hcju/proofwikips huggingface dataset:
Definition:Binomial Coefficient/Historical Note
The binomial coefficients have been known about since at least the ancient Greeks and Romans, who were familiar with them for small values of $k$. See the historical note to Pascal's Triangle for further history.
Definition:Binomial Coefficient/Integers/Definition 2
Let $n \in \Z_{\ge 0}$ and $k \in \Z$. The number of different ways $k$ objects can be chosen (irrespective of order) from a set of $n$ objects is denoted: :$\dbinom n k$ This number $\dbinom n k$ is known as a '''binomial coefficient'''.
Definition:Binomial Coefficient/Integers/Definition 3
Let $n \in \Z_{\ge 0}$ and $k \in \Z$. Then the '''binomial coefficient''' $\dbinom n k$ is defined as the coefficient of the term $a^k b^{n - k}$ in the expansion of $\paren {a + b}^n$.
Definition:Geometric Mean/Mean Proportional/General Definition
In the language of {{AuthorRef|Euclid}}, the terms of a (finite) geometric sequence of positive integers between (and not including) the first and last terms are called '''mean proportionals'''.
Definition:Geometric Sequence of Integers in Lowest Terms
Let $G_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers. Let $r$ be the common ratio of $G_n$. Let $S$ be the set of all such geometric sequence: :$S = \left\{{G: G}\right.$ is a geometric sequence of integers whose common ratio is $\left.{r}\right\}$ Then $G_n$ is in '''lowest terms''' if the absolute values of the terms of $G_n$ are the smallest, term for term, of all the elements of $S$: :$\forall Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n} \in S: \forall j \in \set {0, 1, \ldots, n}: \size {a_j} \le \size {b_j}$
Definition:Geometric Sequence/Finite
A '''finite geometric sequence''' is a geometric sequence with a finite number of terms .
Definition:Geometric Sequence/Finite/Extremes
Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a finite geometric sequence. The '''extremes''' of $G_n$ are the terms $a_0$ and $a_n$ of $G_n$.
Definition:Geometric Sequence/Initial Term
Let $G = \left\langle{a_0, a_1, \ldots}\right\rangle$ be a geometric sequence. The '''initial term''' of $G_n$ is the term $a_0$. The same definition applies to a finite geometric sequence $G_n = \sequence {a_0, a_1, \ldots, a_n}$.
Definition:Geometric Sequence/Finite/Final Term
Let $G_n = \sequence {a_0, a_1, \ldots, a_n}$ be a finite geometric sequence. The '''final term''' of $G_n$ is the term $a_n$.
Definition:Geometric Sequence/Integers
A '''geometric sequence of integers''' is a finite geometric sequence whose terms are all integers.
Definition:Unit (One)/Naturally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $S^*$ be the zero complement of $S$. By Zero Complement is Not Empty, $S^*$ is not empty. Therefore, by axiom $(NO4)$, it has a smallest element for $\preceq$. This smallest element is called '''one''' and denoted $1$.
Definition:Commensurable/Notation
There appears to be no universally acknowledged symbol to denote commensurability. {{AuthorRef|Thomas Little Heath|Thomas L. Heath}} in his edition of {{BookLink|Euclid: The Thirteen Books of The Elements: Volume 3|Sir Thomas L. Heath|ed = 2nd|edpage = Second Edition}} makes the following suggestions: :$(1): \quad$ To denote that $A$ is commensurable or commensurable in length with $B$:  :::$A \mathop{\frown} B$ :$(2): \quad$ To denote that $A$ is commensurable in square with $B$:  :::$A \mathop{\frown\!\!-} B$ :$(3): \quad$ To denote that $A$ is incommensurable or incommensurable in length with $B$:  :::$A \mathop{\smile} B$ :$(4): \quad$ To denote that $A$ is incommensurable in square with $B$:  :::$A \mathop{\smile\!\!-} B$ This convention may be used on  if accompanied by a note which includes a link to this page.
Definition:Apotome of Medial/Whole
The real number $a$ is called the '''whole''' of the apotome of a medial.
Definition:Apotome of Medial/Annex
The real number $b$ is called the '''annex''' of the apotome of a medial.
Definition:Apotome of Medial/Terms
The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Apotome of Medial/Whole}} === Annex === {{Definition:Apotome of Medial/Annex}}
Definition:Minor (Euclidean)/Whole
The real number $a$ is called the '''whole''' of the minor.
