-axiom}} {{axiom | q = \mathbf A \left({S, P}\right)         | lc= The universal affirmative:         | ml= \forall x: S \left({x}\right)         | mo= \implies         | mr= P \left({x}\right) }} {{axiom | q = \mathbf E \left({S, P}\right)         | lc= The universal negative:         | ml= \forall x: S \left({x}\right)         | mo= \implies         | mr= \neg P \left({x}\right) }} {{axiom | q = \mathbf I \left({S, P}\right)         | lc= The particular affirmative:         | ml= \exists x: S \left({x}\right)         | mo= \land          | mr= P \left({x}\right) }} {{axiom | q = \mathbf O \left({S, P}\right)         | lc= The particular negative:         | ml= \exists x: S \left({x}\right)         | mo= \land          | mr= \neg P \left({x}\right) }} {{end-axiom}} Then: :$\mathbf A \left({S, P}\right)$ and $\mathbf E \left({S, P}\right)$ are both false {{iff}}: :$\mathbf I \left({S, P}\right)$ and $\mathbf O \left({S, P}\right)$ are both true.


Law of Simple Conversion of I

Consider the particular affirmative categorical statement ''Some $S$ is $P$'': :$\map {\mathbf I} {S, P}: \exists x: \map S x \land  \map P x$ Then ''Some $P$ is $S$'': :$\map {\mathbf I} {P, S}$


Law of Simple Conversion of E

Consider the universal negative categorical statement ''No $S$ is $P$'': :$\mathbf E \left({S, P}\right): \forall x: S \left({x}\right) \implies \neg P \left({x}\right)$ Then ''No $P$ is $S$'': :$\mathbf E \left({P, S}\right)$


Conversion per Accidens

Consider the categorical statements: :$\map {\mathbf A} {S, P}: \quad$ The universal affirmative: $\forall x: \map S x \implies \map P x$ :$\map {\mathbf I} {P, S}: \quad$ The particular affirmative: $\exists x: \map P x \land \map S x$ Then: :$\map {\mathbf A} {S, P} \implies \map {\mathbf I} {P, S}$ {{iff}}: :$\exists x: \map S x$ Using the symbology of predicate logic: :$\exists x: \map S x \iff \paren {\paren {\forall x: \map S x \implies \map P x} \implies \paren {\exists x: \map P x \land \map S x} }$ This law has the traditional name '''conversion per accidens of $\mathbf A$'''. Thus the $\mathbf A$ form '''converts per accidens''' to the $\mathbf I$ form.


Number of Standard Instances of Categorical Syllogism

There are $256$ distinct standard instances of the categorical syllogism.


No Valid Categorical Syllogism contains two Negative Premises

No categorical syllogism of which both premises are negative categorical statements is valid.


Valid Patterns of Categorical Syllogism

The following categorical syllogisms are valid: :$\begin{array}{rl} \text{I} & AAA \\ \text{I} & AII \\ \text{I} & EAE \\ \text{I} & EIO \\ * \text{I} & AAI \\ * \text{I} & EAO \\ \end{array} \qquad \begin{array}{rl} \text{II} & EAE \\ \text{II} & AEE \\ \text{II} & AOO \\ \text{II} & EIO \\ * \text{II} & EAO \\ * \text{II} & AEO \\ \end{array} \qquad \begin{array}{rl} \dagger \text{III} & AAI \\ \text{III} & AII \\ \text{III} & IAI \\ \dagger \text{III} & EAO \\ \text{III} & EIO \\ \text{III} & OAO \\ \end{array} \qquad \begin{array}{rl} \S \text{IV} & AAI \\ \text{IV} & AEE \\ \dagger \text{IV} & EAO \\ \text{IV} & EIO \\ \text{IV} & IAI \\ * \text{IV} & AEO \\ \end{array}$ In the above: :$\text{I}, \text{II}, \text{III}, \text{IV}$ denote the four figures of the categorical syllogisms :$A, E, I, O$ denote the universal affirmative, universal negative, particular affirmative and particular negative respectively: see Shorthand for Categorical Syllogism :Syllogisms marked $*$ require the assumption that $\exists x: \map S x$, that is, that there exists an object fulfilling the secondary predicate :Syllogisms marked $\dagger$ require the assumption that $\exists x: \map M x$, that is, that there exists an object fulfilling the middle predicate :Syllogisms marked $\S$ require the assumption that $\exists x: \map P x$, that is, that there exists an object fulfilling the primary predicate


Distributed Term of Conclusion of Valid Categorical Syllogism is Distributed in Premise

Let $T$ be a term of the conclusion $C$ of a valid categorical syllogism $Q$. Let $T$ be distributed in $C$. Then $T$ is also distributed in whichever premise of $Q$ in which it appears.


Middle Term of Valid Categorical Syllogism is Distributed at least Once

Let $M$ be the middle term of a valid categorical syllogism $Q$. Then $M$ is distributed in at least one of the premises of $Q$ in which it appears.


Conclusion of Valid Categorical Syllogism is Negative iff one Premise is Negative

The conclusion of a valid categorical syllogism is negative {{iff}} one of the premises is also negative.


No Valid Categorical Syllogism contains two Particular Premises

Let $Q$ be a valid categorical syllogism. Then at least one of the premises of $Q$ is universal.


No Valid Categorical Syllogism with Particular Premise has Universal Conclusion

Let $Q$ be a valid categorical syllogism. Let one of the premises of $Q$ be particular. Then the conclusion of $Q$ is also particular.


Elimination of all but 48 Categorical Syllogisms as Invalid

Of the $256$ different types of categorical syllogism, all but $48$ can immediately be identified as invalid by consideration of the Rules of Quantity and the Rules of Quality.


Conjunction implies Disjunction

:$\vdash \paren {p \land q} \implies \paren {p \lor q}$


Angle Bisector Vector

Let $\mathbf u$ and $\mathbf v$ be vectors of non-zero length. Let $\left\Vert{\mathbf u}\right\Vert$ and $\left\Vert{\mathbf v}\right\Vert$ be their respective lengths. Then $\left\Vert{\mathbf u}\right\Vert \mathbf v + \left\Vert{\mathbf v}\right\Vert \mathbf u$ is the angle bisector of $\mathbf u$ and $\mathbf v$.


Injection from Finite Set to Itself is Surjection/Corollary

Let $S$ be a finite set. Let $f: S \to S$ be an injection. Then $f$ is a permutation.


Surjection from Finite Set to Itself is Permutation

Let $S$ be a finite set. Let $f: S \to S$ be an surjection. Then $f$ is a permutation.


Order of Power of Group Element

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $g \in G$ be an element of $G$ such that: :$\order g = n$ where $\order g$ denotes the order of $g$. Then: :$\forall m \in \Z: \order {g^m} = \dfrac n {\gcd \set {m, n} }$ where $\gcd \set {m, n}$ denotes the greatest common divisor of $m$ and $n$.


Existence of Group of Finite Order

Let $n \in \Z_{>0}$. Then there exists at least one group whose order is $n$.


Vitali Theorem

There exists a set of real numbers which is not Lebesgue measurable.


Finite Number of Groups of Given Finite Order

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then there exists a finite number of types of group of order $n$.


Nu of Prime Number is 1

Let $p$ be a prime number. Then: :$\map \nu p = 1$ where $\nu$ denotes the $\nu$ function: the number of types of group of a given order.


Image of Subset under Relation equals Union of Images of Elements

Let $S$ and $T$ be sets. Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Let $X \subseteq S$ be a subset of $S$. Then: :$\displaystyle \RR \sqbrk X = \bigcup_{x \mathop \in X} \map \RR x$ where: :$\RR \sqbrk X$ is the image of the subset $X$ under $\RR$ :$\map \RR x$ is the image of the element $x$ under $\RR$.


Preimage of Subset under Relation equals Union of Preimages of Elements

Let $S$ and $T$ be sets. Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Let $\RR^{-1} \subseteq T \times S$ be the inverse relation to $\RR$ Let $Y \subseteq T$ be a subset of $T$. Then: :$\RR^{-1} \sqbrk Y = \displaystyle \bigcup_{y \mathop \in Y} \map {\RR^{-1} } y$ where: :$\RR^{-1} \sqbrk Y$ is the preimage of the subset $Y$ under $\RR$ :$\RR^{-1} \sqbrk Y$ is the preimage of the element $y$ under $\RR$.


Preimage of Subset under Mapping equals Union of Preimages of Elements

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, defined as: :$f^{-1} = \set {\tuple {t, s}: \map f s = t}$ Let $Y \subseteq T$ be a subset of $T$. Then: :$\displaystyle f^{-1} \sqbrk Y = \bigcup_{y \mathop \in Y} \map {f^{-1} } y$ where: :$f^{-1} \sqbrk Y$ is the preimage of the subset $Y$ under $f$ :$\map {f^{-1} } y$ is the preimage of the element $y$ under $f$.


Image of Domain of Mapping is Image Set

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. The image of $S$ is the image set of $f$: :$f \sqbrk S = \Img f$


Image of Singleton under Mapping

Let $f: S \to T$ be a mapping. Then the image of an element of $S$ is equal to the image of a singleton containing that element, the singleton being a subset of $S$: :$\forall s \in S: \set {\map f s} = f \sqbrk {\set s}$


Image of Subset under Mapping equals Union of Images of Elements

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $X \subseteq S$ be a subset of $S$. Then: :$\displaystyle f \sqbrk X = \bigcup_{x \mathop \in X} \map f x$ where: :$f \sqbrk X$ is the image of the subset $X$ under $f$ :$\map f x$ is the image of the element $x$ under $f$.


Mapping Images are Disjoint only if Domains are Disjoint

Let $S$ and $T$ be sets. Let: :$f \sqbrk S \cap f \sqbrk T = \O$ where $f \sqbrk S$ denotes the image set of $S$. Then: :$S \cap T = \O$


Image of Relation is Domain of Inverse Relation

Let $\RR \subseteq S \times T$ be a relation. Let $\RR^{-1} \subseteq T \times S$ be the inverse of $\RR$. Then: :$\Img \RR = \Dom {\RR^{-1} }$ That is, the image of a relation is the domain of its inverse.


Domain of Relation is Image of Inverse Relation

Let $\RR \subseteq S \times T$ be a relation. Let $\RR^{-1} \subseteq T \times S$ be the inverse of $\RR$. Then: :$\Dom \RR = \Img {\RR^{-1} }$ That is, the domain of a relation is the image of its inverse.


Image of Set Difference under Injection

Let $f: S \to T$ be a mapping. Let $S_1$ and $S_2$ be subsets of $S$. Let $S_1 \setminus S_2$ denote the set difference between $S_1$ and $S_2$. Then: :$\forall S_1, S_2 \subseteq S: f \left[{S_1}\right] \setminus f \left[{S_2}\right] = f \left[{S_1 \setminus S_2}\right]$ {{iff}} $f$ is an injection.


De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Union

:$\displaystyle \map \complement {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \map \complement {S_i}$


De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection

:$\displaystyle \map \complement {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \map \complement {S_i}$


Closure of Intersection and Symmetric Difference imply Closure of Set Difference

Let $\mathcal R$ be a system of sets such that for all $A, B \in \mathcal R$: : $(1): \quad A \cap B \in \mathcal R$ : $(2): \quad A * B \in \mathcal R$ where $\cap$ denotes set intersection and $*$ denotes set symmetric difference. Then: :$\forall A, B \in \mathcal R: A \setminus B \in \mathcal R$ where $\setminus$ denotes set difference.


Closure of Intersection and Symmetric Difference imply Closure of Union

Let $\mathcal R$ be a system of sets such that for all $A, B \in \mathcal R$: : $(1): \quad A \cap B \in \mathcal R$ : $(2): \quad A * B \in \mathcal R$ where $\cap$ denotes set intersection and $*$ denotes set symmetric difference. Then: :$\forall A, B \in \mathcal R: A \cup B \in \mathcal R$ where $\cup$ denotes set union.


