hcju/proofwikips huggingface dataset:
Intersection of Empty Set/Paradoxical Implications
Although it appears counter-intuitive, the reasoning is sound. This result is therefore classed as a veridical paradox. {{AuthorRef|Paul R. Halmos}} declares, in Section $5$ of his {{BookLink|Naive Set Theory|Paul R. Halmos}} of $1960$ that: :''There is no profound problem here; it is merely a nuisance to be forced always to be making qualifications and exceptions just because some set somewhere along some construction might turn out to be empty. There is nothing to be done about this; it is just a fact of life.'' However, later in that same work (in Section $9$, in the context of indexed families of sets) he says that: :''... an empty intersection does not make sense.'' a sentiment which is repeated in the $2008$ collaboration with {{AuthorRef|Steven Givant}}, {{BookLink|Introduction to Boolean Algebras|Paul Halmos}}, p. $457$. In {{BookReference|Topology|2000|James R. Munkres|ed = 2nd|edpage = Second Edition}}, the author recognizes the result, but does not adopt it: :''If one has a given large set $X$ that is specified at the outset of the discussion to be one's "universe of discourse," and one considers only subsets of $X$ throughout, it is reasonable to let $\displaystyle \bigcap_{A \mathop \in \AA} A = X$ when $\AA$ is empty. Not all mathematicians follow this convention, however. To avoid difficulty, we shall not define the intersection when $\AA$ is empty.'' {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty}} accepts this result, but cautiously: :''A natural mnemonic for these extreme cases is that $\bigcap \SS$ "grows larger" as $\SS$ "grows smaller", and $\bigcup \SS$ grows smaller as $\SS$ grows smaller. No other convention is possible, but the case $\SS = \O$ will often be treated redundantly by itself in definitions and proofs, as a reminder of the null case.'' {{BookReference|Set Theory|1999|András Hajnal|author2 = Peter Hamburger}} dismiss it casually: :''As usual, we adopt the convention that in case $A = \O$ the expression $\bigcap A$ is defined only in case we work with the subsets of an underlying set $X$. In this case we put $\bigcap A = X$.'' and {{BookReference|Programming, Games and Transportation Networks|1965|Claude Berge|author2 = A. Ghouila-Houri}} do not even acknowledge that there may be a problem in the first place: :''In the case where $I = \O$, we have ::$\displaystyle \bigcap_{i \mathop \in I} A_i = X$;'' :''this moreover is the only case where $X$ plays a role; in fact, if $I$ is not empty, clearly we have: ::$\displaystyle \bigcap_{i \mathop \in I} A_i = \set {x \mid x \in A_i \text { for every } i \in I}$.
Ordered Pair/Kuratowski Formalization/Motivation
The only reason for the Kuratowski formalization of ordered pairs: :$\tuple {a, b} = \set {\set a, \set {a, b} }$ is so their existence can be justified in the strictures of the axiomatic set theory, in particular Zermelo-Fraenkel set theory (ZF). Once that has been demonstrated, there is no need to invoke it again. The fact that this formulation allows that: :$\tuple {a, b} = \tuple {c, d} \iff a = c, b = d$ is its stated aim. The fact that $\set {a, b} \in \tuple {a, b}$ is an unfortunate side-effect brought about by means of the definition. It would be possible to add another axiom to ZF or ZFC specifically to allow for ordered pairs to be defined, and in some systems of axiomatic set theory this is what is done.
Heine-Borel Theorem/Real Line/Closed and Bounded Set
Let $F$ be a closed and bounded real set. Let $C$ be a set of open real sets. Let $C$ be a cover of $F$. Then there is a finite subset of $C$ that covers $F$.
Cardinality of Set of All Mappings from Empty Set
Let $T$ be a set. Let $T^\O$ denote the set of all mappings from $\O$ to $S$. Then: :$\card {T^\O} = 1$ where $\card {T^\O}$ denotes the cardinality of $\O^S$.
Cardinality of Set of All Mappings to Empty Set
Let $S$ be a set. Let $\O^S$ be the set of all mappings from $S$ to $\O$. Then: :$\card {\O^S} = \begin{cases} 1 & : S = \O \\  0 & : S \ne \O \end{cases}$ where $\card {\O^S}$ denotes the cardinality of $\O^S$.