Definition:Minor (Euclidean)/Annex
The real number $b$ is called the '''annex''' of the minor.
Definition:Minor (Euclidean)/Terms
The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:Minor (Euclidean)/Whole}} === Annex === {{Definition:Minor (Euclidean)/Annex}}
Definition:That which produces Medial Whole with Rational Area/Whole
The real number $a$ is called the '''whole''' of the straight line which produces with a rational area a medial whole.
Definition:That which produces Medial Whole with Rational Area/Annex
The real number $b$ is called the '''annex''' of the straight line which produces with a rational area a medial whole.
Definition:That which produces Medial Whole with Rational Area/Terms
The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:That which produces Medial Whole with Rational Area/Whole}} === Annex === {{Definition:That which produces Medial Whole with Rational Area/Annex}}
Definition:That which produces Medial Whole with Medial Area/Whole
The real number $a$ is called the '''whole''' of the straight line which produces with a medial area a medial whole.
Definition:That which produces Medial Whole with Medial Area/Annex
The real number $b$ is called the '''annex''' of the straight line which produces with a medial area a medial whole.
Definition:That which produces Medial Whole with Medial Area/Terms
The '''terms''' of $a - b$ are the elements $a$ and $b$. === Whole === {{Definition:That which produces Medial Whole with Medial Area/Whole}} === Annex === {{Definition:That which produces Medial Whole with Medial Area/Annex}}
Definition:Piecewise Continuously Differentiable Function/Definition 2
$f$ is '''piecewise continuously differentiable''' : :there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that: ::$f$ is continuously differentiable on $\closedint {x_{i − 1} } {x_i}$, where the derivative at $x_{i − 1}$ understood as right-handed and the derivative at $x_i$ understood as left-handed, for every $i \in \set {1, \ldots, n}$.
Definition:Right Angle/Perpendicular/Plane
{{EuclidDefinition|book = XI|def = 3|name = Line at Right Angles to Plane}} :400px In the above diagram, the line $AB$ has been constructed so as to be a '''perpendicular''' to the plane containing the straight lines $CD$ and $EF$.
Definition:Right Angle/Perpendicular/Plane to Plane
{{EuclidDefinition|book = XI|def = 4|name = Plane at Right Angles to Plane}} :400px In the above diagram, the two planes have been constructed so as to make lines perpendicular to their common section perpendicular to each other. Thus the two planes are perpendicular to each other.
Definition:Pi/Definition 1
Take a circle in a plane whose circumference is $C$ and whose radius is $r$. Then $\pi$ can be defined as $\pi = \dfrac C {2r}$.
Definition:Pi/Definition 2
The real functions sine and cosine can be shown to be periodic. The number $\pi$ is defined as the real number such that: :the period of both sine and cosine is $2 \pi$.
Definition:Piecewise Continuously Differentiable Function/Definition 1
$f$ is '''piecewise continuously differentiable''' : :$(1): \quad f$ is continuous :$(2): \quad$ there exists a finite subdivision $\set {x_0, \ldots, x_n}$ of $\closedint a b$, $x_0 = a$ and $x_n = b$, such that: ::$(2.1): \quad f$ is continuously differentiable on $\openint {x_{i - 1}} {x_i}$ for every $i \in \set {1, \ldots, n}$ ::$(2.2): \quad$ the one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i - 1}}^+} \map {f'} x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map {f'} x$ exist for every $i \in \set {1, \ldots, n}$.
Definition:Parallelepiped/Height
:400px The '''height''' of a parallelepiped is the length of the perpendicular from the plane of the base to the plane opposite. In the above diagram, $h$ is the '''height''' of the parallelepiped whose base is $AB$.
Definition:Similar Situation
Two similar solid figures are said to be in a '''similar situation'''  corresponding surfaces are similarly inclined and when corresponding edges are parallel.
Definition:Solid Geometry
'''Solid geometry''' is the study of geometric figures in three dimensions. As such it is a misnomer, because the figures themselves under study are strictly speaking the surfaces of such figures.
Definition:Group Theory
'''Group Theory''' is a branch of abstract algebra which studies groups and other related algebraic structures.
Definition:Abstract Algebra
'''Abstract algebra''' is a branch of mathematics which studies algebraic structures and algebraic systems. It can be roughly described as the study of sets equipped with operations.
