Complement of Closed Interval Defined by Absolute Value

:$\set {x \in \R: \size {\xi - x} > \delta} = \R \setminus \closedint {\xi - \delta} {\xi + \delta}$ where: :$\closedint {\xi - \delta} {\xi + \delta}$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$ :$\setminus$ denotes the set difference operator.


Size of y-1 lt n and Size of y+1 gt 1 over n

Let $T_n \subseteq \R$ be the subset of the set of real numbers $\R$ defined as: :$T_n = \set {y: \size {y - 1} < n \land \size {y + 1} > \dfrac 1 n}$ Then: :$T_n = \openint {1 - n} {-1 - \dfrac 1 n} \cup \openint {-1 + \dfrac 1 n} {1 + n}$


Real Part as Mapping is Surjection

Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Re z$ where $\map \Re z$ denotes the real part of $z$. Then $f$ is a surjection.


Imaginary Part as Mapping is Surjection

Let $f: \C \to \R$ be the projection from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Im z$ where $\map \Im z$ denotes the imaginary part of $z$. Then $f$ is a surjection.


Condition for Mapping from Quotient Set to be Well-Defined

Let $S$ and $T$ be sets. Let $\RR$ be an equivalence relation on $S$. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $S / \RR$ be the quotient set of $S$ induced by $\RR$. Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$. Then: :there exists a mapping $\phi: S / \RR \to T$ such that $\phi \circ q_\RR = f$ {{iff}}: :$\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$ ::$\begin {xy} \xymatrix@L + 2mu@ + 1em {  S \ar[r]^*{f}    \ar[d]_*{q_\RR} & T \\ S / \RR \ar@{-->}[ur]_*{\phi} } \end {xy}$


Condition for Mapping from Quotient Set to be Surjection

Let the mapping $\phi: S / \RR \to T$ defined as: :$\phi \circ q_\RR = f$ be well-defined. Then: :$\phi$ is a surjection {{iff}}: :$f$ is a surjection.


Condition for Mapping from Quotient Set to be Injection

Let the mapping $\phi: S / \RR \to T$ defined as: :$\phi \circ q_\RR = f$ be well-defined. Then: :$\phi$ is an injection {{iff}}: :$\forall x, y \in S: \tuple {x, y} \in \RR \iff \map f x = \map f y$


Mapping from Quotient Set when Defined is Unique

Let the mapping $\phi: S / \RR \to T$ defined as: :$\phi \circ q_\RR = f$ be well-defined. Then $\phi$ is unique.


Conditions for Commutative Diagram on Quotient Mappings between Mappings

Let $A$ and $B$ be sets. Let $\RR_S$ and $\RR_T$ be equivalence relations on $S$ and $T$ respectively. Let $f: S \to T$ be a mapping from $S$ to $T$. Let $S / \RR_S$ and $T / \RR_T$ be the quotient sets of $S$ and $T$ induced by $\RR_S$ and $\RR_T$ respectively. Let $q_S: S \to S / \RR_S$ and $q_T: T \to T / \RR_T$ be the quotient mappings induced by $\RR_S$ and $\RR_T$ respectively. Then a mapping $g: S / \RR_S \to T / \RR_T$ exists such that: :$q_T \circ f = g \circ q_S$ {{iff}}: :$\forall x, y \in S: x \mathrel {\RR_S} y \implies \map f x \mathrel {\RR_T} \map f y$ ::$\begin {xy} \xymatrix@L + 2mu@ + 1em {  S \ar[r]^*{f}    \ar[d]_*{q_S} & T \ar[d]^*{q_T} \\ S / \RR_S \ar@{-->}[r]_*{g} & T / \RR_T } \end {xy}$


Subset of Finite Dimensional Normed Vector Space is Compact iff Closed and Bounded/Sufficient Condition

Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. Let $K \subset X$ be a compact subset. Then $K$ is closed and bounded.


Factors of Group Direct Product are not Subgroups

Let $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$ be groups. Let $\struct {G \times H, \circ}$ be the group direct product of $\struct {G, \circ_1}$ and $\struct {H, \circ_2}$. Then neither $\struct {G, \circ_1}$ nor $\struct {H, \circ_2}$ is a subgroup of $\struct {G \times H, \circ}$.


Additive Group of Complex Numbers is Direct Product of Reals with Reals

Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Then the direct product $\struct {\R, +} \times \struct {\R, +}$ is isomorphic with $\struct {\C, +}$.


Imaginary Numbers under Addition form Group

Let $\II$ denote the set of complex numbers of the form $0 + i y$ That is, let $\II$ be the set of all wholly imaginary numbers. Then the algebraic structure $\struct {\II, +}$ is a group.


Imaginary Numbers under Multiplication do not form Group

Let $\II$ denote the set of complex numbers of the form $0 + i y$ for $y \in \R_{\ne 0}$. That is, let $\II$ be the set of all wholly imaginary non-zero numbers. Then the algebraic structure $\struct {\II, \times}$ is not a group.


Set of Isometries in Complex Plane under Composition forms Group

Let $S$ be the set of all complex functions $f: \C \to \C$ which preserve distance when embedded in the complex plane. That is: :$\size {\map f a - \map f b} = \size {a - b}$ Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$. Then $\struct {S, \circ}$ is a group.


Set of Affine Mappings on Real Line under Composition forms Group

Let $S$ be the set of all real functions $f: \R \to \R$ of the form: :$\forall x \in \R: \map f x = r x + s$ where $r \in \R_{\ne 0}$ and $s \in \R$ Let $\struct {S, \circ}$ be the algebraic structure formed from $S$ and the composition operation $\circ$. Then $\struct {S, \circ}$ is a group.


Arbitrary Cyclic Group of Order 4

Let $S = \set {1, 2, 3, 4}$. Consider the algebraic structure $\struct {S, \circ}$ given by the Cayley table: :$\begin{array}{r|rrrr} \circ & 2 & 3 & 4 & 1 \\ \hline 2 & 2 & 3 & 4 & 1  \\ 3 & 3 & 4 & 1 & 2 \\ 4 & 4 & 1 & 2 & 3 \\ 1 & 1 & 2 & 3 & 4 \\ \end{array}$ Then $\struct {S, \circ}$ is a group. Specifically, $\struct {S, \circ}$ is the cyclic group of order $4$.


Factors of Sums of Powers of 100,000/General Result

All integers $n$ of the form: :$n = \displaystyle \sum_{k \mathop = 0}^m 10^{r k}$ for $m \in \Z_{> 0}$ are composite for $r \ge 2$. The only exceptions are $r = 2^k, m = 1$ for some $k \in \N$, and $r = m + 1 =$ some odd prime, where $n$ could be prime.


Odd Integers under Addition do not form Subgroup of Integers

Let $S$ denote the set of odd integers. Then $\struct {S, +}$ is not a subgroup of the additive group of integers $\struct {\Z, +}$.


Equivalence of Definitions of Closed Set in Normed Vector Space

{{TFAE|def = Closed Set in Normed Vector Space|view = Closed Set|context = Normed Vector Space|contextview = Normed Vector Spaces}} Let $V = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $F \subseteq X$.


Cosets of Positive Reals in Multiplicative Group of Complex Numbers

Let $S$ be the positive real axis of the complex plane: :$S = \set {z \in \C: z = x + 0 i, x \in \R_{>0} }$ Consider the algebraic structure $\struct {S, \times}$ as a subgroup of the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$. The cosets of $\struct {S, \times}$ are the sets of the form: :$\set {z \in \C: \exists r \in \R_{>0}: z = r e^{i \theta}}$ for some $\theta \in \hointr 0 {2 \pi}$ That is, the sets of all complex numbers with a constant argument.


Morphism from Multiplicative Group of Complex Numbers to Unit Circle

Let $\struct {\C_{\ne 0}, \times}$ denote the multiplicative group of complex numbers. Let $f: \C_{\ne 0} \to \C_{\ne 0}$ be the mapping defined as: :$\forall z \in \C_{\ne 0}: \map f z = \dfrac z {\cmod z}$ where $\cmod z$ denotes the modulus of $z$. Then $f$ is an endomorphism on $\struct {\C_{\ne 0}, \times}$ whose kernel is the positive real axis: :$\set {z \in \C: z = x + 0 i, x \in \R_{>0} }$ and whose image is the unit circle: :$\set {z \in \C: \cmod z = 1}$


Leigh.Samphier/Sandbox/P-adic Expansion Representative of P-adic Number is Unique

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\mathbf a$ be an equivalence class in $\Q_p$. Let $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ be $p$-adic expansions that represent $\mathbf a$. Then: :$(1) \quad m = k$ :$(2) \quad \forall i \ge m : d_i = e_i$ That is, the $p$-adic expansions $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ are identical.


Rational Numbers are Dense Subfield of P-adic Numbers

Let $p$ be a prime number. Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Let $\phi: \Q \to \Q_p$ be the mapping defined by: :$\map \phi r = \sequence {r, r, r, \dotsc} + \NN$ where $\sequence {r, r, r, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {r, r, r, \dotsc}$. Then: :$\Q$ is isometrically isomorphic to $\map \phi \Q$ which is a dense subfield of $\Q_p$.


Graph is 0-Regular iff Edgeless

Let $G$ be a graph. Then $G$ is $0$-regular graph {{iff}} $G$ is edgeless.


Edgeless Graph of Order n has n Components

Let $N_n$ denote the edgeless graph with $n$ vertices. Then $N_n$ has $n$ components.


Complete Graph of Order 1 is Edgeless

The complete graph $K_1$ of order $1$ is the edgeless graph $N_1$.


Edgeless Graph of Order 1 is Tree

Let $N_1$ denote the edgeless graph with $1$ vertex. Then $N_1$ is a tree.


Edgeless Graph of Order Greater than 1 is Forest

Let $N_n$ denote the edgeless graph with $n$ vertices such that $n > 1$. Then $N_n$ is a forest.


Size of Complete Graph

Let $K_n$ denote the complete graph of order $n$ where $n \ge 0$. The size of $K_n$ is given by: :$\size {K_n} = \dfrac {n \paren {n - 1} } 2$


Field Operations of P-adic Numbers as Quotient of Cauchy Sequences

Let $p$ be a prime number. Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. Then the field operations on $\Q_p$ are defined by: :$+ : \quad \forall \sequence {x_n} + \NN, \sequence {y_n} + \NN \in \CC \, \big / \NN$: :::$\quad \paren {\sequence {x_n} + \NN} + \paren {\sequence {y_n} + \NN} = \sequence {x_n + y_n} + \NN$ :$\circ : \quad \forall \sequence {x_n} + \NN, \sequence {y_n} + \NN \in \CC \, \big / \NN$:  :::$\quad \paren {\sequence {x_n} + \NN} \paren {\sequence {y_n} + \NN} = \sequence {x_n y_n} + \NN$


Size of Graph/Examples/Order 3 Graphs

Let $G$ be a simple graph of order $3$. Then it is possible for $G$ to have a size of $0$, $1$, $2$ or $3$. Examples of each are presented below: :600px


Simple Graph where All Vertices and All Edges are Adjacent

Let $G$ be a simple graph in which: :every vertex is adjacent to every other vertex and: :every edge is adjacent to every other edge. Then $G$ is of order no greater than $3$.


Simple Graph whose Vertices all Incident but Edges not Adjacent

Let $G = \struct {V, E}$ be a simple graph such that: :every vertex is incident with at least one edge :no two edges are adjacent to each other. Then $G$ has an even number of vertices.


