Inverse of Vandermonde Matrix/Eisinberg Formula

Let: {{begin-eqn}} {{eqn | l = \prod_{k \mathop = 1}^n \paren {x - x_k}       | r = a_nx^n + \sum_{m \mathop = 0}^{n-1} a_m x^m       | c = Polynomial expansion in powers of $x$ }} {{eqn | r = x^n + \sum_{m \mathop = 0}^{n - 1} \paren {-1}^{n - m} \map {e_{n-m} } {x_1, \ldots, x_n} \, x^m       | c =  Viete's Formulas and {{Defof|Elementary Symmetric Function}} }} {{eqn | l = W_n        | r =  \begin{bmatrix}   1         &  x_1      & \cdots & x_1^{n-1} \\   1         & x_2       & \cdots & x_2^{n-1} \\ \vdots      & \vdots    & \ddots & \vdots    \\   1         & x_1^{n-1} & \cdots & x_n^{n-1} \\ \end{bmatrix}       | c = {{Defof|Vandermonde Matrix}} of Order $n$ }} {{end-eqn}} Let $W_n$ have a matrix inverse $W_n^{-1} = \begin {bmatrix} d_{ij} \end {bmatrix}$.  Let $a_n = \map {e_0} {x_1, \ldots, x_n} = 1$. Then: {{begin-eqn}} {{eqn | n = 1       | l = d_{ij}       | r = \dfrac {\displaystyle \sum_{k \mathop = 0}^{n - i} a_{i + k} \, x_j^k} {\displaystyle \prod_{m \mathop = 1, m \mathop \ne j }^n \paren {x_j - x_m} }       | c = for $i, j = 1, \ldots, n$ }} {{eqn | r = \dfrac {\displaystyle \sum_{k \mathop = 0}^{n - i} \paren {-1}^{n - i - k} \map {e_{n - i - k} } {x_1, \ldots, x_n} \, x_j^k} {\displaystyle \prod_{m \mathop = 1, m \mathop \ne j }^n \paren {x_j - x_m} }       | c = for $i, j = 1, \ldots, n$ }} {{end-eqn}}


Linear First Order ODE/dy = f(x) dx

Let $f: \R \to \R$ be an integrable real function. The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \map f x$ has the general solution: :$y = \displaystyle \int \map f x \rd x + C$ where $\displaystyle \int \map f x \rd x$ denotes the primitive of $f$.


Linear First Order ODE/dy = f(x) dx/Initial Condition

Consider the linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = \map f x$ subject to the initial condition: :$y = y_0$ when $x = x_0$ $(1)$ has the particular solution: :$y = y_0 + \displaystyle \int_{x_0}^x \map f \xi \rd \xi$ where $\displaystyle \int \map f x \rd x$ denotes the primitive of $f$.


Solution by Integrating Factor/Examples/y' - 3y = sin x

The linear first order ODE: :$\dfrac {\d y} {\d x} - 3 y = \sin x$ has the general solution: :$y = \dfrac 1 {10} \paren {3 \sin x - \cos x} + C e^{3 x}$


Solution to Linear First Order ODE with Constant Coefficients

A linear first order ODE with constant coefficients in the form: :$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$ has the general solution: :$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$


Solution to Linear First Order ODE with Constant Coefficients/With Initial Condition

Consider the linear first order ODE with constant coefficients in the form: :$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$ with initial condition $\tuple {x_0, y_0}$ Then $(1)$ has the particular solution: :$\displaystyle y = e^{-a x} \int_{x_0}^x e^{a \xi} \map Q \xi \rd \xi + y_0 e^{a \paren {x - x_0} }$


Solution by Integrating Factor/Examples/y' + y = x^-1

Consider the linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} + y = \dfrac 1 x$ with the initial condition $\tuple {1, 0}$. This has the particular solution: :$y = \displaystyle e^{-x} \int_1^x \dfrac {e^\xi \rd \xi} \xi$


General Solution equals Particular Solution plus Complementary Function

Consider the linear first order ODE with constant coefficients: :$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$ The general solution to $(1)$ consists of: :the particular solution to $(1)$ for which the arbitrary constant is $0$ plus: :the complementary function to $(1)$.


Derivation of Auxiliary Equation to Constant Coefficient LSOODE

Consider the linear Second Order ODE with Constant Coefficients: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + p \dfrac {\d y} {\d x} + q y = \map R x$ and its auxiliary equation: :$(2): \quad m^2 + p m + q = 0$ The fact that the solutions of $(2)$ dictate the general solution of $(1)$ can be derived.


Linear Second Order ODE/y'' + 2 y' + 2 y = 0

The second order ODE: :$(1): \quad y'' + 2 y' + 2 y = 0$ has the general solution: :$y = e^{-x} \paren {A \cos x + B \sin x}$


Linear Second Order ODE/y'' + 2 y' + 2 y = 0/Verification

The equation: :$(1): \quad y = e^{-x} \paren {A \cos x + B \sin x}$ is a set of solutions to the second order ODE: :$y'' + 2 y' + 2 y = 0$


Linear Second Order ODE/y'' - 7 y' - 5 y = x^3 - 1

The second order ODE: :$(1): \quad y'' - 7 y' - 5 y = x^3 - 1$ has the general solution: :$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x} + \dfrac 1 {625} \paren {-125 x^3 + 525 x^2 - 1620 x + 2603}$


Linear Second Order ODE/y'' - 7 y' - 5 y = 0

The second order ODE: :$(1): \quad y'' - 7 y' - 5 y = 0$ has the general solution: :$y = C_1 \, \map \exp {\paren {\dfrac 7 2 + \dfrac {\sqrt {69} } 2} x} + C_2 \, \map \exp {\paren {\dfrac 7 2 - \dfrac {\sqrt {69} } 2} x}$


Convergent Sequence is Cauchy Sequence/Normed Vector Space

Let $\struct{X, \norm{\,\cdot\,} }$ be a normed vector space. Every convergent sequence in $X$ is a Cauchy sequence.


Linear Second Order ODE/y'' - 2 y' - 5 y = 0

The second order ODE: :$(1): \quad y'' - 2 y' - 5 y = 0$ has the general solution: :$y = C_1 \, \map \exp {\paren {1 + \sqrt 6} x} + C_2 \, \map \exp {\paren {1 - \sqrt 6} x}$


Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x

The second order ODE: :$(1): \quad y'' - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$ has the general solution: :$y = C_1 \, \map \exp {\paren {1 + \sqrt 6} x} + C_2 \, \map \exp {\paren {1 - \sqrt 6} x} + \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$


Linear Second Order ODE/y'' - 2 y' - 5 y = 2 cos 3 x - sin 3 x/Particular Solution

The second order ODE: :$(1): \quad y'' - 2 y' - 5 y = 2 \cos 3 x - \sin 3 x$ has a particular solution: :$y_p = \dfrac 1 {116} \paren {\sin 3 x - 17 \cos 3 x}$


Linear Second Order ODE/y'' + 4 y = 3 sin 2 x

The second order ODE: :$(1): \quad y'' + 4 y = 3 \sin 2 x$ has the general solution: :$y = C_1 \sin k x + C_2 \cos k x - \dfrac 3 4 x \cos 2 x$


Cauchy Sequence is Bounded/Normed Vector Space

Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space. Every Cauchy sequence in $X$ is bounded.


Linear Second Order ODE/y'' - 4 y' - 5 y = 0

The second order ODE: :$(1): \quad y'' - 4 y' - 5 y = 0$ has the general solution: :$y = C_1 e^{5 x} + C_2 e^{-x}$


Linear Second Order ODE/y'' - 4 y' - 5 y = x^2/y(0) = 1, y'(0) = -1

Consider the second order ODE: :$(1): \quad y'' - 4 y' - 5 y = x^2$ whose initial conditions are: :$y = 1$ when $x = 0$ :$y' = -1$ when $x = 0$ $(1)$ has the particular solution: :$y = \dfrac {e^{5 x} } {375} + \dfrac {4 e^{-x} } 3 - \dfrac {x^2} 5 + \dfrac {8 x} {25} - \dfrac {42} {125}$


P-adic Unit has Norm Equal to One

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ denote the $p$-adic integers. Let $x \in \Q_p$. Then x is a $p$-adic unit {{iff}} $\norm x_p = 1$


P-adic Number times P-adic Norm is P-adic Unit

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p^\times$ be the $p$-adic units. Let $a \in \Q_p$. Then there exists $n \in \Z$ such that: :$p^n a \in \Z_p^\times$ where :$p^n = \norm a_p$


Solution to Simultaneous Homogeneous Linear First Order ODEs with Constant Coefficients

Consider the system of linear first order ordinary differential equations with constant coefficients: {{begin-eqn}} {{eqn | n = 1       | l = \dfrac {\d y} {\d x} + a y + b z       | r = 0 }} {{eqn | n = 2       | l = \dfrac {\d x} {\d z} + c y + d z       | r = 0 }} {{end-eqn}} The general solution to $(1)$ and $(2)$ consists of the linear combinations of the following: {{begin-eqn}} {{eqn | l = y       | r = A_1 e^{k_1 x} }} {{eqn | l = z       | r = B_1 e^{k_1 x} }} {{end-eqn}} and: {{begin-eqn}} {{eqn | l = y       | r = A_2 e^{k_2 x} }} {{eqn | l = z       | r = B_2 e^{k_2 x} }} {{end-eqn}} where $A_1 : B_1 = A_2 : B_2 = r$ where $r$ is either of the roots of the quadratic equation: :$\paren {k + a} \paren {k + d} - b c = 0$


Trivial Solution of Homogeneous Linear 1st Order ODE

The homogeneous linear first order ODE: :$\dfrac {\d y} {\d x} + \map Q x y = 0$ has the particular solution: :$\map y x = 0$ that is, the zero constant function. This particular solution is referred to as the '''trivial solution'''.


Simultaneous Homogeneous Linear First Order ODEs/Examples/y' - 3y + 2z = 0, y' + 4y - z = 0

Consider the system of linear first order ordinary differential equations with constant coefficients: {{begin-eqn}} {{eqn | n = 1       | l = \dfrac {\d y} {\d x} - 3 y + 2 z       | r = 0 }} {{eqn | n = 2       | l = \dfrac {\d x} {\d z} + 4 y - z       | r = 0 }} {{end-eqn}} The general solution to $(1)$ and $(2)$ consists of the linear combinations of the following: {{begin-eqn}} {{eqn | l = y       | r = C_1 e^{5 x} + C_2 e^{-x} }} {{eqn | l = z       | r = -C_1 e^{5 x} + 2 C_2 e^{-x} }} {{end-eqn}}


Particular Solution of Constant Coefficient Linear nth Order ODE

Consider the linear $n$th order ODE with constant coefficients: :$(1): \quad \displaystyle \sum_{k \mathop = 0}^n a_k \dfrac {\d^k y} {d x^k} = \map R x$ Let $(1)$ have the following $n$ initial conditions: :$(2): \quad y = y_0, \dfrac {\d y} {\d x} = y_1, \dotsc, \dfrac {\d^{n - 1} y} {\d x^{n - 1} } = y_{n - 1}$ when $x = x_0$. Then there exists exactly one particular solution of $(1)$ which satisfies $(2)$.


Particular Solution of System of Constant Coefficient Linear 1st Order ODEs

Consider the system of linear first order ordinary differential equations with constant coefficients: {{begin-eqn}} {{eqn | n = 1       | l = \dfrac {\d y} {\d x} + a y + b z       | r = 0 }} {{eqn | n = 2       | l = \dfrac {\d x} {\d z} + c y + d z       | r = 0 }} {{end-eqn}} Let $(1)$ and $(2)$ have the following $n$ initial conditions: :$(3): \quad y = y_0, z = z_0$ when $x = x_0$. Then there exists exactly one particular solution of $(1)$ and $(2)$ which satisfies $(3)$.