Definition:Algebra (Mathematical Branch)
'''Algebra''' is the branch of mathematics which studies the techniques of manipulation of objects and expressions.
Definition:Parallelepiped/Opposite Face
The '''opposite face''' of the face $F$ of a parallelepiped $P$ is the face of $P$ which is parallel to $F$. In the above example, the pairs of parallel planes are: :Face $ABCD$ is opposite $HGFE$ :Face $ADEH$ is opposite $BCFG$ :Face $ABGH$ is opposite $DCFE$
Definition:Prism/Height
The '''height''' of a prism is the length of the perpendicular between the bases of the prism. In the above diagram, the distance $h$ is the '''height''' of the prism $AJ$.
Definition:Prism/Opposite
Let $P$ be a prism whose bases are $P_1$ and $P_2$. The '''opposite face''' of the base $P_1$ is the base $P_2$. In the above example, the base $ABCDE$ is opposite $FGHIJ$.
Definition:Prism/Height/Euclidean Variant
:400px Although the height of a prism is generally understood to be the length of the perpendicular joining opposite faces, {{AuthorRef|Euclid}} was inconsistent in his usage in {{BookLink|The Elements|Euclid}}. In his {{EuclidPropLink|book = XI|prop = 39|title = Prisms of equal Height with Parallelogram and Triangle as Base}}, he defines the base of one prism as being one of the opposite parallel faces, but of the other he defines the base as being an arbitrary one of the parallelograms. Having defined the base in this manner, the '''height''' is then defined as being the height of one of the opposite parallel faces whose base is the edge which intersects the base so defined. Using this definition, the distance $h$ in the above diagram is the '''height''' of the prism $PQRSTU$.
Definition:Prism/Lateral Face
The '''lateral faces''' of a prism are the parallelogramic faces which join the bases. In the above diagram, the faces $ABGF, BCHG, CDIH, DEJI$ and $EAFJ$ are the '''lateral faces''' of the prism.
Definition:Triangular Prism
:250px A '''triangular prism''' is a prism whose bases are triangles.
Definition:Natural Numbers/Von Neumann Construction
Let $\omega$ denote the minimal infinite successor set. The '''natural numbers''' can be defined as the elements of $\omega$. Following Definition 2 of $\omega$, this amounts to defining the '''natural numbers''' as the finite ordinals. In terms of the empty set $\O$ and successor sets, we thus define: {{begin-eqn}} {{eqn | l = 0       | o = :=       | r = \O = \set {}       | c =  }} {{eqn | l = 1       | o = :=       | r = 0^+ = 0 \cup \set 0 = \set 0       | c =  }} {{eqn | l = 2       | o = :=       | r = 1^+ = 1 \cup \set 1 = \set {0, 1}       | c =  }} {{eqn | l = 3       | o = :=       | r = 2^+ = 2 \cup \set 2 = \set {0, 1, 2}       | c =  }} {{eqn | o = \vdots       | c =  }} {{eqn | l = n + 1       | o = :=       | r = n^+ = n \cup \set n       | c =  }} {{end-eqn}}
Definition:Piecewise Continuous Function/One-Sided Limits
$f$ is '''piecewise continuous with one-sided limits''' : :there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that, for all $i \in \set {1, 2, \ldots, n}$: ::$(1): \quad f$ is continuous on $\openint {x_{i − 1} } {x_i}$ ::$(2): \quad$ the one-sided limits $\displaystyle \lim_{x \mathop \to {x_{i − 1} }^+} \map f x$ and $\displaystyle \lim_{x \mathop \to {x_i}^-} \map f x$ exist.
Definition:Piecewise Continuous Function/Bounded
$f$ is a '''bounded piecewise continuous function''' : :there exists a finite subdivision $\set {x_0, x_1, \ldots, x_n}$ of $\closedint a b$, where $x_0 = a$ and $x_n = b$, such that: ::$(1): \quad$ for all $i \in \set {1, 2, \ldots, n}$, $f$ is continuous on $\openint {x_{i − 1} } {x_i}$  ::$(2): \quad$ $f$ is bounded on $\closedint a b$.