Size of Star Graph

Let $S_n$ denote the star graph of order $n$ where $n > 0$. The size of $S_n$ is given by: :$\size {S_n} = n - 1$


Smallest Simple Graph with One Vertex Adjacent to All Others

Let $G = \struct {V, E}$ be a simple graph of order $n$. Let $G$ be the simple graph with the smallest size such that one vertex is adjacent to all other vertices of $G$. Then $G$ is the star graph of order $n$ and is of size $n - 1$.


Heine-Borel Theorem/Normed Vector Space

Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. A subset $K \subset X$ is compact {{iff}} $K$ is closed and bounded.


Heine-Borel Theorem/Necessary Condition

Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. Let $K \subseteq X$ be closed and bounded . Then $K$ is a compact subset.


Heine-Borel Theorem/Sufficient Condition

Let $\struct {X, \norm {\,\cdot\,}}$ be a finite-dimensional normed vector space. Let $K \subset X$ be a compact subset. Then $K$ is closed and bounded.


Maximum Number of Arcs in Digraph

Let $D_n$ be a digraph of order $n$ such that $n \ge 1$. Let $D_n$ have the greatest number of arcs of all digraphs of order $n$. The number of arcs in $D$ is given by: :$\size {D_n} = n \paren {n - 1}$


Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $x \in \Q_p: x \ne 0$. Then: :$\exists v \in \Z: \norm x_p = p^{-v}$ === Lemma === {{:Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma}}


Leigh.Samphier/Sandbox/P-adic Norm of p-adic Number is Power of p/Lemma

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. Let $\sequence {x_n}$ be a Cauchy sequence in $\struct{\Q, \norm {\,\cdot\,}_p}$ such that $\sequence {x_n}$ does not converge to $0$. Then: :$\exists v \in \Z: \displaystyle \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$


Cycle Graph is Hamiltonian

Let $n \in \Z$ be an integer such that $n \ge 3$. Let $C_n$ denote the cycle graph of order $n$. Then $C_n$ is Hamiltonian.


Maximum Degree of Vertex in Simple Graph

Let $G = \struct {V, E}$ be a simple graph. Let $\card V$ denote the order of $G$. Then no vertex of $G$ has a degree higher than $\card V - 1$.


Degrees of Vertices determine Order and Size of Graph

Let $G = \struct {V, E}$ be a simple graph. Let the degrees of each of the vertices of $G$ be given. Then it is possible to determine both the order $\card V$ and size $\card E$ of $G$.


Order and Size of Graph do not determine Degrees of Vertices

Let $G = \struct {V, E}$ be a simple graph. Let both the order $\card V$ and size $\card E$ of $G$ be given. Then it is not always possible to determine the degrees of each of the vertices of $G$.


Size of Regular Graph in terms of Degree and Order

Let $G = \struct {V, E}$ be an $r$-regular graph of order $p$. Let $q$ denote the size of $G$. Then: :$q = \dfrac {p r} 2$ when such an $r$-regular graph exists. If an $r$-regular graph of order $p$ does exist, then $p r$ is an even integer.


Same Degrees of Vertices does not imply Graph Isomorphism

Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs such that: :$\card {\map V G} = \card {\map V H}$ where $\card {\map V G}$ denotes the order of $G$. Let $\phi: G \to H$ be a mapping which preserves the degrees of the vertices: :$\forall v \in \map V G: \map {\deg_H} {\map \phi v} = \map {\deg_G} v$ Then it is not necessarily the case that $\phi$ is an isomorphism.


Isomorphism Classes for Order 2 Simple Graphs

There are $2$ equivalence classes for simple graphs of order $2$ under graph isomorphism: :the edgeless graph of order $2$ and :the complete graph of order $2$.


Isomorphism Classes for Order 4 Size 3 Simple Graphs

There are $3$ equivalence classes for simple graphs of order $4$ and size $3$ under isomorphism: :400px


Differential Equation governing First-Order Reaction

Let a substance decompose spontaneously in a '''first-order reaction'''. The differential equation which governs this reaction is given by: :$-\dfrac {\d x} {\d t} = k x$ where: :$x$ determines the quantity of substance at time $t$. :$k \in \R_{>0}$.


Formula for Radiocarbon Dating

Let $Q$ be a quantity of a sample of dead organic material (usually wood) whose time of death is to be determined. Let $t$ years be the age of $Q$ which is to be determined. Let $r$ denote the ratio of the quantity of carbon-14 remaining in $Q$ after time $t$ to the quantity of carbon-14 in $Q$ at the time of its death. Then the number of years that have elapsed since the death of $Q$ is given by: :$t = -8060 \ln r$


Leigh.Samphier/Sandbox/Representative of P-adic Sum

Let $p$ be any prime number. Let $\Q_p$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. and $\norm {\,\cdot\,}_p$ denotes the p-adic norm on the rationals $\Q$. Let $x, y$ be any left cosets of $\Q_p$. Let $\sequence{x_n}$ and $\sequence{y_n}$ be any repesentatives of $x$ and $y$ respectively. Then: :the sequence $\sequence{x_n + y_n}$ is a repesentative of $x + y$


Leigh.Samphier/Sandbox/Representative of P-adic Product

Let $p$ be any prime number. Let $\Q_p$ be the $p$-adic numbers as a quotient of Cauchy sequences. That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where: :$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ :$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$. and $\norm {\,\cdot\,}_p$ denotes the p-adic norm on the rationals $\Q$. Let $x, y$ be any left cosets of $\Q_p$. Let $\sequence{x_n}$ and $\sequence{y_n}$ be any repesentatives of $x$ and $y$ respectively. Then: :the sequence $\sequence{x_n y_n}$ is a repesentative of $x y$


Leigh.Samphier/Sandbox/Complete Normed Division Ring is Completion of Dense Subring

Let $\struct {R, \norm {\, \cdot \,} }$ be a complete normed division ring. Let $\struct {S, \norm {\, \cdot \,}}$ be a dense normed division subring of $\struct {R, \norm {\, \cdot \,}}$. Then: :$\struct {R, \norm {\, \cdot \,} }$ is a completion of $\struct {S, \norm {\, \cdot \,}}$ where the inclusion mapping $i : S \to R$ is the required distance-preserving ring monomorphism.


Leigh.Samphier/Sandbox/Distance-Preserving Homomorphism Preserves Norm

Let $\struct {R_1, \norm {\, \cdot \,}_1 }, \struct {R_2, \norm {\, \cdot \,}_2 }$ be normed division rings. Let $\phi: R_1 \to R_2$ be a distance-preserving ring homomorphism. Then: :$\forall x \in R_1 : \norm{\map \phi x}_2 = \norm x_1$


Leigh.Samphier/Sandbox/Element of Completion is Limit of Sequence in Normed Division Ring

Let $\struct {R_1, \norm {\, \cdot \,}_1 }, \struct {R_2, \norm {\, \cdot \,}_2 }$ be normed division rings. Let $\struct {R_2, \norm {\, \cdot \,}_2 }$ be a completion of $\struct {R_1, \norm {\, \cdot \,}_1 }$ with distance-preserving ring monomorphism $\phi: R_1 \to R_2$. Then for all $x \in R_2$, there exists a sequence $\sequence{x_n}$ in $R_1$: :$x = \displaystyle \lim_{n \mathop \to \infty} \map \phi {x_n}$


Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion

Let $\struct {R_1, \norm {\, \cdot \,}_1 }$ be a normed division ring. Let $\struct {R_2, \norm {\, \cdot \,}_2 }$ be a normed division ring completion of $\struct {R_1, \norm {\, \cdot \,}_1 }$ with distance-preserving ring monomorphism $\phi: R_1 \to R_2$. Then for all $x \in R_2$, there exists a sequence $\sequence{x_n}$ in $R_1$: :$x = \displaystyle \lim_{n \mathop \to \infty} \map \phi {x_n}$ and :$\norm x_2 = \displaystyle \lim_{n \mathop \to \infty} \norm {x_n}_1$


Leigh.Samphier/Sandbox/Normed Division Ring Determines Norm on Completion/Corollary

Let $\struct {R, \norm {\, \cdot \,} }$ be a complete normed division ring. Let $\struct {S, \norm {\, \cdot \,}}$ be a dense normed division subring of $\struct {R, \norm {\, \cdot \,}}$. Then for all $x \in R$, there exists a sequence $\sequence{x_n}$ in $S$: :$x = \displaystyle \lim_{n \mathop \to \infty} x_n$ and :$\norm x = \displaystyle \lim_{n \mathop \to \infty} \norm {x_n}$


Leigh.Samphier/Sandbox/Inclusion Mapping on Normed Division Subring is Distance Preserving Monomorphism

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {S, \norm {\, \cdot \,}_S}$ be a normed division subring of $\struct {R, \norm {\, \cdot \,}}$. Then the inclusion mapping $i : S \to R$ is a distance-preserving ring monomorphism.


Period of Reciprocal of Prime

Consider the decimal expansion of the reciprocal $\dfrac 1 p$ of a prime $p$. If $p \nmid a$, the decimal expansion of $\dfrac 1 p$ is periodic in base $a$ and its period of recurrence is the order of $a$ modulo $p$. If $p \divides a$, the decimal expansion of $\dfrac 1 p$ in base $a$ terminates.


Maximum Period of Reciprocal of Prime

Let $p$ be a prime number such that $p$ is not a divisor of $10$. The period of recurrence of the reciprocal of $p$ when expressed in decimal notation is less than or equal to $p - 1$.


Maximum Abscissa for Loop of Folium of Descartes

Consider the folium of Descartes defined in parametric form as: :$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$ :500px The point on the loop at which the $x$ value is at a maximum occurs when $t = \sqrt [3] {\dfrac 1 2}$, corresponding to the point $P$ defined as: :$P = \tuple {2^{2/3} a, 2^{1/3} a}$


Condition for Denesting of Square Root/Lemma

Let $a, b, c, d \in \Q_{\ge 0}$. Suppose $\sqrt b \notin \Q$. Then: :$\sqrt {a + \sqrt b} = \sqrt {c + \sqrt d} \implies a = c, b = d$


First Order ODE/y' + y = 0

The first order ODE: :$\dfrac {\d y} {\d x} + y = 0$ has the general solution: :$y = C e^{-x}$ where $C$ is an arbitrary constant.


Order-Preserving Mapping Not Always Semilattice Homomorphism

Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices. Let $\preceq_1$ be the ordering on $S$ defined by: :$a \preceq_1 b \iff \paren {a \circ b} = b$ Let $\preceq_2$ be the ordering on $T$ defined by: :$x \preceq_2 y \iff \paren {x * y} = y$ Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be an order-preserving mapping. Then: : $\phi: \struct {S, \circ} \to \struct {T, *}$ may not be a semilattice homomorphism


Factorial as Product of Consecutive Factorials/Lemma 2

Let $n \in \N$. Then $\paren {2 n - 2}! \, \paren {2 n - 1}! > \paren {3 n - 1}!$ for all $n \ge 7$.


Factorial as Product of Consecutive Factorials/Lemma 1

Let $n \in \N$. Then $\paren {2 n - 1}! \, \paren {2 n}! > \paren {3 n - 1}!$ for all $n > 1$.


Entire Function with Bounded Real Part is Constant

Let $f : \C \to \C$ be an entire function.  Let the real part of $f$ be bounded. That is, there exists a positive real number $M$ such that:  :$\cmod {\map \Re {\map f z} } < M$ for all $z \in \C$, where $\map \Re {\map f z}$ denotes the real part of $\map f z$. Then $f$ is constant.