Linear First Order ODE/y' - y = e^x/y(0) = 0

Consider the linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} - y = e^x$ subject to the initial condition: :$\map y 0 = 0$ $(1)$ has the particular solution: :$y = x e^x$


Linear Second Order ODE/y'' + 4 y' + 5 y = 0

The second order ODE: :$(1): \quad y'' + 4 y' + 5 y = 0$ has the general solution: :$y = e^{-2 x} \paren {C_1 \cos x + C_2 \sin x}$


Linear Second Order ODE/y'' - 5 y' + 6 y = 0

The linear second order ODE: :$(1): \quad y'' - 5 y' + 6 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{3 x}$


Linear Second Order ODE/y'' - 5 y' + 6 y = cos x + sin x

The second order ODE: :$(1): \quad y'' - 5 y' + 6 y = \cos x + \sin x$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{3 x} + \dfrac {\cos x} 5$


Lindelöf's Lemma/Lemma

Let $C$ be a set of open real sets. Then there is a countable subset $D$ of $C$ such that: :$\displaystyle \bigcup_{O \mathop \in D} O = \bigcup_{O \mathop \in C} O$


Linear Second Order ODE/y'' - 4 y = x^2 - 3 x - 4

The second order ODE: :$(1): \quad y'' - 4 y = x^2 - 3 x - 4$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-2 x} - \dfrac {x^2} 4 + \dfrac {3 x} 4 + \dfrac 7 8$


Linear Second Order ODE/y'' + 2 y' + y = x exp -x

The second order ODE: :$(1): \quad y'' + 2 y' + y = x e^{-x}$ has the general solution: :$y = e^{-x} \paren {C_1 + C_2 x + \dfrac {x^3} 6}$


Angular Momentum Commutation Rules

Let $J_x$, $J_y$ and $J_z$ denote the angular momentum operators. Then: {{begin-eqn}} {{eqn | l = \sqbrk {J_x, J_y}       | r = i J_z }} {{eqn | l = \sqbrk {J_y, J_z}       | r = i J_x }} {{eqn | l = \sqbrk {J_z, J_x}       | r = i J_y }} {{end-eqn}} where $\sqbrk {\, \cdot, \cdot \,}$ denotes the commutator operator.


P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ be a $p$-adic expansion. Let $\mathbf a$ be the equivalence class in $\Q_p$ containing $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$. Let $l$ be the index of the first non-zero coefficient in the $p$-adic expansion: :$l = \min \set {i: i \ge m \land d_i \ne 0}$ Then: :$\norm {\mathbf a}_p = p^{-l}$


Eventually Constant Sequence Converges to Constant

:$\displaystyle \lim_{n \mathop \to \infty} x_n = \lambda$


Non-Zero Complex Numbers under Multiplication form Group

Let $\C_{\ne 0}$ be the set of complex numbers without zero, that is: :$\C_{\ne 0} = \C \setminus \set 0$ The structure $\struct {\C_{\ne 0}, \times}$ is a group.


Real Numbers under Addition form Group

Let $\R$ be the set of real numbers. The structure $\struct {\R, +}$ is an infinite abelian group.


General Linear Group is not Abelian

Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\GL {n, K}$ be the general linear group of order $n$ over $K$. Then $\GL {n, K}$ is not an abelian group.


Group of Unitary Matrices under Multiplication is not Abelian

Let $n > 1$ be a natural number. Then the group of unitary matrices $\map U n$ is not abelian.


Closure in Subspace/Corollary 1

Let $W \subseteq S$ and let $\map {\cl_T} W$ denote the closure of $W$ in $T$. Let $\map {\cl_H} {W \cap H}$ denote the closure of $W \cap H$ in $T_H$. Then: :$\map {\cl_H} {W \cap H} \subseteq \map {\cl_T} W \cap H$


Complex Numbers form Vector Space over Themselves

The set of complex numbers $\C$, with the operations of addition and multiplication, forms a vector space.


Quaternions form Vector Space over Themselves

The set of quaternions $\H$, with the operations of addition and multiplication, forms a vector space.


Definition:Differential Operator

Let $A$ be a mapping from a function space $\FF_1$ to another function space $\FF_2$. Let $f \in \FF_2$ be a real function such that $f$ is the image of $u \in \FF_1$ that is:　$f = A \sqbrk u$ A '''differential operator''' is represented as a linear combination, finitely generated by $u$ and its derivatives containing higher degree such as :$\displaystyle \map P {x, D} = \sum _{\size \alpha \mathop \le m} \map {a_\alpha} x D^\alpha$ where: :$\alpha = \set {\alpha_1, \alpha_2, \dotsc \alpha_n}$ is a set of non-negative integers forming a multi-index :$\size \alpha = \alpha_1 + \alpha_2 + \dotsb + \alpha_n$ is the length of $\alpha$ :the $\map {a_\alpha} x$ are real functions on a open domain in a real cartesian space of $n$ dimensions :$D^\alpha = D_1^{\alpha_1} D_2^{\alpha_2} \dotsm D_n^{\alpha_n}$. {{Proofread|This definition copied from Wikipedia and made more or less coherent. Please correct as necessary.}} Category:Definitions/Operator Theory Category:Definitions/Differential Calculus dkbpexztmfvuwqouudtsa0vqyna6v7q


Simple Events are Mutually Exclusive

Let $\EE$ be an experiment. Let $e_1$ and $e_2$ be distinct simple events in $\EE$. Then $e_1$ and $e_2$ are mutually exclusive.


Non-Trivial Event is Union of Simple Events

Let $\EE$ be an experiment. Let $e$ be an event in $\EE$ such that $e \ne \O$. That is, such that $e$ is non-trivial. Then $e$ can be expressed as the union of a set of simple events in $\EE$.


Sample Space is Union of All Distinct Simple Events

Let $\EE$ be an experiment. Let $\Omega$ denote the sample space of $\EE$. Then $\Omega$ is the union of the set of simple events in $\EE$.


Equivalence of Definitions of Probability Measure

Let $\EE$ be an experiment. {{TFAE|def = Probability Measure}}


Product Space is Product in Category of Topological Spaces

Let $\mathbf{Top}$ be the category of topological spaces. Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\struct{X, \tau}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Then $\struct{X, \tau}$ is the product of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ in $\mathbf{Top}$.


Strictly Positive Rational Numbers are Closed under Addition

:$\forall a, b \in \Q_{>0}: a + b \in \Q_{>0}$


Strictly Positive Rational Numbers are Closed under Multiplication

:$\forall a, b \in \Q_{>0}: a b \in \Q_{>0}$


Set of Rationals Greater than Root 2 has no Smallest Element

Let $B$ be the set of all positive rational numbers $p$ such that $p^2 > 2$. Then $B$ has no smallest element.


Rational Number Not in Cut is Greater than Element of Cut

Let $\alpha$ be a cut. Let $p \in \alpha$. Let $q \in \Q$ such that $q \notin \alpha$. Then $q > p$.


Natural Basis of Tychonoff Topology

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Then the natural basis on $X$ is the set $\BB$ of cartesian products of the form $\displaystyle \prod_{i \mathop \in I} U_i$ where: :for all $i \in I : U_i \in \tau_i$ :for all but finitely many indices $i : U_i = X_i$


Natural Basis of Tychonoff Topology/Finite Product

Let $n \in \N$. For all $k \in \set {1, \ldots, n}$, let $\struct {X_k, \tau_k}$ be topological spaces. Let $\displaystyle X = \prod_{k \mathop = 1}^n X_k$ be the cartesian product of $X_1, \ldots, X_n$. Then the natural basis on $X$ is: :$\BB = \set {\displaystyle \prod_{k \mathop = 1}^n U_k : \forall k : U_k \in \tau_k}$


Box Topology may not be Coarsest Topology such that Projections are Continuous

Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the box topology on $X$. For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ Then $\tau$ may not be the coarsest topology on $X$ for which the projections $\family{\pr_i} _{i \mathop \in I}$ are continuous.


Box Topology may not form Categorical Product in the Category of Topological Spaces

Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the box topology on $X$. Then $\tau$ may not be the categorical product in the category of topological spaces.


Ordering on Cuts satisfies Trichotomy Law

Let $\alpha$ and $\beta$ be cuts. Then exactly one of the following applies: {{begin-eqn}} {{eqn | n = 1       | l = \alpha       | o = <       | r = \beta }} {{eqn | n = 2       | l = \alpha       | r = \beta }} {{eqn | n = 3       | l = \alpha       | o = >       | r = \beta }} {{end-eqn}} where $<$ and so $>$ denote the strict ordering of cuts: :$\alpha < \beta \iff \exists p \in \Q: p \in \alpha, p \notin \beta$ Hence the ordering of cuts $\le$ is a total ordering.


Ordering on Cuts is Transitive

Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let: {{begin-eqn}} {{eqn | n = 1       | l = \alpha       | o = <       | r = \beta }} {{eqn | n = 2       | l = \beta       | o = <       | r = \gamma }} {{end-eqn}} where $<$ denotes the strict ordering of cuts: :$\alpha < \beta \iff \exists p \in \Q: p \in \alpha, p \notin \beta$ Then: :$\alpha < \gamma$ Hence the ordering of cuts $\le$ is a transitive relation.


Ordering on Cuts is Total

Let $\CC$ denote the set of cuts. Let $<$ denote the strict ordering on cuts defined as: :$\forall \alpha, \beta \in \CC: \alpha < \beta \iff \exists p \in \Q: p \in \alpha, p \notin \beta$ Then $<$ is a (strict) total ordering on $\CC$.


Sum of Cuts is Cut

Let $\alpha$ and $\beta$ be cuts. Let $\gamma$ be the set of all rational numbers $r$ such that: :$\exists p \in \alpha, q \in \beta: r = p + q$ Then $\gamma$ is also a cut. Thus the operation of addition on the set of cuts is closed.


Addition of Cuts is Commutative

Let $\alpha$ and $\beta$ be cuts. Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$. Then: :$\alpha + \beta = \beta + \alpha$


Addition of Cuts is Associative

Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$. Then: :$\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$


Identity Element for Addition of Cuts

Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Then: :$\alpha + 0^* = \alpha$ where $+$ denotes the operation of addition of cuts.


Existence of Upper and Lower Numbers of Cut whose Difference equal Given Rational

Let $\alpha$ be a cut. Let $r \in \Q_{>0}$ be a (strictly) positive rational number. Then there exist rational numbers $p$ and $q$ such that: :$p \in \alpha, q \notin \alpha$ :$q - p = r$ such that $q$ is not the smallest upper number of $\alpha$.


Existence of Unique Inverse Element for Addition of Cuts

Let $\alpha$ be a cut. Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Then there exists a unique cut $\beta$ such that: :$\alpha + \beta = 0^*$ where $+$ denotes the operation of addition of cuts.


Ordering on Cuts is Compatible with Addition of Cuts

Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let the operation of $\alpha + \beta$ be the sum of $\alpha$ and $\beta$. Let $\beta < \gamma$ denotes the strict ordering on cuts defined as: :$\beta < \gamma \iff \exists p \in \Q: p \in \beta, p \notin \gamma$ Then: :$\beta < \gamma \implies \alpha + \beta < \alpha + \gamma$


Ordering on Cuts is Compatible with Addition of Cuts/Corollary

Let $0^*$ denote the rational cut associated with the (rational) number $0$. If: :$\alpha > 0^*$ and $\gamma > 0^*$ then: :$\alpha + \gamma > 0^*$


Existence of Unique Difference between Cuts

Let $\alpha$ and $\beta$ be cuts. Then there exists exactly one cut $\gamma$ such that: :$\alpha + \gamma = \beta$


Set of Cuts under Addition forms Abelian Group

Let $\CC$ denote the set of cuts. Let $\struct {\CC, +}$ denote the algebraic structure formed from $\CC$ and the operation $+$ of addition of cuts. Then $\struct {\CC, +}$ forms an abelian group.


