hcju/proofwikips huggingface dataset:
Definition:Convergent Net
Let $\left({X, \tau}\right)$ be a topological space, and let $\left({I, \leq}\right)$ be a directed set. Let $\left({x_i}\right)_{i \in I}$ be a net. The net $\left({x_i}\right)$ is said to '''converge to''' $x \in X$, denoted $\displaystyle x_i \to x$ or $\lim x_i = x$, iff: :$\forall U \in \tau: x \in U \implies \exists i_0 \in I: \forall i \ge i_0: x_i \in U$ That is, for every open $U$ with $x \in U$, there exists an $i_0 \in I$ such that forall $i \ge i_0$, $x_i \in U$. If $x_i \to x$, then $x$ is called a '''limit (point) of $\left({x_i}\right)$'''. A net $\left({x_i}\right)_{i \in I}$ is called '''convergent''' if there is an $x \in X$ such that $x_i \to x$. If such an $x$ does not exist, the net is said to be '''divergent'''. === Cluster Point === The net $\left({x_i}\right)$ is said to '''cluster at''' $x \in X$, denoted $\displaystyle x_i \mathop{\longrightarrow}_{\text{cl}} x$, iff: :$\forall U \in \tau, i_0 \in I: x \in U \implies \exists i \ge i_0: x_i \in U$ That is, for every open $U$ with $x \in U$, and for every $i_0 \in I$, there is an $i \ge i_0$ such that $x_i \in U$. If $\displaystyle x_i \mathop{\longrightarrow}_{\text{cl}} x$, then $x$ is called a '''cluster point of $\left({x_i}\right)$'''.
Definition:Precede
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$ such that $a \preceq b$. Then '''$a$ precedes $b$'''.
Definition:Strictly Precede
=== Definition 1 === {{:Definition:Strictly Precede/Definition 1}} === Definition 2 === {{:Definition:Strictly Precede/Definition 2}}
Definition:Strictly Succeed
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a \prec b$. That is, let $a$ strictly precede $b$. Then '''$b$ strictly succeeds $a$'''. This can be expressed symbolically as: :$b \succ a$
Definition:Succeed
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b \in S$ such that $a \preceq b$. Then '''$b$ succeeds $a$'''.
Definition:Between
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b, c \in S$ such that $a \preceq b$ and $b \preceq c$. Then '''$b$ is between $a$ and $c$'''.
Definition:Strictly Between
Let $\left({S, \preceq}\right)$ be an ordered set. Let $a, b, c \in S$ such that $a \prec b$ and $b \prec c$. That is, such that: :$a \preceq b$ and $a \ne b$ :$b \preceq c$ and $b \ne c$ Then '''$b$ is strictly between $a$ and $c$'''.
Definition:Circular Proof
A '''circular proof''' is a proof $A$ in terms of another proof $B$ such that the truth value of $B$ depends (either directly or indirectly) on the truth value of $A$. A '''circular proof''' is valid, as every statement implies itself.  The problem of a '''circular proof''' is that if the truth of a conclusion is [http://en.wiktionary.org/wiki/dubious dubious], premises that are equally dubious provide no reason to support the conclusion.
Definition:Idempotent Semigroup
An '''idempotent semigroup''' is a semigroup whose operation is idempotent. That is, a semigroup $\struct {S, \circ}$ is '''idempotent''' : :$\forall x \in S: x \circ x = x$
Definition:Commutative Semigroup with respect to Equivalence Relation
Let $C$ be a class. Let $\thickapprox$ be an equivalence relation on $C$. Let $\left({C, \cdot}\right)$ be a large magma. Then $\left({C, \cdot}\right)$ is a '''commutative semigroup with respect to $\thickapprox$''' iff: :it is a semigroup with respect to $\thickapprox$ :$\forall x, y \in C: x \cdot y \thickapprox y \cdot x$
Definition:Bounded Linear Transformation
Let $H, K$ be Hilbert spaces. Let $A: H \to K$ be a linear transformation. Then $A$ is a '''bounded linear transformation'''  :$\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_K \le c \left\Vert{h}\right\Vert_H$
Definition:Space of Bounded Linear Transformations
Let $H, K$ be Hilbert spaces. Let $\Bbb F \in \set {\R, \C}$ be the ground field of $K$. Then the '''space of bounded linear transformations from $H$ to $K$''', $\map B {H, K}$, is the set of all bounded linear transformations: :$\map B {H, K} := \set {A: H \to K: A \text{ linear}, \norm A < \infty}$ endowed with pointwise addition and ($\F$)-scalar multiplication. Then $\map B {H, K}$ is a vector space over $\Bbb F$. Furthermore, let $\norm {\,\cdot\,}$ denote the norm on bounded linear transformations. Then $\norm{\,\cdot\,}$ is a norm on $\map B {H, K}$, and it even turns $\map B {H, K}$ into a Banach space. These results are proved in Space of Bounded Linear Transformations is Banach Space. === Space of Bounded Linear Operators === When $H = K$, one denotes $\map B H$ for $\map B {H, K}$. In line with the definition of linear operator, $\map B H$ is called the '''space of bounded linear operators on $H$'''.