Intersection Distributes over Union/Family of Sets/Corollary
Let $I$ and $J$ be indexing sets. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$. Then: :$\displaystyle \bigcup_{\tuple {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$ where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.
Cartesian Product of Unions/General Result
Let $I$ and $J$ be indexing sets. Let $\family {A_i}_{i \mathop \in I}$ and $\family {B_j}_{j \mathop \in J}$ be families of sets indexed by $I$ and $J$ respectively. Then: :$\displaystyle \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j} = \bigcup_{\tuple {i, j} \mathop \in I \times J} \paren {A_i \times B_j}$ where: :$\displaystyle \bigcup_{i \mathop \in I} A_i$ denotes the union of $\family {A_i}_{i \mathop \in I}$ and so on :$\times$ denotes Cartesian product.
Preimages All Exist iff Surjection/Corollary
:$\forall B \subseteq T, B \ne \O: f^{-1} \sqbrk B \ne \O$ : :$f$ is a surjection where $f^{-1} \sqbrk B$ denotes the preimage of $B \subseteq T$.
Image of Preimage under Relation is Subset
Let $\mathcal R \subseteq S \times T$ be a relation. Then: :$B \subseteq T \implies \paren {\mathcal R \circ \mathcal R^{-1} } \sqbrk B \subseteq B$ where: :$\mathcal R \sqbrk B$ denotes the image of $B$ under $\mathcal R$ :$\mathcal R^{-1} \sqbrk B$ denotes the preimage of $B$ under $\mathcal R$ :$\mathcal R \circ \mathcal R^{-1}$ denotes composition of $\mathcal R$ and $\mathcal R^{-1}$.
Inverse of Direct Image Mapping does not necessarily equal Inverse Image Mapping
Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation. Let $\mathrel R^\to$ be the direct image mapping of $\mathcal R$. Let $\mathrel R^\gets$ be the inverse image mapping of $\mathcal R$. Then it is not necessarily the case that: :$\paren {\mathrel R^\to}^{-1} = \mathcal R^\gets$ where $\paren {\mathrel R^\to}^{-1}$ denote the inverse of $\mathrel R^\to$. That is, the inverse of the direct image mapping of $\mathcal R$ does not always equal the inverse image mapping of $\mathcal R$.
Singleton of Power Set less Empty Set is Minimal Subset
Let $S$ be a set which is non-empty. Let $\mathcal C = \mathcal P \left({S}\right) \setminus \varnothing$, that is, the power set of $S$ without the empty set. Let $x \in S$. Then $\left\{{x}\right\}$ is a minimal element of the ordered structure $\left({\mathcal C, \subseteq}\right)$.
Natural Numbers under Multiplication form Semigroup
Let $\N$ be the set of natural numbers. Let $\times$ denote the operation of multiplication on $\N$. The structure $\struct {\N, \times}$ forms a semigroup.
Bernoulli's Hanging Chain Problem
Consider a uniform chain $C$ whose physical properties are as follows: :$C$ is of length $l$ :The mass per unit length of $C$ is $m$ :$C$ is of zero stiffness. Let $C$ be suspended in a vertical line from a fixed point and otherwise free to move. Let $C$ be slightly disturbed in a vertical plane from its position of stable equilibrium. Let $\map y t$ be the horizontal displacement at time $t$ from its position of stable equilibrium of a particle of $C$ which is a vertical distance $x$ from its point of attachment. The $2$nd order ODE describing the motion of $y$ is: :$\dfrac {\d^2 y} {\d t^2} = g \paren {l - x} \dfrac {\d^2 y} {\d x^2} - g \dfrac {\d y} {\d x}$
Real Line Continuity by Inverse of Mapping
Let $f$ be a real function. Let the domain of $f$ be open. Let $f^{-1}$ be the inverse of $f$. Then $f$ is continuous : :for every open real set $O$ that overlaps with the image of $f$, the preimage $f^{-1} \left [{O}\right]$ is open.
Trisecting the Angle/Neusis Construction
Let $\alpha$ be an angle which is to be trisected. This can be achieved by means of a neusis construction.