Product of Positive Cuts is Positive Cut

Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$ and $\beta$ be cuts such that $\alpha \ge 0^*$ and $\beta \ge 0^*$, where $\ge$ denotes the ordering on cuts. Let $\gamma$ be the set of all rational numbers $r$ such that either: :$r < 0$ or: :$\exists p \in \alpha, q \in \beta: r = p q$ where $p \ge 0$ and $q \ge 0$. Then $\gamma$ is also a cut. Thus the operation of multiplication on the set of positive cuts is closed.


Product of Cuts is Cut

Let $\alpha$ and $\beta$ be cuts. Let $\alpha \beta$ denote the product of cuts. Then $\alpha \beta$ is also a cut. Thus the operation of multiplication on the set of cuts is closed.


Factorisation of z^n+a

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $a \in \C$ be a complex number. Then: :$z^n + a = \displaystyle \prod_{k \mathop = 0}^{n - 1} \paren {z - \alpha_k b}$ where: :$\alpha_k$ are the complex $n$th roots of negative unity :$b$ is any complex number such that $b^n = a$.


Multiplication of Cuts is Commutative

Let $\alpha$ and $\beta$ be cuts. Let the $\alpha \beta$ be the product of $\alpha$ and $\beta$. Then: :$\alpha \beta = \beta \alpha$


Multiplication of Cuts is Associative

Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let $\alpha \beta$ denote the product of $\alpha$ and $\beta$. Then: :$\paren {\alpha \beta} \gamma = \alpha \paren {\beta \gamma}$


Natural Basis of Tychonoff Topology/Lemma 1

:$\SS \subseteq \BB$


Natural Basis of Tychonoff Topology/Lemma 2

:$\forall B_1, B_2 \in \BB : B_1 \cap B_2 \in \BB$


Natural Basis of Tychonoff Topology/Lemma 3

:$\displaystyle \forall B \in \BB : B = \bigcap_{j \mathop \in J} \pr_j^{-1} \sqbrk {U_j}$ where: :$\displaystyle B = \prod_{i \mathop \in I} U_i$ :$J = \set{j \in I : U_i \ne X_i}$ is finite.


Multiplication of Cuts Distributes over Addition

Let $\alpha$, $\beta$ and $\gamma$ be cuts. Let: :$\alpha + \beta$ denote the sum of $\alpha$ and $\beta$. :$\alpha \beta$ denote the product of $\alpha$ and $\beta$. Then: :$\alpha \paren {\beta + \gamma} = \alpha \beta + \alpha \gamma$


Product of Cut with Zero Cut equals Zero Cut

Let $\alpha$ be a cut. Let $0^*$ denote the rational cut associated with the (rational) number $0$. Then: :$\alpha 0^* = 0^*$ where $\alpha 0^*$ denote the product of $\alpha$ and $0^*$.


Product of Cuts is Zero Cut iff Either Factor equals Zero Cut

Let $\alpha$ and $\beta$ be cuts. Let $0^*$ denote the rational cut associated with the (rational) number $0$. Then: :$\alpha \beta = 0^*$ {{iff}}: :$\alpha = 0^*$ or $\beta = 0^*$ where $\alpha \beta$ denotes the product of $\alpha$ and $\beta$.


Cut Associated with 1 is Identity for Multiplication of Cuts

Let $\alpha$ be a cut. Let $1^*$ denote the rational cut (rational) number $1$. Then: :$\alpha 1^* = \alpha$ where $\alpha 1^*$ denote the product of $\alpha$ and $1^*$.


Multiplication of Positive Cuts preserves Ordering

Let $0^*$ denote the rational cut associated with the (rational) number $0$. Let $\alpha$, $\beta$ and $\gamma$ be cuts such that: :$0^* < \alpha < \beta$ :$0^* < \gamma$ where $<$ denotes the strict ordering on cuts. Then :$\alpha \gamma < \beta \gamma$ where $\alpha \gamma$ denotes the product of $\alpha$ and $\gamma$.


Existence of Unique Inverse Element for Multiplication of Cuts

Let $0^*$ be the rational cut associated with the (rational) number $0$: :$0^* = \set {r \in \Q: r < 0}$ Let $\alpha$ be a cut such that $\alpha \ne 0^*$. Then for every cut $\beta$, there exists a unique cut $\gamma$ such that: :$\alpha \gamma = \beta$ where $\alpha \gamma$ denotes the operation of product of $\alpha$ and $\gamma$. In this context, $\gamma$ can be expressed as $\beta / \alpha$.


Sum of Rational Cuts is Rational Cut

Let $p \in\ Q$ and $q \in \Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* + q^* = \paren {p + q}^*$ Thus the operation of addition on the set of rational cuts is closed.


Ordering of Rational Cuts preserves Ordering of Associated Rational Numbers

Let $p \in\ Q$ and $q \in \Q$ be rational numbers. Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$. Then: :$p^* < q^* \iff p < q$ where $p^* < q^*$ denotes the strict ordering on cuts defined as: :$\beta < \gamma \iff \exists p \in \Q: p \in \beta, p \notin \gamma$


Exists Rational Cut Between two Cuts

Let $\alpha$ and $\beta$ be cuts. Let $\alpha < \beta$, where $<$ denotes the strict ordering on cuts. Then there exists a rational cut $r^*$ associated with the rational number $r$ such that: :$\alpha < r^* < \beta$


Condition for Rational Cut to be Less than Given Cut

Let $\alpha$ be a cut. Let $p^*$ be the rational cut associated with a rational number $p$. Then: :$p \in \alpha$ {{iff}}: :$p^* < \alpha$ where $<$ denotes the strict ordering on cuts.


Continuous Mapping to Topological Product/General Result

Let $X$ be a topological space. Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$. Let $\displaystyle Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \mathop \in I}$.  For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ Let $f$ be a mapping from $X$ to $Y$.  Then $f$ is continuous {{iff}} $\pr_i \circ f$ is continuous for all $i \in I$.


Set of Cuts forms Ordered Field

Let $\CC$ denote the set of cuts. Let $\struct {\CC, +, \times, \le}$ denote the ordered structure formed from $\CC$ and: :the operation $+$ of addition of cuts :the operation $\times$ of multiplication of cuts :the ordering $\le$ of cuts. Then $\struct {\CC, +, \times, \le}$ is an ordered field.


Box Topology contains Tychonoff Topology

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the Tychonoff topology on $X$. Let $\tau'$ be the box topology on $X$. Then: :$\tau \subseteq \tau'$


Ordered Field of Rational Cuts is Isomorphic to Rational Numbers

Let $\struct {\RR, +, \times, \le}$ denote the ordered field of rational cuts. Let $\struct {\Q, +, \times, \le}$ denote the field of rational numbers. Then $\struct {\RR, +, \times, \le}$ and $\struct {\Q, +, \times, \le}$ are isomorphic.


Supremum of Subset of Real Numbers May or May Not be in Subset

Let $S \subset \R$ be a proper subset of the set $\R$ of real numbers. Let $S$ admit a supremum $M$. Then $M$ may or may not be an element of $S$.


Uniqueness of Positive Root of Positive Real Number/Positive Exponent

Let $x \in \R$ be a real number such that $x > 0$. Let $n \in \Z$ be an integer such that $n > 0$. Then there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.


Basis Condition for Coarser Topology

Let $S$ be a set. Let $\BB_1$ and $\BB_2$ be two bases on $S$. Let $\tau_1$ and $\tau_2$ be the topologies generated by $\BB_1$ and $\BB_2$ respectively. If $\BB_1$ and $\BB_2$ satisfy: :$\forall U \in \BB_1 : \exists \AA \subseteq \BB_2: U = \bigcup \AA$ then $\tau_1$ is coarser than $\tau_2$.


Basis Condition for Coarser Topology/Corollary 2

If $\BB_1 \subseteq \BB_2$ then $\tau_1$ is coarser than $\tau_2$.


Continuous Mapping on Finer Domain and Coarser Codomain Topologies is Continuous

Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces. Let $f: \struct {X, \tau_1} \to \struct {Y, \tau_2}$ be a continuous mapping. Let $\tau'_1$ be a finer topology on $X$ than $\tau_1$, that is, $\tau_1 \subseteq \tau'_1$. Let $\tau'_2$ be a coarser topology on $Y$ than $\tau_2$, that is, $\tau'_2 \subseteq \tau_2$. Then: :$f: \struct {X, \tau'_1} \to \struct {Y, \tau'_2}$ is a continuous mapping.


Projection from Box Topology is Continuous

Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Let $\tau$ be the box topology on $X$. For each $i \in I$, let $\pr_i: X \to X_i$ denote the $i$th projection on $X$: :$\forall \family {x_j}_{j \mathop \in I} \in X: \map {\pr_i} {\family {x_j}_{j \mathop \in I} } = x_i$ Then $\pr_i: \struct{X,\tau} \to \struct{X_i,\tau_i}$ is continuous for all $i \in I$.


Domain Topology Contains Initial Topology iff Mappings are Continuous

Let $\struct{Y, \tau}$ be a topological space. Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be a family of topological spaces. Let $\family {f_i}_{i \mathop \in I}$ be a family of mappings $f_i : Y \to X_i$. Let $\tau'$ be the initial topology on $Y$ with respect to $\family {f_i}_{i \mathop \in I}$. Then: :$\tau' \subseteq \tau$ {{iff}} $\forall i \in I : f_i: \struct{Y, \tau} \to \struct{X_i, \tau_i}$ is $\tuple{\tau, \tau_i}$-continuous.


Final Topology Contains Codomain Topology iff Mappings are Continuous

Let $\struct{Y, \tau}$ be a topological space. Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be a family of topological spaces. Let $\family {f_i}_{i \mathop \in I}$ be a family of mappings $f_i : X_i \to Y$. Let $\tau'$ be the final topology on $Y$ with respect to $\family {f_i}_{i \mathop \in I}$. Then: :$\tau \subseteq \tau'$ {{iff}} $\forall i \in I : f_i: \struct{X_i, \tau_i} \to \struct{Y, \tau}$ is continuous.


Equivalence of Definitions of Final Topology

Let $X$ be a set. Let $I$ be an indexing set. Let $\family {\struct {Y_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces indexed by $I$. Let $\family {f_i: Y_i \to X}_{i \mathop \in I}$ be an indexed family of mappings indexed by $I$. {{TFAE|def = Final Topology}}


Condition for Trivial Relation to be Mapping

Let $S$ and $T$ be sets. Let $\RR = S \times T$ be the trivial relation in $S$ to $T$. Then $\RR$ is a mapping {{iff}} either: :$(2): \card S = 0$ or: :$(1): \card T = 1$ where $\card {\, \cdot \,}$ denotes cardinality.


Relation to Empty Set is Mapping iff Domain is Empty

Let $S$ be a set Let $S \times \O$ denote the cartesian product of $S$ with the empty set $\O$. Let $\RR \subseteq S \times \O$ be a relation in $S$ to $\O$. Then $\RR$ is a mapping {{iff}} $S = \O$.


Equivalence of Definitions of Initial Topology/Definition 1 Implies Definition 2

Let: :$\SS = \set{\map {f_i^{-1}} U: i \in I, U \in \tau_i} \subseteq \map \PP X$ where $\map {f_i^{-1}} U$ denotes the preimage of $U$ under $f_i$. Let $\tau$ be the topology on $X$ generated by the subbase $\SS$.


Equivalence of Definitions of Initial Topology/Definition 2 Implies Definition 1

Let $\tau$ be the coarsest topology on $X$ such that each $f_i: X \to Y_i$ is $\tuple{\tau, \tau_i}$-continuous. Let: :$\SS = \set {\map {f_i^{-1} } U: i \in I, U \in \tau_i} \subseteq \map \PP X$ where $\map {f_i^{-1} } U$ denotes the preimage of $U$ under $f_i$.