Complement of Limit Inferior is Limit Superior of Complements

Let $\left\{{E_n : n \in \N}\right\}$ be a sequence of sets. Then: :$\displaystyle \complement \left({\liminf_{n \mathop \to \infty} \ E_n}\right) = \limsup_{n \mathop \to \infty} \ \complement \left({E_n}\right)$ where $\liminf$ and $\limsup$ denote the limit inferior and limit superior, respectively.


Sigma-Ring is Closed under Countable Intersections

Let $\mathcal R$ be a $\sigma$-ring. Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N} \in \mathcal R$ be a sequence of sets in $\mathcal R$. Then: :$\displaystyle \bigcap_{n \mathop = 1}^\infty A_n \in \mathcal R$


Pappus's Hexagon Theorem

Let $A, B, C$ be a set of collinear points. Let $a, b, c$ be another set of collinear points. Let $X, Y, Z$ be the points of intersection of each of the straight lines $Ab$ and $aB$, $Ac$ and $aC$, and $Bc$ and $bC$. Then $X, Y, Z$ are collinear points.


Two Planes have Line in Common

Two distinct planes have exactly one (straight) line in common.


Principle of Duality in the Plane

Let $P$ be a theorem of projective geometry proven using the propositions of incidence. Let $Q$ be the statement created from $P$ by interchanging: :$(1): \quad$ the terms '''point''' and '''(straight) line''' :$(2): \quad$ the terms '''collinear''' (of points) and '''concurrent''' (of lines) :$(3): \quad$ the terms '''lie on''' and '''intersect at''' and so on. Then $Q$ is also a theorem of projective geometry.


Principle of Duality in Space

Let $P$ be a theorem of projective geometry proven using the propositions of incidence. Let $Q$ be the statement created from $P$ by interchanging: :$(1) \quad$ the terms '''point''' and '''plane''' :$(2) \quad$ the terms '''lie on''' and '''intersect at''' and so on. Then $Q$ is also a theorem of projective geometry.


Desargues' Theorem

Let $\triangle ABC$ and $\triangle A'B'C'$ be triangles lying in the same or different planes. Let the lines $AA'$, $BB'$ and $CC'$ intersect in the point $O$. Then $BC$ meets $B'C'$ in $L$, $CA$ meets $C'A'$ in $M$ and $AB$ meets $A'B'$ in $N$, where $L, M, N$ are collinear.


Domain of Composite Mapping

Let $S_1, S_2, S_3$ be sets. Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings. Let $f_2 \circ f_1: S_1 \to S_3$ be the composite mapping of $f_1$ and $f_2$. Then: :$\Dom {f_1} = \Dom {f_2 \circ f_1}$ where $\Dom {f_1}$ denotes the domain of $f_1$.


Union of Functions Theorem

Let $X$ be a set. Let $\set {X_i: i \in \N}$ be an exhausting sequence of sets on $X$. For each $i \in \N$, let $g_i: X_i \to Y$ be a mapping such that: :$g_{i + 1} \restriction X_i = g_i$ where $g_{i + 1} \restriction X_i$ denotes the restriction of $g_{i + 1}$ to $g_i$. Then: :$\displaystyle \bigcup \set {g_i: i \in \N}$ is a mapping from $X$ to $Y$.


Inductive Definition of Sequence

Let $X$ be a set. Let $h \in \N$. Let $a_i \in X$ for all $i \in \set {1, 2, \ldots, h}$. Let $S$ be the set of all finite sequences whose codomains are in $X$. Let $G: S \to X$ be a mapping. Then there is a unique sequence $f$ whose codomain is in $X$ such that: :$f_i = \begin{cases} a_i & : i \in \set {1, 2, \ldots, h} \\ \map G {f_1, f_2, \ldots, f_{i - 1} } & : i \ge h + 1 \end{cases}$


Model of Root of Propositional Tableau is Model of Branch

Let $\left({T, \mathbf H, \Phi}\right)$ be a propositional tableau. Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a boolean interpretation such that: :$v \models_{\mathrm{BI}} \mathbf H$ that is, such that $v$ is a model for the root $\mathbf H$ of $T$. Then there exists a branch $\Gamma$ of $T$ such that: :$v \models_{\mathrm{BI}} \Phi \left[{\Gamma}\right]$ where $\Phi \left[{\Gamma}\right]$ denotes the image of $\Gamma$ under $\Phi$.


Antireflexive and Transitive Relation is Antisymmetric

Let $\mathcal R \subseteq S \times S$ be a relation which is not null. Let $\mathcal R$ be antireflexive and transitive. Then $\mathcal R$ is also antisymmetric.


Tableau Confutation implies Unsatisfiable

Let $\mathbf H$ be a collection of WFFs of propositional logic. Suppose there exists a tableau confutation of $\mathbf H$. Then $\mathbf H$ is unsatisfiable for boolean interpretations.


Finished Branch Lemma

Let $\Gamma$ be a finished branch of a propositional tableau $\left({T, \mathbf H, \Phi}\right)$. Let $v$ be a boolean interpretation such that: :$v \models_{\mathrm{BI}} \mathbf A$ for every basic WFF $\mathbf A$ that occurs along $\Gamma$. Then: :$v \models_{\mathrm{BI}} \Phi \left[{\Gamma}\right]$ where $\Phi \left[{\Gamma}\right]$ is the image of $\Gamma$ under $\Phi$.


Diagonal Relation is Serial

Let $S$ be a set. Let $\Delta_S$ be the diagonal relation on $S$. Then $\Delta_S$ is a serial relation.


Serial Relation is not Null

Let $S$ be a set such that $S \ne \varnothing$. Let $\mathcal R$ be a serial relation on $S$. Then $\mathcal R$ is not a null relation.


Transitive Relation is Antireflexive iff Asymmetric

Let $\mathcal R \subseteq S \times S$ be a relation which is not null. Let $\mathcal R$ be transitive. Then $\mathcal R$ is antireflexive iff $\mathcal R$ is asymmetric.


Symmetric and Antisymmetric Relation is Transitive

Let $S$ be a set. Let $\RR \subseteq S \times S$ be a relation in $S$ which is both symmetric and antisymmetric. Then $\RR$ is transitive.


Relation is Reflexive and Coreflexive iff Diagonal

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a relation on $S$. Then $\mathcal R$ is reflexive and coreflexive iff: : $\mathcal R = \Delta_S$ where $\Delta_S$ is the diagonal relation.


Symmetric Preordering is Equivalence Relation

Let $\RR \subseteq S \times S$ be a preordering on a set $S$. Let $\RR$ also be symmetric. Then $\RR$ is an equivalence relation on $S$.


Antisymmetric Preordering is Ordering

Let $\mathcal R \subseteq S \times S$ be a preordering on a set $S$. Let $\mathcal R$ also be antisymmetric. Then $\mathcal R$ is an ordering on $S$.


Relation is Antisymmetric and Reflexive iff Intersection with Inverse equals Diagonal Relation

Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. Then $\mathcal R$ is both antisymmetric and reflexive iff: :$\mathcal R \cap \mathcal R^{-1} = \Delta_S$ where $\Delta_S$ denotes the diagonal relation.


Reflexive and Transitive Relation is Idempotent

Let $\RR \subseteq S \times S$ be a relation on a set $S$. Let $\RR$ be both reflexive and  transitive. Then $\RR$ is idempotent, in the sense that: :$\RR \circ \RR = \RR$ where $\circ$ denotes composition of relations.


Equivalence of Definitions of Ordering/Proof 2

The following definitions of ordering are equivalent:


Ordered Tuple/Equality

Let: :$(1): \quad \sequence {a_m} = \tuple {a_1, a_2, \ldots, a_m}$ and :$(2): \quad \sequence {b_n} = \tuple {b_1, b_2, \ldots, b_n}$ be ordered tuples for some $m, n \in \N_{>0}$. Then: :$\sequence {a_m} = \sequence {b_n} \iff n = m \land \forall j \in \N^*_n: a_j = b_j$


Image of Domain of Relation is Image Set

Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation. The image of the domain of $\mathcal R$ is the image set of $\mathcal R$: :$\mathcal R \left [{\operatorname{Dom} \left({\mathcal R}\right)}\right] = \operatorname{Im} \left ({\mathcal R}\right)$ where $\operatorname{Im} \left ({\mathcal R}\right)$ is the image of $\mathcal R$.


Condition for Relation to be Transitive and Antitransitive

Let $S$ be a set. Let $\RR \subseteq S \times S$ be a relation in $S$. Then: : $\RR$ is both transitive and antitransitive {{iff}}: : $\neg \paren {\exists x, y, z \in S: x \mathrel {\RR} y \land y \mathrel {\RR} z}$


Transitive and Antitransitive Relation is Asymmetric

Let $S$ be a set. Let $\mathcal R \subseteq S \times S$ be a relation in $S$. Let $\mathcal R$ be both transitive and antitransitive. Then $\mathcal R$ is asymmetric.


Valid Syllogism in Figure I needs Affirmative Minor Premise and Universal Major Premise

Let $Q$ be a valid categorical syllogism in Figure $\text I$. Then it is a necessary condition that: :The major premise of $Q$ be a universal categorical statement and :The minor premise of $Q$ be an affirmative categorical statement.


Valid Syllogism in Figure II needs Negative Conclusion and Universal Major Premise

Let $Q$ be a valid categorical syllogism in Figure $\text{II}$. Then it is a necessary condition that: :The major premise of $Q$ be a universal categorical statement and :The conclusion of $Q$ be a negative categorical statement.


Valid Syllogism in Figure III needs Particular Conclusion and if Negative then Negative Major Premise

Let $Q$ be a valid categorical syllogism in Figure $\text {III}$. Then it is a necessary condition that: :The conclusion of $Q$ be a particular categorical statement and: :If the conclusion of $Q$ be a negative categorical statement, then so is the major premise of $Q$.


Valid Syllogisms in Figure IV

Let $Q$ be a valid categorical syllogism in Figure $\text {IV}$. Then it is a necessary condition that: :$(1): \quad$ Either: :: the major premise of $Q$ be a negative categorical statement :or: :: the minor premise of $Q$ be a universal categorical statement :or both. :$(2): \quad$ If the conclusion of $Q$ be a negative categorical statement, then the major premise of $Q$ be a universal categorical statement. :$(3): \quad$ If the conclusion of $Q$ be a universal categorical statement, then the minor premise of $Q$ be a negative categorical statement.


Elimination of all but 24 Categorical Syllogisms as Invalid

Of the $256$ different types of categorical syllogism, all but $24$ can be identified as invalid. These are the $24$ patterns which may still be valid: :$\begin{array}{rl} \text{I} & AAA \\ \text{I} & AII \\ \text{I} & EAE \\ \text{I} & EIO \\ \text{I} & AAI \\ \text{I} & EAO \\ \end{array} \qquad \begin{array}{rl} \text{II} & EAE \\ \text{II} & AEE \\ \text{II} & AOO \\ \text{II} & EIO \\ \text{II} & EAO \\ \text{II} & AEO \\ \end{array} \qquad \begin{array}{rl} \text{III} & AAI \\ \text{III} & AII \\ \text{III} & IAI \\ \text{III} & EAO \\ \text{III} & EIO \\ \text{III} & OAO \\ \end{array} \qquad \begin{array}{rl} \text{IV} & AAI \\ \text{IV} & AEE \\ \text{IV} & EAO \\ \text{IV} & EIO \\ \text{IV} & IAI \\ \text{IV} & AEO \\ \end{array}$


Extension of Contradictory Branch is Contradictory

Let $T$ be a propositional tableau. Let $\Gamma$ be a contradictory branch of $T$. Let $\Gamma'$ be an extension of $\Gamma$. Then $\Gamma'$ is also contradictory.


Finished Propositional Tableau has Finished Branch or is Confutation

Let $\struct {T, \mathbf H, \Phi}$ be a finished propositional tableau. Then one of the following holds: :$T$ has a finished branch :$T$ is a tableau confutation.


Definition:P-Series

Let $p \in \C$ be a complex number. The series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^p} = 1 + \frac 1 {2^p} + \frac 1 {3^p} + \frac 1 {4^p} + \dotsb$ is known as a '''$p$-series'''.


