hcju/proofwikips huggingface dataset:
Product of Absolute Values of Integers
Let $a, b \in \Z$ be integers. Let $\size a$ denote the absolute value of $a$: :$\size a = \begin {cases} a & : a \ge 0 \\ -a : a < 0 \end {cases}$ Then: :$\size a \times \size b = \size {a \times b}$
Displacement of Particle under Force
Let $P$ be a particle of constant mass $m$. Let the position of $P$ at time $t$ be specified by the position vector $\mathbf r$. Let a force applied to $P$ be represented by the vector $\mathbf F$. Then the motion of $P$ can be given by the differential equation: :$\mathbf F = m \dfrac {\d^2 \mathbf r} {\d t^2}$ or using Newtonian notation: :$\mathbf F = m \ddot {\mathbf r}$
1-Sequence Space is Separable
$\ell^1$ space is a separable space.
Primitive of Root of Function under Half its Derivative
Let $f$ be a real function which is integrable. Then: :$\ds \int \frac {\map {f'} x} {2 \sqrt {\map f x} } \rd x = \sqrt {\map f x} + C$ where $C$ is an arbitrary constant.
Primitive of Hyperbolic Secant Function/Arctangent of Half Hyperbolic Tangent Form
:$\ds \int \sech x \rd x =  2 \map \arctan {\tanh \dfrac x 2} + C$
First Order ODE/y' + 2 x y = 1
The first order ODE: :$y' + 2 x y = 1$ has the general solution: :$y = e^{-{x^2} } \ds \int_a^x e^{t^2} \rd t$ where $a$ is an arbitrary constant.
Electric Potential over Conducting Surface is Constant
Let $S$ be a conducting surface. The electric potential $V$ over $S$ is constant. This can be expressed using the Laplacian: :$\nabla^2 V = 0$ and is thus seen to satisfy Laplace's equation.
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 4
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 4 Iff Definition 5
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 3 Iff Definition 7
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Base Axiom/Definition 1 Iff Definition 3
Let $S$ be a finite set. Let $\mathscr B$ be a non-empty set of subsets of $S$. {{TFAE|def=Base Axiom (Matroid)|view = Matroid Base Axiom}}
P-Norm is Norm/P-Sequence Space
The $p$-norm on the $p$-sequence space is a norm.
P-Norm is Norm/Complex Numbers
The $p$-norm on the complex numbers is a norm.
P-Norm is Norm/Real Numbers
The $p$-norm on the real numbers is a norm.
Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space
Let $\ell^\infty$ be the space of bounded sequences. Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers. Let $\paren +$ be the pointwise addition on the ring of sequences. Let $\paren {\, \cdot \,}$ be the pointwise multiplication on the ring of sequences. Then $\struct {\ell^\infty, +, \, \cdot \,}_\C$ is a vector space.
Zero Vector has no Direction
A zero vector has no direction.
Power Set is Closed under Countable Unions
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A_n \in \powerset S: n = 1, 2, \ldots: \ds \bigcup_{n \mathop = 1}^\infty A_n \in \powerset S$
Probability of Union of Disjoint Events is Sum of Individual Probabilities
Let $\EE$ be an experiment. Let $\struct {\Omega, \Sigma, \Pr}$ be a probability measure on $\EE$. Then: :$\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$
Probability Measure on Finite Sample Space
Let $\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$ be a finite set. Let $\Sigma$ be a $\sigma$-algebra on $\Omega$. Let $p_1, p_2, \ldots, p_n$ be non-negative real numbers such that: :$p_1 + p_2 + \cdots + p_n = 1$ Let $Q: \Sigma \to \R$ be the mapping defined as: :$\forall A \in \Sigma: \map Q A = \ds \sum_{i: \omega_i \in A} p_i$ Then $\struct {\Omega, \Sigma, Q}$ constitutes a probability space. That is, $Q$ is a probability measure on $\struct {\Omega, \Sigma}$.