Definition:Axis of Solid Figure
=== Axis of Cone === :300px {{:Definition:Axis of Cone}} === Axis of Cylinder === :300px {{:Definition:Axis of Cylinder}} === Axis of Sphere === {{:Definition:Axis of Sphere}}
Definition:Apex
The '''apex''' of a geometric figure is the point which is distinguished from the others by dint of it being furthest away from its base. Not all figures have a discernible '''apex'''; for example, parallelograms, prisms and parallelepipeds do not. === Apex of Triangle === :300px {{:Definition:Apex of Triangle}} === Apex of Isosceles Triangle === :300px {{:Definition:Apex of Isosceles Triangle}} === Apex of Cone === :300px {{:Definition:Apex of Cone}} === Apex of Pyramid === :400px {{:Definition:Apex of Pyramid}}
Definition:Base of Geometric Figure
The '''base''' of a geometric figure is a specific part of that figure which is distinguished from the remainder of that figure and placed (actually or figuratively) at the '''bottom''' of a depiction or visualisation. In some cases the '''base''' is truly qualitiatively different from the rest of the figure. In other cases the '''base''' is selected arbitrarily as one of several parts of the figure which may equally well be so chosen.
Definition:Quadrant
A '''quadrant''' is a sector of a circle whose angle is a right angle.
Definition:Minimal Infinite Successor Set/Definition 3
{{Refactor|Based on Takeuti}} The '''minimal infinite successor set''' $\omega$ is defined as: :$\omega := \set {x \in \On: \paren {x \cup \set x} \subseteq K_I}$ where: :$K_I$ is the class of all non-limit ordinals :$\On$ is the class of all ordinals.
Definition:Minimal Infinite Successor Set/Definition 1
Let $S$ be an infinite successor set. The '''minimal infinite successor set''' $\omega$ is the infinite successor set given by: :$\omega := \displaystyle \bigcap \set {S' \subseteq S: \text{$S'$ is an infinite successor set} }$ that is, $\omega$ is the intersection of every infinite successor set which is a subset of $S$.
Definition:Equiangular Geometric Figures
Two geometric figures are '''equiangular (with each other)''' when the angles of each pair of their corresponding vertices are equal.
Definition:Tetrahedron/Base
One of the faces of a tetrahedron can be chosen arbitrarily, distinguished from the others and identified as the '''base''' of the tetrahedron. In the above diagram, $ABC$ is the '''base''' of the tetrahedron $ABCD$.
Definition:Tetrahedron/Apex
Once the base of a tetrahedron has been identified, the vertex which does not lie on the base is called the '''apex''' of the tetrahedron. In the above diagram, given that the base of the tetrahedron $ABCD$ is the triangle $ABC$, the '''apex''' is $D$.
Definition:Polyhedron/Face
The '''faces''' of a polyhedron are the polygons which contain it.
Definition:Ludolphine Number
The '''Ludolphine number''' is the expression of the value of $\pi$ to $35$ decimal places: :$3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \ldots$
Definition:Sign of Number
The '''sign''' of a number is the symbol indicating whether it is: : positive, denoted by the symbol $+$ or: : negative, denoted by the symbol $-$. Hence a number's '''sign''' has evolved to define the fact of the number being positive or negative independently of the symbol itself. Thus: :the '''sign''' of $3.14159$ is positive and :the '''sign''' of $-75$ is negative.
Definition:Piecewise Continuous Function/Improper Integrals
$f$ is '''piecewise continuous with improper integrals''' : :there exists a finite subdivision $\left\{{x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that for all $i \in \left\{ {1, 2, \ldots, n}\right\}$: ::$(1): \quad f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$  ::$(2): \quad$ the improper integrals $\displaystyle \int_{ {x_{i - 1} }^+}^{ {x_i}^-} f \left({x}\right) \rd x$ all exist.
Definition:Commutative/Elements
Let $\circ$ be a binary operation. Two elements $x, y$ are said to '''commute''' : :$x \circ y = y \circ x$
Definition:Commutative/Operation
Let $\struct {S, \circ}$ be an algebraic structure. Then $\circ$ is '''commutative on $S$''' : :$\forall x, y \in S: x \circ y = y \circ x$
Definition:Commutative Division Ring
A '''commutative division ring''' is a division ring $\left({R, +, \circ}\right)$ in which the ring product $\circ$ is commutative.
Definition:Commutative/Algebraic Structure
Let $\left({S, \circ}\right)$ be an algebraic structure whose operation $\circ$ is a commutative operation. Then $\left({S, \circ}\right)$ is a '''commutative (algebraic) structure'''.
Definition:Operation/Operand
An '''operand''' is one of the objects on which an operation generates its new object.
