hcju/proofwikips huggingface dataset:
Characterization of Boundary by Basis
Let $T = \struct {S, \tau}$ be a topological space. Let $\BB \subseteq \tau$ be a basis. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then $x \in \partial A$ : :for every $U \in \BB$: ::if $x \in U$ ::then $A \cap U \ne \O$ and $\relcomp S A \cap U \ne \O$ where: :$\relcomp S A = S \setminus A$ denotes the complement of $A$ in $S$ :$\partial A$ denotes the boundary of $A$ in $T$.
Union of Interiors and Boundary Equals Whole Space
Let $T = \struct {S, \tau}$ be a topological space. Let $A$ be a subset of $T$. Then: :$S = \Int A \cup \partial A \cup \Int {A'}$ where: :$A' = S \setminus A$ denotes the complement of $A$ relative to $S$ :$\Int A$ denotes the interior of $A$ :$\partial A$ denotes the boundary of $A$.
Characterization of Derivative by Open Sets
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then :$x \in A'$ : :for every open set $U$ of $T$: ::if $x \in U$ ::then there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \ne y$ where :$A'$ denotes the derivative of $A$.
Derivative is Included in Closure
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then :$A' \subseteq A^-$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$.
Closure Equals Union with Derivative
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Then: :$A^- = A \cup A'$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$.
Derivative of Subset is Subset of Derivative
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$, $B$ be subsets of $S$. Then :$A \subseteq B \implies A' \subseteq B'$ where $A'$ denotes the derivative of $A$ in $T$.
Derivative of Union is Union of Derivatives
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$, $B$ be subsets of $S$. Then :$\left({A \cup B}\right)' = A' \cup B\,'$ where :$A'$ denotes the derivative of $A$.
Derivative of Derivative is Subset of Derivative in T1 Space
Let $T = \left({S, \tau}\right)$ be a $T_1$ topological space. Let $A$ be a subset of $S$. Then :$A'' \subseteq A'$ where :$A'$ denotes the derivative of $A$
Closure of Derivative is Derivative in T1 Space
Let $T = \left({S, \tau}\right)$ be a $T_1$ topological space. Let $A$ be a subset of $S$. Then :$A'^- = A'$ where :$A'$ denotes the derivative of $A$ :$A^-$ denotes the closure of $A$.
Union of Derivatives is Subset of Derivative of Union
Let $T = \struct {S, \tau}$ be a topological space. Let: :$\FF \subseteq \powerset S$ be a set of subsets of $S$ where $\powerset S$ denotes the power set of $S$. Then: :$\ds \bigcup_{A \mathop \in \FF} A' \subseteq \paren {\bigcup_{A \mathop \in \FF} A}'$ where $A'$ denotes the derivative of $A$.
Point is Isolated iff not Accumulation Point
Let $T = \left({S, \tau}\right)$ be a topological space. Let $H \subseteq S$. Let $x \in H$. Then: :$x$ is an isolated point in $H$ : :$x$ is not an accumulation point of $H$
Dense-in-itself iff Subset of Derivative
Let $T$ be a topological space. Let $A \subseteq T$. Then: :$A$ is dense-in-itself : :$A \subseteq A'$ where :$A'$ denotes the derivative of $A$.
Closure of Dense-in-itself is Dense-in-itself in T1 Space
Let $T$ be a $T_1$ topological space. Let $A \subseteq T$.  Let $A$ be dense-in-itself. Then the closure $A^-$ of $A$ is also dense-in-itself.
Equivalence of Definitions of Weight of Topological Space
Let $T$ be a topological space. Let $\mathbb B$ be the set of all bases of $T$. The following definitions of the weight of $T$ are equivalent:
Difference of Two Powers/Examples/Difference of Two Cubes
:$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$
Existence of Subfamily of Cardinality not greater than Weight of Space and Unions Equal
Let $T$ be a topological space. Let $\mathcal F$ be a set of open sets of $T$. There exists a subset $\mathcal G \subseteq \mathcal F$ such that: :$\displaystyle \bigcup \mathcal G = \bigcup \mathcal F$ and: :$\left\vert{\mathcal G}\right\vert \leq w \left({T}\right)$ where: :$w \left({T}\right)$ denotes the weight of $T$ :$\left\vert{\mathcal G}\right\vert$ denotes the cardinality of $\mathcal G$.