Squaring the Circle by Compass and Straightedge Construction is Impossible
There is no compass and straightedge construction to allow a square to be constructed whose area is equal to that of a given circle.
Difference between Distances from Point on Hyperbola to Foci is Constant
Let $K$ be a hyperbola. Let $F_1$ and $F_2$ be the foci of $K$. Let $P$ be an arbitrary point on $K$. Then the distance from $P$ to $F_1$ minus the distance from $P$ to $F_2$ is constant for all $P$ on $K$.
Time Taken for Body to Fall at Earth's Surface
Let an object $m$ be released above ground from a point near the Earth's surface and allowed to fall freely. Let $m$ fall a distance $s$ in time $t$. Then: :$s = \dfrac 1 2 g t^2$ or: :$t = \sqrt {\dfrac {2 s} g}$ where $g$ is the Acceleration Due to Gravity at the height through which $m$ falls. It is supposed that the distance $s$ is small enough that $g$ can be considered constant throughout.
Length of Chord of Circle
Let $C$ be a circle of radius $r$. Let $AB$ be a chord which joins the endpoints of the arc $ADB$. Then: :$AB = 2 r \sin \dfrac \theta 2$ where $\theta$ is the angle subtended by $AB$ at the center of $C$.
Spherical Triangles with Same Angles are Congruent
Two triangles on the surface of a given sphere which have the same angles are congruent.
Sum of Angles of Spherical Triangle
The sum of the angles of a spherical triangle is between $\pi$ and $3 \pi$ radians.
Ptolemy's Theorem
Let $ABCD$ be a cyclic quadrilateral. Then: :$AB \times CD + AD \times BC = AC \times BD$
Spherical Law of Sines
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$
Spherical Law of Cosines
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Then: :$\cos a = \cos b \cos c + \sin b \sin c \cos A$
Spherical Law of Cosines/Angles
:$\cos A = -\cos B \cos C + \sin B \sin C \cos a$
Spherical Law of Tangents
:$\dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} } = \dfrac {\tan \frac 1 2 \paren {a + b} } {\tan \frac 1 2 \paren {a - b} }$
Cosine of Half Angle for Spherical Triangles
:$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} }$ where $s = \dfrac {a + b + c} 2$.
Cosine of Half Side for Spherical Triangles
:$\cos \dfrac a 2 = \sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} }$ where $S = \dfrac {A + B + C} 2$.
Napier's Rules for Right Angled Spherical Triangles
:410px Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$. Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively. Let either angle $\angle C$ or side $c$ be a right angle. Let the remaining parts of $\triangle ABC$ be arranged in a circle as above: :for $\angle C$ a right angle, the '''interior''' :for $c$ a right angle, the '''exterior''' where the symbol $\Box$ denotes a right angle. Let one of the parts of this circle be called a '''middle part'''. Let the two neighboring parts of the '''middle part''' be called '''adjacent parts'''. Let the remaining two parts be called '''opposite parts'''.
Equation of Circle/Cartesian
:$\paren {x - a}^2 + \paren {y - b}^2 = R^2$
Equation of Circle/Parametric
:$x = a + R \cos t, \ y = b + R \sin t$
Equation of Circle/Cartesian/Corollary 1
The equation: :$A \paren {x^2 + y^2} + B x + C y + D = 0$ is the equation of a circle with radius $R$ and center $\tuple {a, b}$, where: :$R = \dfrac 1 {2 A} \sqrt {B^2 + C^2 - 4 A D}$ :$\tuple {a, b} = \tuple {\dfrac {-B} {2 A}, \dfrac {-C} {2 A} }$ provided: :$A > 0$ :$B^2 + C^2 \ge 4 A D$
Equation of Circle/Cartesian/Corollary 2
The equation of a circle with radius $R$ whose center is at the origin expressed in Cartesian coordinates is: :$x^2 + y^2 = R^2$
Distance Formula/3 Dimensions
The distance $d$ between two points $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$ in a Cartesian space of 3 dimensions is: :$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2 + \paren {z_1 - z_2}^2}$
Equation of Sphere/Rectangular Coordinates
:$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$
Equation of Sphere/Rectangular Coordinates/Corollary
The equation of a sphere with radius $R$ whose center is at the origin expressed in Cartesian coordinates is: :$x^2 + y^2 + z^2 = R^2$
Equation of Conic Section
The general conic section can be expressed in Cartesian coordinates in the form: :$a x^2 + b x y + c y^2 + d x + e y + f = 0$ for some $a, b, c, d, e, f \in \R$.