Mass of Mole of Isotope of Element

Let $S$ be a substance made up entirely of a particular isotope $Q$ of a particular element. Let one atom of $Q$ contain $n$ neutrons and $p$ protons. Then one mole of $S$ has a mass of approximately $n + p$ grams.


Mass of Mole of Substance

Let $S$ be a substance with molecular weight $W_S$. Then one mole of $S$ has a mass of $W_S$ grams.


Linear Second Order ODE/y'' = y'

The second order ODE: :$(1): \quad y'' = y'$ has the general solution: :$y = A_1 e^x + A_2$


First Order ODE/x y' = 2 y

The first order ODE: :$x y' = 2 y$ has the general solution: :$y = C x^2$ where $C$ is an arbitrary constant.


Singleton in Normed Vector Space is Closed

Let $X$ be a vector space over $\R$ or $\C$. Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x \in X$ be a singleton. Then $x$ is closed.


Singleton Set is Nowhere Dense in Rational Space

Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then every singleton subset of $\Q$ is nowhere dense in $\struct {\Q, \tau_d}$.


Real Function of Two Variables represents Surface in Cartesian 3-Space

Let $S$ and $T$ be subsets of the set of real numbers $\R$. Let $f: S \times T \to \R$ be a real function of two variables. Then the locus of $f$ describes a surface embedded in the Cartesian space $\R^3$.


Cauchy's Lemma (Number Theory)

Let $a$ and $b$ be odd positive integers.  Suppose $a$ and $b$ satisfy:  :$b^2 < 4 a$ :$3 a < b^2 + 2 b + 4$ Then there exist non-negative integers $s, t, u, v$ such that: {{begin-eqn}} {{eqn | l = a       | r = s^2 + t^2 + u^2 + v^2 }} {{eqn | l = b       | r = s + t + u + v }} {{end-eqn}}


Finite Subset of Normed Vector Space is Closed

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $F \subseteq X$ be finite. Then $F$ is closed in $M$.


Primitive of x fourth by Cosine of a x

:$\ds \int x^4 \cos a x \rd x = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12 \sin a x} {a^3} x^2 - \frac {24 \cos a x} {a^4} x + \frac {24 \sin a x} {a^5} + C$


Primitive of x sixth by Cosine of a x

:$\displaystyle \int x^6 \cos a x \rd x = \frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^5 - \frac {30 \sin a x} {a^3} x^4 - \frac {120 \cos a x} {a^4} x^3 + \frac {360 \sin a x} {a^5} x^2 + \frac {720 \cos a x} {a^6} x - \frac {720 \sin a x} {a^7} + C$


Fourier Series/Sixth Power of x over Minus Pi to Pi

:$\displaystyle x^6 = \frac {\pi^6} 7 + \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 + 1440} {n^6} \cos n \pi \cos n x$


Derivative of x to the a x

:$\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$


Integer as Sum of Polygonal Numbers/Lemma 1

Let $n, m \in \N_{>0}$ such that $m \ge 3$. Let $n < 116 m$. Then $n$ can be expressed as a sum of at most $m + 2$ polygonal numbers of order $m + 2$.


Integer as Sum of Polygonal Numbers/Lemma 3

Let $n, m, r \in \R_{>0}$. Suppose $\dfrac n m > 1$. Let $b \in \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$. Define: :$a = 2 \paren {\dfrac {n - b - r} m} + b = \paren {1 - \dfrac 2 m} b + 2 \paren {\dfrac {n - r} m}$ Then $a, b$ satisfy: :$b^2 < 4 a$ :$3 a < b^2 + 2 b + 4$


Integer as Sum of Polygonal Numbers/Lemma 2

Let $n, m \in \R_{>0}$ such that $\dfrac n m \ge 1$. Define $I$ to be the open real interval: :$I = \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$ Then: :For $\dfrac n m \ge 116$, the length of $I$ is greater than $4$.


Integer as Sum of Three Odd Squares

Let $r$ be a positive integer. Then: :$r \equiv 3 \pmod 8$ {{iff}}: :$r$ is the sum of $3$ odd squares. {{explain|Could this be an {{iff}} proof? Could also be named more precisely.</br>It is now. I referred to Integer as Sum of Three Squares when naming this. Could be classified as a corollary}}


Join Semilattice Ordered Subset Not Always Subsemilattice

Let $\struct {S, \circ}$ be a semilattices. Let $\preceq$ be the ordering on $S$ defined by: :$a \preceq b \iff \paren {a \circ b} = b$ Let $T$ be a subset of $S$. Let the ordered subset $\struct{T, \preceq \restriction_T}$ be a join semilattice. Let $\vee$ be the binary operation on $S$ defined by: :for all $a, b \in S$, $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$. Then: :$\struct{T, \vee}$ may not be a subsemilattice of $\struct {S, \circ}$.


Closure of Subset of Closed Set of Topological Space is Subset

Let $T$ = $\struct {S, \tau}$ be a topological space. Let $F$ be a closed set of $T$. Let $H \subseteq F$ be a subset of $F$. Let $H^-$ denote the closure of $H$. Then $H^- \subseteq F$.


Number of Partial Derivatives of Order n

Let $u = \map f {x_1, x_2, \ldots, x_m}$ be a function of the $m$ independent variables $x_1, x_2, \ldots, x_m$. There are $m^n$ partial derivatives of $u$ of order $n$.


Normed Vector Space is Open in Itself

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the set $X$ is an open set of $M$.


Normed Vector Space is Open in Itself/Proof 1

Let $M = \struct{X, \norm {\, \cdot \,}}$ be a normed vector space. Then the set $X$ is an open set of $M$.


Normed Vector Space is Open in Itself/Proof 2


Empty Set is Closed in Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the empty set $\O$ is closed in $M$.


Empty Set is Open in Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the empty set $\O$ is an open set of $M$.


Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent

Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.  Then $\sequence {f_n}$ is uniformly Cauchy on $S$ {{iff}} $\sequence {f_n}$ converges uniformly on $S$.


Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition

Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$. Let $\sequence {f_n}$ be uniformly Cauchy on $S$.  Then $\sequence {f_n}$ is uniformly convergent on $S$.


Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Necessary Condition

Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$. Let $\sequence {f_n}$ be uniformly convergent on $S$. Then $\sequence {f_n}$ is uniformly Cauchy on $S$.


Open Ball of Point Inside Open Ball/Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct{X, \norm {\, \cdot \,}}$. Let $y \in \map {B_\epsilon} x$. Then: : $\exists \delta \in \R: \map {B_\delta} y \subseteq \map {B_\epsilon} x$ That is, for every point in an open $\epsilon$-ball in a normed vector space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.


Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function

Let $S \subseteq \R$. Let $x \in S$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$ converging uniformly to $f : S \to \R$. Let $f_n$ be continuous at $x$ for all $n \in \N$. Then $f$ is continuous at $x$.


Open Ball is Open Set/Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x \in X$. Let $\epsilon \in \R_{>0}$. Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball of $x$ in $M$. Then $\map {B_\epsilon} x$ is an open set of $M$.


Finite Intersection of Open Sets of Normed Vector Space is Open

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $U_1, U_2, \ldots, U_n$ be open in $M$. Then $\displaystyle \bigcap_{i \mathop = 1}^n U_i$ is open in $M$. That is, a finite intersection of open subsets is open.


Union of Open Sets of Normed Vector Space is Open

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. The union of a set of open sets of $M$ is open in $M$.


Definite Integral of Limit of Uniformly Convergent Sequence of Integrable Functions

Let $a, b \in \R$ with $a < b$.  Let $\sequence {f_n}$ be a sequence of Riemann integrable real functions $\closedint a b \to \R$ converging uniformly to $f : \closedint a b \to \R$. Then $f$ is integrable and:  :$\displaystyle \int_a^b \map f x \rd x = \lim_{n \to \infty} \int_a^b \map {f_n} x \rd x$


Intersection of Closed Sets is Closed/Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then the intersection of an arbitrary number of closed sets of $M$ (either finitely or infinitely many) is itself closed.


Normed Vector Space is Closed in Itself

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Then $X$ is closed in $M$.


Infinite Series of Functions is Uniformly Convergent iff Sequence of Partial Sums is Uniformly Cauchy

Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$. Then the infinite series:  :$\displaystyle \sum_{n \mathop = 1}^\infty f_n$ converges uniformly on $S$ {{iff}} for all $\varepsilon \in \R_{> 0}$ there exists $N \in \N$ such that: :$\displaystyle \size {\sum_{k \mathop = m + 1}^n \map {f_k} x} < \varepsilon$ for all $x \in S$ and $n > m > N$.


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

:$T_1 \cap T_2 = \overline {\overline T_1 \cup \overline T_2}$


LCM of 3 Integers in terms of GCDs of Pairs of those Integers/Lemma

Let $a, b, c \in \Z_{>0}$ be strictly positive integers. Then: :$\gcd \set {\gcd \set {a, b}, \gcd \set {a, c} } = \gcd \set {a, b, c}$


Events One of Which equals Intersection

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$. Let $A$ and $B$ be such that: :$A \cap B = A$ Then whenever $A$ occurs, it is always the case that $B$ occurs as well.


Union of Event with Complement is Certainty

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A \in \Sigma$ be an events of $\EE$, so that $A \subseteq \Omega$. Then: :$A \cup \overline A = \Omega$ where $\overline A$ is the complementary event to $A$. That is, $A \cup \overline A$ is a certainty.


Equivalence of Definitions of Adherent Point/Definition 1 iff Definition 2

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. {{TFAE|def = Adherent Point|view = adherent point of $A$}}


Equivalence of Definitions of Adherent Point/Definition 1 iff Definition 3


Compact Sets in Fortissimo Space

A set in Fortissimo space is compact {{iff}} it is finite.


Uniformly Convergent Series of Continuous Functions Converges to Continuous Function

Let $S \subseteq \R$. Let $x \in S$. Let $\sequence {f_n}$ be a sequence of real functions. Let $f_n$ be continuous at $x$ for all $n \in \N$. Let the infinite series:  :$\displaystyle \sum_{n \mathop = 1}^\infty f_n$ be uniformly convergent to a real function $f : S \to \R$. Then $f$ is continuous at $x$.


Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary

Let $S \subseteq \R$. Let $\sequence {f_n}$ be a sequence of real functions. Let $f_n$ be continuous for all $n \in \N$. Let the infinite series: :$\displaystyle \sum_{n \mathop = 1}^\infty f_n$ be uniformly convergent to a real function $f : S \to \R$. Then $f$ is continuous.


Euclidean Space is Banach Space/Proof 1

Let $m$ be a positive integer. Then the Euclidean space $\R^m$, along with the Euclidean norm, forms a Banach space over $\R$.


Euclidean Space is Banach Space/Proof 2


Monotone Function is of Bounded Variation

Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a monotone function. Then $f$ is of bounded variation.


Norm Equivalence is Equivalence

Let $X$ be a vector space. Let $\norm {\, \cdot \,}_a$ and $\norm {\, \cdot \,}_b$ be equivalent norms on $X$. Denote this relation by $\sim$:  :$\norm {\, \cdot \,}_a \sim \norm {\, \cdot \,}_b$. Then $\sim$ is an equivalence relation.