Set of Singletons is Smallest Basis of Discrete Space
Let $T = \left({S, \tau}\right)$ be a discrete topological space. Let $\mathcal B = \left\{{\left\{{x}\right\} : x \in S}\right\}$. Then $\mathcal B$ is the smallest basis of $T$. That is: :$\mathcal B$ is a basis of $T$ and: :for every basis $\mathcal C$ of $T$, $\mathcal B \subseteq \mathcal C$.
Cardinality of Set of Singletons
Let $S$ be a set. Let $T = \left\{ {\left\{{x}\right\}: x \in S}\right\}$ be the set of all singletons of elements of $S$. Then: :$\left\vert T \right\vert = \left\vert S \right\vert$ where $\left\vert S \right\vert$ denotes the cardinality of $S$.
Basis has Subset Basis of Cardinality equal to Weight of Space
Let $T = \struct {X, \tau}$ be a topological space. Let $\BB$ be a basis of $T$. Then there exists a basis $\BB_0$ of $T$ such that :$\BB_0 \subseteq \BB$ and $\card {\BB_0} = \map w T$ where: :$\card {\BB_0}$ denotes the cardinality of $\BB_0$ :$\map w T$ denotes the weight of $T$.
Cardinality of Union not greater than Product
Let $\FF$ be a set of sets. Let: :$\size \FF \le \mathbf m$ where  :$\size \FF$ denotes the cardinality of $\FF$ :$\mathbf m$ is cardinal number (possibly infinite). Let: :$\forall A \in \FF: \size A \le \mathbf n$ where :$\mathbf n$ is cardinal number (possibly infinite). Then: :$\ds \size {\bigcup \FF} \le \size {\mathbf m \times \mathbf n} = \mathbf m \mathbf n$
Image of Mapping of Intersections is Smallest Basis
Let $T = \left({X, \tau}\right)$ be a topological space. Let $f:X \to \tau$ be a mapping such that :$\forall x \in X: \left({x \in f \left({x}\right) \land \forall U \in \tau: x \in U \implies f \left({x}\right) \subseteq U}\right)$. Then the image $\operatorname{Im} \left({f}\right)$ is subset of every basis of $T$.
Cardinality of Image of Mapping of Intersections is not greater than Weight of Space
Let $T = \struct {X, \tau}$ be a topological space. Let $f: X \to \tau$ be a mapping such that: :$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$ Then the cardinality of the image of $f$ is no greater than the weight of $T$: $\card {\Img f} \le \map w T$.
Finite Weight Space has Basis equal to Image of Mapping of Intersections
Let $T = \struct {X, \tau}$ be a topological space with finite weight. Then there exist a basis $\BB$ of $T$ and a mapping $f:X \to \tau$ such that: :$\BB = \Img f$ and :$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$ where $\Img f$ denotes the image of $f$.
Rubik's Cube has 54 Facets
Let $S$ be the set of facets of Rubik's cube. Then the cardinality of $S$ is given by: :$\card S = 54$ That is: :A Rubik's cube has $54$ facets.
Equivalence of Definitions of Symmetric Difference/(3) iff (5)
Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}
Equivalence of Definitions of Symmetric Difference/(2) iff (5)
Let $S$ and $T$ be sets. {{TFAENocat|def = Symmetric Difference|view = symmetric difference $S * T$ between $S$ and $T$}}
Even and Odd Integers form Partition of Integers
The odd integers and even integers form a partition of the integers.
Analog between Logic and Set Theory
The concepts of set theory have directly corresponding concepts in logic: :{| border = "1" |- ! style="padding: 2px 10px" | Set Theory ! style="padding: 2px 10px" | Logic |- | align="left" style="padding: 2px 10px"| Set: $S, T$ | align="left" style="padding: 2px 10px"| Statement: $p, q$ |- | align="left" style="padding: 2px 10px"| Union: $S \cup T$ | align="left" style="padding: 2px 10px"| Disjunction: $p \lor q$ |- | align="left" style="padding: 2px 10px"| Intersection: $S \cap T$ | align="left" style="padding: 2px 10px"| Conjunction: $p \land q$ |- | align="left" style="padding: 2px 10px"| Subset: $S \subseteq T$ | align="left" style="padding: 2px 10px"| Conditional: $p \implies q$ |- | align="left" style="padding: 2px 10px"| Symmetric Difference: $S * T$ | align="left" style="padding: 2px 10px"| Exclusive Or: $p \oplus q$ |- | align="left" style="padding: 2px 10px"| Complement: $\relcomp {} S$ | align="left" style="padding: 2px 10px"| Logical Not: $\lnot p$ |- | align="left" style="padding: 2px 10px"| Set Equality: $S = T$ | align="left" style="padding: 2px 10px"| Biconditional: $p \iff q$ |- | align="left" style="padding: 2px 10px"| Venn Diagram | align="left" style="padding: 2px 10px"| Truth Table |}
Cardinality of Set is Finite iff Set is Finite
Let $A$ be a set. $\left\vert{A}\right\vert$ is finite  $A$ is finite where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.