Definition:Semigroup with respect to Equivalence Relation
Let $C$ be a class. Let $\thickapprox$ be an equivalence relation on $C$. Let $\left({C, \cdot}\right)$ be a large magma. Then $\left({C, \cdot}\right)$ is a '''semigroup with respect to $\thickapprox$'''  iff: :$\forall x, y,z \in C: \left( x \cdot y \right)\cdot z \thickapprox x \cdot \left(y \cdot z\right)$
Definition:Ground Field
Let $K$ be a field. Let $V$ be a vector space over $K$. Then $K$ is referred to as the '''ground field''' of $V$.
Definition:Pointwise Scalar Multiplication of Mappings
Let $\struct {R, +_R, \times_R}$ be a ring, and let $\struct {S, \circ}_R$ be an $R$-algebraic structure. Let $X$ be a non-empty set, and let $S^X$ be the set of all mappings from $X$ to $S$. Then '''pointwise ($R$)-scalar multiplication''' on $S^X$ is the binary operation $\circ: R \times S^X \to S^X$ (the $\circ$ is the same as for $S$) defined by: :$\forall \lambda \in R: \forall f \in S^X: \forall x \in X: \map {\paren {\lambda \circ f} } x := \lambda \circ \map f x$
Definition:Degenerate Case
A '''degenerate case''' is a specific manifestation of a particular type of object being included in another, usually simpler, type of object. {{WIP|Include links to the various types of degenerate objects that we have already defined.}}
Definition:Sesquilinear Form
Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. A '''sesquilinear form''' is a function $u: H \times K \to \Bbb F$ subject to: :$(1): \qquad \forall \alpha \in \Bbb F, h_1, h_2 \in H, k \in K: u \left({\alpha h_1 + h_2, k}\right) = \alpha u \left({h_1, k}\right) + u \left({h_2, k}\right)$ :$(2): \qquad \forall \alpha \in \Bbb F, h \in H, k_1, k_2 \in K: u \left({h, \alpha k_1 + k_2}\right) = \bar{\alpha} u \left({h, k_1}\right) + u \left({h, k_2}\right)$ That is, $u$ is linear in the first argument, and conjugate linear in the second. If $\Bbb F = \R$, then a sesquilinear form is the same as a bilinear map.
Definition:Equidistance
Define the following coordinates in the $xy$-plane: :$a = \tuple {x_1, x_2}$ :$b = \tuple {y_1, y_2}$ :$c = \tuple {z_1, z_2}$ :$d = \tuple {u_1, u_2}$ where $a, b, c, d \in \R^2$ 400px :$a b \equiv c d \dashv \vdash \paren {\paren {x_1 - y_1}^2 + \paren {x_2 - y_2}^2 = \paren {z_1 - u_1}^2 + \paren {z_2 - u_2}^2}$
Definition:Bounded Sesquilinear Form
Let $H, K$ be Hilbert spaces over $\Bbb F \in \left\{{\R, \C}\right\}$. Let $u: H \times K \to \Bbb F$ be a sesquilinear form. Then $u$ is said to be a '''bounded sesquilinear form''', or to be '''bounded''', iff: :$\exists M \in \R: \forall h \in H, k \in K: \left\vert{u \left({h, k}\right)}\right\vert \le M \left\Vert{h}\right\Vert_H \left\Vert{k}\right\Vert_K$ A constant $M$ satisfying the above is called a '''bound''' for $u$.
Definition:Adjoint Linear Transformation
Let $H, K$ be Hilbert spaces. Let $A \in \map B {H, K}$ be a bounded linear transformation. Let $B \in \map B {K, H}$ be the unique bounded linear transformation provided by Existence and Uniqueness of Adjoint. Then $B$ is called the '''adjoint''' of $A$, and denoted $A^*$.