Continuous Non-Negative Real Function with Zero Integral is Zero Function

Let $a, b$ be real numbers with $a < b$.  Let $f : \closedint a b \to \R$ be a continuous function.  Let:  :$\map f x \ge 0$ for all $x \in \closedint a b$. Let:  :$\displaystyle \int_a^b \map f x \rd x = 0$ Then $\map f x = 0$ for all $x \in \closedint a b$.


Sequential Continuity is Equivalent to Continuity in the Reals/Sufficient Condition

Let $A \subseteq \R$ be a subset of the real numbers. Let $c \in A$. Let $f : A \to \R$ be a real function. Then if $f$ is continuous at $c$: :for each sequence $\sequence {x_n}$ in $A$ that converges to $c$, the sequence $\sequence {\map f {x_n} }$ converges to $\map f c$.


Sequential Continuity is Equivalent to Continuity in the Reals/Necessary Condition


Metric Closure and Topological Closure of Subset are Equivalent

Let $M = \struct{A, d}$ be a metric space. Let $T = \struct{A, \tau}$ be the topological space with the topology induced by $d$. Let $H \subseteq A$. Then: :the metric closure of $H$ in $M$ equals the topological closure of $H$ in $T$


Set together with Condensation Points is not necessarily Closed

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $\CC$ denote the set of condensation points of $H$. Then it is not necessarily the case that $H \cup \CC$ is a closed set of $T$.


Basis Test for Limit Point

Let $T = \struct {S, \tau}$ be a topological space. Let $\BB$ be a synthetic basis of $T$. Let $H \subseteq S$. Then $x \in S$ is a limit point of $H$ {{iff}}: :$\forall U \in \BB : x \in U$ satisfies $H \cap U \setminus \set x \ne \O$


Basis Test for Adherent Point

Let $T = \struct {S, \tau}$ be a topological space. Let $\BB$ be a synthetic basis of $T$. Let $H \subseteq S$. Then $x \in S$ is an adherent point of $H$ {{iff}}: :$\forall U \in \BB : x \in U$ satisfies $H \cap U \ne \O$


Set of Liouville Numbers is Uncountable

The set of Liouville numbers is uncountable.


Isolated Point in Metric Space iff Isolated Point in Topological Space

Let $M = \struct {A, d}$ be a metric space. Let $T = \struct {A, \tau}$ be the topological space with the topology induced by $d$. Let $H \subseteq A$. Let $x \in H$ Then: :$x$ is an isolated point of $H$ in $M$ {{iff}} $x$ is an isolated point of $H$ in $T$


Limit Point in Metric Space iff Limit Point in Topological Space

Let $M = \struct {A, d}$ be a metric space. Let $T = \struct {A, \tau}$ be the topological space with the topology induced by $d$. Let $H \subseteq A$. Then: :$x \in H$ is a limit point in $M$ {{iff}} $x$ is a limit point in $T$


Sum of Functions of Bounded Variation is of Bounded Variation

Let $a, b$ be real numbers with $a < b$. Let $f, g : \closedint a b \to \R$ be functions of bounded variation. Let $V_f$ and $V_g$ be the total variations of $f$ and $g$ respectively.  Then $f + g$ is of bounded variation with: :$V_{f + g} \le V_f + V_g$ where $V_{f + g}$ denotes the total variation of $f + g$.


Open Sets in Vector Spaces with Equivalent Norms Coincide

Let $M_a = \struct {X, \norm {\, \cdot \, }_a}$ and $M_b = \struct {X, \norm {\, \cdot \,}_b}$ be normed vector spaces. Let $U \subseteq X$ be an open set in $M_a$. Suppose, $\norm {\, \cdot \, }_a$ and $\norm {\, \cdot \,}_b$ are equivalent norms, i.e. $\norm {\, \cdot \, }_a \sim \norm {\, \cdot \,}_b$. Then $U$ is also open in $M_b$.


Local Basis Test for Isolated Point

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $x \in H$. Let $\BB_x$ be a local basis of $x$. Then $x$ is an isolated point of $H$ {{iff}}: :$\exists U \in \BB_x : U \cap H = \set x$


Local Basis Test for Limit Point

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $x \in S$. Let $\BB_x$ be a local basis of $x$. Then $x \in S$ is a limit point of $H$ {{iff}}: :$\forall U \in \BB_x : H \cap U \setminus \set x \ne \O$


Local Basis Test for Adherent Point

Let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $x \in S$. Let $\BB_x$ be a local basis of $x$. Then $x \in S$ is an adherent point of $H$ {{iff}}: :$\forall U \in \BB_x : H \cap U \ne \O$


Closure of Subset of Metric Space is Closed

Let $M = \left({A, d}\right)$ be a metric space. Let $H \subseteq A$ be a subset of $A$. Let $H^-$ denote the closure of $H$. Then $H^-$ is a closed set of $M$.


Convergent Sequences in Vector Spaces with Equivalent Norms Coincide

Let $M_a = \struct {X, \norm {\, \cdot \, }_a}$ and $M_b = \struct {X, \norm {\, \cdot \,}_b}$ be normed vector spaces. Let $\sequence {x_n}_{n \mathop \in \N}$ be an convergent sequence in $M_a$. Suppose, $\norm {\, \cdot \, }_a$ and $\norm {\, \cdot \,}_b$ are equivalent norms, i.e. $\norm {\, \cdot \, }_a \sim \norm {\, \cdot \,}_b$. Then $\sequence {x_n}_{n \mathop \in \N}$ is also convergent in $M_b$.


Open Mapping is not necessarily Closed Mapping

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $f: T_1 \to T_2$ be a mapping which is not a bijection. Let $f$ be an open mapping. Then it is not necessarily the case that $f$ is also a closed mapping.


Equivalence of Definitions of Limit Point of Filter Basis

{{TFAE|def = Limit Point of Filter Basis}} Let $T = \struct {S, \tau}$ be a topological space. Let $\FF$ be a filter on the underlying set $S$ of $T$. Let $\BB$ be a filter basis of $\FF$.


Richert's Theorem

Let $S = \set {s_1, s_2, \dots}$ be an infinite set of (strictly) positive integers, with the property: :$s_n < s_{n + 1}$ for every $n \in \N$ Suppose there exists some integers $N, k$ such that every integer $n$ with $N < n \le N + s_{k + 1}$: :$n$ can be expressed as a sum of distinct elements in $\set {s_1, s_2, \dots, s_k}$ :$s_{i + 1} \le 2 s_i$ for every $i \ge k$ Then for any $n > N$, $n$ can be expressed as a sum of distinct elements in $S$.


Number as Sum of Distinct Primes

For $n \ne 1, 4, 6$, $n$ can be expressed as the sum of distinct primes.


Differentiable Function of Bounded Variation may not have Bounded Derivative

Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a continuous function of bounded variation. Let $f$ be differentiable on $\openint a b$. Then $f'$ is not necessarily bounded.


Existence of Urysohn Function does not guarantee Normal Space

Let $T = \struct {S, \tau}$ be a regular space. Let $T$ have the property that: :For all closed sets $A, B \subseteq S$ of $T$ such that $A \cap B = \O$, there exists an Urysohn function for $A$ and $B$. Then it is not necessarily the case that $T$ is a normal space.


Existence of Compact Space which Satisfies No Separation Axioms

There exists at least one example of a compact space for which none of the Tychonoff separation axioms are satisfied.


Everywhere Dense iff Interior of Complement is Empty

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subset S$. Then $A$ is everywhere dense {{iff}}: :$\paren {\relcomp S A}^\circ = \O$ where $A^\circ$ is the interior of $A$.


Greatest Set is Unique

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. Then the greatest set of $\TT$, if it exists, must be unique.


Greatest Set may not Exist

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. The greatest set of $\TT$ may not exist.


Smallest Set is Unique

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. Then the smallest set of $\TT$, if it exists, must be unique.


Smallest Set may not Exist

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$. The smallest set of $\TT$ may not exist.


Mapping is Surjection if its Direct Image Mapping is Surjection

Let $f: S \to T$ be a mapping. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$. Let $f^\to$ be a surjection. Then $f: S \to T$ is also a surjection.


Nested Sequences in Complete Metric Space not Tending to Zero may be Disjoint

Let $M = \struct {A, d}$ be a complete metric space. Let $\family {S_k}_{k \mathop \in \N}$ be a nested sequence of closed balls in $M$. Let the radii of $\family {S_k}_{k \mathop \in \N}$ be convergent in $M$, but not to zero. Then it is not necessarily the case that their intersection $\displaystyle \bigcap S_k$ is non-empty.


Union of Interiors is Subset of Interior of Union

Let $T$ be a topological space. Let $\H$ be a set of subsets of $T$. That is, let $\H \subseteq \powerset T$ where $\powerset T$ is the power set of $T$. Then the union of the interiors of the elements of $\H$ is a subset of the interior of the union of $\H$. :$\displaystyle \bigcup_{H \mathop \in \H} H^\circ \subseteq \paren {\bigcup_{H \mathop \in \H} H}^\circ $


Closed Ball is Closed/Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $x \in X$. Let $\epsilon \in \R_{> 0}$. Let $\map {B_\epsilon^-} x$ be the closed $\epsilon$-ball of $x$ in $M$. Then $\map {B_\epsilon^-} x$ is a closed set of $M$.


Unit Sphere is Closed/Normed Vector Space

Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\Bbb S := \set {x \in X : \norm {x} = 1}$ be a unit sphere in $M$. Then $\Bbb S$ is closed in $M$.


Continued Fraction Expansion of Euler's Number/Proof 1/Lemma

:For $n \in \Z , n \ge 0$: {{begin-eqn}} {{eqn | l = A_n       | r = q_{3 n} e - p_{3 n} }} {{eqn | l = B_n       | r = p_{3 n + 1} - q_{3 n + 1} e }} {{eqn | l = C_n       | r = p_{3 n + 2} - q_{3 n + 2} e }} {{end-eqn}}


Limit Points in Particular Point Space/Subset

Let $U \subseteq S$ such that $p \in U$. Let $x \in S$ such that $x \ne p$. Then $x$ is a limit point of $U$.


Limit Points in Closed Extension Space/Subset

Let $U \subseteq S^*_p$ such that $p \in U$. Let $x \in S$. Then $x$ is a limit point of $U$.


Convergent Sequence in Particular Point Space

Let $T = \struct {S, \tau_p}$ be a particular point space. Let $\sequence {a_i}$ be a convergent sequence in $T$. Except for a finite number of indices, the terms of $\sequence {a_i}$ for which $a_i \ne p$ are all equal.


Accumulation Points for Sequence in Particular Point Space

Let $T = \struct {S, \tau_p}$ be a particular point space. Let $\sequence {a_i}$ be an infinite sequence in $T$. Let $\beta$ be an accumulation point of $\sequence {a_i}$. Then $\beta$ is such that an infinite number of terms of $\sequence {a_i}$ are equal either to $\beta$ or to $p$.


Absolute Value of Absolutely Continuous Function is Absolutely Continuous

Let $I \subseteq \R$ be a real interval. Let $f : I \to \R$ be an absolutely continuous function. Then $\size f$ is absolutely continuous.


Limit Points in Particular Point Space/Subset/Proof 2

Let $T = \struct {S, \tau_p}$ be a particular point space. {{:Limit Points in Particular Point Space/Subset}}


Limit Points in Particular Point Space/Subset/Proof 1


Sum of Absolutely Continuous Functions is Absolutely Continuous

Let $I \subseteq \R$ be a real interval. Let $f, g : I \to \R$ be absolutely continuous functions. Then $f + g$ is absolutely continuous.