Alternating Harmonic Series is Conditionally Convergent

The alternating harmonic series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \cdots$ is conditionally convergent.


Manipulation of Absolutely Convergent Series/Permutation

If $\pi: \N \to \N$ is a permutation of $N$, then: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty a_{\map \pi n}$


Manipulation of Absolutely Convergent Series/Characteristic Function

Let $A \subseteq \N$. Then: :$\displaystyle \sum_{n \mathop = 1}^\infty a_n \map {\chi_A} n = \sum_{n \mathop \in A} a_n$ where $\chi_A$ is the characteristic function of $A$.


Manipulation of Absolutely Convergent Series/Scale Factor

Let $c \in \R$, or $c \in \C$. Then: :$\displaystyle c \sum_{n \mathop = 1}^\infty a_n = \sum_{n \mathop = 1}^\infty c a_n$


Cauchy Product of Absolutely Convergent Series

Let $\displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 0}^\infty b_n$ be two real series that are absolutely convergent. Then the Cauchy product of $\displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $\displaystyle \sum_{n \mathop = 0}^\infty b_n$ is absolutely convergent.


Binomial Coefficient with Self minus One

:$\forall n \in \N_{>0}: \dbinom n {n - 1} = n$


Rising Sum of Binomial Coefficients/Marginal Cases

=== $n = 0$ === When $n = 0$ we have: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^m \binom j 0       | r = \binom 0 0 + \binom 1 0 + \binom 2 0 + \cdots \binom m 0       | c =  }} {{eqn | r = 1 + 1 + \cdots + 1       | c = from $0$ to $m$ }} {{eqn | r = m + 1       | c = as there are $m+1$ of them }} {{eqn | r = \binom {m+1} 1       | c =  }} {{end-eqn}} So the theorem holds for $n = 0$. {{qed}} === $n = 1$ === When $n = 1$ we have: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^m \binom {1 + j} 1       | r = \binom 1 1 + \binom 2 1 + \binom 3 1 + \cdots \binom {m + 1} 1       | c =  }} {{eqn | r = 1 + 2 + \cdots + \paren {m + 1}       | c =  }} {{eqn | r = \frac {\paren {m + 1} \paren {m + 2} } 2       | c = Closed Form for Triangular Numbers }} {{eqn | r = \binom {m + 2} 2       | c = {{Defof|Binomial Coefficient}} }} {{end-eqn}} So the theorem holds for $n = 1$. {{qed}}


Uniformly Convergent Series of Continuous Functions is Continuous

Let $\sequence {f_n}$ be a sequence of real functions. Let each of $\sequence {f_n}$ be continuous on the interval $\hointr a b$. {{explain|Investigation needed as to whether there is a mistake in {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition}} -- should it actually be a closed interval?}} Let the series: :$\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$ be uniformly convergent for all $x \in \closedint a b$. Then $f$ is continuous on $\hointr a b$.


Definite Integral of Uniformly Convergent Series of Continuous Functions

Let $\sequence {f_n}$ be a sequence of real functions. Let each of $\sequence {f_n}$ be continuous on the interval $\hointr a b$. {{explain|Investigation needed as to whether there is a mistake in {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition}} -- should it actually be a closed interval?}} Let the series: :$\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$ be uniformly convergent for all $x \in \closedint a b$. Then: :$\displaystyle \int_a^b \map f x \rd x = \sum_{n \mathop = 1}^\infty \int_a^b \map {f_n} x \rd x$


Derivative of Uniformly Convergent Series of Continuously Differentiable Functions

Let $\sequence {f_n}$ be a sequence of real functions. Let each of $\sequence {f_n}$ be continuously differentiable on the interval $\closedint a b$. Let the series: :$\displaystyle \map f x := \sum_{n \mathop = 1}^\infty \map {f_n} x$ be pointwise convergent for all $x \in \closedint a b$. Let the series: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac \d {\d x} \map {f_n} x$ be uniformly convergent for all $x \in \closedint a b$. Then: :$\displaystyle \frac \d {\d x} \map f x := \sum_{n \mathop = 1}^\infty \frac \d {\d x} \map {f_n} x$


Power Series Converges Uniformly within Radius of Convergence

Let $\displaystyle S := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Let $\rho \in \R$ such that $0 \le \rho < R$. Then $S$ is uniformly convergent on $D = \set {x: \size {x - \xi} \le \rho}$.


Power Series Converges to Continuous Function

Let $\displaystyle \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Then $\map f x$ is a continuous function on $\set {x: \size {x - \xi} < R}$.


Power Series is Termwise Integrable within Radius of Convergence

Let $\displaystyle \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Then: :$\displaystyle \int_a^b \map f x \rd x = \sum_{n \mathop = 0}^\infty \int_a^b a_n x^n \rd x = \sum_{n \mathop = 0}^\infty a_n \frac {x^{n + 1} } {n + 1}$


Power Series is Termwise Differentiable within Radius of Convergence

Let $\displaystyle \map f x := \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$ be a power series about a point $\xi$. Let $R$ be the radius of convergence of $S$. Then: :$\displaystyle \frac \d {\d x} \map f x = \sum_{n \mathop = 0}^\infty \frac \d {\d x} a_n x^n = \sum_{n \mathop = 1}^\infty  n a_n x^{n - 1}$


Power Series Expansion for Logarithm of 1 + x

The Newton-Mercator series defines the  natural logarithm function as a power series expansion: {{begin-eqn}} {{eqn | l = \map \ln {1 + x}       | r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n }} {{eqn | r = x - \frac {x^2} 2 + \frac {x^3} 3 - \frac {x^4} 4 + \cdots }} {{end-eqn}} valid for all $x \in \R$ such that $-1 < x \le 1$.


Newton-Mercator Series/Examples/2

The Newton-Mercator Series for $x = 1$ converges to the natural logarithm of $2$: {{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n       | r = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb       | c =  }} {{eqn | r = \ln 2       | c =  }} {{end-eqn}} This real number is known as Mercator's constant.


Power Series Expansion for Sine Function

The sine function has the power series expansion: {{begin-eqn}} {{eqn | l = \sin x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac {x^7} {7!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.


Power Series Expansion for Cosine Function

The cosine function has the power series expansion: {{begin-eqn}} {{eqn | l = \cos x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots       | c =  }} {{end-eqn}} valid for all $x \in \R$.


Spacing Limit Theorem

Let $X_{\left({i}\right)}$ be the $i$th ordered statistic of $N$ samples from a continuous random distribution with density function $f_X \left({x}\right)$. Then the spacing between the ordered statistics given $X_{\left({i}\right)}$ converges in distribution to exponential for sufficiently large sampling according to: :$N \left({X_{\left({i + 1}\right)} - X_{\left({i}\right)} }\right) \xrightarrow D \exp \left({ \dfrac 1 { f \left({ X_{\left({i}\right)} }\right)} }\right)$ as $N \to \infty$ for $i = 1, 2, 3, \dotsc, N-1$.


Union of Closures of Singleton Rationals is Rational Space

Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$. Let $B_\alpha$ denote the singleton containing the rational number $\alpha$. Then the union of the closures in the set of real numbers $\R$ of all $B_\alpha$ is $\Q$: :$\displaystyle \bigcup_{\alpha \mathop \in \Q} \map \cl {B_\alpha} = \Q$


Closure of Union of Singleton Rationals is Real Number Line

Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$. Let $B_\alpha$ be the singleton containing the rational number $\alpha$. Then the closure in the set of real numbers $\R$ of the union of all $B_\alpha$ is $\R$ itself: :$\displaystyle \map \cl {\bigcup_{\alpha \mathop \in \Q} B_\alpha} = \R$


Closure of Rational Numbers is Real Numbers

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology. Then: :$\Q^- = \R$ where $\Q^-$ denotes the closure of $\Q$.


Real Number is Closed in Real Number Line

Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Let $\alpha \in \R$ be a real number. Then $\set \alpha$ is closed in $\struct {\R, \tau}$.


Intersection of Exteriors of Singleton Rationals is Irrationals

Let $\struct {\Q, \tau_d}$ be the rational number space under the usual (Euclidean) topology $\tau_d$. Let $B_\alpha$ be the singleton containing the rational number $\alpha$. Then: :$\displaystyle \bigcap_{\alpha \mathop \in \Q} B_\alpha^e = \R \setminus \Q$ where $B_\alpha^e$ denotes the exterior of $B_\alpha$ in $\R$.


Rational Numbers form F-Sigma Set in Reals

Let $\Q$ be the set of rational numbers. Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Then $\Q$ is a $F_\sigma$ set in $\R$.


Set of Rational Numbers is not Closed in Reals

Let $\Q$ be the set of rational numbers. Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Then $\Q$ is not closed in $\R$.


Set of Rational Numbers is not G-Delta Set in Reals

Let $\Q$ be the set of rational numbers. Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Then $\Q$ is not a $G_\delta$ set in $\R$.


Eigenvalue of Matrix Powers

Let $A$ be a square matrix. Let $\lambda$ be an eigenvalue of $A$ and $\mathbf v$ be the corresponding eigenvector. Then: :$A^n \mathbf v = \lambda^n \mathbf v$ holds for each positive integer $n$.


Closure of Intersection of Rationals and Irrationals is Empty Set

Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Let $\Q$ be the set of rational numbers. Then: :$\paren {\Q \cap \paren {\R \setminus \Q} }^- = \O$ where: :$\R \setminus \Q$ denotes the set of irrational numbers :$\paren {\Q \cap \paren {\R \setminus \Q} }^-$ denotes the closure of $\Q \cap \paren {\R \setminus \Q}$.


Intersection of Closures of Rationals and Irrationals is Reals

Let $\struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Let $\Q$ be the set of rational numbers. Then: :$\Q^- \cap \paren {\R \setminus \Q}^- = \R$ where: :$\R \setminus \Q$ denotes the set of irrational numbers :$\Q^-$ denotes the closure of $\Q$.


Irrational Numbers form Metric Space

Let $\mathbb I = \R \setminus \Q$ be the set of all irrational numbers. Let $d: \mathbb I \times \mathbb I \to \R$ be defined as: :$\map d {x_1, x_2} = \size {x_1 - x_2}$ where $\size x$ is the absolute value of $x$. Then $d$ is a metric on $\mathbb I$ and so $\struct {\mathbb I, d}$ is a metric space.


Rational Number Space is Completely Normal

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is a completely normal space.


Irrational Number Space is Completely Normal

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is a completely normal space.


Rational Number Space is Paracompact

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is paracompact.


Euclidean Plus Metric is Metric

Let $\R$ be the set of real numbers. Let $d: \R \times \R \to \R$ be the Euclidean plus metric: :$\map d {x, y} := \size {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j}} - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Then $d$ is indeed a metric.


Open Ball in Euclidean Plus Metric is Subset of Equivalent Ball in Euclidean Metric

Let $\R$ be the set of real numbers. Let $d: \R \times \R \to \R$ be the Euclidean plus metric: :$\map d {x, y} := \size {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Let $d': \R \times \R \to \R$ be the Euclidean metric. Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number. Let $\map {B_\epsilon} {p; d}$ be an open $\epsilon$-ball of $p$ in $\R$ on $d$. Let $\map {B'_\epsilon} {p; d'}$ be an open $\epsilon$-ball of $p$ in $\R$ on $d'$. Then: :$\map {B_\epsilon} {p; d} \subseteq \map {B'_\epsilon} {p; d'}$


Irrational Number Space is Complete Metric Space

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is a complete metric space.


Rational Number Space is Meager

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is meager.


Rational Number Space is Separable

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is separable.


Underlying Set of Topological Space is Everywhere Dense

Let $T = \struct {S, \tau}$ be a topological space. Then the underlying set $S$ of $T$ is everywhere dense in $T$.


Irrational Number Space is Separable

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is separable.


Irrational Number Space is Second-Countable

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is second-countable.


Compact Set of Rational Numbers is Nowhere Dense

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Let $S \subseteq \Q$ be a compact set of $\Q$. Then $S$ is nowhere dense in $\Q$.