Definition:Upper Semilattice on Classical Set
Let $\struct {S, \le}$ be an ordered set with the property that: :$\forall x, y \in S: \sup \set {x, y} \in S$ where $\sup$ denotes supremum. Then $\struct {S, \vee}$ is called an '''upper semilattice''', where $\vee: S \times S \to S$ is defined by: :$x \vee y := \sup \set {x, y}$ An '''upper semilattice''' hence is a particular kind of algebraic structure.
Definition:Tarski's Axioms
'''Tarski's Axioms''' are a series of axioms whose purpose is to provide a rigorous basis for the definition of Euclidean geometry entirely within the framework of first order logic. In the following: : $\equiv$ denotes the relation of equidistance. : $\mathsf{B}$ denotes the relation of betweenness. : $=$ denotes the relation of equality. The axioms are as follows:
Definition:Directed Line Segment
A '''directed line segment''' is a line segment endowed with the additional property of direction. It is often used in the context of applied mathematics to represent a vector quantity. {{expand|Perhaps the above statement should also be expanded to allow a D.L.S. to be defined as a '''vector quantity''' applied at a particular point. There is a danger (as pointed out on the Definition:Vector Quantity page) of implying  / believing that a vector, in general, is applied at a particular point, for example usually the origin. Thus, this page allows the opportunity to consider a definition of an object which consists of a vector "rooted" at a particular point, as a convenient fiction for what is actually happening in the context of physics.}} {{stub|needs a picture It may be worthwhile to point out that this can be formalized with an ordered pair. Establish connection with Definition:Affine Space}}
Definition:Vector (Euclidean Space)
A vector is defined as an element of a vector space. We have that $\R^n$, with the operations of vector addition and scalar multiplication, form a real vector space. Hence a '''vector in $\R^n$''' is defined as any element of $\R^n$. {{expand|While it is possible to identify a vector by a tuple in $\R^n$ we need to explain that the vector is not the point.}} === $\R^2$: Plane Vector === {{:Definition:Vector (Euclidean Space)/Plane Vector}} === $\R^3$: Space Vector === {{:Definition:Vector (Euclidean Space)/Space Vector}}
Definition:Engineering Notation
=== Euclidean 2-space === Define the ordered 2-tuples: {{begin-eqn}} {{eqn | l = \mathbf i       | r = \left\langle{1, 0}\right\rangle }} {{eqn | l = \mathbf j       | r = \left\langle{0, 1}\right\rangle }} {{end-eqn}} From Standard Ordered Basis is Basis, we have that any vector in $\R^2$ can be represented by: :$c_1 \mathbf i + c_2 \mathbf j$ where $c_1, c_2 \in \R$. This way of presenting vectors is called '''engineering notation'''. === Euclidean 3-space === Define the ordered 3-tuples: {{begin-eqn}} {{eqn | l = \mathbf i       | r = \left\langle{1, 0, 0}\right\rangle }} {{eqn | l = \mathbf j       | r = \left\langle{0, 1, 0}\right\rangle }} {{eqn | l = \mathbf k       | r = \left\langle{0, 0, 1}\right\rangle }} {{end-eqn}} By the same logic as the above definition, we can write any vector in $\R^3$ as: :$c_1 \mathbf i + c_2 \mathbf j + c_3 \mathbf k$ where $c_1, c_2, c_3 \in \R$. Note that $\mathbf i$ and $\mathbf j$ take on a different meaning in $3$-space than in $2$-space. === Euclidean $n$-space === In higher dimensions, rather than writing $\mathbf l, \mathbf m, \mathbf n$, and so on, the convention is to use: {{begin-eqn}} {{eqn | l = \mathbf e_1       | r = \left\langle{1, 0, 0, \ldots, 0, 0}\right\rangle }} {{eqn | l = \mathbf e_2       | r = \left\langle{0, 1, 0, \ldots, 0, 0}\right\rangle }} {{eqn | o = \vdots }} {{eqn | l = \mathbf e_n       | r = \left\langle{0, 0, 0, \ldots, 0, 1}\right\rangle }} {{end-eqn}} Then any vector in $\R^n$ can be expressed as: :$c_1 \mathbf e_1 + c_2 \mathbf e_2 + \cdots + c_n \mathbf e_n$ where $c_1, c_2, \cdots, c_n \in \R$. This convention is also frequently seen for $2$-space and $3$-space.