Graph of Quadratic describes Parabola
The locus of the equation defining a quadratic: :$y = a x^2 + b x + c$ describes a parabola.
Graph of Quadratic describes Parabola/Corollary 1
The locus of the equation of the square function: :$y = x^2$ describes a parabola.
Natural Numbers under Addition form Commutative Semigroup
The algebraic structure $\left({\N, +}\right)$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative semigroup.
Gödel's Incompleteness Theorems/First/Corollary
If $T$ is both consistent and complete, it does not contain minimal arithmetic.
Monomorphism that is Split Epimorphism is Split Monomorphism
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a morphism in $\mathbf C$ such that $f$ is a monomorphism and a split epimorphism. Then $f: C \to D$ is a split monomorphism. {{explain|What are $C$ and $D$?}}
Epimorphism that is Split Monomorphism is Split Epimorphism
Let $\mathbf C$ be a metacategory. Let $f: C \to D$ be a epimorphism and a split monomorphism. Then $f: C \to D$ is a split epimorphism.
Westwood's Puzzle/Proof 1
:500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$.
Westwood's Puzzle/Proof 2
:500px Take any rectangle $ABCD$ and draw the diagonal $AC$. Inscribe a circle $GFJ$ in one of the resulting triangles $\triangle ABC$. Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle. Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABCD$.
Vector Cross Product Operator is Bilinear
Let $\mathbf u$, $\mathbf v$ and $\mathbf w$ be vectors in a vector space $\mathbf V$ of $3$ dimensions: {{begin-eqn}} {{eqn | l = \mathbf u       | r = u_i \mathbf i + u_j \mathbf j + u_k \mathbf k }} {{eqn | l = \mathbf v       | r = v_i \mathbf i + v_j \mathbf j + v_k \mathbf k }} {{eqn | l = \mathbf w       | r = w_i \mathbf i + w_j \mathbf j + w_k \mathbf k }} {{end-eqn}} where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of $\mathbf V$. Let $c$ be a real number. Then: : $\left({c \mathbf u + \mathbf v}\right) \times \mathbf w = c \left({ \mathbf u \times \mathbf w}\right) + \mathbf v \times \mathbf w$
Natural Numbers have No Proper Zero Divisors
Let $\N$ be the natural numbers. Then for all $m, n \in \N$: :$m \times n = 0 \iff m = 0 \lor n = 0$ That is, $\N$ has no proper zero divisors.
Diagonals of Rhombus Bisect Angles
Let $OABC$ be a rhombus. Then: :$(1): \quad OB$ bisects $\angle AOC$ and $\angle ABC$ :$(2): \quad AC$ bisects $\angle OAB$ and $\angle OCB$ 400px
Divergent Sequence with Finite Number of Terms Deleted is Divergent
Let $\left({X, d}\right)$ be a metric space. Let $\left \langle {x_k} \right \rangle$ be a sequence in $X$. Let $\left \langle {x_k} \right \rangle$ be divergent. Let a finite number of terms be deleted from $\left \langle {x_k} \right \rangle$. Then the resulting subsequence is divergent.
Tensor Product is Module
Let $R$ be a ring. Let $M$ be a $R$-right module. Let $N$ be a $R$-left module. Then: :$T = \displaystyle \bigoplus_{s \mathop \in M \mathop \times N} R s$ is a left module.
Supremum of Subset of Real Numbers is Arbitrarily Close
Let $A \subseteq \R$ be a subset of the real numbers. Let $b$ be a supremum of $A$. Let $\epsilon \in \R_{>0}$. Then: :$\exists x \in A: b − x < \epsilon$
Tensor with Zero Element is Zero in Tensor
Let $R$ be a ring. Let $M$ be a right $R$-module. Let $N$ be a left $R$-module. Let $M \otimes_R N$ denote their tensor product. Then: :$0\otimes_R n = m \otimes_R 0 = 0 \otimes_R 0$ is the zero in $M \otimes_R N$.