Topology Generated by Closed Sets
Let $X$ be a set. Let $\mathcal F$ be a set of subsets of $X$. Suppose that :$\varnothing \in \mathcal F$ and :for every subsets $A$ and $B$ of $X$ if $A, B \in \mathcal F$, then $A \cup B \in \mathcal F$ and :for every subset $\mathcal G \subseteq \mathcal F$, $\bigcap \mathcal G \in \mathcal F$ and :$\tau = \left\{{\complement_X \left(A\right): A \in \mathcal F}\right\}$. Then: :$T = \left( {X, \tau} \right)$ is topological space and :for every subset $A$ of $X$, $A$ is closed in $T$  $A \in \mathcal F$.
Equivalence of Definitions of Countably Infinite Set
Let $S$ be a set. {{TFAE|def = Countably Infinite Set}}
Set of Odd Integers is Countably Infinite
Let $\Bbb O$ be the set of odd integers. Then $\Bbb O$ is countably infinite.
Unique Readability for Polish Notation
Let $\AA$ be an alphabet. Then Polish notation for $\AA$ has the unique readability property.
Product of Rook Matrices is Rook Matrix
Let $\mathbf A$ and $\mathbf B$ be rook matrices. Their product $\mathbf {A B}$ is also a rook matrix.
Topology Defined by Basis
Let $S$ be a set. Let $\mathcal B$ be a set of subsets of $S$. Suppose that :$(B1): \quad \forall A_1, A_2 \in \mathcal B: \forall x \in A_1 \cap A_2: \exists A \in \mathcal B: x \in A \subseteq A_1 \cap A_2$ :$(B2): \quad \forall x \in X: \exists A \in \mathcal B: x \in A$ ::$\tau = \left\{{\bigcup \mathcal G: \mathcal G \subseteq \mathcal B}\right\}$ Then: :$T = \left({S, \tau}\right)$ is a topological space :$\mathcal B$ is a basis of $T$.
Matrix is Non-Invertible iff Product with Non-Zero Vector is Zero
Let $\mathbf A$ be a square matrix of order $n$. Then $\mathbf A$ is non-invertible if there exists a vector $\mathbf v$ of $n$ such that: :$\mathbf v \ne \mathbf 0$ :$\mathbf A \mathbf v = \mathbf 0$ where $\mathbf 0$ is the zero vector.
Equivalence of Definitions of Integer Congruence
Let $m \in \Z_{> 0}$. {{TFAE|def = Congruence (Number Theory)/Integers|view = congruence modulo $m$}}
Krull's Theorem
Let $R$ be a non-null ring with unity. Then $R$ has a maximal ideal.
NAND as Disjunction of Negations
: $p \uparrow q \dashv \vdash \neg p \lor \neg q$
Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication
: $\vdash \left({p \lor p}\right) \implies p$
Rule of Addition/Sequent Form/Formulation 1/Form 2/Proof 2
:$q \vdash p \lor q$
Rule of Addition/Sequent Form/Formulation 2/Form 2
:$\vdash q \implies \left({p \lor q}\right)$
Hilbert Proof System Instance 2 is Consistent
Instance 2 of the Hilbert proof systems $\mathscr H_2$ is consistent.
Set of Local Minimum is Countable
Let $X$ be a subset of $\R$. The set: :$\leftset {x \in X: x}$ is local minimum in $\rightset X$ is countable.
Set of Pairwise Disjoint Intervals is Countable
Let $X$ be a subset of $\mathcal P \left({\R}\right)$ such that: :$(1): \quad X$ is pairwise disjoint: ::::$\forall A,B \in X: A \ne B \implies A \cap B = \varnothing$. :$(2): \quad$ every element of $X$ contains an open interval: ::::$\forall A \in X: \exists x, y \in \R: x < y \land \left({x \,.\,.\, y}\right) \subseteq A$. Then $X$ is countable.