Definition:Vector (Linear Algebra)
Let $V$ be a vector space. Any element $v$ of $V$ is called a '''vector'''.
Definition:Self-Adjoint Operator
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then $A$ is said to be '''self-adjoint''' or '''hermitian''' iff: :$A = A^*$ That is, if it equals its adjoint $A^*$. {{namedfor|Charles Hermite}}
Definition:Normal Operator
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then $A$ is said to be '''normal''' : :$A^* A = A A^*$ where $A^*$ denotes the adjoint of $A$.
Definition:Real Part (Linear Operator)
Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then the '''real part of $A$''' is the self-adjoint operator: :$\operatorname{Re} A := \dfrac 1 2 \left({A + A^*}\right)$ The real part of $A$ may be denoted by $\operatorname{Re} \left({A}\right)$, $\operatorname{re} \left({A}\right)$ or $\Re \left({A}\right)$. This resembles the notation for the real part of a complex number.
Definition:Imaginary Part (Linear Operator)
Let $H$ be a Hilbert space over $\C$. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then the '''imaginary part of $A$''' is the self-adjoint operator: :$\operatorname{Im} A := \dfrac 1 {2i} \left({A - A^*}\right)$ The imaginary part of $A$ may be denoted by $\operatorname{Im} \left({A}\right)$, $\operatorname{im} \left({A}\right)$ or $\Im \left({A}\right)$. This resembles the notation for the imaginary part of a complex number.
Definition:Absurd
'''Absurd''' is a word used mainly in logic meaning either '''meaningless''', '''contradictory''' or '''internally inconsistent'''.
Definition:Idempotent Operator
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Then $A$ is said to be '''idempotent''', or an '''idempotent operator''', iff $A^2 = A$.
Definition:Projection (Hilbert Spaces)
Let $H$ be a Hilbert space. Let $P \in \map B H$ be an idempotent operator. Then $P$ is said to be a '''projection''' : :$\ker P = \paren {\Img P}^\perp$ where: :$\ker P$ denotes the kernel of $P$ :$\Img P$ denotes the image of $P$ :$\perp$ denotes orthocomplementation.
Definition:Complementary Idempotent
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be an idempotent operator. Then the '''complementary idempotent (operator) of $A$''' is the bounded linear operator $I - A$, where $I$ is the identity operator on $H$. The name is appropriate, by Complementary Idempotent is Idempotent and the fact that $I - \left({I - A}\right) = A$. === Complementary Projection === If $A$ is a projection, $I - A$ is called the '''complementary projection'''. This name is justified by Complementary Projection is Projection.
Definition:Closed Linear Subspace
Let $V$ be a topological vector space. A linear subspace of $V$ that is closed is called a '''closed linear subspace'''.
Definition:Internal Hilbert Space Direct Sum
Let $H$ be a Hilbert space. Let $\left\{{M_i : i \in I}\right\}$ be an $I$-indexed collection of pairwise orthogonal subspaces of $H$. Then the '''internal (Hilbert space) direct sum''' of the $M_i$ is their closed linear span $\vee_i M_i$. It is denoted by $\bigoplus_i M_i$, or $\displaystyle \bigoplus_{i \in I} M_i$ if the set $I$ is to be stressed. When $I$ is finite, by Closed Linear Subspaces Closed under Setwise Addition, have that: :$\displaystyle \bigoplus_{i \in I} M_i = \sum_{i \in I} M_i$, where $\displaystyle \sum$ signifies setwise addition.
Definition:Orthogonal Difference
Let $H$ be a Hilbert space. Let $M, N$ be closed linear subspaces of $H$. Then the '''orthogonal difference of $M$ and $N$''', denoted $M \ominus N$, is the set $M \cap N^\perp$. It is in fact a closed linear subspace of $H$, as proven on Orthogonal Difference is Closed Linear Subspace.
Definition:Maximal/Set
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\mathcal T \subseteq \powerset S$ be a subset of $\powerset S$. Let $\struct {\mathcal T, \subseteq}$ be the ordered set formed on $\mathcal T$ by $\subseteq$ considered as an ordering. Then $T \in \mathcal T$ is a '''maximal set''' of $\mathcal T$  $T$ is a maximal element of $\struct {\mathcal T, \subseteq}$. That is: :$\forall X \in \mathcal T: T \subseteq X \implies T = X$
Definition:Invariant Subspace
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $M$ be a closed linear subspace of $H$. Then $M$ is said to be an '''invariant subspace for $A$''' iff $h \in M \implies Ah \in M$. That is, if $AM \subseteq M$.