Primitive of Arcsecant of x over a/Formulation 1
:$\displaystyle \int \operatorname{arcsec} \frac x a \ \mathrm d x = \begin{cases} \displaystyle x \operatorname{arcsec} \frac x a - a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : 0 < \operatorname{arcsec} \dfrac x a < \dfrac \pi 2 \\ \displaystyle x \operatorname{arcsec} \frac x a + a \ln \left({x + \sqrt {x^2 - a^2} }\right) + C & : \dfrac \pi 2 < \operatorname{arcsec} \dfrac x a < \pi \\ \end{cases}$
Cross Product of Vector with Itself is Zero
Let $\mathbf x$ be a vector in a vector space of $3$ dimensions: : $\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$ Then: :$\mathbf x \times \mathbf x = \mathbf 0$ where $\times$ denotes vector cross product.
Reciprocal of Riemann Zeta Function
For $\Re \left({z}\right) > 1$: :$\displaystyle \frac 1 {\zeta \left({z}\right)} = \sum_{k \mathop = 1}^\infty \frac{\mu \left({k}\right)} {k^z}$ where: : $\zeta$ is the Riemann zeta function : $\mu$ is the Möbius function.
Condition for Agreement of Family of Mappings
Let $\left({A_i}\right)_{i \mathop \in I}, \left({B_i}\right)_{i \mathop \in I}$ be families of non empty sets. Let $\left({f_i}\right)_{i \mathop \in I}$ be a family of mappings such that: :$\forall i \in I: f_i \in \mathcal F \left({A_i, B_i}\right)$ {{explain|Clarify: what is $\mathcal F \left({A_i, B_i}\right)$? From the context it can be understood as being the set of all mappings $f_i: A_i \to B_i$ but this is just a guess. The domain and range of each of the $f_i$ could help with being explicitly defined.<br/>Note also that I have taken the liberty of exchanging the difficult-to-read $\mathscr F$ with the more eye-friendly $\mathcal F$.}} We have that: :$\displaystyle \bigcup_{i \mathop \in I} f_i \in \mathcal F \left({\bigcup_{i \mathop \in I} A_i, \bigcup_{i \mathop \in I} B_i }\right)$ : :$\displaystyle \forall i, j \in I: \operatorname{Dom} f_i \cap \operatorname{Dom} f_j \ne \varnothing \implies \left({\forall a \in \left({\operatorname{Dom} f_i \cap \operatorname{Dom} f_j}\right), \left({a, b}\right) \in f_i \implies \left({a, b}\right) \in f_j}\right)$
Supremum of Set of Real Numbers is at least Supremum of Subset
Let $S$ be a set of real numbers. Let $S$ have a supremum. Let $T$ be a non-empty subset of $S$. Then $\sup T$ exists and: :$\sup T \le \sup S$
Supremum of Subset of Union Equals Supremum of Union
Let $S$ be a non-empty real set. Let $S$ have a supremum. Let $\set {S_i: i \in \set {1, 2, \ldots, n} }$, $n \in \N_{>0}$, be a set of non-empty subsets of $S$. Let $\bigcup S_i = S$. Then there exists a $j$ in $\set {1, 2, \ldots, n}$ such that: :$\sup S_j = \sup S$
Condition for Ideal to be Total Ring
Let $\left({A, +, \circ}\right)$ be a commutative ring with unity. Let $I$ be an ideal of $A$ such that the quotient ring $A / I$ is a field. Let $J$ be an ideal of $A$ such that $I \subsetneq J$. Then: :$A = J$
Area between Two Non-Intersecting Chords
Let $AB$ and $CD$ be two chords of a circle whose center is at $O$ and whose radius is $r$. :400px :400px Let $\alpha$ and $\theta$ be respectively the measures in radians of the angles $\angle COD$ and $\angle AOB$. Then the area $\mathcal A$ between the two chords is given by: : $\mathcal A = \dfrac {r^2} 2 \left({\theta - \sin \theta - \alpha + \sin \alpha}\right)$ if $O$ is not included in the area, and: : $\mathcal A = r^2 \left({\pi - \dfrac 1 2 \left({\theta - \sin \theta + \alpha - \sin \alpha}\right)}\right)$ if $O$ is included in the area.