Norms on Finite-Dimensional Real Vector Space are Equivalent

Norms on finite-dimensional real vector space are equivalent.


Integral of Distribution Function

Let $\struct {X, \Sigma, \mu}$ be a measure space and $f$ be a $\mu$-measurable function. Let $p > 0, r \geq 0$. For $\lambda > 0$, let $E_\lambda = \set {x \in X: \size {\map f x} > \lambda}$, so that $\map m \lambda = \map \mu {E_\lambda}$ is the distribution function of $f$. Then: :$\displaystyle \int_0^\infty p \lambda^{p - 1} \int_{E_\lambda} \size f^r \rd \mu \rd \lambda = \int_X \size f^{p + r} \rd \mu$ and in particular: :$\displaystyle \int_0^\infty p \lambda^{p - 1} \map m \lambda \rd \lambda = \int_X \size f^p \rd \mu$


Either-Or Topology is Compact

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a compact space.


Compact Space is Lindelöf

Every compact space is Lindelöf.


Limit Point of Underlying Set of Sequence of Reciprocals and Reciprocals + 1

Let $\sequence {a_n}$ denote the sequence defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $S$ denote the set of terms of $\sequence {a_n}$ considered as a subset of $\struct {\R, \tau_d}$. Then $0$ is a limit point of $S$.


Omega-Accumulation Point of Underlying Set of Sequence of Reciprocals and Reciprocals + 1

Let $\sequence {a_n}$ denote the sequence defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $S$ denote the set of terms of $\sequence {a_n}$ considered as a subset of $\struct {\R, \tau_d}$. Then $0$ is an $\omega$-accumulation point of $S$.


Accumulation Point of Sequence of Reciprocals and Reciprocals + 1

Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $\sequence {a_n}$ denote the sequence in $\struct {\R, \tau}$ defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Then $0$ is an accumulation point of $\sequence {a_n}$.


Zero is not a Limit Point of Sequence of Reciprocals and Reciprocals + 1

Let $\struct {\R, \tau}$ denote the real number line under the usual (Euclidean) topology. Let $\sequence {a_n}$ denote the sequence in $\struct {\R, \tau}$ defined as: {{begin-eqn}} {{eqn | l = a_n       | r = \begin {cases} \dfrac 2 {n + 1} & : \text {$n$ odd} \\ 1 + \dfrac 2 n & : \text {$n$ even} \end {cases}       | c =  }} {{eqn | r = \sequence {\dfrac 1 1, 1 + \dfrac 1 1, \dfrac 1 2, 1 + \dfrac 1 2, \dfrac 1 3, 1 + \dfrac 1 3, \dotsb}       | c =  }} {{end-eqn}} Then $0$ is not a limit point of $\sequence {a_n}$.


1 plus Power of 2 is not Perfect Power except 9

The only solution to: :$1 + 2^n = a^b$ is: :$\tuple {n, a, b} = \tuple {3, 3, 2}$ for positive integers $n, a, b$ with $b > 1$.


Fermat Number is not Perfect Power

There exist no Fermat numbers which are perfect powers.


Uncountable Closed Ordinal Space is Countably Compact

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is a countably compact space.


Sum of Euler Numbers by Binomial Coefficients Vanishes

$\forall n \in \Z_{>0}: \displaystyle \sum_{k \mathop = 0}^n \binom {2 n} {2 k} E_{2 k} = 0$ where $E_k$ denotes the $k$th Euler number. == Corollary == Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = -\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}       | c =  }} {{eqn | r = -\paren {\binom {2 n} 0 E_0 + \binom {2 n} 2 E_2 + \binom {2 n} 4 E_4 + \cdots + \binom {2 n} {2 n - 2} E_{2 n - 2} }       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number.


Uncountable Open Ordinal Space is Countably Compact

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is a countably compact space.


Uncountable Open Ordinal Space is not Metacompact

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a metacompact space.


Uncountable Open Ordinal Space is not Paracompact

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a paracompact space.


Uncountable Open Ordinal Space is not Lindelöf

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a Lindelöf space.


Uncountable Open Ordinal Space is not Sigma-Compact

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is not a $\sigma$-compact space.


Uncountable Closed Ordinal Space is Lindelöf

Let $\Omega$ denote the first uncountable ordinal. Let $\closedint 0 \Omega$ denote the closed ordinal space on $\Omega$. Then $\closedint 0 \Omega$ is a Lindelöf space.


Uncountable Open Ordinal Space is Sequentially Compact

Let $\Omega$ denote the first uncountable ordinal. Let $\hointr 0 \Omega$ denote the open ordinal space on $\Omega$. Then $\hointr 0 \Omega$ is a sequentially compact space.


Sum of Euler Numbers by Binomial Coefficients Vanishes/Corollary

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: {{begin-eqn}} {{eqn | l = E_{2 n}       | r = -\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}       | c =  }} {{eqn | r = -\paren {\binom {2 n} 0 E_0 + \binom {2 n} 2 E_2 + \binom {2 n} 4 E_4 + \cdots + \binom {2 n} {2 n - 2} E_{2 n - 2} }       | c =  }} {{end-eqn}} where $E_n$ denotes the $n$th Euler number.


Integer to Power of Multiple of Order/Corollary

Then $\map \phi n$ is a multiple of $c$, where $\map \phi n$ is the Euler phi function of $n$.


Divisor of Fermat Number/Euler's Result

Then $m$ is in the form: :$k \, 2^{n + 1} + 1$ where $k \in \Z_{>0}$ is an integer.


Divisor of Fermat Number/Refinement by Lucas

Let $n \ge 2$. Then $m$ is in the form: :$k \, 2^{n + 2} + 1$


Definition:Double Pointed Real Number Line

Let $T_\R = \struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$. Let $T = T_\R \times T_D$ be theproduct space of $T_\R$ and $T_D$. $T$ is known as the '''double pointed real number line'''.


Set Closure is Smallest Closed Set/Normed Vector Space

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $S$ be a subset of $X$: :$S \subseteq X$ Let $S^-$ be the closure of $S$. Then $S^-$ is the smallest closed set which contains $S$.


Bounded Real Function may not be of Bounded Variation

Let $a, b$ be real numbers with $a < b$. Let $f : \closedint a b \to \R$ be a bounded function. Then $f$ is not necessarily of bounded variation.


Element is Loop iff Member of Closure of Empty Set

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $x \in S$. Then: :$x$ is a loop {{iff}} $x \in \map \sigma \O$ where $\map \sigma \O$ denotes the closure of the empty set.


Singleton is Dependent implies Rank is Zero/Corollary

:$x$ is a loop {{iff}} $\map \rho {\set x} = 0$


Superset of Dependent Set is Dependent/Corollary

Let $A \subseteq S$. Let $x \in A$. If $x$ is a loop then $A$ is dependent.


Closure of Subset contains Loop

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $x$ be a loop of $M$. Let $A \subseteq S$. Then: :$x \in \map \sigma A$ where $\map \sigma A$ denotes the closure of $A$.


Element is Loop iff Singleton is Circuit

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $x \in S$. Then: :$x$ is a loop {{iff}} $\set x$ is a circuit


Element is Member of Base iff Not Loop

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $\mathscr B$ denote the set of all bases of $M$. Let $x \in S$. Then: :$\exists B \in \mathscr B: x \in B$ {{iff}} $x$ is not a loop


Parallel Relationship is Transitive

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $x, y, z \in S : x \ne y, x \ne z, y \ne z$. If $x$ is parallel to $y$ and $y$ is parallel to $z$ then $x$ is parallel to $z$.


Distinct Matroid Elements are Parallel iff Each is in Closure of Other

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$. Let $x, y \in S : x \ne y$. Then $x$ is parallel to $y$ {{iff}}: :$(1)\quad x$ and $y$ are not loops :$(2)\quad x \in \map \sigma {\set y}$ :$(3)\quad y \in \map \sigma {\set x}$


Closure of Subset Contains Parallel Elements

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\sigma: \powerset S \to \powerset S$ denote the closure operator of $M$. Let $A \subseteq S$. Let $x, y \in S$. If $x \in \map \sigma A$ and $y$ is parallel to $x$ then: :$y \in \map \sigma A$


Loop Belongs to Every Flat

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $x \in S$. If $x$ is a loop and $A$ is a flat subset then $x \in A$.


Parallel Elements Depend on Same Subsets

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $x, y \in S$. Let $x$ be parallel to $y$. Then: :$x$ depends on $A$ {{iff}} $y$ depends on $A$


Absolutely Continuous Real Function is Uniformly Continuous

Let $I \subseteq \R$ be a real interval. Let $f : I \to \R$ be an absolutely continuous function. Then $f$ is uniformly continuous.


Closure of Convex Subset in Normed Vector Space is Convex

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $C \subseteq X$ be a convex subset of $X$. Let $C^-$ be the closure of $C$. Then $C^- \subseteq X$ is also a convex subset of $X$.


Minimal Number of Distinct Prime Factors for Integer to have Abundancy Index Exceed Given Number

Let $r \in \R$. Let $\mathbb P^-$ be the set of prime numbers with possibly finitely many numbers removed. Define: :$M = \min \set {m \in \N: \displaystyle \prod_{i \mathop = 1}^m \frac {p_i} {p_i - 1} > r}$ where $p_i$ is the $i$th element of $\mathbb P^-$, ordered by size. Then $M$ satisfies: :$(1): \quad$ Every number formed with fewer than $M$ distinct prime factors in $\mathbb P^-$ has abundancy index less than $r$ :$(2): \quad$ There exists some number formed with $M$ distinct prime factors in $\mathbb P^-$ with abundancy index at least $r$ So $M$ is the minimal number of distinct prime factors in $\mathbb P^-$ a number must have for it to have abundancy index at least $r$. For $r$ an integer greater than $1$: If $\mathbb P^-$ is taken to be the set of all prime numbers, the values of $M$ are: :$2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, \cdots$ {{OEIS|A005579}} This theorem shows that this sequence is a subsequence of the sequence A256969 in the OEIS, only differing by an offset. If we require the numbers to be odd, we remove $2$ from $\mathbb P^-$. The sequence of values of $M$ are: :$3, 8, 21, 54, 141, 372, 995, 2697, 7397, 20502, \cdots$ {{OEIS|A005580}}


1 plus Square is not Perfect Power

The equation: :$x^p = y^2 + 1$ has no solution in the integers for $x, y, p > 1$.


1 plus Perfect Power is not Prime Power except for 9

The only solution to: :$x^m = y^n + 1$ is: :$\tuple {x, m, y, n} = \tuple {3, 2, 2, 3}$ for positive integers $x, y, m, n > 1$, and $x$ is a prime number. This is a special case of Catalan's Conjecture.


Field is Principal Ideal Domain

Let $F$ be a field. Then $F$ is a principal ideal domain.


Complex Vector Space is Vector Space

Let $\C$ denote the set of complex numbers. Then the complex vector space $\C^n$ is a vector space.


Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition

Let $\struct {X, \norm {\, \cdot \,}}$ is a normed vector space. Let $D \subseteq X$ be a subset of $X$. Let $D^-$ be the closure of $D$. Then $D$ is dense iff $D^- = X$.


Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition


Modulus of Limit/Normed Vector Space

Let $\struct {X, \norm { \, \cdot \, } }$ be a normed vector space. Let $\sequence {x_n}$ be a convergent sequence in $R$ to the limit $x$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = x$. Then :$\displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = \norm x$


Matrix Entrywise Addition forms Abelian Group

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$. Then $\struct {\map {\MM_R} {m, n}, +}$, where $+$ is matrix entrywise addition, is a group.


Definition:Negative Matrix/General Group

Let $\struct {G, \cdot}$ be a group. Let $\map {\MM_G} {m, n}$ denote the $m \times n$ matrix space over $\struct {G, \cdot}$. Let $\mathbf A = \sqbrk a_{m n}$ be an element of $\struct {\map {\MM_G} {m, n}, \circ}$, where $\circ$ is the Hadamard product. Then the '''negative (matrix) of $\mathbf A = \sqbrk a_{m n}$''' is denoted and defined as: :$-\mathbf A := \sqbrk {a^{-1} }_{m n}$ where $a^{-1}$ is the inverse element of $a \in G$.


Negative Matrix is Inverse for Matrix Entrywise Addition over Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$. Let $\mathbf A$ be an element of $\map {\MM_R} {m, n}$. Let $-\mathbf A$ be the negative of $\mathbf A$. Then $-\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition.


Zero Matrix is Identity for Matrix Entrywise Addition over Ring

Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. Let $\mathbf 0_R = \sqbrk {0_R}_{m n}$ be the zero matrix of $\map {\MM_R} {m, n}$. Then $\mathbf 0_R$ is the identity element for matrix entrywise addition.


Properties of Matrix Entrywise Addition over Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {R, +, \circ}$. Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$. Let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation of matrix entrywise addition satisfies the following properties: :$+$ is closed on $\map {\MM_R} {m, n}$ :$+$ is associative on $\map {\MM_R} {m, n}$ :$+$ is commutative on $\map {\MM_R} {m, n}$.


Matrix Entrywise Addition is Associative

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems. For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is associative on $\map \MM {m, n}$. That is: :$\paren {\mathbf A + \mathbf B} + \mathbf C = \mathbf A + \paren {\mathbf B + \mathbf C}$ for all $\mathbf A$, $\mathbf B$ and $\mathbf C$ in $\map \MM {m, n}$.


Matrix Entrywise Addition over Ring is Commutative

Let $\struct {R, +, \circ}$ be a ring. Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$. For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is commutative on $\map {\MM_R} {m, n}$. That is: :$\mathbf A + \mathbf B = \mathbf B + \mathbf A$ for all $\mathbf A$ and $\mathbf B$ in $\map {\MM_R} {m, n}$.


Zero Matrix is Identity for Matrix Entrywise Addition

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems. Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$. Then $\mathbf 0$ is the identity element for matrix entrywise addition.


Matrix Scalar Product is Associative

Let $\Bbb F$ denote one of the standard number systems. Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$. For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$. The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense: For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$: :$\lambda \paren {\mu \mathbf A} = \paren {\lambda \mu} \mathbf A$


Matrix Scalar Product Distributes over Number Addition

Let $\Bbb F$ denote one of the standard number systems. Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$. For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$. The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense: For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$: :$\paren {\lambda + \mu} \mathbf A = \lambda \mathbf A + \mu \mathbf A$


Negative Matrix is Inverse for Matrix Entrywise Addition

Let $\Bbb F$ denote one of the standard number systems. Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$. Let $\mathbf A$ be an element of $\map \MM {m, n}$. Let $-\mathbf A$ be the negative of $\mathbf A$. Then $-\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition.


Zero Matrix is Zero for Matrix Multiplication

Let $\struct {R, +, \times}$ be a ring. Let $\mathbf A$ be a matrix over $R$ of order $m \times n$ Let $\mathbf 0$ be a zero matrix whose order is such that either: :$\mathbf 0 \mathbf A$ is defined or: :$\mathbf A \mathbf 0$ is defined or both. Then: :$\mathbf 0 \mathbf A = \mathbf 0$ or: :$\mathbf A \mathbf 0 = \mathbf 0$ whenever they are defined. The order of $\mathbf 0$ will be according to the orders of the factor matrices.


Unit Matrix is Identity for Matrix Multiplication

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map {\MM_R} n$ denote the metric space of square matrices of order $n$ over $R$. Let $\mathbf I_n$ denote the unit matrix of order $n$: Then: :$\forall \mathbf A \in \map {\MM_R} n: \mathbf A \mathbf I_n = \mathbf A = \mathbf I_n \mathbf A$ That is, the unit matrix $\mathbf I_n$ is the identity element for (conventional) matrix multiplication over $\map {\MM_R} n$.


Unit Matrix is Identity for Matrix Multiplication/Left

Let $\map {\MM_R} {m, n}$ denote the $m \times n$ metric space over $R$. Let $I_m$ denote the unit matrix of order $m$. Then: :$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf I_m \mathbf A  = \mathbf A$


Unit Matrix is Identity for Matrix Multiplication/Right

Let $\map {\MM_R} {m, n}$ denote the $m \times n$ metric space over $R$. Let $I_n$ denote the unit matrix of order $n$. Then: :$\forall \mathbf A \in \map {\MM_R} {m, n}: \mathbf A \mathbf I_n  = \mathbf A$


Inverse of Square Matrix over Field is Unique

Let $\Bbb F$ be a field, usually one of the standard number fields $\Q$, $\R$ or $\C$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\map \MM n$ denote the matrix space of order $n$ square matrices over $\Bbb F$. Let $\mathbf B$ be an inverse matrix of $\mathbf A$. Then $\mathbf B$ is the only inverse matrix of $\mathbf A$.


Rank of Empty Set is Zero

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho : \powerset S \to \Z$ be the rank function of $M$. Then: :$\map \rho \O = 0$


Rank Function is Increasing

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\rho: \powerset S \to \Z$ be the rank function of $M$. Let $A, B \subseteq S$ be subsets of $S$ such that $A \subseteq B$. Then: :$\map \rho A \le \map \rho B$


Inverse of Transpose of Matrix is Transpose of Inverse

Let $\mathbf A$ be a matrix over a field. Let $\mathbf A^\intercal$ denote the transpose of $\mathbf A$. Let $\mathbf A$ be an invertible matrix. Then $\mathbf A^\intercal$ is also invertible and: :$\paren {\mathbf A^\intercal}^{-1} = \paren {\mathbf A^{-1} }^\intercal$ where $\mathbf A^{-1}$ denotes the inverse of $\mathbf A$.


Max yields Supremum of Parameters/General Case

Let $x_1, x_2, \dots ,x_n \in S$ for some $n \in \N_{>0}$. Then: :$\max \set {x_1, x_2, \dotsc, x_n} = \sup \set {x_1, x_2, \dotsc, x_n}$


Max of Subfamily of Operands Less or Equal to Max

Let $\struct {S, \preceq}$ be a totally ordered set. Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$. Let $\set{k_1, k_2, \dotsc, k_m} \subseteq \set{1, 2, \dotsc, n}$ Then: :$\max \set {x_{k_1}, x_{k_2}, \dotsc, x_{k_m}} \preceq \max \set {x_1, x_2, \dotsc, x_n}$ where: :$\max$ denotes the max operation


Elementary Matrix corresponding to Elementary Row Operation

Let $\mathbf I$ denote the unit matrix of order $m$ over a field $K$. Let $e$ be an elementary row operation on $\mathbf I$. Let $\mathbf E$ be the elementary row matrix of order $m$ uniquely defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Let $r_k$ denote the $k$th row of $\mathbf I$ for $1 \le k \le m$.


Elementary Matrix corresponding to Elementary Row Operation/Scale Row

Let $e$ be the elementary row operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply row $k$ of $\mathbf I$ by $\lambda$         | m = r_k \to \lambda r_k }} {{end-axiom}}


Elementary Matrix corresponding to Elementary Row Operation/Scale Row and Add

Let $e$ be the elementary row operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times row $j$ to row $i$         | m = r_i \to r_i + \lambda r_j }} {{end-axiom}}


Elementary Matrix corresponding to Elementary Row Operation/Exchange Rows

Let $e$ be the elementary row operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Interchange rows $i$ and $j$         | m = r_i \leftrightarrow r_j }} {{end-axiom}}


Power Set of Singleton

Let $x$ be an object. Then the power set of the singleton $\set x$ is: :$\powerset {\set x} = \set {\O, \set x}$


Row Operation to Clear First Column of Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$. Then there exists a row operation to convert $\mathbf A$ into another $m \times n$ matrix $\mathbf B = \sqbrk b_{m n}$ with the following properties: :$(1): \quad$ Except possibly for element $b_{1 1}$, all the elements of column $1$ are $0$ :$(2): \quad$ If $b_{1 1} \ne 0$, then $b_{1 1} = 1$. This process is referred to as '''clearing the first column'''.


Matrix is Row Equivalent to Echelon Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be a matrix of order $m \times n$ over a field $F$. Then $A$ is row equivalent to an echelon matrix of order $m \times n$.


Singleton is Independent implies Rank is One

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $x \in S$. Let $\set x$ be independent. Then: :$\map \rho {\set x} = 1$ where $\rho$ denotes the rank function of $M$.


Singleton is Dependent implies Rank is Zero

:$\map \rho {\set x} = 0$


System of Simultaneous Equations may have No Solution

Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is empty.


System of Simultaneous Equations may have Unique Solution

Let $S$ be a system of simultaneous equations. Then it is possible that $S$ may have a solution set which is a singleton.


Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent {{iff}} $X$ is a Banach space.


Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Necessary Condition

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$. Suppose $X$ is a Banach space. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.


Absolutely Convergent Series is Convergent iff Normed Vector Space is Banach/Sufficient Condition

Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space. Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $X$. Suppose $\ds \sum_{n \mathop = 1}^\infty a_n$ is convergent. Then $X$ is a Banach space.


Trivial Solution to System of Homogeneous Simultaneous Linear Equations is Solution

Let $S$ be a '''system of homogeneous simultaneous linear equations''': :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$ Consider the trivial solution to $A$: :$\tuple {x_1, x_2, \ldots, x_n}$ such that: :$\forall j \in \set {1, 2, \ldots, n}: x_j = 0$ Then the trivial solution is indeed a solution to $S$.


Sine of Integer Multiple of Argument/Formulation 3

{{begin-eqn}} {{eqn | l = \sin n \theta       | r = \sin \theta \cos^{n - 1} \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 1} \theta} {\cos^{n - 1} \theta} }       | c =  }} {{eqn | r = \sin \theta \cos^{n - 1} \theta \sum_{k \mathop = 0}^{n - 1} \frac {\cos k \theta} {\cos^k \theta}       | c =  }} {{end-eqn}}


Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations

Let $S$ be a system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$. Let $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ be obtained from $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ by means of an elementary row operation. Let $S'$ be the system of simultaneous linear equations of which $\begin {pmatrix} \mathbf A' & \mathbf b' \end {pmatrix}$ is the augmented matrix. Then $S$ and $S'$ are equivalent.


Existence of Inverse Elementary Row Operation

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be an elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is an elementary row operation which always exists and is unique.


Superset of Dependent Set is Dependent

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $A, B \subseteq S$ such that $A \subseteq B$ If $A$ is a dependent subset then $B$ is a dependent subset.