Set is Countable if Cardinality equals Cardinality of Countable Set
Let $X, Y$ be sets. Let: : $\left\vert{X}\right\vert = \left\vert{Y}\right\vert$ where $\left\vert{X}\right\vert$ denotes the cardinality of $X$. If $X$ is countable then $Y$ is countable.
Double Negation/Double Negation Introduction/Sequent Form
{{:Double Negation/Double Negation Introduction/Sequent Form/Formulation 1}}
Double Negation/Double Negation Elimination/Proof Rule
:If we can conclude $\neg \neg \phi$, then we may infer $\phi$.
Double Negation/Double Negation Elimination/Formulation 1/Sequent Form
{{:Double Negation/Double Negation Elimination/Sequent Form/Formulation 1}}
Biconditional Elimination/Also known as
Some sources refer to the Biconditional Elimination as the rule of '''Biconditional-Conditional'''.
Principle of Non-Contradiction/Explanation
The '''Principle of Non-Contradiction''' can be expressed in natural language as follows: :A statement can not be both true and not true at the same time. This means: if we have managed to deduce that a statement is both true and false, then the sequence of deductions show that the pool of assumptions upon which the sequent rests contains assumptions which are mutually contradictory. Thus it provides a means of eliminating a logical not from a sequent.
Principle of Non-Contradiction/Proof Rule
:If we can conclude both $\phi$ and $\neg \phi$, we may infer a contradiction.
Solutions of Pythagorean Equation/Primitive
The set of all primitive Pythagorean triples is generated by: :$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$ where: :$m, n \in \Z_{>0}$ are (strictly) positive integers :$m \perp n$, that is, $m$ and $n$ are coprime :$m$ and $n$ are of opposite parity :$m > n$
Solutions of Pythagorean Equation/General
Let $x, y, z$ be a solution to the Pythagorean equation. Then $x = k x', y = k y', z = k z'$, where: :$\tuple {x', y', z'}$ is a primitive Pythagorean triple :$k \in \Z: k \ge 1$
Goldbach Conjecture implies Goldbach's Marginal Conjecture
Suppose the Goldbach Conjecture holds: :Every even integer greater than $2$ is the sum of two primes. Then Goldbach's Marginal Conjecture follows: :Every integer greater than $5$ can be written as the sum of three primes.
Infinite Number of Chen Primes
There exists an infinite number of Chen primes. That is, there exists an infinite number of pairs: :$\set {p, p + 2}$ where: :$p$ is a prime :$p + 2$ is either a prime or a semiprime.
Congruent Integers are of same Quadratic Character
Let $p$ be an odd prime. Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$. Let $a \equiv b \pmod p$. Then $a$ and $b$ have the same quadratic character.
Weight of Sorgenfrey Line is Continuum
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $\map w T = \mathfrak c$ where :$\map w T$ denotes the weight of $T$ :$\mathfrak c$ denotes continuum, the cardinality of real numbers.
Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure
Let $\mathcal B$ be a set of subsets of $\R$, the set of all real numbers. Let: :$\left\vert{\mathcal B}\right\vert < \mathfrak c$ where :$\left\vert{\mathcal B}\right\vert$ denotes the cardinality of $\mathcal B$ :$\mathfrak c = \left\vert{\R}\right\vert$ denotes continuum. Then: :$\exists x, y \in \R: x < y \land \left[{x \,.\,.\, y}\right) \notin \left\{{\bigcup \mathcal G: \mathcal G \subseteq \mathcal B}\right\}$
Cardinality of Basis of Sorgenfrey Line not greater than Continuum
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Let :$\BB = \set {\hointr x y: x, y \in \R \land x < y}$ be the basis of $T$. Then $\card \BB \le \mathfrak c$ where :$\card \BB$ denotes the cardinality of $\BB$ :$\mathfrak c = \card \R$ denotes the continuum.
Construction of Regular Prime p-Gon Exists iff p is Fermat Prime
Let $p$ be a prime number. Then there exists a compass and straightedge construction for a regular $p$-gon  $p$ is a Fermat prime.