Even Natural Numbers are Infinite

The set of even natural numbers is infinite.


Set of Points on Line Segment is Infinite

The set of points on a line segment is infinite.


Finite Sets are Comparable

Let $S$ and $T$ be finite sets. Then $S$ and $T$ are comparable by size.


Equivalence of Definitions of Final Topology/Definition 1 Implies Definition 2

Let: :$\tau = \set{U \subseteq X: \forall i \in I: \map {f_i^{-1}} U \in \tau_i} \subseteq \powerset X$


Equivalence of Definitions of Final Topology/Definition 2 Implies Definition 1

Let $\tau$ be the finest topology on $X$ such that each $f_i: Y_i \to X$ is $\tuple{\tau_i, \tau}$-continuous.


Strictly Positive Integers have same Cardinality as Natural Numbers

Let $\Z_{>0} := \set {1, 2, 3, \ldots}$ denote the set of strictly positive integers. Let $\N := \set {0, 1, 2, \ldots}$ denote the set of natural numbers. Then $\Z_{>0}$ has the same cardinality as $\N$.


Power Set of Natural Numbers is Cardinality of Continuum

Let $\N$ denote the set of natural numbers. Let $\powerset \N$ denote the power set of $\N$. Let $\card {\powerset \N}$ denote the cardinality of $\powerset \N$. Then $\card {\powerset \N}$ is the cardinality of the continuum.


Continuum Hypothesis is Independent of ZFC

The Continuum Hypothesis can be neither proved nor disproved from the axioms of either Zermelo-Fraenkel set theory (ZF) or ZFC.


Axiom of Choice is Independent of ZF

The Axiom of Choice can be neither proved nor disproved from the axioms of Zermelo-Fraenkel set theory.


Empty Set can be Derived from Comprehension Principle

The '''empty set''' can be formed by application of the comprehension principle. Hence the '''empty set''' can be derived as a valid object in Frege set theory.


Power Set can be Derived using Comprehension Principle

Let $a$ be a set. By application of the comprehension principle, the power set $\powerset a$ can be formed. Hence the '''power set''' $\powerset a$ can be derived as a valid object in Frege set theory.


Set Union can be Derived using Comprehension Principle

Let $a$ be a set of sets. By application of the comprehension principle, the union $\bigcup a$ can be formed. Hence the '''union''' $\bigcup a$ can be derived as a valid object in Frege set theory.


Set of Natural Numbers can be Derived using Comprehension Principle

Let $\N$ denote the set of natural numbers. By application of the comprehension principle, $\N$ can be derived as a valid object in Frege set theory.


Exists Subset which is not Element

Let $S$ be a set. Then there exists at least one subset of $S$ which is not an element of $S$.


Box Topology may not be Coarsest Topology such that Projections are Continuous/Lemma

Let $\struct{X, \tau}$ be a topological space. Let $U \in \tau$ such that $U \neq \O$ and $U \neq X$. Let: :$Y = \displaystyle \prod_{n \mathop \in \N } X = X \times X \times X \times \ldots$ be the countable Cartesian product of $\family {X}_{n \in \N}$. Let $\tau_T$ be the Tychonoff topology on $Y$. Let $\tau_b$ be the box topology on $Y$. Let: :$V = \displaystyle \prod_{n \mathop \in \N } U = U \times U \times U \times \ldots$ be the countable Cartesian product of $\family {U}_{n \in \N}$. Then: :$V$ is an element of the box topology $\tau_b$ :$V$ is not an element of the Tychonoff topology $\tau_T$


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. 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.


Class is Subclass of Universal Class

Let $V$ denote the universal class. Let $A$ be a class. Then $A$ is a subclass of $V$.


Not Every Class is a Set

Let $A$ be a class. Then it is not necessarily the case that $A$ is also a set.


Class has Subclass which is not Element

Let $A$ be a class. Then $A$ has at least one subclass $B$ which is not an element of $A$.


Basic Universe is Supercomplete

Let $V$ be a basic universe. Then $V$ is supercomplete.


Basic Universe is not Set

Let $V$ be a basic universe. Then $V$ is not a set.


Empty Class is Subclass of All Classes

The empty class is a subclass of all classes.


Empty Class is Supercomplete

The empty class is supercomplete.


Basic Universe is not Empty

Let $V$ be a basic universe Then $V$ is not the empty class.


Singleton Class can be Formed from Set

Let $V$ be a basic universe. Let $a \in V$ be a set. Then the singleton class $\set a$ can be formed, which is a subclass of $V$.


Singleton Class of Empty Set is Supercomplete

Let $\O$ denote the empty set. Then the singleton $\set \O$ is supercomplete.


Singleton Classes are Equal iff Sets are Equal

Let $a$ and $b$ be sets. Let $\set a$ and $\set b$ denote the singleton classes of $a$ and $b$. Then: :$\set a = \set b \iff a = b$


Doubleton Class can be Formed from Two Sets

Let $V$ be a basic universe. Let $a, b \in V$ be sets. Then the doubleton class $\set {a, b}$ can be formed, which is a subclass of $V$.


Doubleton Class of Equal Sets is Singleton Class

Let $V$ be a basic universe. Let $a, b \in V$ be sets. Consider the doubleton class $\set {a, b}$. Let $a = b$. Then: :$\set {a, b} = \set a$ where $\set a$ denotes the singleton class of $a$.


Equivalence of Definitions of Axiom of Pairing

The following formulations of the '''axiom of pairing''' in the context of '''axiomatic set theory''' are equivalent:


Singleton Class of Set is Set

Let $x$ be a set. Then the singleton class $\set x$ is likewise a set.


Basic Universe has Infinite Number of Elements

Let $V$ be a basic universe. Then $V$ has an infinite number of elements.


Equivalence of Definitions of Axiom of Pairing for Classes

The following formulations of the '''axiom of pairing''' in the context of '''class theory''' are equivalent:


Equality of Ordered Pairs/Lemma

Let $\set {a, b}$ and $\set {a, d}$ be doubletons such that $\set {a, b} = \set {a, d}$. Then: :$b = d$


Equality of Ordered Pairs/Necessary Condition

Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs such that $\tuple {a, b} = \tuple {c, d}$. Then $a = c$ and $b = d$.


Equality of Ordered Pairs/Sufficient Condition

Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs. Let $a = c$ and $b = d$. Then: :$\tuple {a, b} = \tuple {c, d}$


Intersection of Class Exists and is Unique

Let $V$ be a basic universe. Let $A \subseteq V$ be a class. Let $\displaystyle \bigcap A$ denote the intersection of $A$. Then $\displaystyle \bigcap A$ is guaranteed to exist and is unique.


Intersection of Non-Empty Class is Set

Let $V$ be a basic universe. Let $A \subseteq V$ be a non-empty class. Let $\displaystyle \bigcap A$ denote the intersection of $A$. Then $\displaystyle \bigcap A$ is a set.


Intersection of Empty Set/Class Theory

Let $V$ be a basic universe. Let $\O$ denote the empty class. Then the intersection of $\O$ is $V$: :$\displaystyle \bigcap \O = V$


Union of Subclass is Subset of Union of Class

Let $A$ and $B$ be classes. Let $\displaystyle \bigcup A$ and $\displaystyle \bigcup B$ denote the union of $A$ and union of $B$ respectively. Let $A$ be a subclass of $B$: :$A \subseteq B$ Then $\displaystyle \bigcup A$ is a subset of $\displaystyle \bigcup B$: :$\displaystyle \bigcup A \subseteq \displaystyle \bigcup B$


Union of Transitive Class is Subset

Let $A$ be a transitive class. Let $\displaystyle \bigcup A$ denote the union of $A$. Then: :$\displaystyle \bigcup A \subseteq A$


Union of Class is Subset implies Class is Transitive

Let $A$ be a class. Let $\displaystyle \bigcup A$ denote the union of $A$. Let: :$\displaystyle \bigcup A \subseteq A$ Then $A$ is transitive.


Union of Transitive Class is Transitive

Let $A$ be a class. Let $\displaystyle \bigcup A$ denote the union of $A$. Let $A$ be transitive. Then $\displaystyle \bigcup A$ is also transitive.


Union of Class is Transitive if Every Element is Transitive

Let $A$ be a class. Let $\bigcup A$ denote the union of $A$. Let $A$ be such that every element of $A$ is transitive. Then $\bigcup A$ is also transitive.


Set Difference with Set Difference is Union of Set Difference with Intersection/Corollary

:$T \setminus \paren {S \setminus T} = T$


Power Set Exists and is Unique

Let $V$ be a basic universe. Let $x \in V$ be a set. Let $\powerset x$ denote the power set of $x$. Then $\powerset x$ is guaranteed to exist and is unique.


Element of Class is Subset of Union of Class

Let $A$ be a class. Let $\displaystyle \bigcup A$ denote the union of $A$. Let $x \in A$. Then: :$x \subseteq \displaystyle \bigcup A$


Set equals Union of Power Set

Let $x$ be a set of sets. Let $\powerset x$ denote the power set of $x$. Let $\displaystyle \map \bigcup {\powerset x}$ denote the union of $\powerset x$. Then: :$x = \displaystyle \map \bigcup {\powerset x}$


Cartesian Product Exists and is Unique

Let $A$ and $B$ be classes. Let $A \times B$ be the '''cartesian product''' of $A$ and $B$. Then $A \times B$ exists and is unique.


Cartesian Product of Sets is Set

Let $V$ be a basic universe. Let $A$ and $B$ be sets in $V$. Then $A \times B$ is also a set.


Real Function with Positive Derivative is Increasing

If $\forall x \in \openint a b: \map {f'} x \ge 0$, then $f$ is increasing on $\closedint a b$.


Real Function with Strictly Positive Derivative is Strictly Increasing

If $\forall x \in \openint a b: \map {f'} x > 0$, then $f$ is strictly increasing on $\closedint a b$.


Continuous Real Function Differentiable on Borel Set

Let $\map \BB {\R, \size {\, \cdot \,} }$ be the Borel Sigma-Algebra on $\R$ with the usual topology. Let $f: \R \to \R$ be a continuous real function. Let $\map D f$ be the set of all points at which $f$ is differentiable. Then $\map D f$ is a Borel Set with respect to $\map \BB {\R, \size {\, \cdot \,} }$.


Limit with Rational Epsilon and Delta

Let $\openint a b$ be an open real interval. Let $c \in \openint a b$. Let $f: \openint a b \setminus \set c \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon > 0 \in \Q_{>0}: \exists \delta \in \Q_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$ Then the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$.


Chain Rule for Probability

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$. The '''conditional probability of $A$ given $B$''' is: :$\map \Pr {A \mid B} = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$


Limit with Epsilon Powers of 2

Let $\openint a b$ be an open real interval. Let $c \in \openint a b$. Let $f: \openint a b \setminus \set c \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall n > 0 \in \N: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < 2^{-n} $ Then the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$.


Characterization of Probability Density Function

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $\Omega_X = \Img X$, the image of $X$. Let the '''probability density function''' of $X$ is the mapping $f_X: \R \to \closedint 0 1$ be defined as: :$\forall x \in \R: \map {f_X} x = \begin {cases} \displaystyle \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$ Suppose that the cumulative distribution function of $X$ defines a continuously differentiable real function $F_X: x \mapsto \map \Pr {X \le x}$. Then the '''probability density function''' of $X$ is the mapping $f_X: \R \to \closedint 0 1$ satisfies: :$\dfrac{\d}{\d x} \map{F_X}{x} = \map {f_X} x$.


Abel's Test

Let $\displaystyle \sum a_n$ be a convergent real series. Let $\sequence {b_n}$ be a decreasing sequence of positive real numbers. Then the series $\displaystyle \sum a_n b_n$ is also convergent.


Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2

Let $S$ be a set. Let $\BB$ be a synthetic basis on $S$. Let $\tau$ be the topology on $S$ generated by the synthetic basis $\mathcal B$: :$\tau = \set{\bigcup \AA: \AA \subseteq \BB}$ Then: :$\forall U \subseteq S: U \in \tau \iff U = \bigcup \set {B \in \BB: B \subseteq U}$


Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 3

Let $S$ be a set. Let $\mathcal B$ be a synthetic basis on $S$. Let $\tau$ be the topology on $S$ generated by the synthetic basis $\mathcal B$: :$\tau = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$ Then: :$\forall U \subseteq S: U \in \tau \iff \forall x \in U: \exists B \in \mathcal B: x \in B \subseteq U$


Projection from Product Topology is Open and Continuous

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the Tychonoff topology on $S$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Then both $\pr_1$ and $\pr_2$ are open and continuous.


Projection from Product Topology is Open and Continuous/General Result

Let $\family {T_i}_{i \mathop \in I} = \family {\struct{S_i, \tau_i}}_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space. Let $\tau$ denote the Tychonoff topology on $S$. Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\pr_i$ is open and continuous for all $i \in I$.


Projection from Product Topology is Continuous/General Result

Let $\family {T_i}_{i \mathop \in I} = \family {\struct{S_i, \tau_i}}_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space. Let $\tau$ denote the Tychonoff topology on $S$. Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\pr_i$ is continuous for all $i \in I$.


Projection from Product Topology is Open/General Result

Let $\family {T_i}_{i \mathop \in I} = \family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space. Let $\tau$ denote the Tychonoff topology on $S$. Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\pr_i$ is open for all $i \in I$.


Product Space Basis Induced from Factor Space Bases

Let $\family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\BB_\alpha$ be a basis for the topology $\tau_\alpha$ for each $\alpha \in I$. Let $\struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let $\BB$ be the set of all cartesian products of the form $\displaystyle \prod_{\alpha \mathop \in I} U_\alpha$ where: :for all but finitely many indices $\alpha : U_\alpha = S_\alpha$ :for all $\alpha \in I : U_\alpha \ne S_\alpha \implies U_\alpha \in \BB_\alpha$ Then $\BB$ is a basis for the topology on the product space $\struct{S, \tau}$.


Existence and Uniqueness of Domain of Relation

Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Then the domain $\Dom \RR$ of $\RR$ exists and is unique.


Existence and Uniqueness of Image of Relation

Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Then the image $\Img \RR$ of $\RR$ exists and is unique.


Union of Union of Relation is Union of Domain with Image

Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\Dom \RR$ denote the domain of $\RR$. Then: :$\map \bigcup {\bigcup \RR} = \Dom \RR \cup \Img \RR$ where: :$\bigcup \RR$ denotes the union of $\RR$ :$\Dom \RR$ denotes the domain of $\RR$ :$\Img \RR$ denotes the image of $\RR$.


Domain of Relation is Subset of Union of Union of Relation

Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\Dom \RR$ denote the domain of $\RR$. Then: :$\Dom \RR \subseteq \map \bigcup {\bigcup \RR}$ where $\bigcup \RR$ denotes the union of $\RR$.


Image of Relation is Subset of Union of Union of Relation

Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\Img \RR$ denote the image of $\RR$. Then: :$\Img \RR \subseteq \map \bigcup {\bigcup \RR}$ where $\bigcup \RR$ denotes the union of $\RR$.


Relation is Set implies Domain and Image are Sets

Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\RR$ be a set. Then $\Dom \RR$ and $\Img \RR$ are also sets.


Definition:Mapping/Class Theory

Let $V$ be a basic universe. A '''mapping''' $f$ in the context of Class Theory is a relation such that: :$f \subseteq V \times V$: :$\forall x \in \Dom f: \exists! y \in \Img f: \tuple {x, y} \in f$ That is, for every $x$ in the domain of $f$, there exists exactly one $y$ in the image of $f$ such that $\tuple {x, y} \in f$.


Set is Transitive iff Subset of Power Set

A set $S$ is transitive {{iff}}: :$S \subseteq \powerset S$ where $\powerset S$ denotes the power set of $S$.


Universal Class less Set is not Transitive

Let $V$ be a basic universe. Let $a \in V$ be a set. Then: :$V \setminus \set a$ is not a transitive class where $\setminus$ denotes class difference.


Equivalence of Formulations of Axiom of Infinity for Zermelo Universe

The following formulations of the '''Axiom of Infinity''' in the context of '''class theory''' are equivalent:


Inductive Construction of Natural Numbers fulfils Peano's Axioms

Let $P$ denote the set of natural numbers by definition as an inductive set. Then $P$ fulfils Peano's axioms.


Natural Number is Transitive Set

Let $n$ be a natural number. Then $n$ is a transitive set.


Natural Number is Ordinary Set

Let $n$ be a natural number. Then $n$ is an ordinary set.


Product Space Local Basis Induced from Factor Spaces Local Bases

Let $\family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let $x = \family{x_\alpha} \in S$. Let $\BB_\alpha$ be a local basis for $x_\alpha$ in the topological space $\struct {S_\alpha, \tau_\alpha}$ for each $\alpha \in I$. Let $\BB_x$ be the set of all cartesian products of the form $\displaystyle \prod_{\alpha \mathop \in I} U_\alpha$ where: :for all but finitely many indices $\alpha : U_\alpha = S_\alpha$ :for all $\alpha \in I : U_\alpha \ne S_\alpha \implies U_\alpha \in \BB_\alpha$ Then $\BB_x$ is a local basis for $x$ in the product space $\struct{S, \tau}$.


Natural Numbers cannot be Elements of Each Other

Let $m$ and $n$ be natural numbers. Then it cannot be the case that both $m \in n$ and $n \in m$.


Inductive Construction of Natural Numbers fulfils Peano's Axiom of Injectivity

Let $P$ denote the set of natural numbers by definition as an inductive set. Then $P$ fulfils: :{{PeanoAxiom|3}} where $s$ denotes the successor mapping.


Element of Natural Number is Natural Number

Let $n$ be a natural number. Let $m \in n$. Then $m$ is also a natural number.


Natural Number is Superset of its Union

Let $n \in \N$ be a natural number as defined by the von Neumann construction. Then: :$\bigcup n \subseteq n$


Subspace of Product Space Homeomorphic to Factor Space/Product with Singleton

Let $T_1$ and $T_2$ be non-empty topological spaces. Let $b \in T_2$. Let $T_1 \times T_2$ be the product space of $T_1$ and $T_2$. Let $T_2 \times T_1$ be the product space of $T_2$ and $T_1$. Then: :$T_1$ is homeomorphic to the subspace $T_1 \times \set b$ of $T_1 \times T_2$ :$T_1$ is homeomorphic to the subspace $\set b \times T_1$ of $T_2 \times T_1$


Set of Natural Numbers Equals its Union

Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$. Then: :$\bigcup \omega = \omega$


Set of Natural Numbers Equals Union of its Successor

Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$. Then: :$\bigcup \omega^+ = \omega$


Equivalence of Definitions of Minimally Inductive Class

Let $A$ be a class. Let $g$ be a mapping on $A$. {{TFAE|def = Minimally Inductive Class under General Mapping|view = minimally inductive class under $g$}}


Principle of General Induction

Let $M$ be a class. Let $g: M \to M$ be a mapping on $M$. Let $M$ be minimally inductive under $g$. Let $P: M \to \set {\T, \F}$ be a propositional function on $M$. Suppose that: :$(1): \quad \map P \O = \T$ :$(2): \quad \forall x \in M: \map P x = \T \implies \map P {\map g x} = \T$ Then: :$\forall x \in M: \map P x = \T$


Von Neumann Construction of Natural Numbers is Minimally Inductive

Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction. $\omega$ is a minimally inductive class under the successor mapping.


Cartesian Product of Subsets/Family of Subsets

Let $\family {S_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\family {T_i}_{i \mathop \in I}$ be a family of sets. Let $T = \displaystyle \prod_{i \mathop \in I} T_i$ be the Cartesian product of $\family {T_i}_{i \mathop \in I}$. Then: :$\paren{\forall i \in I: T_i \subseteq S_i} \implies T \subseteq S$.


Double Induction Principle

Let $M$ be a class. Let $g: M \to M$ be a mapping on $M$. Let $M$ be a minimally inductive class under $g$. Let $\RR$ be a relation on $M$ which satisfies: {{begin-axiom}} {{axiom | n = \text D_1         | q = \forall x \in M         | m = \map \RR {x, \O} }} {{axiom | n = \text D_2         | q = \forall x, y \in M         | m = \map \RR {x, y} \land \map \RR {y, x} \implies \map \RR {x, \map g y} }} {{end-axiom}} Then $\map \RR {x, y}$ holds for all $x, y \in M$.


Double Induction Principle/Lemma

Let $x$ be a right normal element of $M$ with respect to $\RR$. Then $x$ is also a left normal element of $M$ with respect to $\RR$.


Cartesian Product of Subsets/Family of Nonempty Subsets

Let $T_i \ne \O$ for all $i \in I$. Then: :$T \subseteq S \iff \forall i \in I: T_i \subseteq S_i$.


Projection is Injection iff Factor is Singleton/Family of Sets

Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\pr_j: S \to S_j$ be the $j$th projection on $S$. Then $\pr_j$ is an injection {{iff}} $S_i$ is a singleton for all $i \in I \setminus \set j$.


Progressing Function Lemma

Let $A$ be a class. Let $g$ be a progressing mapping on $A$. Let $\RR$ be the relation defined as: :$\map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$ where $\lor$ denotes disjunction (inclusive "or"). Then: {{begin-axiom}} {{axiom | n = 1         | q = \forall y \in \Dom g         | ml= \map \RR {y, \O} }} {{axiom | n = 2         | q = \forall x, y \in \Dom g         | ml= \map \RR {x, y} \land \map \RR {y, x}         | mo= \implies         | mr= \map \RR {x, \map g y} }} {{end-axiom}} where $\land$ denotes conjunction ("and").


Minimally Inductive Class under Progressing Mapping induces Nest

Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then $M$ is a nest in which: :$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$


Sandwich Principle

Let $A$ be a class. Let $g: A \to A$ be a mapping on $A$ such that: :for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$. Then: :$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$ That is, there is no element $y$ of $A$ such that: :$x \subset y \subset \map g x$ where $\subset$ denotes a proper subset.


Sandwich Principle/Corollary 1

Let: :$x \subset y$ where $\subset$ denotes a proper subset. Then: :$\map g x \subseteq y$


Sandwich Principle/Corollary 2

Let $g$ be a progressing mapping. Let $x \subseteq y$. Then: :$\map g x \subseteq \map g y$


Class under Progressing Mapping such that Elements are Sandwiched is Nest

Let $A$ be a class. Let $g: A \to A$ be a progressing mapping on $A$ such that: :$\forall x, y \in A: \map g x \subseteq y \lor y \subseteq x$ Then $A$ is a nest: :$\forall x, y \in A: x \subseteq y \lor y \subseteq x$


Characteristics of Minimally Inductive Class under Progressing Mapping

Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then for all $x, y \in M$:


Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle

:$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$


Characteristics of Minimally Inductive Class under Progressing Mapping/Image of Proper Subset is Subset

:$x \subset y \implies \map g x \subseteq y$


Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets

:$x \subseteq y \implies \map g x \subseteq \map g y$


Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element

Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then every non-empty bounded subset of $M$ has a greatest element.


Projection is Injection iff Factor is Singleton/Family of Sets/Necessary Condition

Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\pr_j: S \to S_j$ be the $j$th projection on $S$. Let $\pr_j$ be an injection. Then $S_i$ is a singleton for all $i \in I \setminus \set j$.