Compact Set of Irrational Numbers is Nowhere Dense

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Let $S \subseteq \R \setminus \Q$ be a compact set of $\R \setminus \Q$. Then $S$ is nowhere dense in $\R \setminus \Q$.


Rational Number Space is not Locally Compact Hausdorff Space

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is not a locally compact Hausdorff Space.


Irrational Number Space is not Locally Compact Hausdorff Space

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is not a locally compact Hausdorff Space.


Rationals plus Irrational are Everywhere Dense in Irrationals

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Let $x \in \R \setminus \Q$ be an arbitrary irrational number. Let $S_x$ be the set defined as: :$S_x := \set {x + q: q \in \Q}$ Then $S_x$ is everywhere dense in $\struct {\R \setminus \Q, \tau_d}$.


Rational Number Space is Totally Separated

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is totally separated.


Irrational Number Space is Totally Separated

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is totally separated.


Rational Number Space is Dense-in-itself

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is dense-in-itself.


Irrational Number Space is Dense-in-itself

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is dense-in-itself.


Irrational Number Space is not Scattered

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is not scattered.


Rational Number Space is Zero Dimensional

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is zero dimensional.


Irrational Number Space is Zero Dimensional

Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is zero dimensional.


Open Unit Interval on Rational Number Space is Bounded but not Compact

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then: :$\openint 0 1 \cap \Q$ is totally bounded but not compact where $\openint 0 1$ is the open unit interval.


Closure of Integer Reciprocal Space

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n: n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then: :$A^- = A \cup \set 0$ where $A^-$ denotes the closure of $A$ in $\R$.


Zero is Limit Point of Integer Reciprocal Space

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then $0$ is the only limit point of $A$ in $\R$.


Zero is Omega-Accumulation Point of Integer Reciprocal Space Union with Closed Interval

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \cup \closedint 2 3$ where $\closedint 2 3$ is a closed interval of $\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is an $\omega$-accumulation point of $B$ in $\R$.


Zero is not Condensation Point of Integer Reciprocal Space Union with Closed Interval

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Let $B$ be the uncountable set: :$B := A \cup \closedint 2 3$ where $\closedint 2 3$ is a closed interval of $\R$. $2$ and $3$ are to all intents arbitrary, but convenient. Then $0$ is not a condensation point of $B$ in $\R$.


Integer Reciprocal Space contains Cauchy Sequence with no Limit Point

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space under the usual (Euclidean) topology. Then $A$ has a Cauchy sequence which has no limit point in $A$.


Integer Reciprocal Space with Zero is not Locally Connected

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is not locally connected.


Discrete Space is Locally Connected

Let $T = \struct {S, \tau}$ be a discrete topological space. Then $T$ is locally connected.


Integer Reciprocal Space with Zero is Totally Separated

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is totally separated.


Quasicomponents of Integer Reciprocal Space with Zero are Single Points

Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then the quasicomponents of $A$ are singletons.


Interior of Union of Adjacent Open Intervals

Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A = A^\circ$ where $A^\circ$ is the interior of $A$.


Interior of Closure of Interior of Union of Adjacent Open Intervals

Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A^{\circ - \circ} = A^{- \circ} = \openint a c$ where: :$A^\circ$ is the interior of $A$ :$A^-$ is the closure of $A$.


Closure of Interior of Closure of Union of Adjacent Open Intervals

Let $a, b, c \in R$ where $a < b < c$. Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A^{- \circ -} = A^{\circ -} = A^- = \closedint a c$ where: :$A^\circ$ is the interior of $A$ :$A^-$ is the closure of $A$.


Exterior of Exterior of Union of Adjacent Open Intervals

Let $A$ be the union of the two adjacent open intervals: :$A := \openint a b \cup \openint b c$ Then: :$A^{ee} = \openint a c$ where $A^e$ is the exterior of $A$.


Open Real Interval is Regular Open

Let $\struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology. Let $\openint a b$ be an open interval of $\R$. Then $\openint a b$ is regular open in $\struct {\R, \tau_d}$.


Closure of Open Real Interval is Closed Real Interval

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.   Let $\openint a b$ be an open interval of $\R$. Then the closure of $\openint a b$ is the closed interval $\closedint a b$.


Limit Points of Open Real Interval

Let $\struct {\R, \tau_d}$ be the real number line under the usual (Euclidean) topology.   Let $\openint a b$ be an open interval of $\R$. Then the limit points of $\openint a b$ are: :all the points in $\openint a b$ as well as: :the points $a$ and $b$.


Interior of Closed Real Interval is Open Real Interval

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.   Let $\closedint a b$ be a closed interval of $\R$. Then the interior of $\closedint a b$ is the open interval $\left({a, b}\right)$.


Union of Regular Open Sets is not necessarily Regular Open

Let $T = \struct {S, \tau}$ be a topological space. Let $U$ and $V$ be regular open sets of $T$. Then $U \cup V$ is not also necessarily a regular open set of $T$.


Closure of Intersection may not equal Intersection of Closures

Let $T = \struct {S, \tau}$ be a topological space. Let $H_1$ and $H_2$ be subsets of $S$. Let ${H_1}^-$ and ${H_2}^-$ denote the closures of $H_1$ and $H_2$ respectively. Then it is not necessarily the case that: :$\paren {H_1 \cap H_2}^- = {H_1}^- \cap {H_2}^-$


Closure of Empty Set is Empty Set

Let $T = \left({S, \tau}\right)$ be a topological space. Then the closure of the empty set $\varnothing$ in $T$ is $\varnothing$.


Closed Real Interval is Regular Closed

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\closedint a b$ be a closed interval of $\R$. Then $\closedint a b$ is regular closed in $\struct {\R, \tau_d}$.


Closed Real Interval is Closed in Real Number Line

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\closedint a b$ be a closed interval of $\R$. Then $\closedint a b$ is closed (in the topological sense) in $\struct {\R, \tau_d}$.


Intersection of Regular Closed Sets is not necessarily Regular Closed

Let $T = \struct {S, \tau}$ be a topological space. Let $U$ and $V$ be regular closed sets of $T$. Then $U \cap V$ is not also necessarily a regular closed set of $T$.


Interior of Union is not necessarily Union of Interiors

Let $T = \struct {S, \tau}$ be a topological space. Let $H_1$ and $H_2$ be subsets of $S$. Let ${H_1}^\circ$ and ${H_2}^\circ$ denote the interiors of $H_1$ and $H_2$ respectively. Then it is not necessarily the case that: :$\left({H_1 \cup H_2}\right)^\circ = {H_1}^\circ \cup {H_2}^\circ$


Expectation of Exponential Distribution

Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$ Then the expectation of $X$ is given by: :$\expect X = \beta$


Von Mangoldt Equivalence

For $n \in \N_{>0}$, let $\Lambda \left({n}\right)$ be the von Mangoldt function. Then: :$\displaystyle \lim_{N \to \infty} \frac 1 N \sum_{n \mathop = 1}^N \Lambda \left({n}\right) = 1$ is logically equivalent to the Prime Number Theorem.


Zeta Equivalence to Prime Number Theorem

Let $\map \zeta z$ be the Riemann $\zeta$ function. The Prime Number Theorem is logically equivalent to the statement that the average of the first $N$ coefficients of $\dfrac {\zeta'} {\zeta}$ tend to $-1$ as $N$ goes to infinity. {{explain|What does $z$ range over, and what does it mean by "first $N$ coefficients" of $\dfrac {\zeta'} {\zeta}$?}}


Substitution Rule for Matrices

Let $\mathbf A$ be a square matrix of order $n$. Then: :$(1): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{j k} = a_{i k}$ :$(2): \quad \ds \sum_{j \mathop = 1}^n \delta_{i j} a_{k j} = a_{k i}$ where: :$\delta_{i j}$ is the Kronecker delta :$a_{j k}$ is element $\tuple {j, k}$ of $\mathbf A$.


Matrix is Invertible iff Determinant has Multiplicative Inverse/Necessary Condition

Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be an invertible square matrix of order $n$. Let $\mathbf B = \mathbf A^{-1}$ be the inverse of $\mathbf A$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Then: :$\map \det {\mathbf B} = \dfrac 1 {\map \det {\mathbf A} }$


Product of Orthogonal Matrix with Transpose is Identity

Let $\mathbf Q$ be an orthogonal matrix. Then: :$\mathbf Q \mathbf Q^\intercal = \mathbf I = \mathbf Q^\intercal \mathbf Q$ where: :$\mathbf Q^\intercal$ is the transpose of $\mathbf Q$ :$\mathbf I$ is a unit (identity) matrix


Determinant of Orthogonal Matrix is Plus or Minus One

Let $\mathbf Q$ be an orthogonal matrix. Then: :$\det \mathbf Q = \pm 1$ where $\det \mathbf Q$ is the determinant of $\mathbf Q$.


Product of Proper Orthogonal Matrices is Proper Orthogonal Matrix

Let $\mathbf P$ and $\mathbf Q$ be proper orthogonal matrices. Let $\mathbf P \mathbf Q$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$. Then $\mathbf P \mathbf Q$ is a proper orthogonal matrix.


Product of Orthogonal Matrices is Orthogonal Matrix

Let $\mathbf P$ and $\mathbf Q$ be orthogonal matrices. Let $\mathbf P \mathbf Q$ be the (conventional) matrix product of $\mathbf P$ and $\mathbf Q$. Then $\mathbf P \mathbf Q$ is an orthogonal matrix.


Internal Group Direct Product is Injective/General Result

Let $G$ be a group whose identity is $e$. Let $\sequence {H_n}$ be a sequence of subgroups of $G$. Let $\displaystyle \phi_n: \prod_{j \mathop = 1}^n H_j \to G$ be a mapping defined by: :$\displaystyle \map {\phi_n} {h_1, h_2, \ldots, h_n} = \prod_{j \mathop = 1}^n h_j$ Then $\phi_n$ is injective {{iff}}: :$\forall i, j \in \set {1, 2, \ldots, n}: i \ne j \implies H_i \cap H_j = \set e$ That is, {{iff}} $\sequence {H_n}$ is a sequence of independent subgroups.


Trace of Matrix Product

Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$. Let $\mathbf A \mathbf B$ be the (conventional) matrix product of $\mathbf A$ and $\mathbf B$. Then: :$\ds \map \tr {\mathbf A \mathbf B} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n a_{i j} b_{j i}$ where $\map \tr {\mathbf A \mathbf B}$ denotes the trace of $\mathbf A \mathbf B$. Using the Einstein summation convention, this can be expressed as: :$\map \tr {\mathbf A \mathbf B} = a_{i j} b_{j i}$


Trace of Matrix Product/General Result

Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_m$ be square matrices of order $n$. Let $\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$ be the (conventional) matrix product of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_m$. Then: :$(1): \quad \ds \map \tr {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m} = \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} \cdots \map {a_{m - 1} } {i_{m - 1}, i_m} \map {a_m} {i_m, i_1}$ where: :$\map {a_1} {i_1, i_2}$ (for example) denotes the element of $\mathbf A_1$ whose indices are $i_1$ and $i_2$ :$\map \tr {\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m}$ denotes the trace of $\mathbf A_1 \mathbf A_2 \cdots \mathbf A_m$. In $(1)$, the Einstein summation convention is used, with the implicit understanding that a summation is performed over each of the indices $i_1$ to $i_m$.


Product of Finite Sequence of Matrices

Let $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_n$ be matrices. Let the order of $\mathbf A_j$ be $d_j \times d_{j + 1}$. Let $\displaystyle \mathbf C := \prod_{j \mathop = 1}^n \mathbf A_j = \mathbf A_1 \mathbf A_2 \cdots \mathbf A_n$ be the (conventional) matrix product of $\mathbf A_1, \mathbf A_2, \ldots, \mathbf A_n$. Then: :$(1): \quad \displaystyle \map c {i_1, i_{n + 1} } = \sum_{i_n \mathop = 1}^{d_n} \dotsm \sum_{i_3 \mathop = 1}^{d_3} \sum_{i_2 \mathop = 1}^{d_2} \map {a_1} {i_1, i_2} \map {a_2} {i_2, i_3} \dotsm \map {a_{n - 1} } {i_{n - 1}, i_n} \map {a_n} {i_n, i_{n + 1} }$ where: :$\map {a_1} {i_1, i_2}$ (for example) denotes the element of $\mathbf A_1$ whose indices are $i_1$ and $i_2$ :the order of $\mathbf C$ is $d_1 \times d_{n+1}$.