Cardinalities form Inequality implies Difference is Nonempty
Let $X, Y$ be sets. Let :$\left\vert{X}\right\vert < \left\vert{Y}\right\vert$ where $\left\vert{X}\right\vert$ denotes the cardinality of $X$. Then: :$Y \setminus X \ne \varnothing$
Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum
Let $\BB$ be a set of subsets of $\R$. Let: :$\size \BB < \mathfrak c$ where :$\size \BB$ denotes the cardinality of $\BB$ :$\mathfrak c = \size \R$ denotes continuum. Let :$X = \leftset {x \in \R: \exists U \in \set {\bigcup \GG: \GG \subseteq \BB}: x}$ is local minimum in $\rightset U$ Then: :$\size X < \mathfrak c$
Slope of Secant
Let $f: \R \to \R$ be a real function. Let the graph of $f$ be depicted on a Cartesian plane. :400px Let $AB$ be a secant of $f$ where: :$A = \tuple {x, \map f x}$ :$A = \tuple {x + h, \map f {x + h} }$ Then the slope of $AB$ is given by: :$\dfrac {\map f {x + h} - \map f x} h$
Derivative of Curve at Point
Let $f: \R \to \R$ be a real function. Let the graph $G$ of $f$ be depicted on a Cartesian plane. Then the derivative of $f$ at $x = \xi$ is equal to the tangent to $G$ at $x = \xi$.
Derivative of Square Function
Let $f: \R \to \R$ be the square function: :$\forall x \in \R: \map f x = x^2$ Then the derivative of $f$ is given by: :$\map {f'} x = 2 x$
Countable iff Cardinality not greater than Aleph Zero
Let $X$ be set. $X$ is countable : $\left\vert{X}\right\vert \leq \aleph_0$ where: :$\left\vert{X}\right\vert$ denotes the cardinality of $X$ :$\aleph_0 = \left\vert{\N}\right\vert$ by Aleph Zero equals Cardinality of Naturals.
Aleph Zero equals Cardinality of Naturals
$\aleph_0 = \left\vert{\N}\right\vert$ where :$\aleph$ denotes the aleph mapping, :$\left\vert{\N}\right\vert$ denotes the cardinality of $\N$.
Body under Constant Acceleration/Velocity after Time
: $\mathbf v = \mathbf u + \mathbf a t$
Body under Constant Acceleration/Distance after Time
:$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$
Body under Constant Acceleration/Velocity after Distance
:$\mathbf v \cdot \mathbf v = \mathbf u \cdot \mathbf u + 2 \mathbf a \cdot \mathbf s$
Body in Free Fall moves in Parabolic Path
A body in free fall above the surface of the Earth follows a path approximating to a parabola.
Aleph Zero is less than Continuum
$\aleph_0 < \mathfrak c$ where :$\aleph$ denotes the aleph mapping, :$\mathfrak c$ denotes continuum, the cardinality of real numbers.
Cardinality of Set less than Cardinality of Power Set
Let $X$ be a set. Then: :$\left\vert{X}\right\vert < \left\vert{\mathcal P \left({X}\right)}\right\vert$ where :$\left\vert{X}\right\vert$ denotes the cardinality of $X$, :$\mathcal P \left({X}\right)$ denotes the power set of $X$.
Volume of Solid of Revolution
Let $f: \R \to \R$ be a real function which is integrable on the interval $\closedint a b$. Let the points be defined: :$A = \tuple {a, \map f a}$ :$B = \tuple {b, \map f b}$ :$C = \tuple {b, 0}$ :$D = \tuple {a, 0}$ Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and the curve defined by $\set {\map f x: a \le x \le b}$. Let the solid of revolution $S$ be generated by rotating $ABCD$ around the $x$-axis (that is, $y = 0$). Then the volume $V$ of $S$ is given by: :$\displaystyle V = \pi \int_a^b \paren {\map f x}^2 \rd x$
Continuum equals Cardinality of Power Set of Naturals
$\mathfrak c = \card {\powerset \N}$ where :$\powerset \N$ denotes the power set of $\N$ :$\card {\powerset \N}$ denotes the cardinality of $\powerset \N$ :$\mathfrak c = \card \R$ denotes the continuum.
Acceleration is Second Derivative of Displacement with respect to Time
The '''acceleration''' $\mathbf a$ of a body $M$ is the second derivative of the displacement $\mathbf s$ of $M$ from a given point of reference with respect to time $t$: :$\mathbf a = \dfrac {\d^2 \mathbf s} {\d t^2}$
Equation of Catenary/Formulation 2
The '''catenary''' is described by the equation: :$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$ where $a$ is a constant. The lowest point of the chain is at $\tuple {0, a}$.
Cardinality of Power Set is Invariant
Let $X, Y$ be sets. Let $\card X = \card Y$ where $\card X$ denotes the cardinality of $X$. Then: :$\card {\powerset X} = \card {\powerset Y}$ where $\powerset X$ denotes the power set of $X$.