Definition:Reducing Subspace
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator. Let $M$ be a closed linear subspace of $H$; denote by $M^\perp$ its orthocomplement. Then $M$ is said to be a '''reducing subspace for $A$''' iff both $M$ and $M^\perp$ are invariant subspaces for $A$. That is, if $AM \subseteq M$ and $A M^\perp \subseteq M^\perp$.
Definition:Matrix Notation (Bounded Linear Operator)
Let $H$ be a Hilbert space. Let $A \in B \left({H}\right)$ be a bounded linear operator on $H$. Let $M$ be a closed linear subspace of $H$; denote by $M^\perp$ its orthocomplement. Observe that $M \oplus M^\perp \cong H$ by Direct Sum of Subspace and Orthocomplement. The associated isomorphism $U$ induces a bounded linear operator $A'$ on $M \oplus M^\perp$. $A'$ can be written as the matrix:  :$\begin{pmatrix}  W & X \\  Y & Z \end{pmatrix}$  where $W \in B \left({M}\right)$, $X \in B \left({M^\perp, M}\right)$, $Y \in B \left({M, M^\perp}\right)$, $Z \in B \left({M^\perp}\right)$. In the above matrix, linear transformations $W, X, Y, Z$ are uniquely determined by the requirement that: :$\begin{pmatrix} Wm + Xm^\perp \\ Ym + Zm^\perp\end{pmatrix} = \begin{pmatrix}  W & X \\  Y & Z \end{pmatrix} \begin{pmatrix}m \\m^\perp \end{pmatrix} = U^{-1} \left({Ah}\right)$ whenever $h = U \left({m, m^\perp}\right)$. By abuse of notation, one writes $A$ instead of $A'$ and calls it the '''matrix notation for $A$ (with respect to $M$)'''.
Definition:Inverse (Bounded Linear Transformation)
Let $H, K$ be Hilbert spaces. Let $A \in \map B {H, K}$ be a bounded linear transformation. An '''inverse''' for $A$ is a bounded linear transformation $A^{-1} \in \map B {K, H}$ satisfying: :$AA^{-1} = I_K$ :$A^{-1}A = I_H$ where $I_K, I_H$ denote the identity operators on $K, H$, respectively. If such a $A^{-1}$ exists, $A$ is said to be an '''invertible (bounded) linear transformation'''. The operation assigning $A^{-1}$ to $A$ is referred to as '''inverting'''. === Invertible Bounded Linear Operator === When $H = K$, the notation simplifies considerably, and $A$ is said to be a '''invertible (bounded) linear operator'''.
Definition:Coprime/GCD Domain
Let $\struct {D, +, \times}$ be a GCD domain. Let $U \subseteq D$ be the group of units of $D$. Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$ Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime'''  $d \in U$. That is, two elements of a GCD domain are '''coprime'''  their greatest common divisor is a unit of $D$.
Definition:Coprime/Euclidean Domain
Let $\struct {D, +, \times}$ be a Euclidean domain. Let $U \subseteq D$ be the group of units of $D$. Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$ Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime'''  $d \in U$. That is, two elements of a Euclidean domain are '''coprime'''  their greatest common divisor is a unit of $D$.
Definition:Coprime/Integers
Let $a$ and $b$ be integers such that $b \ne 0$ and $a \ne 0$ (that is, they are both non-zero). Let $\gcd \set {a, b}$ denote the greatest common divisor of $a$ and $b$. Then $a$ and $b$ are '''coprime'''  $\gcd \set {a, b} = 1$.
Definition:Monotonicity
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a monotone mapping. Then whether $\phi$ is increasing or decreasing is known as the '''monotonicity of $\phi$'''.  That is: :if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({x}\right) \mathop{\preceq_2} \phi \left({y}\right)$, then the '''monotonicity''' of $\phi$ is increasing :if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({x}\right) \mathop{\prec_2} \phi \left({y}\right)$, then the '''monotonicity''' of $\phi$ is strictly increasing :if $\phi$ is such that $x \mathop{\preceq_1} y \implies \phi \left({y}\right) \mathop{\preceq_2} \phi \left({x}\right)$, then the '''monotonicity''' of $\phi$ is decreasing :if $\phi$ is such that $x \mathop{\prec_1} y \implies \phi \left({y}\right) \mathop{\prec_2} \phi \left({x}\right)$, then the '''monotonicity''' of $\phi$ is strictly decreasing.