Characteristic of Galois Field is Prime

Let $\GF$ be a Galois field.  Then the characteristic of $\GF$ is a prime number.


Finite Ring with No Proper Zero Divisors is Field

Let $\struct {R, +, \circ}$ be a finite non-null ring with no proper zero divisors. Then $R$ is a field.


Equivalence of Definitions of Dot Product

{{TFAE|def = Dot Product}} Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^n$.


Determinant of Kronecker Delta Elements

Let $\lambda$ and $\pi$ be permutations on $\left\{{1, 2, 3}\right\}$. Let: : $\left({i, j, k}\right) = \left({\lambda \left({1}\right), \lambda \left({2}\right), \lambda \left({3}\right)}\right)$ : $\left({r, s, t}\right) = \left({\pi \left({1}\right), \pi \left({2}\right), \pi \left({3}\right)}\right)$ Then: :$\begin{vmatrix}   \delta_{ir} & \delta_{is} & \delta_{it} \\   \delta_{jr} & \delta_{js} & \delta_{jt} \\   \delta_{kr} & \delta_{ks} & \delta_{kt} \end{vmatrix} = \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right)$ where: : $\delta_{ir}$ denotes the Kronecker delta : $\begin{vmatrix} \cdot \end{vmatrix}$ denotes a determinant : $\operatorname{sgn} \left({i, j, k}\right)$ is the sign of the permutation $\left({i, j, k}\right)$ of the set $\left\{{1, 2, 3}\right\}$.


Row Equivalent Matrix for Homogeneous System has same Solutions/Corollary

:$\set {\mathbf x: \mathbf A \mathbf x = \mathbf 0} = \set {\mathbf x: \map {\mathrm {ref} } {\mathbf A} \mathbf x = \mathbf 0}$ where $\map {\mathrm {ref} } {\mathbf A}$ is the reduced echelon form of $\mathbf A$.


Continuous Mapping is Sequentially Continuous/Corollary

Let $f$ be continuous (everywhere) on $X$. Then $f$ is sequentially continuous on $X$.


Product of Complex Numbers in Polar Form

Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form. Then: :$z_1 z_2 = r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} }$


Subtraction of Complex Numbers

Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers. The subtraction operation on $z_1$ and $z_2$ is: :$z_1 - z_2 = \paren {a_1 - a_2} + i \paren {b_1 - b_2}$


Division of Complex Numbers

Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers such that $z_2 \ne 0$. The operation of division is performed on $z_1$ by $z_2$ as follows: :$\dfrac {z_1} {z_2} = \dfrac {a_1 a_2 + b_1 b_2} {a_2^2 + b_2^2} + i \dfrac {a_2 b_1 - a_1 b_2} {a_2^2 + b_2^2}$


De Moivre's Formula/Proof 1

Let $z \in \C$ be a complex number expressed in complex form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \map \cos {\omega x} + i \, \map \sin {\omega x}$


Division of Complex Numbers in Polar Form

Let $z_1 := \polar {r_1, \theta_1}$ and $z_2 := \polar {r_2, \theta_2}$ be complex numbers expressed in polar form, such that $z_2 \ne 0$. Then: :$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \paren {\map \cos {\theta_1 - \theta_2} + i \map \sin {\theta_1 - \theta_2} }$ or: :$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} \map \cis {\theta_1 - \theta_2}$


Roots of Complex Number

Let $z := \polar {r, \theta}$ be a complex number expressed in polar form, such that $z \ne 0$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then the $n$th roots of $z$ are given by: :$z^{1 / n} = \set {r^{1 / n} \paren {\map \cos {\dfrac {\theta + 2 \pi k} n} + i \, \map \sin {\dfrac {\theta + 2 \pi k} n} }: k \in \set {0, 1, 2, \ldots, n - 1} }$ There are $n$ distinct such $n$th roots.


Zeroth Power of Real Number equals One

Let $a \in \R_{>0}$ be a (strictly) positive real number. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$a^0 = 1$


Exponential of One

: $\exp 1 = e$ where $e$ is Euler's number: $e = 2.718281828\ldots$


Exponential of Zero

: $\exp 0 = 1$


Difference of Logarithms

:$\log_b x - \log_b y = \map {\log_b} {\dfrac x y}$


==== Form 1 ==== {{:Change of Base of Logarithm/Base 10 to Base e/Form 1}} ==== Form 2 ==== {{:Change of Base of Logarithm/Base 10 to Base e/Form 2}}


==== Form 1 ==== {{:Change of Base of Logarithm/Base e to Base 10/Form 1}} ==== Form 2 ==== {{:Change of Base of Logarithm/Base e to Base 10/Form 2}}


Power of Identity is Identity

Let $\struct {M, \circ}$ be a monoid whose identity element is $e$. Then: :$\forall n \in \Z: e^n = e$


Tangent Exponential Formulation/Formulation 2

:$\tan z = \dfrac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }$


Cotangent Exponential Formulation

:$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$


Cosecant Exponential Formulation

Let $z$ be a complex number. Let $\csc z$ denote the cosecant function and $i$ denote the imaginary unit: $i^2 = -1$. Then: :$\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$


Secant Exponential Formulation

Let $z$ be a complex number. Let $\sec z$ denote the secant function and $i$ denote the imaginary unit: $i^2 = -1$. Then: :$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$


Periodicity of Complex Exponential Function

For all $k \in \Z$: :$e^{i \paren {\theta + 2 k \pi} } = e^{i \theta}$


Product of Complex Numbers in Exponential Form

Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form. Then: :$z_1 z_2 = r_1 r_2 e^{i \paren {\theta_1 + \theta_2} }$


Division of Complex Numbers in Exponential Form

Let $z_1 := r_1 e^{i \theta_1}$ and $z_2 := r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form. Then: :$\dfrac {z_1} {z_2} = \dfrac {r_1} {r_2} e^{i \paren {\theta_1 - \theta_2} }$


De Moivre's Formula/Exponential Form

:$\forall \omega \in \C: \paren {r e^{i \theta} }^\omega = r^\omega e^{i \omega \theta}$


Z/(m)-Module Associated with Ring of Characteristic m

Let $\left({R,+,*}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the characteristic of $R$ be $m$. Let $\left({\Z_m, +_m, \times_m}\right)$ be the ring of integers modulo $m$. Let $\circ$ be the mapping from $\Z_m \times R$ to $R$ defined as: :$\forall \left[\!\left[a\right]\!\right]_m \in \Z_m: \forall x \in R: \left[\!\left[a\right]\!\right]_m \circ x = a \cdot x$ where $\left[\!\left[a\right]\!\right]_m$ is the residue class of $a$ modulo $m$ and $a \cdot x$ is the $a$th power of $x$. Then $\left({R, +, \circ}\right)_{\Z_m}$ is a unitary $\Z_m$-module.


Sum of Squares of Hyperbolic Secant and Tangent

:$\sech^2 x + \tanh^2 x = 1$ where $\sech$ and $\tanh$ are hyperbolic secant and hyperbolic tangent.


Difference of Squares of Hyperbolic Cotangent and Cosecant

:$\coth^2 x - \csch^2 x = 1$ where $\coth$ and $\csch$ are hyperbolic cotangent and hyperbolic cosecant.


Hyperbolic Cosecant Function is Odd

Let $\csch: \C \to \C$ be the hyperbolic cosecant function on the set of complex numbers. Then $\csch$ is odd: :$\map \csch {-x} = -\csch x$


Hyperbolic Secant Function is Even

:$\map \sech {-x} = \sech x$


Hyperbolic Cotangent Function is Odd

Let $\coth: \C \to \C$ be the hyperbolic cotangent function on the set of complex numbers. Then $\coth$ is odd: :$\map \coth {-x} = -\coth x$


Hyperbolic Sine of Sum

:$\map \sinh {a + b} = \sinh a \cosh b + \cosh a \sinh b$


Hyperbolic Cosine of Sum

:$\map \cosh {a + b} = \cosh a \cosh b + \sinh a \sinh b$


Hyperbolic Tangent of Sum

:$\map \tanh {a + b} = \dfrac {\tanh a + \tanh b} {1 + \tanh a \tanh b}$


Hyperbolic Cotangent of Sum

:$\map \coth {a + b} = \dfrac {\coth a \coth b + 1} {\coth b + \coth a}$


Double Angle Formulas/Hyperbolic Sine

:$\sinh 2 x = 2 \sinh x \cosh x$


Double Angle Formulas/Hyperbolic Cosine

:$\cosh 2 x = \cosh^2 x + \sinh^2 x$


Double Angle Formulas/Hyperbolic Tangent

: $\tanh 2 x = \dfrac {2 \tanh x} {1 + \tanh^2 x}$


Half Angle Formulas/Hyperbolic Sine

{{begin-eqn}} {{eqn | l = \sinh \frac x 2       | r = +\sqrt {\frac {\cosh x - 1} 2}       | c = for $x \ge 0$ }} {{eqn | l = \sinh \frac x 2       | r = -\sqrt {\dfrac {\cosh x - 1} 2}       | c = for $x \le 0$ }} {{end-eqn}}


Half Angle Formulas/Hyperbolic Cosine

:$\cosh \dfrac x 2 = +\sqrt {\dfrac {\cosh x + 1} 2}$


Half Angle Formulas/Hyperbolic Tangent

{{begin-eqn}} {{eqn | l = \tanh \frac x 2       | r = +\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }       | c = for $x \ge 1$ }} {{eqn | l = \tanh \frac x 2       | r = -\sqrt {\frac {\cosh x - 1} {\cosh x + 1} }       | c = for $x \le 1$ }} {{end-eqn}}


Half Angle Formulas/Hyperbolic Tangent/Corollary 2

For $x \ne 0$: :$\tanh \dfrac x 2 = \dfrac {\cosh x - 1} {\sinh x}$


Triple Angle Formulas/Hyperbolic Sine

:$\sinh 3 x = 3 \sinh x + 4 \sinh^3 x$


Triple Angle Formulas/Hyperbolic Cosine

: $\cosh 3 x = 4 \cosh^3 x - 3 \cosh x$


Triple Angle Formulas/Hyperbolic Tangent

: $\tanh \left({3 x}\right) = \dfrac {3 \tanh x + \tanh^3 x} {1 + 3 \tanh^2 x}$


Quadruple Angle Formulas/Hyperbolic Sine

: $\sinh \left({4 x}\right) = 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x$


Quadruple Angle Formulas/Hyperbolic Cosine

: $\cosh \left({4 x}\right) = 8 \cosh^4 x - 8 \cosh^2 x + 1$


Quadruple Angle Formulas/Hyperbolic Tangent

: $\tanh \left({4 x}\right) = \dfrac {4 \tanh x + 4 \tanh^3 x} {1 + 6 \tanh^2 x + \tanh^4 x}$


Power Reduction Formulas/Hyperbolic Sine Squared

:$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$


Power Reduction Formulas/Hyperbolic Cosine Squared

:$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$


Power Reduction Formulas/Hyperbolic Sine to 4th

:$\sinh^4 x = \dfrac {3 - 4 \cosh 2 x + \cosh 4 x} 8$


Prosthaphaeresis Formulas/Hyperbolic Sine plus Hyperbolic Sine

:$\sinh x + \sinh y = 2 \sinh \left({\dfrac {x + y} 2}\right) \cosh \left({\dfrac {x - y} 2}\right)$


Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine

:$\sinh x - \sinh y = 2 \map \cosh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$


Prosthaphaeresis Formulas/Hyperbolic Cosine plus Hyperbolic Cosine

:$\cosh x + \cosh y = 2 \map \cosh {\dfrac {x + y} 2} \map \cosh {\dfrac {x - y} 2}$


Simpson's Formulas/Hyperbolic Sine by Hyperbolic Sine

:$\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$


Simpson's Formulas/Hyperbolic Cosine by Hyperbolic Cosine

:$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$


Simpson's Formulas/Hyperbolic Sine by Hyperbolic Cosine

:$\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$


Graph of Hyperbolic Cosine Function


Inverse Hyperbolic Sine of Reciprocal equals Inverse Hyperbolic Cosecant

Everywhere that the function is defined: :$\map {\sinh^{-1} } {\dfrac 1 x} = \csch^{-1} x$ where $\sinh^{-1}$ and $\csch^{-1}$ denote inverse hyperbolic sine and inverse hyperbolic cosecant respectively.