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Rank Axioms
Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers. {{TFAE|def=Rank Axioms (Matroid)|view=Matroid Rank Axioms}}
Length of Element of Arc in Orthogonal Curvilinear Coordinates
Let $\tuple {q_1, q_2, q_3}$ denote a set of orthogonal curvilinear coordinates. Let the relation between those orthogonal curvilinear coordinates and Cartesian coordinates be expressed as: {{begin-eqn}} {{eqn | l = x       | r = \map x {q_1, q_2, q_3} }} {{eqn | l = y       | r = \map y {q_1, q_2, q_3} }} {{eqn | l = z       | r = \map z {q_1, q_2, q_3} }} {{end-eqn}} where $\tuple {x, y, z}$ denotes the Cartesian coordinates. The length $\d l$ of a small arc is given by: :$\d l = {h_1}^2 {\d q_1}^2 + {h_2}^2 {\d q_2}^2 + {h_3}^2 {\d q_3}^2$ where: :${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$
Laplacian of Function in Orthogonal Curvilinear Coordinates
Let $\map \psi {q_1, q_2, q_3}$ denote a real-valued function embedded in an orthogonal curvilinear coordinate system. Then the Laplacian of $\psi$ can be expressed as: :$\nabla^2 \psi = \dfrac 1 {h_1 h_2 h_3} \paren {\map {\dfrac \partial {\partial q_1} } {\dfrac {h_2 h_3} {h_1} \dfrac {\partial \psi} {\partial q_1} } + \map {\dfrac \partial {\partial q_2} } {\dfrac {h_3 h_1} {h_2} \dfrac {\partial \psi} {\partial q_2} } + \map {\dfrac \partial {\partial q_3} } {\dfrac {h_1 h_2} {h_3} \dfrac {\partial \psi} {\partial q_3} } }$ where: :${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$
Pressure of Gas for Density and Temperature
The pressure, density and temperature of a body of gas are linked by the equation: :$p = k \rho \paren {1 + \alpha t}$ where: :$p$ denotes the pressure of the gas :$\rho$ denotes the density of the gas :$t$ denotes thetemperature of the gas :$k$ and $\alpha$ are constants.
Leigh.Samphier/Sandbox/Matroid Satisfies Circuit Axioms
Let $S$ be a finite set. Let $\mathscr C$ be a non-empty set of subsets of $S$. Then $\mathscr C$ is the set of circuits of a matroid on $S$  $\mathscr C$ satisfies the circuit axioms: {{: Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1}}
Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms
Let $S$ be a finite set. Let $\mathscr C$ be a non-empty set of subsets of $S$. {{TFAE|def=Circuit Axioms (Matroid)|view=Matroid Circuit Axioms}}
Leigh.Samphier/Sandbox/Rank of Matroid Circuit is One Less Than Cardinality
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $C \subseteq S$ be a circuit of $M$. Let $\rho: \powerset S \to \Z$ denote the rank function of $M$. Then: :$\map \rho C = \card C -1$
Leigh.Samphier/Sandbox/Bound for Cardinality of Matroid Circuit
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $C \subseteq S$ be a circuit of $M$. Let $\rho: \powerset S \to \Z$ denote the rank function of $M$. Then: :$\card C \le \map \rho S + 1$
Leigh.Samphier/Sandbox/Matroid with No Circuits Has Single Base
Let $M = \struct {S, \mathscr I}$ be a matroid with no circuits. Then: :$S$ is the only base on $M$.
Leigh.Samphier/Sandbox/Proper Subset of Matroid Circuit is Independent
Let $M = \struct {S, \mathscr I}$ be a matroid. Let $C \subseteq S$ be a circuit of $M$. Then: :every proper subset $A$ of $C$ is independent
Supremum Norm is Norm/Space of Bounded Sequences
The supremum norm on the space of bounded sequences is a norm.
Parallelogram Law for Vector Subtraction
Let $\mathbf u$ and $\mathbf v$ be vectors. Consider a parallelogram, two of whose adjacent sides represent $\mathbf y$ and $\mathbf v$ (in magnitude and direction). :400px Then the diagonal of the parallelogram connecting the terminal points of $\mathbf u$ and $\mathbf v$ represents the magnitude and direction of $\mathbf u - \mathbf v$, the difference of $\mathbf u$ and $\mathbf v$.