Definition:Continuation of Woset
Let $\left({S, \preccurlyeq}\right)$ be a woset. Let $\left({T, \preccurlyeq}\right)$ be a set with an ordering such that: :$(1): \quad T$ is an initial segment of $S$ :$(2): \quad$ The ordering of the elements of $T$ is the same as their ordering in $S$. Then $S$ is a '''continuation''' of $T$.
Definition:Addition/Summand
Let $a + b$ denote the operation of addition on two objects. The objects $a$ and $b$ are known as the '''summands''' of $a + b$.
Definition:Between (Geometry)
Define the following coordinates in the $xy$-plane: {{begin-eqn}} {{eqn | l = a       | r = \left({x_1, x_2}\right) }} {{eqn | l = b       | r = \left({y_1, y_2}\right) }} {{eqn | l = c       | r = \left({z_1, z_2}\right) }} {{end-eqn}} where $a, b, c \in \R^2$. Let: {{begin-eqn}} {{eqn | l = \Delta x_1       | r = x_3 - x_2 }} {{eqn | l = \Delta x_2       | r = x_2 - x_1 }} {{eqn | l = \Delta y_1       | r = y_2 - y_1 }} {{eqn | l = \Delta y_2       | r = y_3 - y_2 }} {{end-eqn}} Then: :File:Betweenness(Analytic Def'n).png :$\mathsf{B}abc \dashv \vdash \left({\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2}\right) \land$ :$\left({0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}\right)$
Definition:Ordering Induced by Injection
Let $\left({T, \le}\right)$ be an ordered set, and let $S$ be a set. Let $f: S \to T$ be an injection. Define $\le_f$ as the '''ordering induced by $f$ on $S$''' by: :$\forall s_1, s_2 \in S: s_1 \le_f s_2 \iff f \left({s_1}\right) \le f \left({s_2}\right)$ That $\le_f$ is in fact an ordering is shown on Ordering Induced by Injection is Ordering.
Definition:Dominate (Analysis)
Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$. Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$. Then $\left \langle {a_n} \right \rangle$ '''dominates''' $\left \langle {z_n} \right \rangle$ iff: :$\forall n \in \N: \left|{z_n}\right| \le a_n$   dpeqz6xo6qbc2m40xp0hhgcrjj7wbro
Definition:Vertex Cut
Let $G$ be a graph. A '''vertex cut''' of $G$ is a set of vertices $W \subseteq V \left({G}\right)$ such that the vertex deletion $G \setminus W$ is disconnected.
Definition:K-Connected
Let $G$ be a graph. Let $k \in \Z_{>0}$. Then $G$ is '''$k$-connected''' : : $\left\lvert{V \left({G}\right) }\right\rvert > k$ : $G$ is connected : $G$ has no vertex cut $W$ with $\left\lvert{W}\right\rvert < k$ Informally: : $G$ is connected and has more than $k$ vertices : There is no vertex cut of $G$ containing less than $k$ vertices.
Definition:Connectivity
Let $G$ be a graph. Then $\kappa \left({G}\right)$, the '''connectivity''' of $G$, is the smallest $k \in \Z_{>0}$ such that $G$ is $k$-connected.  If $G$ is disconnected, or if $G$ has fewer than 2 vertices, then $\kappa \left({G}\right) = 0$.  8n9tlxeii5l0iw6ad6hzbwkzdkislw1
Definition:Vertex Deletion
Let $G = \left({V, E}\right)$ be an (undirected) graph. Let $W \subseteq V$ be a set of vertices of $G$. Then the '''graph obtained by deleting $W$ from $G$''', denoted by $G - W$, is the subgraph induced by $V \setminus W$. Alternatively: : $G - W = \left({V \setminus W, \left\{{e \in E : e \cap W = \varnothing}\right\}}\right)$ Informally, $G - W$ is the graph obtained from $G$ by removing all vertices in $W$ and all edges incident to those vertices. If $W$ is a singleton such that $W = \left\{{v}\right\}$, then $G - W$ may be expressed $G - v$.
Definition:Minimum Degree
Let $G = \left({V, E}\right)$ be a graph. Then the '''minimum degree''' of $G$ is: : $\delta\left({G}\right) = \min\left\{{\deg_G\left({v}\right) : v \in V}\right\}$ That is, it is the minimum degree of all the vertices of $G$.  pdfmm4z4qjqku87z11orh4durbra7v7