Inverse Hyperbolic Cosine of Reciprocal equals Inverse Hyperbolic Secant

Everywhere that the function is defined: :$\map {\cosh^{-1} } {\dfrac 1 x} = \sech^{-1} x$ where $\sinh^{-1}$ and $\csch^{-1}$ denote inverse hyperbolic cosine and inverse hyperbolic secant respectively.


Inverse Hyperbolic Tangent of Reciprocal equals Inverse Hyperbolic Cotangent

Everywhere that the function is defined: :$\map {\tanh^{-1} } {\dfrac 1 x} = \coth^{-1} x$ where $\tanh^{-1}$ and $\coth^{-1}$ denote inverse hyperbolic tangent and inverse hyperbolic cotangent respectively.


Inverse Hyperbolic Sine is Odd Function

Let $x \in \R$. Then: :$\map {\sinh^{-1} } {-x} = -\sinh^{-1} x$ where $\map {\sinh^{-1} } {-x}$ denotes the inverse hyperbolic sine function.


Inverse Hyperbolic Tangent is Odd Function

:$\map {\tanh^{-1} } {-x} = -\tanh^{-1} x$


Inverse Hyperbolic Cosecant is Odd Function

Let $x \in \R$. Then: :$\map {\csch^{-1} } {-x} = -\csch^{-1} x$ where $\map {\csch^{-1} } {-x}$ denotes the inverse hyperbolic cosecant function.


Hyperbolic Cosecant in terms of Cosecant

Let $z \in \C$ be a complex number. Then: :$i \csch z = -\csc \paren {i z}$ where: : $\csc$ denotes the cosecant function : $\csch$ denotes the hyperbolic cosecant : $i$ is the imaginary unit: $i^2 = -1$.


Hyperbolic Secant in terms of Secant

Let $z \in \C$ be a complex number. Then: :$\sech z = \sec \paren {i z}$ where: : $\sec$ denotes the secant function : $\sech$ denotes the hyperbolic secant : $i$ is the imaginary unit: $i^2 = -1$.


Hyperbolic Cotangent in terms of Cotangent

Let $z \in \C$ be a complex number. Then: :$\coth z = -\cot \paren {i z}$ where: : $\cot$ denotes the cotangent function : $\coth$ denotes the hyperbolic cotangent : $i$ is the imaginary unit: $i^2 = -1$.


Sine in terms of Hyperbolic Sine

Let $z \in \C$ be a complex number. Then: :$i \sin z = \map \sinh {i z}$ where: :$\sin$ denotes the complex sine :$\sinh$ denotes the hyperbolic sine :$i$ is the imaginary unit: $i^2 = -1$.


Cosine in terms of Hyperbolic Cosine

Let $z \in \C$ be a complex number. Then: :$\cos z = \map \cosh {i z}$ where: :$\cos$ denotes the complex cosine :$\cosh$ denotes the hyperbolic cosine :$i$ is the imaginary unit: $i^2 = -1$.


Tangent in terms of Hyperbolic Tangent

Let $z \in \C$ be a complex number. Then: :$i \tan z = \map \tanh {i z}$ where: :$\tan$ denotes the tangent function :$\tanh$ denotes the hyperbolic tangent :$i$ is the imaginary unit: $i^2 = -1$.


Cosecant in terms of Hyperbolic Cosecant

Let $z \in \C$ be a complex number. Then: :$i \csc = -\csch \paren {i z}$ where: : $\csc$ denotes the cosecant function : $\csch$ denotes the hyperbolic cosecant : $i$ is the imaginary unit: $i^2 = -1$.


Secant in terms of Hyperbolic Secant

Let $z \in \C$ be a complex number. Then: :$\sec z = \map \sech {i z}$ where: :$\sec$ denotes the secant function :$\sech$ denotes the  hyperbolic secant :$i$ is the imaginary unit: $i^2 = -1$.


Cotangent in terms of Hyperbolic Cotangent

Let $z \in \C$ be a complex number. Then: :$i \cot z = -\coth \paren {i z}$ where: : $\cot$ denotes the cotangent function : $\coth$ denotes the hyperbolic cotangent : $i$ is the imaginary unit: $i^2 = -1$.


General Periodicity Property

Let $f: X \to X$ be a periodic function, where $X$ is either the set of real numbers $\R$ or the set of complex numbers $\C$. Let $L$ be a periodic element of $f$. Then: :$\forall n \in \Z: \forall x \in X: \map f x = \map f {x + n L}$ That is, after every distance $L$, the function $f$ repeats itself.


Periodicity of Hyperbolic Sine

Let $k \in \Z$. Then: :$\map \sinh {x + 2 k \pi i} = \sinh x$


Periodicity of Hyperbolic Cosine

Let $k \in \Z$. Then: :$\cosh \left({x + 2 k \pi i}\right) = \cosh x$


Periodicity of Hyperbolic Tangent

Let $k \in \Z$. Then: :$\tanh \left({x + 2 k \pi i}\right) = \tanh x$


Periodicity of Hyperbolic Cosecant

Let $k \in \Z$. Then: :$\operatorname{csch} \left({x + 2 k \pi i}\right) = \operatorname{csch} x$


Periodicity of Hyperbolic Secant

Let $k \in \Z$. Then: :$\map \sech {x + 2 k \pi i} = \sech x$


Periodicity of Hyperbolic Cotangent

Let $k \in \Z$. Then: :$\map \coth {x + 2 k \pi i} = \coth x$


Inverse Sine of Imaginary Number

:$\map {\sin^{-1} } {i x} = i \sinh^{-1} x$


Inverse Hyperbolic Sine of Imaginary Number

:$\sinh^{-1} \left({i x}\right) = i \sin^{-1} x$


Inverse Hyperbolic Tangent of Imaginary Number

:$\map {\tanh^{-1} } {i x} = i \tan^{-1} x$


Inverse Hyperbolic Cotangent of Imaginary Number

:$\map {\coth^{-1} } {i x} = i \cot^{-1} x$


Inverse Cosecant of Imaginary Number

:$\map {\csc^{-1} } {i x} = i \csch^{-1} x$


Inverse Hyperbolic Cosecant of Imaginary Number

:$\map {\csch^{-1} } {i x} = -i \csc^{-1} x$


Arccosecant of Reciprocal equals Arcsine

:$\map \arccsc {\dfrac 1 x} = \arcsin x$


Arcsecant of Reciprocal equals Arccosine

:$\map \arcsec {\dfrac 1 x} = \arccos x$


Arccotangent of Reciprocal equals Arctangent

:$\map \arccot {\dfrac 1 x} = \arctan x$


Solution to Quadratic Equation/Real Coefficients

Let $a, b, c \in \R$. The quadratic equation $a x^2 + b x + c = 0$ has: :Two real solutions if $b^2 - 4 a c > 0$ :One real solution if $b^2 - 4 a c = 0$ :Two complex solutions if $b^2 - 4 a c < 0$, and those two solutions are complex conjugates.


Sum of Roots of Quadratic Equation

Let $P$ be the quadratic equation $a x^2 + b x + c = 0$. Let $\alpha$ and $\beta$ be the roots of $P$. Then: :$\alpha + \beta = -\dfrac b a$


Product of Roots of Quadratic Equation

Let $P$ be the quadratic equation $a x^2 + b x + c = 0$. Let $\alpha$ and $\beta$ be the roots of $P$. Then: :$\alpha \beta = \dfrac c a$


Definition:Discriminant of Polynomial/Cubic Equation

Let: : $Q = \dfrac {3 a c - b^2} {9 a^2}$ : $R = \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}$ The '''discriminant''' of the cubic equation is given by: : $D := Q^3 + R^2$


Ring is Module over Itself

Let $\struct {R, +, \circ}$ be a ring. Then $\struct {R, +, \circ}_R$ is an $R$-module.  If $\struct {R, +, \circ}$ has a unity, then $\struct {R, +, \circ}_R$ is unitary.


Rational Division is Closed

The set of rational numbers less zero is closed under division: :$\forall a, b \in \Q_{\ne 0}: a / b \in \Q_{\ne 0}$


Complex Modulus of Product of Complex Numbers/General Result

Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: : $\cmod {z_1 z_2 \cdots z_n} = \cmod {z_1} \cdot \cmod {z_2} \cdots \cmod {z_n}$


Complex Modulus of Reciprocal of Complex Number

Let $z \in \C$ be a complex number such that $z \ne 0$. Let $\cmod z$ denote the complex modulus of $z$. Then: :$\cmod {\dfrac 1 z} = \dfrac 1 {\cmod z}$


Triangle Inequality/Complex Numbers/Corollary 1

Let $z_1, z_2 \in \C$ be complex numbers. Let $\cmod z$ be the modulus of $z$. Then: : $\cmod {z_1 + z_2} \ge \cmod {z_1} - \cmod {z_2}$


Complex Modulus of Additive Inverse

Let $z \in \C$ be a complex number. Let $-z$ be the negative of $z$: :$z + \paren {-z} = 0$ Then: :$\cmod z = \cmod {\paren {-z} }$ where $\cmod z$ denotes the modulus of $z$.


Even Power of Negative Real Number

Let $x \in \R$ be a real number. Let $n \in \Z$ be an even integer. Then: :$\paren {-x}^n = x^n$


Complex Addition is Closed/Proof 1

The set of complex numbers $\C$ is closed under addition: :$\forall z, w \in \C: z + w \in \C$


Complex Addition is Closed/Proof 2


Cardinality of Cartesian Product/General Result

Let $\displaystyle \prod_{k \mathop = 1}^n S_k$ be the cartesian product of a (finite) sequence of sets $\sequence {S_n}$. Then: :$\displaystyle \card {\prod_{k \mathop = 1}^n S_k} = \prod_{k \mathop = 1}^n \card {S_k}$


Cardinality of Cartesian Product/General Result/Corollary

Let $S^n$ be a cartesian space. Then: :$\card {S^n} = \card S^n$


Product of Complex Numbers in Polar Form/General Result

Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers. Let $z_j = \polar {r_j, \theta_j}$ be $z_j$ expressed in polar form for each $j \in \set {1, 2, \ldots, n}$. Then: :$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$


Complex Roots of Unity are Vertices of Regular Polygon Inscribed in Circle

Let $n \in \Z$ be an integer such that $n \ge 3$. Let $z \in \C$ be a complex number such that $z^n = 1$. Let $U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$ be the set of $n$th roots of unity. Let $U_n$ be plotted on the complex plane. Then the elements of $U_n$ are located at the vertices of a regular $n$-sided polygon $P$, such that: :$(1):\quad$ $P$ is circumscribed by a unit circle whose center is at $\tuple {0, 0}$ :$(2):\quad$ one of those vertices is at $\tuple {1, 0}$.


Equation of Unit Circle in Complex Plane

Consider the unit circle $C$ whose center is at $\tuple {0, 0}$ on the complex plane. Its equation is given by: :$\cmod z = 1$ where $\cmod z$ denotes the complex modulus of $z$.