Powers of 16 Modulo 20

Let $n \in \Z_{> 0}$ be a strictly positive integer. Then: :$16^n \equiv 16 \pmod {20}$


Sine of Integer Multiple of Argument/Formulation 1/Lemma

:For $n \in \Z$: {{begin-eqn}} {{eqn | l = \map \cos {n \theta} \map \sin {\theta}        | r = \map \sin {n \theta} \map \cos {\theta} - \map \sin {\paren {n - 1 } \theta}  }} {{end-eqn}}


Conjugacy Class of Identity is only Conjugacy Class which is Subgroup

Let $G$ be a group. Let $e$ denote the identity of $G$. Let $\conjclass g$ denote the conjugacy class of the element $g$. Then conjugacy class of identity is the only conjugacy class which is a subgroup of $G$: :$\conjclass g < G \iff g = e$


Empty Group Word is Reduced

Let $S$ be a set Let $\epsilon$ be the empty group word on $S$. Then $\epsilon$ is reduced.


Closed Unit Ball is Convex Set

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space. Let $\map {B_1^-} 0$ be a closed unit ball in $X$. Then $\map {B_1^-} 0$ is convex.


Uncountable Sum as Series/Corollary

Let $f: X \to \closedint 0 {+\infty}$ have uncountably infinite support. Then: :$\displaystyle \sum_{x \mathop \in X} \map f x = +\infty$


Identity Matrix from Upper Triangular Matrix

Let $\mathbf A = \sqbrk a_{m n}$ be an upper triangular matrix of order $m \times n$ with no zero diagonal elements. Let $k = \min \set {m, n}$. Then $\mathbf A$ can be transformed into a matrix such that the first $k$ rows and columns form the unit matrix of order $k$.


Simultaneous Linear Equations have Solution iff Ranks of Matrix of Coefficients and Augmented Matrix are Equal

Let $S$ be a system of simultaneous linear equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ Let $S$ be expressed in matrix form as: :$\mathbf A \mathbf x = \mathbf b$ where: :$\mathbf A = \begin {pmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {pmatrix}$,  $\mathbf x = \begin {pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$, $\mathbf b = \begin {pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {pmatrix}$ Then $S$ has at least one solution {{iff}}: :$\map \rho {\mathbf A} = \map \rho {\begin {array} {c|c} \mathbf A & \mathbf b \end {array} }$ where: :$\map \rho {\mathbf A}$ denotes the rank of $\mathbf A$ :$\paren {\begin {array} {c|c} \mathbf A & \mathbf b \end {array} }$ denotes the augmented matrix of $S$.


Simultaneous Linear Equations has Unique Solution iff Rank of Matrix of Coefficients equals Number of Columns

Let $S$ be a system of $m$ simultaneous linear equations in $n$ variables: :$\displaystyle \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ Let $S$ be expressed in matrix form as: :$\mathbf A \mathbf x = \mathbf b$ where: :$\mathbf A = \begin {pmatrix} \alpha_{1 1} & \alpha_{1 2} & \cdots & \alpha_{1 n} \\ \alpha_{2 1} & \alpha_{2 2} & \cdots & \alpha_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{m 1} & \alpha_{m 2} & \cdots & \alpha_{m n} \\ \end {pmatrix}$,  $\mathbf x = \begin {pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$, $\mathbf b = \begin {pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_m \end {pmatrix}$ Then $S$ has exactly one solution {{iff}}: :$\map \rho {\mathbf A} = n$ where $\map \rho {\mathbf A}$ denotes the rank of $\mathbf A$.


Max Equals an Operand

Let $x_1, x_2, \dotsc, x_n \in S$ for some $n \in \N_{>0}$. Then: :$\exists i \in \closedint 1 n : x_i = \max \set {x_1, x_2, \dotsc, x_n}$


Similar Matrices have same Traces

Let $\mathbf A = \sqbrk a_n$ and $\mathbf B = \sqbrk b_n$ be square matrices of order $n$. Let $\mathbf A$ and $\mathbf B$ be similar. Then: :$\map \tr {\mathbf A} = \map \tr {\mathbf B}$ where $\map \tr {\mathbf A}$ denotes the trace of $\mathbf A$.


Cosine of Integer Multiple of Argument/Formulation 2

{{begin-eqn}} {{eqn | l = \cos n \theta       | r = \cos^n \theta \paren {1 - \dbinom n 2 \paren {\tan \theta}^2 + \dbinom n 4 \paren {\tan \theta}^4 - \cdots}       | c =  }} {{eqn | r = \cos^n \theta \sum_{k \mathop \ge 0} \paren {-1}^k \dbinom n {2 k } \paren {\tan^{2 k } \theta}       | c =  }} {{end-eqn}}


Polygamma Reflection Formula

:$\map {\psi_n} z - \paren {-1}^n \map {\psi_n} {1 - z} = -\pi \dfrac {\d^n} {\d z^n} \cot \pi z$


Polygamma Function in terms of Hurwitz Zeta Function

:$\displaystyle \map {\psi_n} z = \paren {-1}^{n + 1} \map \Gamma {n + 1} \map \zeta {n + 1, z}$


Area of Parallelogram from Determinant

Let $OABC$ be a parallelogram in the Cartesian plane whose vertices are located at: {{begin-eqn}} {{eqn | l = O       | r = \tuple {0, 0} }} {{eqn | l = A       | r = \tuple {a, c} }} {{eqn | l = B       | r = \tuple {a + b, c + d} }} {{eqn | l = C       | r = \tuple {b, d} }} {{end-eqn}} The area of $OABC$ is given by: :$\map \Area {OABC} = \begin {vmatrix} a & b \\ c & d \end {vmatrix}$ where $\begin {vmatrix} a & b \\ c & d \end {vmatrix}$ denotes the determinant of order $2$.


Matrix is Invertible iff Rank equals Order

Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$. Then $\mathbf A$ is invertible {{iff}} its rank also equals $n$.


Determinant of Upper Triangular Matrix

Let $\mathbf T_n$ be an upper triangular matrix of order $n$. Let $\map \det {\mathbf T_n}$ be the determinant of $\mathbf T_n$. Then $\map \det {\mathbf T_n}$ is equal to the product of all the diagonal elements of $\mathbf T_n$. That is: :$\displaystyle \map \det {\mathbf T_n} = \prod_{k \mathop = 1}^n a_{k k}$


Cosine of Integer Multiple of Argument/Formulation 3

{{begin-eqn}} {{eqn | l = \cos n \theta       | r = \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta } \cos^n \theta \paren {1 + 1 + \frac {\cos 2 \theta} {\cos^2 \theta} + \frac {\cos 3 \theta} {\cos^3 \theta} + \cdots + \frac {\cos \paren {n - 2} \theta} {\cos^{n - 2} \theta} }       | c =  }} {{eqn | r = \cos \paren {n - 1} \theta \cos \theta + \paren {1 - \sec^2 \theta } \cos^n \theta \sum_{k \mathop = 0}^{n - 2} \frac {\cos k \theta} {\cos^k \theta}       | c =  }} {{end-eqn}}


Determinant of Elementary Row Matrix/Scale Row

Let $e_1$ be the elementary row operation $\text {ERO} 1$: {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \ne 0$, multiply row $k$ by $\lambda$         | m = r_k \to \lambda r_k }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_1$ be the elementary row matrix corresponding to $e_1$. The determinant of $\mathbf E_1$ is: :$\map \det {\mathbf E_1} = \lambda$


Determinant of Elementary Row Matrix/Scale Row and Add

Let $e_2$ be the elementary row operation $\text {ERO} 2$: {{begin-axiom}} {{axiom | n = \text {ERO} 2         | t = For some $\lambda$, add $\lambda$ times row $j$ to row $i$         | m = r_i \to r_i + \lambda r_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_2$ be the elementary row matrix corresponding to $e_2$. The determinant of $\mathbf E_2$ is: :$\map \det {\mathbf E_2} = 1$


Determinant of Elementary Row Matrix/Exchange Rows

Let $e_3$ be the elementary row operation $\text {ERO} 3$: {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \leftrightarrow r_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_3$ be the elementary row matrix corresponding to $e_3$. The determinant of $\mathbf E_3$ is: :$\map \det {\mathbf E_3} = -1$


Elementary Matrix corresponding to Elementary Column Operation

Let $\mathbf I$ denote the unit matrix of order $n$ over a field $K$. Let $e$ be an elementary column operation on $\mathbf I$. Let $\mathbf E$ be the elementary column matrix of order $n$ uniquely defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Let $\kappa_k$ denote the $k$th column of $\mathbf I$ for $1 \le k \le n$.


Elementary Matrix corresponding to Elementary Column Operation/Scale Column

Let $e$ be the elementary column operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $k$ of $\mathbf I$ by $\lambda$         | m = \kappa_k \to \lambda \kappa_k }} {{end-axiom}}


Elementary Matrix corresponding to Elementary Column Operation/Scale Column and Add

Let $e$ be the elementary column operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ECO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $j$ to row $i$         | m = \kappa_i \to \kappa_i + \lambda r_j }} {{end-axiom}}


Elementary Matrix corresponding to Elementary Column Operation/Exchange Columns

Let $e$ be the elementary column operation acting on $\mathbf I$ as: {{begin-axiom}} {{axiom | n = \text {ECO} 3         | t = Interchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}}


Row Operation has Inverse

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists another row operation $\Gamma'$ which transforms $\mathbf B$ back to $\mathbf A$.


Real Numbers with Absolute Value form Normed Vector Space

Let $\R$ be the set of real numbers. Let $\size {\, \cdot \,}$ be the absolute value. Then $\struct {\R, \size {\, \cdot \,}}$ is a normed vector space.


Row Operation is Equivalent to Pre-Multiplication by Product of Elementary Matrices

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists a unique invertible square matrix $\mathbf R$ of order $m$ such that: :$\mathbf R \mathbf A = \mathbf B$ where $\mathbf R$ is the product of a finite sequence of elementary row matrices.


Product of Matrices is Invertible iff Matrices are Invertible

Let $\mathbf A$ and $\mathbf B$ be square matrices of order $n$. Let $\mathbf A \mathbf B$ denote the matrix product of $\mathbf A$ and $\mathbf B$. Let $\mathbf I$ be the $n \times n$ unit matrix. Let $\mathbf A$ and $\mathbf B$ be invertible. Then: :$\mathbf A \mathbf B$ is invertible {{iff}} :both $\mathbf A$ and $\mathbf B$ are invertible.


Square Root of Number Plus or Minus Square Root

Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then:


Elementary Row Matrix is Invertible

Let $\mathbf E$ be an elementary row matrix. Then $\mathbf E$ is invertible.


Existence of Inverse Elementary Column Operation

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be an elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is an elementary column operation which always exists and is unique.


Column Operation has Inverse

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists another column operation $\Gamma'$ which transforms $\mathbf B$ back to $\mathbf A$.


Elementary Column Matrix is Invertible

Let $\mathbf E$ be an elementary column matrix. Then $\mathbf E$ is invertible.


Column Operation is Equivalent to Post-Multiplication by Product of Elementary Matrices

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Then there exists a unique invertible square matrix $\mathbf K$ of order $n$ such that: :$\mathbf A \mathbf K = \mathbf B$ where $\mathbf K$ is the product of a finite sequence of elementary column matrices.