Projection is Injection iff Factor is Singleton/Family of Sets/Sufficient Condition

Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\pr_j: S \to S_j$ be the $j$th projection on $S$. Let $S_i$ be a singleton for all $i \in I \setminus \set {j}$. Then $\pr_j$ is an injection. </onlyinclude>


Product Space of Subspaces is Subspace of Product Space

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle T = \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Let $\family {\struct {Y_i, \upsilon_i} }_{i \mathop \in I}$ be a family of topological spaces such that: :$\forall i \in I : \struct {Y_i, \upsilon_i}$ is a topological subspace of $\struct {X_i, \tau_i}$ Let $\displaystyle S = \struct {Y, \upsilon} = \prod_{i \mathop \in I} \struct {Y_i, \upsilon_i}$ be the product space of $\family {\struct {Y_i, \upsilon_i} }_{i \mathop \in I}$. Let $T_Y = \struct {Y, \tau_Y}$ be the topological subspace of $T$. Then $S = T_Y$.


Difference of Complex Conjugates

Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 - z_2} = \overline {z_1} - \overline {z_2}$


Nonzero natural number is another natural number successor

Let $\N$ be the 0-based natural numbers: :$\N = \left\{{0, 1, 2, \ldots}\right\}$ Let $s: \N \to \N: \map s n = n + 1$ be the successor function. Then: :$\forall n \in \N \setminus \set 0 \paren {\exists m \in \N: \map s m = n }$


Double Angle Formulas/Hyperbolic Functions

=== Double Angle Formula for Hyperbolic Sine === {{:Double Angle Formulas/Hyperbolic Sine}} === Double Angle Formula for Hyperbolic Cosine === {{:Double Angle Formulas/Hyperbolic Cosine}} === Double Angle Formula for Hyperbolic Tangent === {{:Double Angle Formulas/Hyperbolic Tangent}} where $\sinh, \cosh, \tanh$ denote hyperbolic sine, hyperbolic cosine and hyperbolic tangent respectively.


Subspace of Product Space Homeomorphic to Factor Space/Proof 1/Lemma 2

:$\pr_i {\restriction_{Y_i} } = p_i$


Subspace of Product Space Homeomorphic to Factor Space/Proof 1/Lemma 1

:$Y_i = \prod_{j \mathop \in I} Z_j$


Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element

Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x$ be a fixed point of $g$. Then $x$ is the greatest element of $M$.


Closed Class under Progressing Mapping Lemma

Let $N$ be a class which is closed under a progressing mapping $g$. Let $g$ be such that: :$\forall x, y \in N: \map g x \subseteq y \lor y \subseteq x$ :if $\map g x = x$, then $x$ is the greatest element of $N$. Let the following hold: :$A \subseteq N$ is a subclass of $N$ :$x \in N$ is an element of $N$ Let $x$ be: :a proper subset of all elements of $A$ and: :the greatest element of $A$ with that property. Then $\map g x \in A$ and is the smallest element of $A$.


Minimally Inductive Class under Progressing Mapping is Well-Ordered under Inclusion

Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x$ be a fixed point of $g$. Then $M$ is well-ordered under the inclusion relation.


Minimally Closed Class under Progressing Mapping induces Nest

For all $x, y \in N$: :either $\map g x \subseteq y$ or $y \subseteq x$ and $N$ forms a nest: :$\forall x, y \in N: x \subseteq y$ or $y \subseteq x$


Statement of Conditions: {{:Minimally Closed Class under Progressing Mapping/Statement}} Then the following results hold:


Fixed Point of Progressing Mapping on Minimally Closed Class is Greatest Element

$g$ has no fixed point, unless possibly the greatest element, if there is one.


Smallest Element of Minimally Closed Class under Progressing Mapping

$b$ is the smallest element of $N$.


Equivalence of Definitions of Minimally Closed Class

Let $A$ be a class. Let $g$ be a mapping on $A$. {{TFAE|def = Minimally Closed Class|view = minimally closed class under $g$}}


Principle of General Induction/Minimally Closed Class

Let $M$ be a class. Let $g: M \to M$ be a mapping on $M$. Let $b \in M$ such that $M$ is minimally closed under $g$ with respect to $b$. Let $P: M \to \set {\T, \F}$ be a propositional function on $M$. Suppose that: :$(1): \quad \map P b = \T$ :$(2): \quad \forall x \in M: \map P x = \T \implies \map P {\map g x} = \T$ Then: :$\forall x \in M: \map P x = \T$


Double Induction Principle/Minimally Closed Class

Let $M$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $M$ such that $M$ is minimally closed under $g$ with respect to $b$. Let $\RR$ be a relation on $M$ which satisfies: {{begin-axiom}} {{axiom | n = \text D_1         | q = \forall x \in M         | m = \map \RR {x, b} }} {{axiom | n = \text D_2         | q = \forall x, y \in M         | m = \map \RR {x, y} \land \map \RR {y, x} \implies \map \RR {x, \map g y} }} {{end-axiom}} Then $\map \RR {x, y}$ holds for all $x, y \in M$.


Double Induction Principle/Minimally Closed Class/Lemma

Let $x$ be a right normal element of $M$ with respect to $\RR$. Then $x$ is also a left normal element of $M$ with respect to $\RR$.


Sandwich Principle for Minimally Closed Class

Let $N$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$. Then for all $x, y \in N$: :$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$


Image of Proper Subset under Progressing Mapping on Minimally Closed Class

Let $N$ be a class which is closed under a progressing mapping $g$. Let $b$ be an element of $N$ such that $N$ is minimally closed under $g$ with respect to $b$. Then: :$x \subset y \implies \map g x \subseteq y$


Equation of Cornu Spiral/Parametric

Let $K$ be a Cornu spiral embedded in a Cartesian coordinate plane such that the origin coincides with the point at which $s = 0$. Then $K$ can be expressed by the parametric equations: :$\begin {cases} x = a \sqrt 2 \map {\operatorname C} {\dfrac s {a \sqrt 2} } \\ y = a \sqrt 2 \map {\operatorname S} {\dfrac s {a \sqrt 2} } \end {cases}$ where: :$\operatorname C$ denotes the Fresnel cosine integral function :$\operatorname S$ denotes the Fresnel sine integral function.


Sampling Function is its own Fourier Transform

Consider the sampling function $\operatorname {III}: \R \to \R$. Then: :$\map \FF {\operatorname {III} } = \operatorname {III}$ where $\FF$ denotes the Fourier transform.


Even Impulse Pair is Fourier Transform of Cosine Function

Consider the (real) cosine function $\cos: \R \to \R$. Then: :$\map \FF {\cos} = \operatorname {II}$ where: :$\FF$ denotes the Fourier transform :$\operatorname {II}$ denotes the even impulse pair function.


Convolution of Real Function with Rectangle Function

Let $f: \R \to \R$ be a real function. Consider the rectangle function $\Pi: \R \to \R$. Then: :$\forall x \in \R: \map \Pi x * \map f x = \displaystyle \int_{x \mathop - \frac 1 2}^{x \mathop + \frac 1 2} \map f u \rd u$ where $*$ denotes the convolution integral.


Repeated Fourier Transform of Even Function

Let $f: \R \to \R$ be an even real function which is Lebesgue integrable. Let $\displaystyle \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$. Let $\displaystyle \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier transform of $F$. Then: :$\map g t = \map f t$


Repeated Fourier Transform of Odd Function

Let $f: \R \to \R$ be an odd real function which is Lebesgue integrable. Let $\displaystyle \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$. Let $\displaystyle \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier transform of $F$. Then: :$\map g t = -\map f t$


Repeated Fourier Transform of Real Function

Let $f: \R \to \R$ be a real function which is Lebesgue integrable. Let $\displaystyle \map \FF {\map f t} = \map F s = \int_{-\infty}^\infty e^{-2 \pi i s t} \map f t \rd t$ be the Fourier transform of $f$. Let $\displaystyle \map \FF {\map F s} = \map g t = \int_{-\infty}^\infty e^{-2 \pi i t s} \map F s \rd s$ be the Fourier transform of $F$. Then: :$\map g t = \map f {-t}$


Fourier's Theorem/Integral Form

Let $f: \R \to \R$ be a real function which satisfies the Dirichlet conditions on $\R$. Then: :$\dfrac {\map f {t^+} + \map f {t^-} } 2 = \displaystyle \int_{-\infty}^\infty e^{2 \pi i t s} \paren {\int_{-\infty}^\infty e^{-2 \pi i d t} \map f t \rd t} \rd s$ where: :$\map f {t^+}$ and $\map f {t^-}$ denote the limit from above and the limit from below of $f$ at $t$.


Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Lemma 1

:$\map {p_i^\to} {\map{\pr_k^\gets } {V_k} \cap Y_i}$ is open in $\struct{X_i, \tau_i}$


Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Injection

Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $p_i = \pr_i {\restriction_{Y_i}}$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is an injection.


Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Surjection

Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $p_i = \pr_i {\restriction_{Y_i} }$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is a surjection.


Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Continuous Mapping

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.  Let $p_i = \pr_i {\restriction_{Y_i}}$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is continuous.


Subspace of Product Space Homeomorphic to Factor Space/Proof 2/Open Mapping

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Let $z \in X$. Let $i \in I$.  Let $Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$. Let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$.  Let $p_i = \pr_i {\restriction_{Y_i}}$, where $\pr_i$ is the projection from $X$ to $X_i$. Then: :$p_i$ is an open mapping.


Triple Angle Formulas/Sine/Historical Note

The Triple Angle Formula for Sine is often attributed to {{AuthorRef|François Viète}}, although it was in fact discovered by {{AuthorRef|Jamshīd al-Kāshī}} a century or more earlier.


Exponential Distribution in terms of Continuous Uniform Distribution

Let $X \sim \mathrm U \hointl 0 1$ where $\mathrm U \hointl 0 1$ is the continuous uniform distribution on $\hointl 0 1$. Let $\beta$ be a positive real number. Then:  :$-\beta \ln X \sim \Exponential \lambda$ where $\operatorname {Exp}$ is the exponential distribution.


Expectation of Non-Negative Random Variable is Non-Negative

Let $X$ be a random variable.  Let $\map \Pr {X \ge 0} = 1$.  Then $\expect X \ge 0$, where $\expect X$ denotes the expectation of $X$.


Expectation of Non-Negative Random Variable is Non-Negative/Discrete

Let $X$ be a discrete random variable.  Let $\map \Pr {X \ge 0} = 1$.  Then $\expect X \ge 0$, where $\expect X$ denotes the expectation of $X$.


Expectation of Non-Negative Random Variable is Non-Negative/Continuous

Let $X$ be a continuous random variable.  Let $\map \Pr {X \ge 0} = 1$.  Then $\expect X \ge 0$, where $\expect X$ denotes the expectation of $X$.


Expectation of Almost Surely Constant Random Variable

Let $X$ be an almost surely constant random variable. That is, there exists some $c \in \R$ such that:  :$\map \Pr {X = c} = 1$ Then:  :$\expect X = c$


Expectation Preserves Inequality

Let $X$, $Y$ be random variables. Let $\map \Pr {X \ge Y} = 1$. Then:  :$\expect X \ge \expect Y$


Covariance of Random Variable with Itself

Let $X$ be a random variable. Then $\cov {X, X} = \var X$.


Covariance is Symmetric

Let $X$ and $Y$ be random variables. Then $\cov {X, Y} = \cov {Y, X}$.


Covariance of Sums of Random Variables

Let $n$ be a strictly positive integer. Let $\sequence {X_i}_{1 \le i \le n}$, $\sequence {Y_j}_{1 \le j \le n}$ be sequences of random variables. Then:  :$\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, \sum_{j \mathop = 1}^n Y_j} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}$


Expectation of Linear Transformation of Random Variable/Discrete

Let $X$ be a discrete random variable.  Let $a, b$ be real numbers.  Then we have:  :$\expect {a X + b} = a \expect X + b$ where $\expect X$ denotes the expectation of $X$.