Equation of Unit Circle

Let the unit circle have its center at the origin of the Cartesian plane. Its equation is given by: :$x^2 + y^2 = 1$ {{expand|Present it in polar coordinates as well}}


Tautological Consequent

:$p \implies \top \dashv \vdash \top$


Tautological Antecedent

:$\top \implies p \dashv \vdash p$


Contradictory Consequent

:$p \implies \bot \dashv \vdash \neg p$


Contradictory Antecedent

:$\bot \implies p \dashv \vdash \top$


Biconditional with Tautology

:$p \iff \top \dashv \vdash p$


Unique Representation of Complex Number in Spherical Form

Let $\mathcal P$ be the complex plane. Let $\mathbb S$ be the unit sphere which is tangent to $\mathcal P$ at $\tuple {0, 0}$ (that is, where $z = 0$). Let the diameter of $\mathbb S$ perpendicular to $\mathcal P$ through $\tuple {0, 0}$ be $NS$ where $S$ is the point $\tuple {0, 0}$. Let the point $N$ be referred to as the '''north pole''' of $\mathbb S$ and $S$ be referred to as the '''south pole''' of $\mathbb S$. :900px Let $A$ be a point on $P$. Let the line $NA$ be constructed. Then $NA$ passes through exactly one point $A'$ on the surface of $\mathbb S$ apart from $N$. Similarly, let $A'$ be a point on the surface of $\mathbb S$ apart from $N$. Let the line $NA'$ be constructed. Then $NA'$ passes through exactly one point $A$ on $P$.


Exclusive Or with Contradiction

:$p \oplus \bot \dashv \vdash p$


Product of Complex Number with Conjugate by Dot and Cross Product

Let $z_1$ and $z_2$ be complex numbers. Then: :$\overline {z_1} z_2 = \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$z_1 \circ z_2$ denotes the complex dot product of $z_1$ with $z_2$ :$z_1 \times z_2$ denotes the complex cross product of $z_1$ with $z_2$.


Product of Complex Number with Conjugate in Exponential Form

Let $z_1$ and $z_2$ be complex numbers. Then: :$\overline {z_1} z_2 = \cmod {z_1} \, \cmod {z_2} e^{i \theta}$ where: :$\overline {z_1}$ denotes the complex conjugate of $z_1$ :$\cmod {z_1}$ denotes the complex modulus of $z_1$ :$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.


Complex Numbers are Perpendicular iff Dot Product is Zero

Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \ne 0$ and $z_2 \ne 0$. Then $z_1$ and $z_2$ are perpendicular {{iff}}: :$z_1 \circ z_2 = 0$ where $z_1 \circ z_2$ denotes the complex dot product of $z_1$ with $z_2$.


Complex Numbers are Parallel iff Cross Product is Zero

Let $z_1$ and $z_2$ be complex numbers in vector form such that $z_1 \ne 0$ and $z_2 \ne 0$. Then $z_1$ and $z_2$ are parallel {{iff}}: :$z_1 \times z_2 = 0$ where $z_1 \times z_2$ denotes the complex cross product of $z_1$ with $z_2$.


Tautology iff Negation is Unsatisfiable

Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is a tautology iff its negation $\neg \mathbf A$ is unsatisfiable.


Satisfiable iff Negation is Falsifiable

Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is satisfiable iff its negation $\neg \mathbf A$ is falsifiable.


Subset of Satisfiable Set is Satisfiable

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathscr M$-satisfiable set of formulas from $\mathcal L$. Let $\mathcal F'$ be a subset of $\mathcal F$. Then $\mathcal F'$ is also $\mathscr M$-satisfiable.


Satisfiable Set Union Tautology is Satisfiable

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathscr M$-satisfiable set of formulas from $\mathcal L$. Let $\phi$ be a tautology for $\mathscr M$. Then $\mathcal F \cup \left\{{\phi}\right\}$ is also $\mathscr M$-satisfiable.


Unsatisfiable Set Union Formula is Unsatisfiable

Let $\LL$ be a logical language. Let $\mathscr M$ be a formal semantics for $\LL$. Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$. Let $\phi$ be a logical formula. Then $\FF \cup \set \phi$ is also $\mathscr M$-unsatisfiable.


Superset of Unsatisfiable Set is Unsatisfiable

Let $\LL$ be a logical language. Let $\mathscr M$ be a formal semantics for $\LL$. Let $\FF$ be an $\mathscr M$-unsatisfiable set of formulas from $\LL$. Let $\FF'$ be a superset of $\FF$. Then $\FF'$ is also $\mathscr M$-unsatisfiable.


Unsatisfiable Set minus Tautology is Unsatisfiable

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be an $\mathscr M$-unsatisfiable set of formulas from $\mathcal L$. Let $\phi \in \mathcal F$ be a tautology. Then $\mathcal F \setminus \set {\phi}$ is also $\mathscr M$-unsatisfiable.


Semantic Consequence of Set Union Formula

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a set of logical formulas from $\mathcal L$. Let $\phi$ be an $\mathscr M$-semantic consequence of $\mathcal F$. Let $\psi$ be another logical formula. Then: :$\mathcal F \cup \left\{{\psi}\right\} \models_{\mathscr M} \phi$ that is, $\phi$ is also a semantic consequence of $\mathcal F \cup \left\{{\psi}\right\}$.


Semantic Consequence of Superset

Let $\LL$ be a logical language. Let $\mathscr M$ be a formal semantics for $\LL$. Let $\FF$ be a set of logical formulas from $\LL$. Let $\phi$ be an $\mathscr M$-semantic consequence of $\FF$. Let $\FF'$ be another set of logical formulas. Then: :$\FF \cup \FF' \models_{\mathscr M} \phi$ that is, $\phi$ is also a semantic consequence of $\FF \cup \FF'$.


Semantic Consequence of Set minus Tautology

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a set of logical formulas from $\mathcal L$. Let $\phi$ be an $\mathscr M$-semantic consequence of $\mathcal F$. Let $\psi \in \mathcal F$ be a tautology. Then: :$\mathcal F \setminus \left\{{\psi}\right\} \models_{\mathscr M} \phi$ that is, $\phi$ is also a semantic consequence of $\mathcal F \setminus \left\{{\psi}\right\}$.


Set of Literals Satisfiable iff No Complementary Pairs

Let $S$ be a set of literals. Then $S$ is satisfiable {{iff}} it contains no complementary pairs.


Equivalence of Definitions of Exterior Point (Complex Analysis)

{{TFAE|def = Exterior Point (Complex Analysis)|view = exterior point|context = Complex Analysis}} Let $S \subseteq \C$ be a subset of the complex plane. Let $z_0 \in \C$.


Quintuple Angle Formulas/Sine/Corollary

For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$ :$\dfrac {\sin 5 \theta} {\sin \theta} = 16 \cos^4 \theta - 12 \cos^2 \theta + 1$ where $\sin$ denotes sine and $\cos$ denotes cosine.


Complex Multiplication as Geometrical Transformation

Let $z_1 = \left\langle{r_1, \theta_1}\right\rangle$ and $z_2 = \left\langle{r_2, \theta_2}\right\rangle$ be complex numbers expressed in polar form. Let $z_1$ and $z_2$ be represented on the complex plane $\C$ in vector form. Let $z = z_1 z_2$ be the product of $z_1$ and $z_2$. Then $z$ can be interpreted as the result of: : rotating $z_1$ about the origin of $\C$ by $\theta_2$ in the positive direction : multiplying the modulus of $z_1$ by $r_2$.


Argument of Quotient equals Difference of Arguments

Let $z_1$ and $z_2$ be complex numbers. Then: :$\map \arg {\dfrac {z_1} {z_2} } = \map \arg {z_1} - \map \arg {z_1} + 2 k \pi$ where: :$\arg$ denotes the argument of a complex number :$k$ can be $0$, $1$ or $-1$.


Cardinality of Finite Vector Space

Let $V$ be a $K$-vector space. Let $K$ be finite. Let the dimension of $V$ be finite. Then: :$\size V = \size K^{\map \dim V}$


Sum of Roots of Polynomial

Let $P$ be the polynomial equation: : $a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$ such that $a_n \ne 0$. The sum of the roots of $P$ is $-\dfrac {a_{n - 1} } {a_n}$.


Product of Roots of Polynomial

Let $P$ be the polynomial equation: :$a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0 = 0$ such that $a_n \ne 0$. The product of the roots of $P$ is $\dfrac {\paren {-1}^n a_0} {a_n}$.


Sum of Sines of Fractions of Pi

Let $n \in \Z$ such that $n > 1$. Then: :$\displaystyle \sum_{k \mathop = 1}^{n - 1} \sin \frac {2 k \pi} n = 0$


Area of Parallelogram in Complex Plane

Let $z_1$ and $z_2$ be complex numbers expressed as vectors. Let $ABCD$ be the parallelogram formed by letting $AD = z_1$ and $AB = z_2$. Then the area $\AA$ of $ABCD$ is given by: :$\AA = z_1 \times z_2$ where $z_1 \times z_2$ denotes the cross product of $z_1$ and $z_2$.


Area of Triangle in Determinant Form

Let $A = \tuple {x_1, y_1}, B = \tuple {x_2, y_2}, C = \tuple {x_3, y_3}$ be points in the Cartesian plane. The area $\mathcal A$ of the triangle whose vertices are at $A$, $B$ and $C$ is given by: :$\mathcal A = \dfrac 1 2 \size {\paren {\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix} } }$


Equation of Circle in Complex Plane/Formulation 2

Let $\C$ be the complex plane. Let $C$ be a circle in $\C$. Then $C$ may be written as: :$\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$ where: :$\alpha \in \R_{\ne 0}$ is real and non-zero :$\gamma \in \R$ is real :$\beta \in \C$ is complex such that $\cmod \beta^2 > \alpha \gamma$. The curve $C$ is a straight line {{iff}} $\alpha = 0$ and $\beta \ne 0$.


Equation of Line in Complex Plane/Formulation 1

Let $\C$ be the complex plane. Let $L$ be a straight line in $\C$. Then $L$ may be written as: :$\beta z + \overline \beta \overline z  + \gamma = 0$ where $\gamma$ is real and $\beta$ may be complex.


Vertices of Equilateral Triangle in Complex Plane

Let $z_1$, $z_2$ and $z_3$ be complex numbers. Then: : $z_1$, $z_2$ and $z_3$ represent on the complex plane the vertices of an equilateral triangle {{iff}}: :${z_1}^2 + {z_2}^2 + {z_3}^2 = z_1 z_2 + z_2 z_3 + z_3 z_1$


Product of Sines of Fractions of Pi

Let $m \in \Z$ such that $m > 1$. Then: :$\displaystyle \prod_{k \mathop = 1}^{m - 1} \sin \frac {k \pi} m = \frac m {2^{m - 1} }$


Product of Complex Conjugates/General Result

Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers. Let $\overline z$ be the complex conjugate of the complex number $z$. Then: :$\displaystyle \overline {\prod_{j \mathop = 1}^n z_j} = \prod_{j \mathop = 1}^n \overline {z_j}$


Quotient of Complex Conjugates

Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ be the complex conjugate of the complex number $z$. Then: :$\overline {\paren {\dfrac {z_1} {z_2} } } = \dfrac {\paren {\overline {z_1} } } {\paren {\overline {z_2} } }$ for $z_2 \ne 0$.


Condition for Points in Complex Plane to form Parallelogram

Let $A = z_1$, $B = z_2$, $C = z_3$ and $D = z_4$ represent on the complex plane the vertices of a quadrilateral. Then $ABCD$ is a parallelogram {{iff}}: :$z_1 - z_2 + z_3 - z_4 = 0$


Quadrilateral with Bisecting Diagonals is Parallelogram

Let $ABCD$ be a quadrilateral. Let the diagonals of $ABCD$ bisect each other. Then $ABCD$ is a parallelogram.


Medians of Triangle Meet at Point

Let $\triangle ABC$ be a triangle. Then the medians of $\triangle ABC$ meet at a single point. This point is called the centroid of $\triangle ABC$. {{expand|State where that point is, that is $1/3$ the way along each median}}


Midpoints of Sides of Quadrilateral form Parallelogram

Let $\Box ABCD$ be a quadrilateral. Let $E, F, G, H$ be the midpoints of $AB, BC, CD, DA$ respectively. Then $\Box EFGH$ is a parallelogram.


Line from Bisector of Side of Parallelogram to Vertex Trisects Diagonal

Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AD$. Then the point at which the line $BE$ meets $AC$ trisects $AC$.