Existence of Inverse Elementary Row Operation/Scalar Product of Row

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply row $i$ by $\lambda$         | m = r_i \to \lambda r_i }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r_i \to \dfrac 1 \lambda r_i$


Existence of Inverse Elementary Row Operation/Add Scalar Product of Row to Another

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times row $k$ to row $l$         | m = r_k \to r_k + \lambda r_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r'_k \to r'_k - \lambda r'_l$


Existence of Inverse Elementary Row Operation/Exchange Rows

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary row operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Exchange rows $k$ and $l$         | m = r_k \leftrightarrow r_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary row operation: :$e' := r_k \leftrightarrow r_l$ That is: :$e' = e$


Existence of Inverse Elementary Column Operation/Scalar Product of Column

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $k$ by $\lambda$         | m = \kappa_k \to \lambda \kappa_k }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \kappa_k \to \dfrac 1 \lambda \kappa_k$


Existence of Inverse Elementary Column Operation/Add Scalar Product of Column to Another

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ECO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $l$ to column $k$         | m = \kappa_k \to \kappa_k + \lambda \kappa_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \kappa_k \to \kappa_k - \lambda \kappa_l$


Existence of Inverse Elementary Column Operation/Exchange Columns

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\map e {\mathbf A}$ be the elementary column operation which transforms $\mathbf A$ to a new matrix $\mathbf A' \in \map \MM {m, n}$. {{begin-axiom}} {{axiom | n = \text {ECO} 3         | t = Interchange columns $k$ and $l$         | m = \kappa_k \leftrightarrow \kappa_l }} {{end-axiom}} Let $\map {e'} {\mathbf A'}$ be the inverse of $e$. Then $e'$ is the elementary column operation: :$e' := \kappa_k \leftrightarrow \kappa_l$ That is: :$e' = e$


Elementary Column Operations as Matrix Multiplications

Let $e$ be an elementary column operation. Let $\mathbf E$ be the elementary column matrix of order $n$ defined as: :$\mathbf E = e \paren {\mathbf I}$ where $\mathbf I$ is the unit matrix. Then for every $m \times n$ matrix $\mathbf A$: :$e \paren {\mathbf A} = \mathbf A \mathbf E$ where $\mathbf A \mathbf E$ denotes the conventional matrix product.


Square Root of Number Plus Square Root/Proof 1

Let $a$ and $b$ be (strictly) positive real numbers such that $a^2 - b > 0$. Then: {{:Square Root of Number Plus Square Root}}


Square Root of Number Plus Square Root/Proof 2


Square Root of Number Minus Square Root/Proof 1


Exchange of Columns as Sequence of Other Elementary Column Operations

Let $\mathbf A$ be an $m \times n$ matrix. Let $i, j \in \closedint 1 m: i \ne j$ Let $\kappa_k$ denote the $k$th column of $\mathbf A$ for $1 \le k \le n$: :$\kappa_k = \begin {pmatrix} a_{1 k} \\ a_{2 k} \\ \vdots \\ a_{m k} \end {pmatrix}$ Let $e$ be the elementary column operation acting on $\mathbf A$ as: {{begin-axiom}} {{axiom | n = \text {ERO} 3         | t = Interchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}} Then $e$ can be expressed as a finite sequence of exactly $4$ instances of the other two elementary column operations. {{begin-axiom}} {{axiom | n = \text {ERO} 1         | t = For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$         | m = \kappa_i \to \lambda \kappa_i }} {{axiom | n = \text {ERO} 2         | t = For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$         | m = \kappa_i \to \kappa_i + \lambda \kappa_j }} {{end-axiom}}


Determinant of Elementary Column Matrix

Let $\mathbf E$ be an elementary column matrix. The determinant of $\mathbf E$ is as follows:


Determinant of Elementary Column Matrix/Scale Column

Let $e_1$ be the elementary column operation $\text {ECO} 1$: {{begin-axiom}} {{axiom | n = \text {ECO} 1         | t = For some $\lambda \ne 0$, multiply column $k$ by $\lambda$         | m = \kappa_k \to \lambda \kappa_k }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_1$ be the elementary column matrix corresponding to $e_1$. The determinant of $\mathbf E_1$ is: :$\map \det {\mathbf E_1} = \lambda$


Determinant of Elementary Column Matrix/Scale Column and Add

Let $e_2$ be the elementary column operation $\text {ECO} 2$: {{begin-axiom}} {{axiom | n = \text {ECO} 2         | t = For some $\lambda$, add $\lambda$ times column $j$ to column $i$         | m = \kappa_i \to \kappa_i + \lambda \kappa_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_2$ be the elementary column matrix corresponding to $e_2$. The determinant of $\mathbf E_2$ is: :$\map \det {\mathbf E_2} = 1$


Determinant with Column Multiplied by Constant

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\mathbf B$ be the matrix resulting from one column of $\mathbf A$ having been multiplied by a constant $c$. Then: :$\map \det {\mathbf B} = c \map \det {\mathbf A}$ That is, multiplying one column of a square matrix by a constant multiplies its determinant by that constant.


Determinant of Elementary Column Matrix/Exchange Columns

Let $e_3$ be the elementary column operation $\text {ECO} 3$: {{begin-axiom}} {{axiom | n = \text {ECO} 3         | t = Exchange columns $i$ and $j$         | m = \kappa_i \leftrightarrow \kappa_j }} {{end-axiom}} which is to operate on some arbitrary matrix space. Let $\mathbf E_3$ be the elementary column matrix corresponding to $e_3$. The determinant of $\mathbf E_3$ is: :$\map \det {\mathbf E_3} = -1$


Determinant of Rescaling Matrix/Corollary

Let $\mathbf A$ be a square matrix of order $n$. Let $\lambda$ be a scalar. Let $\lambda \mathbf A$ denote the scalar product of $\mathbf A$ by $\lambda$. Then: :$\map \det {\lambda \mathbf A} = \lambda^n \map \det {\mathbf A}$ where $\det$ denotes determinant.


Sequence of Row Operations is Row Operation

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma_1$ be a row operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Let $\Gamma_2$ be a row operation which transforms $\mathbf B$ to another new matrix $\mathbf C \in \map \MM {m, n}$. Then there exists another row operation $\Gamma$ which transforms $\mathbf A$ back to $\mathbf C$ such that $\Gamma$ consists of $\Gamma_1$ followed by $\Gamma_2$.


Sequence of Column Operations is Column Operation

Let $\map \MM {m, n}$ be a metric space of order $m \times n$ over a field $K$. Let $\mathbf A \in \map \MM {m, n}$ be a matrix.  Let $\Gamma_1$ be a column operation which transforms $\mathbf A$ to a new matrix $\mathbf B \in \map \MM {m, n}$. Let $\Gamma_2$ be a column operation which transforms $\mathbf B$ to another new matrix $\mathbf C \in \map \MM {m, n}$. Then there exists another column operation $\Gamma$ which transforms $\mathbf A$ back to $\mathbf C$ such that $\Gamma$ consists of $\Gamma_1$ followed by $\Gamma_2$.


Equivalence of Definitions of Determinant

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. {{TFAE|def = Determinant of Matrix|view = the determinant of $\mathbf A$}}


Intersection With Singleton is Disjoint if Not Element

Let $S$ be a set. Let $\set x$ be the singleton of $x$. Then: :$x \notin S$ {{iff}} $\set x \cap S = \O$


Product of Matrix with Adjugate equals Determinant by Unit Matrix

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\adj {\mathbf A}$ be the determinant of $\mathbf A$. Then: :$\paren {\adj {\mathbf A} } \mathbf A = \map \det {\mathbf A} \mathbf I = \mathbf A \paren {\adj {\mathbf A} }$ where $\mathbf I$ denotes the unit matrix of order $n$.


Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant

Let $\mathbf A = \sqbrk a_n$ be an invertible square matrix of order $n$. Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$. Let $\adj {\mathbf A}$ be the determinant of $\mathbf A$. Then: :$\mathbf A^{-1} = \dfrac 1 {\map \det {\mathbf A} } \cdot \adj {\mathbf A}$ where $\mathbf A^{-1}$ denotes the inverse of $\mathbf A$


Greedy Algorithm yields Maximal Set

Let $\struct{S,\mathscr F}$ be an independence system. Let $w : S \to \R_{\ge 0}$ be a weight function. Then the Greedy Algorithm selects a maximal set $A_0$ in $\mathscr F$.


Greedy Algorithm may not yield Maximum Weight

Let $\struct{S,\mathscr F}$ be an independence system. Let $w : S \to \R_{\ge 0}$ be a weight function. Then the maximal set $A_0 \in \mathscr F$ selected by the Greedy Algorithm may not have maximum weight.


Greedy Algorithm guarantees Maximum Weight iff Matroid

Let $S$ be a finite set. Let $\mathscr I$ be a non-empty set of subsets of $S$. Then $\mathscr I$ is the set of independent subsets of a matroid on $S$ {{iff}}: :$(1) \quad \struct{S, \mathscr I}$ is an independence system :$(2) \quad$ for all non-negative weight functions $w : S \to \R_{\ge 0}$, the Greedy Algorithm selects $A_w \in \mathscr I$: :::::$\forall B \in \mathscr I: \map {w^+} {A_w} \ge \map {w^+} B$ :where $w^+$ denotes the extended weight function of $w$.


Space of Continuous on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication form Vector Space

Let $I := \closedint a b$ be a closed real interval. Let $\map \CC I$ be a space of continuous on closed interval real-valued functions. Let $\struct {\R, +_\R, \times_\R}$ be the field of real numbers. Let $\paren +$ be the pointwise addition of real-valued functions. Let $\paren {\, \cdot \,}$ be the pointwise scalar multiplication of real-valued functions. Then $\struct {\map \CC I, +, \, \cdot \,}_\R$ is a vector space.


Independent Subset is Contained in Base

Let $M = \struct {S, \mathscr I}$ be a matroid. Let $\mathscr B$ denote the set of all bases of $M$. Let $A \in \mathscr I$. Then: :$\exists B \in \mathscr B : A \subseteq B$


Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary

Let $\struct{S, \preceq}$ be a finite ordered set. Let $x \in S$. Then there exists a maximal element $M \in S$ and a minimal element $m \in S$ such that: :$m \preceq x \preceq M$


Equivalent Conditions for Element is Loop

Let $M = \struct{S, \mathscr I}$ be a matroid. Let $\sigma$ denote the closure operator on $M$. Let $\rho$ denote the rank function of $M$. Let $\mathscr B$ denote the set of all bases of $M$. Let $x \in S$. {{TFAE}} :$(1)\quad x$ is a loop :$(2)\quad x \in \map \sigma \O$  :$(3)\quad \map \rho {\set x} = 0$ :$(4)\quad \set x$ is a circuit :$(5)\quad x$ is not an element of any $B \in \mathscr B$


Power Set of Doubleton

Let $x, y$ be distinct objects. Then the power set of the doubleton $\set {x, y}$ is: :$\powerset {\set {x, y}} = \big \{ \O, \set x, \set y, \set {x,y} \big \}$


Doubleton of Elements is Subset

Let $S$ be a set. Let $\set {x,y}$ be the doubleton of distinct $x$ and $y$. Then: :$x, y \in S \iff \set {x,y} \subseteq S$


Sum of Unitary Divisors of Power of Prime

Let $n = p^k$ be the power of a prime number $p$. Then the sum of all positive unitary divisors of $n$ is $1 + n$.


Sum of Unitary Divisors is Multiplicative

Let $\map {\sigma^*} n$ denote the sum of unitary divisors of $n$. Then the function: :$\displaystyle \sigma^*: \Z_{>0} \to \Z_{>0}: \map {\sigma^*} n = \sum_{\substack d \mathop \divides n \\ d \mathop \perp \frac n d} d$ is multiplicative.