Expectation of Linear Transformation of Random Variable/Continuous

Let $X$ be a continuous random variable.  Let $a, b$ be real numbers.  Then we have:  :$\expect {a X + b} = a \expect X + b$ where $\expect X$ denotes the expectation of $X$.


Cauchy's Mean Theorem/Proof of Equality Condition

Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all positive. Let $A_n$ be the arithmetic mean of $x_1, x_2, \ldots, x_n$. Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$. Then: :$A_n = G_n$ {{iff}}: :$\forall i, j \in \set {1, 2, \ldots, n}: x_i = x_j$ That is, {{iff}} all terms are equal. Then:


Fourier Series/Square Wave

600pxthumbrightSquare Wave and $9$th Approximation Let $\map S x$ be the square wave defined on the real numbers $\R$ as: :$\forall x \in \R: \map S x = \begin {cases} 1 & : x \in \openint 0 l \\ -1 & : x \in \openint {-l} 0 \\ \map S {x + 2 l} & : x < -l \\ \map S {x - 2 l} & : x > +l \end {cases}$ Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map S x       | o = \sim       | r = \frac 4 \pi \sum_{r \mathop = 0}^\infty \frac 1 {2 r + 1} {\sin \frac {\pi x} l}       | c =  }} {{eqn | r = \frac 4 \pi \paren {\sin \frac {\pi x} l + \dfrac 1 3 \sin \frac {3 \pi x} l + \dfrac 1 5 \sin \frac {5 \pi x} l + \dotsb}       | c =  }} {{end-eqn}}


Fourier Series/Triangle Wave

600pxthumbrightTriangle Wave and $9$th Approximation <onlyinclude> Let $\map T x$ be the triangle wave defined on the real numbers $\R$ as: :$\forall x \in \R: \map T x = \begin {cases} \size x & : x \in \closedint {-l} l \\ \map T {x + 2 l} & : x < -l \\ \map T {x - 2 l} & : x > +l \end {cases}$ where: :$l$ is a given real constant :$\size x$ denotes the absolute value of $x$. Then its Fourier series can be expressed as: {{begin-eqn}} {{eqn | l = \map T x       | o = \sim       | r = \frac l 2 - \frac {4 l} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} l       | c =  }} {{eqn | r = \frac l 2 - \frac {4 l} {\pi^2} \paren {\cos \dfrac {\pi x} l + \frac 1 {3^2} \cos \dfrac {3 \pi x} l + \frac 1 {5^2} \cos \dfrac {5 \pi x} l + \dotsb}       | c =  }} {{end-eqn}}


Subspace of Product Space Homeomorphic to Factor Space/Product with Singleton/Lemma

:$f$ is a bijection.


Standard Continuous Uniform Distribution in terms of Exponential Distribution

Let $X$ and $Y$ be independent random variables. Let $\beta$ be a strictly positive real number. Let $X$ and $Y$ be random samples from the exponential distribution with parameter $\beta$.  Then: :$\dfrac X {X + Y} \sim \operatorname U \openint 0 1$ where $\operatorname U \openint 0 1$ is the uniform distribution on $\openint 0 1$.


Identity Function is Odd Function

Let $I_\R: \R \to \R$ denote the identity function on $\R$. Then $I_\R$ is an odd function.


Half-Range Fourier Series/Identity Function

Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint 0 \lambda \to \R$ be the identity function on the open real interval $\openint 0 \lambda$: :$\forall x \in \openint 0 \lambda: \map f x = x$ The half-range Fourier series of $f$ over $\openint 0 \lambda$ can be given in the following forms:


Half-Range Fourier Series/Identity Function/Cosine

The half-range Fourier cosine series for $\map f x$ can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} \lambda       | c =  }} {{eqn | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \paren {\cos \dfrac {\pi x} \lambda + \frac 1 {3^2} \cos \dfrac {3 \pi x} \lambda + \frac 1 {5^2} \cos \dfrac {5 \pi x} \lambda + \dotsb}       | c =  }} {{end-eqn}}


Half-Range Fourier Series/Identity Function/Sine

The half-range Fourier sine series for $\map f x$ can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \dfrac {2 \lambda} \pi \displaystyle \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \sin \frac {n \pi x} \lambda       | c =  }} {{eqn | r = \dfrac {2 \lambda} \pi \paren {\sin \dfrac {\pi x} \lambda - \frac 1 2 \sin \dfrac {2 \pi x} \lambda + \frac 1 3 \sin \dfrac {3 \pi x} \lambda - \dotsb}       | c =  }} {{end-eqn}}


Absolute Value Function is Even Function

Let $\size {\, \cdot \,} : \R \to \R$ denote the absolute value function on $\R$: Then $\size {\, \cdot \,}$ is an even function.


Fourier Series/Absolute Value Function over Symmetric Range

Let $\lambda \in \R_{>0}$ be a strictly positive real number. Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the absolute value function on the open real interval $\openint {-\lambda} \lambda$: :$\forall x \in \openint {-\lambda} \lambda: \map f x = \size x$ The Fourier series of $f$ over $\openint {-\lambda} \lambda$ can be given as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} \cos \dfrac {\paren {2 n + 1} \pi x} \lambda       | c =  }} {{eqn | r = \frac \lambda 2 - \frac {4 \lambda} {\pi^2} \paren {\cos \dfrac {\pi x} \lambda + \frac 1 {3^2} \cos \dfrac {3 \pi x} \lambda + \frac 1 {5^2} \dfrac {5 \pi x} \lambda + \dotsb}       | c =  }} {{end-eqn}}


Factor Spaces of Hausdorff Product Space are Hausdorff

Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Let $T$ be a $T_2$ (Hausdorff) space. Then: :for each $\alpha \in I$, $\struct {S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space.


Cartesian Product is Empty iff Factor is Empty/Family of Sets

Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Then: :$S = \O$  {{iff}} $S_i = \O$ for some $i \in I$


Derivative of P-Norm wrt P

Let $p \ge 1$ be a real number. Let $\ell^p$ denote the $p$-sequence space. Let $\mathbf x = \sequence {x_n} \in \ell^p$. Let $\norm {\mathbf x}_p$ be a p-norm. Suppose, $\norm {\mathbf x}_p \ne 0$. Then: :$\displaystyle \dfrac \d {\d p} \norm {\mathbf x}_p = \frac {\norm {\mathbf x}_p} p \paren { \frac {\sum_{n \mathop = 0}^\infty \size {x_n}^p \map \ln {\size {x_n} } } {\norm {\bf x}_p^p} - \map \ln {\norm {\bf x}_p} }$


Extension of Half-Range Fourier Sine Function to Symmetric Range

Let $\map f x$ be a real function defined on the interval $\openint 0 \lambda$. Let $\map f x$ be represented by the half-range Fourier sine series $\map S x$: :$\map f x \sim \map S x = \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$ where for all $n \in \Z_{> 0}$: :$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$ Then $\map S x$ also represents the extension $g: \openint {-\lambda} \lambda \to \R$ of $f$, defined as: :$\forall x \in \openint {-\lambda} \lambda: \map g x = \begin {cases} \map f x & : x > 0 \\ -\map f {-x} & : x < 0 \\ 0 & : x = 0 \end {cases}$


Primitive of x by Logarithm of x squared plus a squared

:$\displaystyle \int x \map \ln {x^2 + a^2} \rd x = \frac {\paren {x^2 + a^2} \map \ln {x^2 + a^2} - x^2} 2 + C$


Equivalence of Definitions of Sets Separated by Neighborhoods

Let $T = \struct {S, \tau}$ be a topological space. {{TFAE|def = Separated by Neighborhoods/Sets|view = Sets Separated by Neighborhoods}}


Equivalence of Definitions of Points Separated by Neighborhoods

Let $T = \struct {S, \tau}$ be a topological space. {{TFAE|def = Points Separated by Neighborhoods}}


Product Space is T3 iff Factor Spaces are T3/Product Space is T3 implies Factor Spaces are T3

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_3$ space. Then for each $\alpha \in I$, $\struct {S_\alpha, \tau_\alpha}$ is a $T_3$ space.


Product Space is T3 iff Factor Spaces are T3/Factor Spaces are T3 implies Product Space is T3

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. For each $\alpha \in I$, let $\struct{S_\alpha, \tau_\alpha}$ be a $T_3$ space. Then $T$ is a $T_3$ space.


Mittag-Leffler Expansion for Cosecant Function/Real Domain

:$\pi \cosec \pi \alpha = \dfrac 1 \alpha + \displaystyle 2 \sum_{n \mathop \ge 1} \paren {-1}^n \dfrac {\alpha} {\alpha^2 - n^2}$


Sum from -m to m of 1 minus Cosine of n + alpha of theta over n + alpha

For $0 < \theta < 2 \pi$: :$\displaystyle \sum_{n \mathop = -m}^m \dfrac {1 - \cos \paren {n + \alpha} \theta} {n + \alpha} = \int_0^\theta \map \sin {\alpha u} \dfrac {\sin \paren {m + \frac 1 2} u \rd u} {\sin \frac 1 2 u}$


Sum over Integers of Cosine of n + alpha of theta over n + alpha

Let $\alpha \in \R$ be a real number which is specifically not an integer. For $0 < \theta < 2 \pi$: :$\displaystyle \dfrac 1 \alpha + \sum_{n \mathop \ge 1} \dfrac {2 \alpha} {\alpha^2 - n^2} = \sum_{n \mathop \in \Z} \dfrac {\cos \paren {n + \alpha} \theta} {n + \alpha}$


Sum of Complex Exponentials of i times Arithmetic Sequence of Angles

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


Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1

:$\displaystyle \sum_{k \mathop = 0}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n \alpha} 2} + i \map \sin {\theta + \frac {n \alpha} 2} } \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} }$


Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2

:$\displaystyle \sum_{k \mathop = 1}^n e^{i \paren {\theta + k \alpha} } = \paren {\map \cos {\theta + \frac {n + 1} 2 \alpha} + i \map \sin {\theta + \frac {n + 1} 2 \alpha} } \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} }$


Sum of Sines of Arithmetic Sequence of Angles/Formulation 1

{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^n \map \sin {\theta + k \alpha}       | r = \sin \theta + \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \frac {\map \sin {\alpha \paren {n + 1} / 2} } {\map \sin {\alpha / 2} } \map \sin {\theta + \frac {n \alpha} 2} }} {{end-eqn}}


Sum of Sines of Arithmetic Sequence of Angles/Formulation 2

{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n \map \sin {\theta + k \alpha}       | r = \map \sin {\theta + \alpha} + \map \sin {\theta + 2 \alpha} + \map \sin {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \map \sin {\theta + \frac {n + 1} 2 \alpha}\frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} } }} {{end-eqn}}


Sum of Cosines of Arithmetic Sequence of Angles/Formulation 2

{{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n \map \cos {\theta + k \alpha}       | r = \map \cos {\theta + \alpha} + \map \cos {\theta + 2 \alpha} + \map \cos {\theta + 3 \alpha} + \dotsb }} {{eqn | r = \map \cos {\theta + \frac {n + 1} 2 \alpha} \frac {\map \sin {n \alpha / 2} } {\map \sin {\alpha / 2} } }} {{end-eqn}}


Weierstrass Approximation Theorem/Lemma 1

:$\displaystyle \sum_{k \mathop = 0}^n k \map {p_{n,k} } t = n t$


Weierstrass Approximation Theorem/Lemma 2

:$\displaystyle \sum_{k \mathop = 0}^n \paren {k - nt}^2 \map {p_{n,k} } t = n t \paren {1 - t}$


Convergent Sequence in Normed Vector Space has Unique Limit

Let $\struct {X, \norm {\,\cdot\,}}$ be a normed vector space. Let $\sequence {x_n}$ be a sequence in $\struct {X, \norm {\,\cdot\,} }$. Then $\sequence {x_n}$ can have at most one limit.


Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Product Space is T3 1/2 implies Factor Spaces are T3 1/2

Let $\mathbb S = \family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_{3 \frac 1 2}$ space. Then for each $\alpha \in I$, $\struct{S_\alpha, \tau_\alpha}$ is a $T_{3 \frac 1 2}$ space.


Product Space is T3 1/2 iff Factor Spaces are T3 1/2/Factor Spaces are T3 1/2 implies Product Space is T3 1/2

Let $\mathbb S = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$ with $S_\alpha \neq \O$ for every $\alpha \in I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. For each $\alpha \in I$, let $\struct{S_\alpha, \tau_\alpha}$ be a $T_{3 \frac 1 2}$ space. Then $T$ is a $T_{3 \frac 1 2}$ space.


Half-Range Fourier Sine Series/Sine of Non-Integer Multiple of x over 0 to Pi

Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer. Let $\map f x$ be the real function defined on $\openint 0 \pi$ as: :$\map f x = \sin \lambda x$ Then its half-range Fourier sine series can be expressed as: {{begin-eqn}} {{eqn | l = \map f x       | o = \sim       | r = \frac {2 \sin \lambda \pi} \pi \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {n \sin n x} {\lambda^2 - n^2} }       | c =  }} {{eqn | r = \frac {2 \sin \lambda \pi} \pi \paren {-\frac {\sin x} {\lambda^2 - 1} + \frac {2 \sin 2 x} {\lambda^2 - 4} - \frac {3 \sin 3 x} {\lambda^2 - 9}  + \frac {4 \sin 4 x} {\lambda^2 - 16} - \dotsb}       | c =  }} {{end-eqn}}


Minimum Rule for Continuous Functions

Let $\struct {S, \tau}$ be a topological space. Let $f, g: S \to \R$ be continuous real-valued functions. Let $\min \set {f, g}: S \to \R$ denote the pointwise minimum of $f$ and $g$. Then: :$\min \set {f, g}$ is continuous.


Min is Half of Sum Less Absolute Difference

For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$: :$\min \set {a, b} = \dfrac 1 2 \paren {a + b - \size {a - b} }$


Continuity Test for Real-Valued Functions

Let $\struct{S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Let $x \in S$. Then $f$ is continuous at $x$ {{iff}}: :$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$


Continuity Test for Real-Valued Functions/Everywhere Continuous

Let $\struct{S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Then $f$ is everywhere continuous {{iff}}: :$\forall x \in S : \forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$


Oesterlé-Masser Conjecture

Let $\epsilon \in \R$ be a strictly positive real number.


Oesterlé-Masser Conjecture/Formulation 1

There exists only a finite number of triples of (strictly) positive integers $\tuple {a, b, c}$ with the conditions: :$a + b = c$ :$a$, $b$ and $c$ are pairwise coprime such that: :$c > \map {\operatorname {rad} } {a b c}^{1 + \epsilon}$ where $\operatorname {rad}$ denotes the radical of an integer.


Definite Integral of Function satisfying Dirichlet Conditions is Continuous

Let $f: \R \to \R$ be a real function defined in the open interval $\openint {-\pi} \pi$. Let $f$ fulfil the Dirichlet conditions in $\openint {-\pi} \pi$. Let $a_0, a_1, \dotsc; b_1, \dotsc$ be the Fourier coefficients of $f$ in $\openint {-\pi} \pi$. Then the real function: :$\map F x = \displaystyle \int_{-\pi}^x \map f t \rd t - \dfrac {a_0} 2 x$ is continuous on $\openint {-\pi} \pi$.


Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 1

Let $G := \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\circ: G \times G \to G$ be the binary operation defined as: :$\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$ {{:Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers}}


Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2


Triangle Inequality/Complex Numbers/Corollary 2

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


Continuity Test for Real-Valued Functions/Necessary Condition

Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Let $x \in S$. Let $f$ be continuous at $x$ Then: :$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$


Continuity Test for Real-Valued Functions/Sufficient Condition

Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R$ be a real-valued function. Let $x \in S$. Let $f$ satisfy: :$\forall \epsilon \in \R_{>0} : \exists U \in \tau : x \in U : \map {f^\to} U \subseteq \openint {\map f x - \epsilon} {\map f x + \epsilon}$ Then $f$ is continuous at $x$


Equivalence of Definitions of Completely Hausdorff Space

{{TFAE|def = Completely Hausdorff Space|view = a completely Hausdorff space}} Let $T = \struct {S, \tau}$ be a topological space.


Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Necessary Condition

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a completely Hausdorff space. Then for each $\alpha \in I$, $\struct {S_\alpha, \tau_\alpha}$ is a completely Hausdorff space.


Metric Space is T4

Let $M = \struct {A, d}$ be a metric space. Then $M$ is $T_4$.


Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff/Sufficient Condition

Let $\SS = \family {\struct {S_\alpha, \tau_\alpha} }$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\SS$. Let each of $\struct {S_\alpha, \tau_\alpha}$ for $\alpha \in I$ be completely Hausdorff spaces. Then $T$ is a completely Hausdorff spaces.


P-adic Expansion Less Intial Zero Terms Represents Same P-adic Number

Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as a quotient of Cauchy sequences. Let $\mathbf a$ be an equivalence class in $\Q_p$. Let $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ be a $p$-adic expansion that represents $\mathbf a$. Let $l$ be the first index $i \ge m$ such that $d_i \ne 0$ Then the series: :$\displaystyle \sum_{i \mathop = l}^\infty d_i p^i$ also represents $\mathbf a$.


Subsequence is Equivalent to Cauchy Sequence

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Let $\sequence {x_{m_n} }$ be a subsequence of $\sequence {x_n}$. Then: :$\displaystyle \lim_{n \mathop \to \infty} {x_n - x_{m_n} } = 0$


Numbers with Square-Free Binomial Coefficients/Lemma

Let $n$ be a (strictly) positive integer.  Let $p$ be a prime number. By Basis Representation Theorem, there is a unique sequence $\sequence {a_j}_{0 \mathop \le j \mathop \le r}$ such that: :$(1): \quad \displaystyle n = \sum_{k \mathop = 0}^r a_k p^k$ :$(2): \quad \displaystyle \forall k \in \closedint 0 r: a_k \in \N_b$ :$(3): \quad r_t \ne 0$ Suppose $r \ge 2$ and $p^2 \nmid \dbinom n m$ for all $0 \le m \le n$.  Then: :$p^{r - 1} \divides \paren {n + 1}$ that is: :$p^{r - 1}$ divides $\paren {n + 1}$.


Derivative of Generating Function/General Result/Corollary

Let the coefficient of $z^n$ extracted from $\map G z$ be denoted: :$\sqbrk {z^n} \map G z := a_n$ Then: :$\sqbrk {z^m} \map G z = \dfrac 1 {m!} \map {G^{\paren m} } 0$ where $G^{\paren m}$ denotes the $m$th derivative of $G$.


Let $m$ be a positive integer. Then: :$\dfrac {\d^m} {\d z^m} \map G z = \displaystyle \sum_{k \mathop \ge 0} \dfrac {\paren {k + m}!} {k!} a_{k + m} z^k$ === Corollary === {{:Derivative of Generating Function/General Result/Corollary}}


Space of Continuous on Closed Interval Real-Valued Functions with Supremum Norm form Banach Space

Let $I = \closedint a b$ be a closed real interval. Let $\map \CC I$ be the space of real-valued functions, continuous on $I$. Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on real-valued functions, continuous on $I$. Then $\struct {\map \CC I, \norm {\,\cdot\,}_\infty}$ is a Banach space.


Inductive Set under Progressing Mapping has Minimally Inductive Subset

Let $A$ be an inductive class under a mapping $g$. Let $A$ be a set. Then there exists some subset $S$ of $A$ such that $S$ is minimally inductive under $g$.


Recurrence Relation for Bell Numbers

Let $B_n$ be the Bell number for $n \in \Z_{\ge 0}$. Then: :$B_{n + 1} = \displaystyle \sum_{k \mathop = 0}^n \dbinom n k B_k$ where $\dbinom n k$ are binomial coefficients.


Natural Number m is Less than n implies n is not Greater than Successor of n

Let $\N$ be the natural numbers. Let $m, n \in \N$. Then: :$m < n \implies m + 1 \le n$


Natural Number Ordering is Preserved by Successor Mapping

Let $\N$ be the natural numbers. Let $m, n \in \N$. Then: :$n \le m \implies n^+ \le m^+$


Non-Empty Bounded Subset of Natural Numbers has Greatest Element

Let $\omega$ be the set of natural numbers defined as the von Neumann construction. Then every non-empty bounded subset of $\omega$ has a greatest element.


Natural Number Less than or Equal to Successor of Another

Let $\N$ be the natural numbers. Let $m, n \in \N$ such that $m \le n^+$. Then either: :$(1): \quad m \le n$ or: :$(2): \quad m = n^+$


Mapping whose Image of Natural Number n is Subset of Image of Successor

Let $f: \N \to A$ be a mapping from the set of natural numbers $\N$ to a class $A$. Let $f$ have the property that: :$\forall n \in \N: \map f n \subseteq \map f {n^+}$ where $n^+$ is the successor of $n$. Then: :$\forall n, m \in N: n \le m \implies \map f n \subseteq \map f m$


Mapping whose Image of Natural Number n is Subset of Image of Successor/Corollary

Let $f$ have the property that: :$\forall n \in \N: \map f n \subsetneq \map f {n^+}$ where $n^+$ is the successor of $n$. Then: :$\forall n, m \in N: n < m \implies \map f n \subsetneq \map f m$


Non-Empty Finite Set of Natural Numbers has Greatest Element

Let $A$ be a non-empty finite set of natural numbers. Then $A$ has a greatest element.


Set of Subsets of Element of Minimally Inductive Class is Finite

Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x \in M$. Let $S$ be the set of all $y \in M$ such that $y \subseteq x$. Then $S$ is finite.


Minimally Inductive Class with Fixed Element is Finite

Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let there exist an element $x \in M$ such that $x = \map g x$. Then $M$ is a finite class.


Set of Natural Numbers is either Finite or Denumerable

Let $S$ be a subset of the natural numbers $\N$. Then $S$ is either finite or denumerable.


Unlike Electric Charges Attract

Let $a$ and $b$ be stationary particles, each carrying an electric charge of $q_a$ and $q_b$ respectively. Let $q_a$ and $q_b$ be of the opposite sign. That is, let $q_a$ and $q_b$ be unlike charges. Then the forces exerted by $a$ on $b$, and by $b$ on $a$, are such as to cause $a$ and $b$ to attract each other.


Value of Vacuum Permittivity

The value of the '''vacuum permittivity''' is calculated as: :$\varepsilon_0 = 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ (farads per metre) with a relative uncertainty of $1 \cdotp 5 \times 10^{-10}$.


Successor Mapping on Natural Numbers has no Fixed Element

Let $\N$ denote the set of natural numbers. Then: :$\forall n \in \N: n + 1 \ne n$


Residue of Fibonacci Number Modulo Fibonacci Number/Lemma

:$F_{m n + 1} \equiv \paren {\begin{cases} F_1 & : m \bmod 4 = 0 \\ F_{n - 1} & : m \bmod 4 = 1 \\ \paren {-1}^n F_1 & : m \bmod 4 = 2 \\ \paren {-1}^n F_{n - 1} & : m \bmod 4 = 3 \end{cases} } \pmod {F_n}$


Complement of Interval Defined by Absolute Value

Let $\xi, \delta \in \R$ be real numbers. Let $\delta > 0$. Then:


Complement of Open Interval Defined by Absolute Value

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