Space of Bounded Sequences with Supremum Norm forms Normed Vector Space
Space of bounded sequences with supremum norm forms a normed vector space.
Riesz's Lemma
Let $X$ be a normed vector space. Let $Y$ be a proper closed linear subspace of $X$.  Let $\alpha \in \openint 0 1$. Then there exists $x_\alpha \in X$ such that:  :$\norm {x_\alpha} = 1$ with: :$\norm {x_\alpha - y} > \alpha$ for all $y \in Y$.
Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact
Let $X$ be a normed vector space. Let $S = \map {S_1} 0$ be the unit sphere centred at $0$ in $X$. Then $X$ is finite dimensional  $S$ is compact.
Finite Dimensional Subspace of Normed Vector Space is Closed
Let $V$ be a normed vector space. Let $W$ be a finite dimensional subspace of $V$.  Then $W$ is closed.
Catalan's Conjecture
The only solution to the Diophantine equation: :$x^a - y^b = 1$ for $a, b > 1$ and $x, y > 0$, is: :$x = 3, a = 2, y = 2, b = 3$
Total Force on Charged Particle from 2 Charged Particles
Let $p_1$, $p_2$ and $p_3$ be charged particles. Let $q_1$, $q_2$ and $q_3$ be the electric charges on $p_1$, $p_2$ and $p_3$ respectively. Let $\mathbf F_{ij}$ denote the force exerted on $q_j$ by $q_i$. Let $\mathbf F_i$ denote the force exerted on $q_i$ by the combined action of the other two charged particles. Then the force $\mathbf F_1$ exerted on $q_1$ by the combined action of $q_2$ and $q_3$ is given by: {{begin-eqn}} {{eqn | l = \mathbf F_1       | r = \mathbf F_{21} + \mathbf F_{31}       | c =  }} {{eqn | r = \dfrac {q_2 q_1} {4 \pi \varepsilon_0 r_{2 1}^3} \mathbf r_{2 1} + \dfrac {q_3 q_1} {4 \pi \varepsilon_0 r_{3 1}^3} \mathbf r_{3 1}       | c =  }} {{end-eqn}} where: :$\mathbf F_{21} + \mathbf F_{31}$ denotes the vector sum of $\mathbf F_{21}$ and $\mathbf F_{31}$ :$\mathbf r_{ij}$ denotes the displacement from $p_i$ to $p_j$ :$r_{ij}$ denotes the distance between $p_i$ and $p_j$ :$\varepsilon_0$ denotes the vacuum permittivity.
Total Force on Charged Particle from Multiple Charged Particles
Let $p_1, p_2, \ldots, p_n$ be charged particles. Let $q_1, q_2, \ldots, q_n$ be the electric charges on $p_1, p_2, \ldots, p_n$ respectively. For all $i$ in $\set {1, 2, \ldots, n}$ where $i \ne j$, let $\mathbf F_{i j}$ denote the force exerted on $q_j$ by $q_i$. For all $i$ in $\set {1, 2, \ldots, n}$, let $\mathbf F_i$ denote the force exerted on $q_i$ by the combined action of all the other charged particles. Then the force $\mathbf F_i$ exerted on $q_i$ by the combined action of all the other charged particles is given by: {{begin-eqn}} {{eqn | l = \mathbf F_i       | r = \sum_{\substack {1 \mathop \le j \mathop \le n \\ i \mathop \ne j} } \mathbf F_{j i}       | c =  }} {{eqn | r = \dfrac 1 {4 \pi \varepsilon_0} \sum_{\substack {1 \mathop \le j \mathop \le n \\ i \mathop \ne j} } \dfrac {q_i q_j} {r_{j i}^3} \mathbf r_{j i}       | c =  }} {{end-eqn}} where: :the summation denotes the vector sum of $\mathbf F_{21}$ and $\mathbf F_{31}$ :$\mathbf r_{ij}$ denotes the displacement from $p_i$ to $p_j$ :$r_{ij}$ denotes the distance between $p_i$ and $p_j$ :$\varepsilon_0$ denotes the vacuum permittivity.
Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm forms Normed Vector Space
Space of Continuously Differentiable on Closed Interval Real-Valued Functions with $C^1$ norm forms a normed vector space.
Ring of Endomorphisms is Ring with Unity
Let $\struct {G, \oplus}$ be an abelian group. Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$. Let $\struct {\mathbb G, \oplus, *}$ denote the '''ring of endomorphisms''' on $\struct {G, \oplus}$. Then $\struct {\mathbb G, \oplus, *}$ is a ring with unity.
Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm is 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 $\map {\CC^1} I$ be the space of real-valued functions, continuously differentiable on $I$. Let $\norm {\, \cdot \,}_{1, \infty}$ be the $\CC^1$ norm. $\struct {\map {\CC^1} I, \norm {\, \cdot \,}_{1, \infty} }$ be the normed space of real-valued functions, continuously differentiable on $I$. Then $\struct {\map {\CC^1} I, \norm {\, \cdot \,}_{1, \infty} }$ is a Banach space.
Set of Order m times n Matrices does not form Ring
Let $m, n \in \N_{>0}$ be non-zero natural numbers such that $m > n$. Let $S$ be the set of all matrices of order $m \times n$. Then the algebraic structure $\struct {S, +, \times}$ is not a ring. Note that $\times$ denotes conventional matrix multiplication.
Set of Order 3 Vectors under Cross Product does not form Ring
Let $S$ be the set of all vectors in a vector space of dimension $3$. Let $\times$ denote the cross product operation. Then the algebraic structure $\struct {S, +, \times}$ is not a ring.
Integers under Subtraction do not form Semigroup
Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction. Then $\struct {\Z, -}$ is not a semigroup.
Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Necessary Condition
Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be the rank function of a matroid on $S$. Then $\rho$ satisfies the rank axioms: {{:Definition:Rank Axioms (Matroid)/Definition 1}}
Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Sufficient Condition
Let $S$ be a finite set. Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers. Let $\rho$ satisfy the rank axioms: {{:Definition:Rank Axioms (Matroid)/Definition 1}} Then $\rho$ is the rank function of a matroid on $S$.
Leigh.Samphier/Sandbox/Independent Subset is Contained in Maximal Independent Subset/Corollary
Let $M = \struct{S, \mathscr I}$ be a matroid. Let $A \subseteq S$. Let $X$ be a maximal independent subset of $A$. Then: :$\card X = \map \rho A$ where $\rho$ is the rank function on $M$.
Motion of Body with Constant Mass
Let $B$ be a body with constant mass $m$ undergoing a force $\mathbf F$. Then the equation of motion of $B$ is given by: :$\mathbf F = m \mathbf a$ where $\mathbf a$ is the acceleration of $B$.
Like Vector Quantities are Multiples of Each Other
Let $\mathbf a$ and $\mathbf b$ be like vector quantities. Then: :$\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$ where: :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$ :$\dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$ denotes the scalar product of $\mathbf b$ by $\dfrac {\size {\mathbf a} } {\size {\mathbf b} }$.
Direct Product Norm is Norm
Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces. Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations. Let $\norm {\tuple {x, y} }$ be the direct product norm. Then $\norm {\tuple {x, y} }$ is a norm on $V$.
Vector Quantity as Scalar Product of Unit Vector Quantity
Let $\mathbf a$ be a vector quantity. Then: :$\mathbf a = \size {\mathbf a} \mathbf {\hat a}$ where: :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$ :$\mathbf {\hat a}$ denotes the unit vector in the direction $\mathbf a$.
Scalar Product of Magnitude by Unit Vector Quantity
Let $\mathbf a$ be a vector quantity. Let $m$ be a scalar quantity. Then: :$m \mathbf a = m \paren {\size {\mathbf a} \hat {\mathbf a} } = \paren {m \size {\mathbf a} } \hat {\mathbf a}$ where: :$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$ :$\hat {\mathbf a}$ denotes the unit vector in the direction $\mathbf a$.
Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors
Let $\mathbf r$ be a vector quantity embedded in space. Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be non-coplanar. Then $\mathbf r$ can be expressed uniquely as the resultant of $3$ vector quantities which are each parallel to one of $\mathbf a$, $\mathbf b$ and $\mathbf c$.
Vectors are Equal iff Components are Equal
Two vector quantities are equal  they have the same components.
Components of Vector in terms of Direction Cosines
Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Let $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ be the direction cosines of $\mathbf r$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively. Let $x$, $y$ and $z$ be the components of $\mathbf r$ in the $\mathbf i$, $\mathbf j$ and $\mathbf k$ directions respectively. Let $r$ denote the magnitude of $\mathbf r$, that is: :$r := \size {\mathbf r}$ Then: {{begin-eqn}} {{eqn | l = x       | r = r \cos \alpha }} {{eqn | l = y       | r = r \cos \beta }} {{eqn | l = z       | r = r \cos \gamma }} {{end-eqn}}
Magnitude of Vector Quantity in terms of Components
Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf r$ be expressed in terms of its components: :$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$ where $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then: :$\size {\mathbf r} = \sqrt {x^2 + y^2 + z^2}$ where $\size {\mathbf r}$ denotes the magnitude of $\mathbf r$.
Components of Zero Vector Quantity are Zero
Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf r$ be expressed in terms of its components: :$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$ Let $\mathbf r$ be the zero vector. Then: :$x = y = z = 0$
Unit Vector in terms of Direction Cosines
Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Let $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ be the direction cosines of $\mathbf r$ with respect to the $x$-axis, $y$-axis and $z$-axis respectively. Let $\mathbf {\hat r}$ denote the unit vector in the direction of $\mathbf r$. Then: :$\mathbf {\hat r} = \paren {\cos \alpha} \mathbf i + \paren {\cos \beta} \mathbf j + \paren {\cos \gamma} \mathbf k$
Unit Vectors of Cartesian 3-Space form Basis
Consider the Cartesian $3$-space $C$. Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively. Then $\set {\mathbf i, \mathbf j, \mathbf k}$ forms a basis for $C$.
Vectors from Sum and Difference
Let $\mathbf a$ and $\mathbf b$ be vector quantities. Let $\mathbf c = \mathbf a + \mathbf b$ and $\mathbf d = \mathbf a - \mathbf b$ be given. Then: {{begin-eqn}} {{eqn | l = \mathbf a       | r = \dfrac 1 2 \paren {\mathbf c + \mathbf d} }} {{eqn | l = \mathbf b       | r = \dfrac 1 2 \paren {\mathbf c - \mathbf d} }} {{end-eqn}}
Leigh.Samphier/Sandbox/Matroid satisfies Rank Axioms/Sufficient Condition/Lemma
:$\forall Y \subseteq S: \map \rho Y \le \card Y$
Dot Product of Perpendicular Vectors
Let $\mathbf a$ and $\mathbf b$ be vector quantities. Let $\mathbf a$ and $\mathbf b$ be perpendicular. Then: :$\mathbf a \cdot \mathbf b = 0$ where $\cdot$ denotes dot product.
Dot Product of Orthogonal Basis Vectors
Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be an orthonormal basis of a vector space $V$. Then: :$\forall i, j \in \set {1, 2, \ldots, n}: \mathbf e_i \cdot \mathbf e_j = \delta_{i j}$ where: :$\mathbf e_i \cdot \mathbf e_j$ denotes the dot product of $\mathbf e_i$ and $\mathbf e_j$ :$\delta_{i j}$ denotes the Kronecker delta.
Second Partial Derivative wrt r of ln (r^2 + s)
:$\dfrac {\partial^2} {\partial r^2} \map \ln {r^2 + s} = \dfrac {2 \paren {s - r^2} } {\paren {r^2 + s}^2}$
Derivative of Square of Tangent
:$\map {\dfrac \d {\d x} } {\tan^2 x} = 2 \tan x \sec^2 x$ when $\cos x \ne 0$.