Supremum of Absolute Value of Difference equals Supremum of Difference
Let $S$ be a non-empty real set. Let $\displaystyle \sup_{x, y \mathop \in S} \paren {x - y}$ exist. Then $\displaystyle \sup_{x, y \mathop \in S} \size {x - y}$ exists and: :$\displaystyle \sup_{x, y \mathop \in S} \size {x - y} = \sup_{x, y \mathop \in S} \paren {x - y}$
Supremum of Sum equals Sum of Suprema
Let $A$ and $B$ be non-empty sets of real numbers. Let $A + B$ be $\set {x + y: x \in A, y \in B}$. Let either $A$ and $B$ have suprema or $A + B$ have a supremum. Then all $\sup A$, $\sup B$, and $\sup \paren {A + B}$ exist and: :$\sup \paren {A + B} = \sup A + \sup B$
Edge is Minimum Weight Bridge iff in All Minimum Spanning Trees
Let $G$ be an undirected network. Let every edge of $G$ have a unique weight. Let $e$ be an edge of $G$. Then $e$ is a bridge of minimum weight in $G$  $e$ belongs to every minimum spanning tree of $G$.
Maximum Weight Edge in all Minimum Spanning Trees is Bridge
Let $G$ be an undirected network. Let every edge of $G$ have a unique weight. Let $e$ be an edge of $G$ that belongs to every minimum spanning tree of $G$. Let $e$ have maximum weight in $G$. Then $e$ is a bridge in $G$.
Scaling Property of Dirac Delta Function
Let $\delta \left({t}\right)$ be the Dirac delta function. Let $a$ be a non zero constant real number. Then: :$\delta \left({a t}\right) = \dfrac {\delta \left({t}\right)} {\left \vert a \right \vert}$
Supremum of Function is less than Supremum of Greater Function
Let $f$ and $g$ be real functions. Let $S$ be a subset of $\operatorname{Dom} \left({f}\right) \cap \operatorname{Dom} \left({g}\right)$. Let $f \left({x}\right) \le g \left({x}\right)$ for every $x \in S$. Let $\displaystyle \sup_{x \mathop \in S} g \left({x}\right)$ exist. Then $\displaystyle \sup_{x \mathop \in S} f \left({x}\right)$ exists and: :$\displaystyle \sup_{x \mathop \in S} f \left({x}\right) \le \sup_{x \mathop \in S} g \left({x}\right)$.
Characterization of Boundary 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 $x \in \operatorname{Fr} A$ : :for every open set $U$ of $T$: ::if $x \in U$ ::then $A \cap U \ne \varnothing$ and $\complement_S \left({A}\right) \cap U \ne \varnothing$ where: :$\complement_S \left({A}\right) = S \setminus A$ denotes the complement of $A$ in $S$ :$\operatorname{Fr} A$ denotes the boundary of $A$.
Characterization of Closure by Open Sets
Let $T = \left({S, \tau}\right)$ be a topological space. Let $A$ be a subset of $S$. Let $x$ be a point of $T$. Let $A^-$ denote the closure of $A$. Then $x \in A^-$ : :for every open set $U$ of $T$: ::$x \in U \implies A \cap U \ne \varnothing$
Summation is Linear/Sum of Summations
:$\displaystyle \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$
Summation is Linear/Scaling of Summations
:$\displaystyle \lambda \sum_{i \mathop = 1}^n x_i = \sum_{i \mathop = 1}^n \lambda x_i$
Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum
Let $f$ be a real function. Let $S$ be a subset of the domain of $f$. Let $\displaystyle \sup_{x \mathop \in S} \set {\map f x}$ and $\displaystyle \inf_{x \mathop \in S} \set {\map f x}$ exist. Then $\displaystyle \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} }$ exists and: :$\displaystyle \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} } = \sup_{x \mathop \in S} \set {\map f x} - \inf_{x \mathop \in S} \set {\map f x}$
Diaconescu-Goodman-Myhill Theorem
The axiom of choice implies the law of excluded middle.