Reals are Isomorphic to Dedekind Cuts
Let $\mathscr D$ be set of all Dedekind cuts of the totally ordered set $\struct {\Q, \le}$. Define a mapping $f: \R \to \mathscr D$ as: :$\forall x \in \R: \map f x = \set {y \in \Q: y < x}$ Then $f$ is a bijection.
Slope of Orthogonal Curves
Let $C_1$ and $C_2$ be curves in a cartesian plane. Let $C_1$ and $C_2$ intersect each other at $P$. Let the slope of $C_1$ and $C_2$ at $P$ be $m_1$ and $m_2$. Then $C_1$ and $C_2$ are orthogonal : :$m_1 = -\dfrac 1 {m_2}$
Orthogonal Trajectories/Concentric Circles
Consider the one-parameter family of curves: :$(1): \quad x^2 + y^2 = c$ Its family of orthogonal trajectories is given by the equation: :$y = c x$
Angle of Tangent to Radius in Polar Coordinates
Let $C$ be a curve embedded in a plane defined by polar coordinates. Let $P$ be the point at $\polar {r, \theta}$. Then the angle $\psi$ made by the tangent to $C$ at $P$ with the radial coordinate is given by: :$\tan \psi = r \dfrac {\d \theta} {\d r}$
Orthogonal Trajectories/Circles Tangent to Y Axis
Consider the one-parameter family of curves: :$(1): \quad x^2 + y^2 = 2 c x$ which describes the loci of circles tangent to the $y$-axis at the origin. Its family of orthogonal trajectories is given by the equation: :$x^2 + y^2 = 2 c y$ which describes the loci of circles tangent to the $x$-axis at the origin. :600px
Orthogonal Trajectories/Parabolas Tangent to X Axis
Consider the one-parameter family of curves of parabolas which are tangent to the $x$-axis at the origin: :$(1): \quad y = c x^2$ Its family of orthogonal trajectories is given by the equation: :$x^2 + 2 y^2 = c$ :600px
Orthogonal Trajectories/Exponential Functions
Consider the one-parameter family of curves of graphs of the exponential function: :$(1): \quad y = c e^x$ Its family of orthogonal trajectories is given by the equation: :$y^2 = -2 x + c$ :600px
Orthogonal Trajectories/Parabolas with Focus at Origin
Consider the one-parameter family of curves of parabolas whose focus is at the origin and whose axis is the $x$-axis: :$(1): \quad y^2 = 4 c \paren {x + c}$ Its family of orthogonal trajectories is given by the equation: :$y^2 = 4 c \paren {x + c}$ :600px
Speed of Body under Free Fall from Height
Let an object $B$ be released above ground from a point near the Earth's surface and allowed to fall freely. Let $B$ fall a distance $s$. Then: :$v = \sqrt {2 g s}$ where: :$v$ is the speed of $B$ after having fallen a distance $s$ :$g$ is the Acceleration Due to Gravity at the height through which $B$ falls. It is supposed that the distance $s$ is small enough that $g$ can be considered constant throughout.
Approximate Motion of Pendulum
Consider a pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$. Let the bob be pulled to one side so that the rod is at a small angle $\alpha$ (less than about $10 \degrees$ or $15 \degrees$) from the vertical and then released. Let $T$ be the period of the pendulum, that is, the time through which the bob takes to travel from one end of its path to the other, and back again. Then: :$T = 2 \pi \sqrt {\dfrac a g}$ where $g$ is the Acceleration Due to Gravity.
Space is Separable iff Density not greater than Aleph Zero
Let $T$ be a topological space. Then: :$T$ is separable  $d \left({T}\right) \leq \aleph_0$ where :$d \left({T}\right)$ denotes the density of $T$, :$\aleph$ denotes the aleph mapping.
Boundary of Empty Set is Empty
Let $T$ be a topological space. Then:  :$\partial_T \O = \O$ where $\partial_T \O$ denotes the boundary in topology $T$ of $\O$.
Escape Velocity of Projectile fired Upwards
Let $P$ be a planet. Let $P$ have an Acceleration Due to Gravity at its surface of $g$. Let $P$ have a radius of $R$. Then the escape velocity of $P$ is given by: :$V = \sqrt {2 g R}$
Union of Boundaries
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B$ be subsets of $S$. Then: :$\partial A \cup \partial B = \partial \left({A \cup B}\right) \cup \partial \left({A \cap B}\right) \cup \left({\partial A \cap \partial B}\right)$ where $\partial A$ denotes the boundary of $A$.