Partial Derivatives of x^y^z
Let: :$u = x^{\paren {y^z} }$ Then: {{begin-eqn}} {{eqn | l = \dfrac {\partial u} {\partial x}       | r = y^z x^{\paren {y^z - 1} } }} {{eqn | l = \dfrac {\partial u} {\partial y}       | r = x^{y^z} z y^{z - 1} \ln x }} {{eqn | l = \dfrac {\partial u} {\partial z}       | r = x^{\paren {y^z} } y^z \ln x \ln y }} {{end-eqn}}
Partial Derivatives of x^u + u^y
Let: :$u = x^u + u^y$ Then: {{begin-eqn}} {{eqn | l = \dfrac {\partial u} {\partial x}       | r = \frac {u^2} {x \paren {1 - u \ln x - y} } }} {{eqn | l = \dfrac {\partial u} {\partial y}       | r = \frac {u \ln u} {1 - u \ln x - y} }} {{end-eqn}}
Third Partial Derivatives of x^y
Let: :$u = x^y$ Then: :$\dfrac {\partial^3 u} {\partial x^2 \partial y} = \dfrac {\partial^3 u} {\partial x \partial y \partial x}$
Number of Terms in Homogeneous Polynomial
The number of terms in a homogeneous polynomial of degree $n$ in $m$ indeterminates is given by: :$N = \dbinom {n + m - 1} n = \dfrac {\paren {n + m - 1}!}{n! \, \paren {m - 1}!}$
Definition:Set
A '''set''' is intuitively defined as any aggregation of objects, called elements, which can be precisely defined in some way or other. We can think of each set as a single entity in itself, and we can denote it (and usually do) by means of a single symbol. That is, ''anything you care to think of'' can be a set. This concept is known as the comprehension principle. However, there are problems with the comprehension principle. If we allow it to be used without any restrictions at all, paradoxes arise, the most famous example probably being Russell's Paradox. Hence some sources define a '''set''' as a ''' 'well-defined' collection of objects''', leaving the concept of what constitutes well-definition to later in the exposition.
Definition:Element
Let $S$ be a set. An '''element of $S$''' is a member of $S$.
Definition:Zermelo-Fraenkel Axioms
=== The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of the Empty Set === {{:Axiom:Axiom of Empty Set/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Specification === {{:Axiom:Axiom of Specification/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity/Set Theory}} === The Axiom of Replacement === {{:Axiom:Axiom of Replacement}} === The Axiom of Foundation === {{:Axiom:Axiom of Foundation}}
Definition:Empty Set
The '''empty set''' is a set which has no elements. That is, $x \in \O$ is false, whatever $x$ is.
Definition:Singleton
A '''singleton''' is a set that contains exactly one element.
Definition:Subset
Let $S$ and $T$ be sets. $S$ is a '''subset''' of a set $T$  all of the elements of $S$ are also elements of $T$. This is denoted: :$S \subseteq T$ That is: :$S \subseteq T \iff \forall x: \paren {x \in S \implies x \in T}$ If the elements of $S$ are not all also elements of $T$, then $S$ is not a '''subset''' of $T$: :$S \nsubseteq T$ means $\neg \paren {S \subseteq T}$
Definition:Set Union
Let $S$ and $T$ be sets. The '''(set) union''' of $S$ and $T$ is the set $S \cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$: :$x \in S \cup T \iff x \in S \lor x \in T$
Definition:Set Intersection
Let $S$ and $T$ be sets. The '''(set) intersection''' of $S$ and $T$ is written $S \cap T$. It means the set which consists of all the elements which are contained in both of $S$ and $T$: :$x \in S \cap T \iff x \in S \land x \in T$
Definition:Set Difference
The '''(set) difference''' between two sets $S$ and $T$ is written $S \setminus T$, and means the set that consists of the elements of $S$ which are not elements of $T$: :$x \in S \setminus T \iff x \in S \land x \notin T$
Definition:Set Equality
Let $S$ and $T$ be sets. === Definition 1 === {{:Definition:Set Equality/Definition 1}} === Definition 2 === {{:Definition:Set Equality/Definition 2}}
Definition:Statement
A '''statement''' is a sentence which has objective and logical meaning.