Sum of Complex Numbers in Exponential Form

Let $z_1 = r_1 e^{i \theta_1}$ and $z_2 = r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form. Let $z_3 = r_3 e^{i \theta_3} = z_1 + z_2$. Then: :$r_3 = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \cos \left({\theta_1 - \theta_2}\right)}$ :$\theta_3 = \map \arctan {\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2} {r_1 \cos \theta_1 + r_2 \cos \theta_2} }$


De Moivre's Formula/Positive Integer Index

Let $z \in \C$ be a complex number expressed in polar form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall n \in \Z_{>0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \map \sin {n x} }$


De Moivre's Formula/Integer Index

Let $z \in \C$ be a complex number expressed in complex form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall n \in \Z: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$


De Moivre's Formula/Negative Integer Index

Let $z \in \C$ be a complex number expressed in complex form: :$z = r \left({\cos x + i \sin x}\right)$ Then: :$\forall n \in \Z_{\le 0}: \paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$


De Moivre's Formula/Rational Index

Let $z \in \C$ be a complex number expressed in complex form: :$z = r \paren {\cos x + i \sin x}$ Then: :$\forall p \in \Q: \paren {r \paren {\cos x + i \sin x} }^p = r^p \paren {\map \cos {p x} + i \, \map \sin {p x} }$


Dot Product Operator is Commutative

:$\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u$


Dot Product with Self is Non-Negative

:$\mathbf u \cdot \mathbf u \ge 0$


Dot Product with Self is Zero iff Zero Vector

:$\mathbf u \cdot \mathbf u = 0 \iff \mathbf u = \mathbf 0$


Vector Cross Product is Anticommutative/Complex

:$\forall z_1, z_2 \in \C: z_1 \times z_2 = -\paren {z_2 \times z_1}$


Polar Form of Complex Conjugate

Let $z := r \left({\cos \theta + i \sin \theta}\right) \in \C$ be a complex number expressed in polar form. Then: :$\overline z = r \left({\cos \theta - i \sin \theta}\right)$ where $\overline z$ denotes the complex conjugate of $z$.


Exponential Form of Complex Conjugate

Let $z := r e^{i \theta} \in \C$ be a complex number expressed in exponential form. Then: :$\overline z = r e^{-i \theta}$ where $\overline z$ denotes the complex conjugate of $z$.


Complex Dot Product in Exponential Form

Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be complex numbers expressed in exponential form. Then: :$z_1 \circ z_2 = r_1 r_2 \, \map \cos {\theta_2 - \theta_1}$ where $z_1 \circ z_2$ denotes the dot product of $z_1$ and $z_2$.


Complex Cross Product in Exponential Form

Let $z_1 := r_1 e^{i \theta_1}, z_2 := r_2 e^{i \theta_2} \in \C$ be complex numbers expressed in exponential form. Then: :$z_1 \times z_2 = r_1 r_2 \map \sin {\theta_2 - \theta_1}$ where $z_1 \times z_2$ denotes the dot product of $z_1$ and $z_2$.


Dot Product Distributes over Addition

:$\paren {\mathbf u + \mathbf v} \cdot \mathbf w = \mathbf u \cdot \mathbf w + \mathbf v \cdot \mathbf w$


Dot Product Operator is Bilinear

:$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$


Dot Product Associates with Scalar Multiplication

:$\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$


Semantic Tableau Algorithm Terminates

Let $\mathbf A$ be a WFF of propositional logic. Then the Semantic Tableau Algorithm for $\mathbf A$ terminates. Each leaf node of the resulting semantic tableau is marked.


Complement of Closed Set in Complex Plane is Open

Let $S \subseteq \C$ be a closed subset of the complex plane $\C$. Then the complement of $S$ in $\C$ is open.


Condition for Quartic with Real Coefficients to have Wholly Imaginary Root

Let $Q$ be the quartic equation: :$(1): \quad z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4 = 0$ such that all of $a_1, a_2, a_3, a_4$ are real numbers. Then $Q$ has a root which is wholly imaginary {{iff}}: :$\text {(a)}: \quad a_3^2 + a_1^2 a_4 = a_1 a_2 a_3$ :$\text {(b)}: \quad a_1 a_3 > 0$


Soundness and Completeness of Semantic Tableaus

Let $\mathbf A$ be a WFF of propositional logic. Let $T$ be a completed semantic tableau for $\mathbf A$. Then $\mathbf A$ is unsatisfiable {{iff}} $T$ is closed.


Semantic Tableau Algorithm is Decision Procedure for Tautologies

The Semantic Tableau Algorithm is a decision procedure for tautologies.


Soundness Theorem for Semantic Tableaus

Let $\mathbf A$ be a WFF of propositional logic. Let $T$ be a completed tableau for $\mathbf A$. Suppose that $T$ is closed. Then $\mathbf A$ is unsatisfiable for boolean interpretations.


Completeness Theorem for Semantic Tableaus

Let $\mathbf A$ be a WFF of propositional logic. Let $\mathbf A$ be unsatisfiable for boolean interpretations. Then every completed tableau for $\mathbf A$ is closed.


Cosine to Power of Even Integer/Proof 2

Let $n \in \Z$ be an even integer. Then: :$\displaystyle \cos^n \theta = \frac 1 {2^{n - 1} } \sum_{k \mathop = 0}^{n / 2} \paren {\binom n k \cos \paren {n - 2 k} \theta}$ That is: :$\cos^n \theta = \dfrac 1 {2^{n - 1} } \paren {\cos n \theta + n \cos \paren {n - 2} \theta + \dfrac {n \paren {n - 1} } 2 \cos \paren {n - 4} \theta + \cdots + \dfrac {n!} {\paren {\paren {\frac n 2}!}^2} \cos \theta}$


Conjugate of Real Polynomial is Polynomial in Conjugate

Let $\map P z$ be a polynomial in a complex number $z$. Let the coefficients of $P$ all be real. Then: :$\overline {\map P z} = \map P {\overline z}$ where $\overline z$ denotes the complex conjugate of $z$.


Let $z_1, z_2 \in \C$ be complex numbers. Let $L$ be a straight line through $z_1$ and $z_2$ in the complex plane. === Formulation 1 === {{:Equation for Line through Two Points in Complex Plane/Formulation 1}} === Parametric Form $1$ === {{:Equation for Line through Two Points in Complex Plane/Parametric Form 1}} === Parametric Form $2$ === {{:Equation for Line through Two Points in Complex Plane/Parametric Form 2}} === Symmetric Form === {{:Equation for Line through Two Points in Complex Plane/Symmetric Form}}


Absolute Value of Components of Complex Number no greater than Root 2 of Modulus

Let $z = x + i y \in \C$ be a complex number. Then: :$\size x + \size y \le \sqrt 2 \cmod z$ where: :$\size x$ and $\size y$ denote the absolute value of $x$ and $y$ :$\cmod z$ denotes the complex modulus of $z$.


Exclusive Or with Factor of Exclusive Or

:$\left({p \oplus q}\right) \oplus q \dashv \vdash p$


Biconditional with Factor of Biconditional

:$\paren {p \iff q} \iff q \dashv \vdash p$


Conditional iff Biconditional of Consequent with Disjunction

:$p \implies q \dashv \vdash q \iff \left({p \lor q}\right)$


Functionally Incomplete Logical Connectives

These sets of logical connectives are ''not'' functionally complete.


Functionally Incomplete Logical Connectives/Conjunction and Disjunction

:$\set {\land, \lor}$: And and Or


Functionally Incomplete Logical Connectives/Negation and Biconditional

:$\set {\neg, \iff}$: Not and Iff


Theory of Set of Formulas is Theory

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F$ be a set of $\mathcal L$-formulas. Let $T \left({\mathcal F}\right)$ be the $\mathcal L$-theory of $\mathcal F$. Then $T \left({\mathcal F}\right)$ is a theory.


Semantic Consequence is Transitive

Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. Let $\mathcal F, \mathcal G$ and $\mathcal H$ be sets of $\mathcal L$-formulas. Suppose that: :$\mathcal F \models_{\mathscr M} \mathcal G$ :$\mathcal G \models_{\mathscr M} \mathcal H$ Then $\mathcal F \models_{\mathscr M} \mathcal H$.


Lemniscate of Bernoulli as Locus in Complex Plane

The locus of $z$ on the complex plane such that: :$\cmod {z - a} \cmod {z - a} = a^2$ is a lemniscate of Bernoulli.


Sum of Cosines of Arithmetic Sequence of Angles

Let $\alpha \in \R$ be a real number such that $\alpha \ne 2 \pi k$ for $k \in \Z$. Then:


Sum of Sines of Arithmetic Sequence of Angles


Product of Tangent and Cotangent

:$\tan \theta \cot \theta = 1$


Product of Cotangents of Fractions of Pi

Let $m \in \Z$ such that $m > 1$. Then: :$\displaystyle \prod_{k \mathop = 1}^{m - 1} \cot \frac {k \pi} {2 m} = 1$


Uncountable Set less Countable Set is Uncountable

Let $S$ be an uncountable set. Let $T \subseteq S$ be a countable subset of $S$. Then: :$S \setminus T$ is uncountable where $\setminus$ denotes set difference.


Irrational Numbers are Uncountably Infinite

The set $\R \setminus \Q$ of irrational numbers is uncountable.


Product of Diagonals from Point of Regular Polygon

Let $A_0, A_1, \ldots, A_{n - 1}$ be the vertices of a regular $n$-gon $P = A_0 A_1 \cdots A_{n - 1}$ which is circumscribed by a unit circle. Then: :$\displaystyle \prod_{k \mathop = 2}^{n - 2} A_0 A_k = \frac {n \csc^2 \frac \pi n} 4$ where $A_0 A_k$ is the length of the line joining $A_0$ to $A_k$.


Condition on Conjugate from Real Product of Complex Numbers

Let $z_1, z_2 \in \C$ be complex numbers such that $z_1 z_2 \in \R_{\ne 0}$. Then: :$\exists p \in \R: z_1 = p \overline {z_2}$ where $\overline {z_2}$ denotes the complex conjugate of $z_2$.


Exponential of Sum

=== Real Numbers === {{:Exponential of Sum/Real Numbers}} === Complex Numbers === {{:Exponential of Sum/Complex Numbers}} == Corollary == {{:Exponential of Sum/Corollary}}


Cayley-Menger Determinant

Let $S$ be a $j$-simplex in $\R^n$. Let the vertices of $S$ be $v_1, v_2, \ldots, v_{j+1}$. Let $B = \left[{\beta_{ij} }\right]$ denote the $\left({j + 1}\right) \times \left({j + 1}\right)$ matrix given by: :$\beta_{ij} = \left\vert{v_i - v_j}\right\vert_2^2$ where $\left\vert{v_i - v_j}\right\vert_2$ is the vector 2-norm of the vector $v_i - v_j$. Then the content $V_j$ is given by: {{begin-eqn}} {{eqn | l = V_j^2 \left({S}\right)         | r = \frac {\left({-1}\right)^{j+1} } {2^j \left({j!}\right)^2} \det C  }} {{end-eqn}} where $C$ is the $\left({j + 2}\right) \times \left({j + 2}\right)$ matrix obtained from $B$ by bordering $B$ with a top row $\left({0, 1, \ldots, 1}\right)$ and a left column $\left({0, 1, \ldots, 1}\right)^\intercal$.


Rule of Commutation/Disjunction/Formulation 2/Forward Implication

:$\vdash \left({p \lor q}\right) \implies \left({q \lor p}\right)$


Provable by Gentzen Proof System iff Negation has Closed Tableau

Let $\mathscr G$ be instance 1 of a Gentzen proof system.


Provable by Gentzen Proof System iff Negation has Closed Tableau/Formula

Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is a $\mathscr G$-theorem iff: :$\neg \mathbf A$ has a closed semantic tableau where $\neg \mathbf A$ is the negation of $\mathbf A$.


Provable by Gentzen Proof System iff Negation has Closed Tableau/Set of Formulas

Let $U$ be a set of WFFs of propositional logic. Then $U$ is a $\mathscr G$-theorem iff: :$\bar U$ has a closed semantic tableau where $\bar U$ is the set comprising the logical complements of all WFFs in $U$.