Definition:Arithmetic
'''Arithmetic''' is the branch of mathematics which concerns the manipulation of numbers, using the operations addition, subtraction, multiplication and division, and the taking of powers.
Definition:Smallest Field
The '''smallest field''' is the set of integers modulo $2$ under modulo addition and modulo multiplication: :$\struct {\Z_2, +_2, \times_2}$ This field has $2$ elements.
Definition:Addition/Modulo Addition/Definition 1
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$ where $\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''addition modulo $m$''' is defined on $\Z_m$ as: :$\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$
Definition:Addition/Modulo Addition/Definition 2
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \left\{{0, 1, \ldots, m-1}\right\}$ The operation of '''addition modulo $m$''' is defined on $\Z_m$ as: :$x +_m y$ equals the remainder after $x + y$ has been divided by $m$.
Definition:Addition/Modulo Addition/Definition 3
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \left\{{0, 1, \ldots, m-1}\right\}$ The operation of '''addition modulo $m$''' is defined on $\Z_m$ as: :$x +_m y := x + y - j m$ where $j$ is the largest integer such that $j m \le x + y$.
Definition:Multiplication/Modulo Multiplication/Definition 1
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$ where $\eqclass x m$ is the residue class of $x$ modulo $m$. The operation of '''multiplication modulo $m$''' is defined on $\Z_m$ as: :$\eqclass a m \times_m \eqclass b m = \eqclass {a b} m$
Definition:Multiplication/Modulo Multiplication/Definition 3
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {0, 1, \ldots, m - 1}$ The operation of '''multiplication modulo $m$''' is defined on $\Z_m$ as: :$x \cdot_m y := x y - k m$ where $k$ is the largest integer such that $k m \le x y$.
Definition:Multiplication/Modulo Multiplication/Definition 2
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$: :$\Z_m = \set {0, 1, \ldots, m - 1}$ The operation of '''multiplication modulo $m$''' is defined on $\Z_m$ as: :$x \cdot_m y$ equals the remainder after $x y$ has been divided by $m$.
Definition:Big Dipper Operation
Let $m, n \in \Z$ be integers such that $m \ge 0, n > 0$. Let $+_{m, n}$ be the binary operation on $\Z_{>0}$ defined as: :$\forall a, b \in \Z_{>0}: a +_{m, n} b = \begin{cases} a + b & : a + b < m \\ a + b - k n & : a + b \ge m \end{cases}$ where $k$ is the largest integer satisfying: :$m + k n \le x + y$ The operation $+_{m, n}$ is known as the '''Big Dipper'''. :600px When the stars of the [http://en.wikipedia.org/wiki/Big_Dipper Big Dipper] are numbered as shown, the sequence: :$1, 1 +_{3, 4} 1, 1 +_{3, 4} 1 +_{3, 4} 1, \ldots$ traces out those stars in the order: :first the handle: $\text{Alkaid}, \text{Mizar}, \text{Alioth}$ then: :round the pan indefinitely: $\text{Megrez}, \text{Dubhe}, \text{Merak}, \text{Phecda}, \text{Megrez}, \ldots$ === Cayley Table === {{:Big Dipper Operation/Cayley Table}}
Definition:Parity Ring
The '''parity ring''' is the ring of two elements which defines the nature of the parity of integers under addition and multiplication: :$\struct {\set {\text{even}, \text{odd} }, +, \times}$
Definition:Group of Gaussian Integer Units
Let $i$ be the imaginary unit: $i = \sqrt {-1}$. Let $U_\C$ be the set of complex numbers defined as: :$U_\C = \set {1, i, -1, -i}$ Let $\times$ denote the operation of complex multiplication. The algebraic structure $\struct {U_\C, \times}$ is the '''group of units of the ring of Gaussian integers'''.
Definition:Ring of Gaussian Integers
The '''ring of Gaussian integers''' $\struct {\Z \sqbrk i, +, \times}$ is the algebraic structure formed from: : the set of Gaussian integers $\Z \sqbrk i$ : the operation of complex addition : the operation of complex multiplication.
Definition:Finite Ordinal
Let $\alpha$ be an ordinal. Then $\alpha$ is said to be '''finite'''  one of the following holds: :$\alpha = \O$ :$\alpha = \beta^+$ for some '''finite ordinal''' $\beta$ where $\O$ denotes the empty set, and $\beta^+$ is the successor ordinal of $\beta$.
Definition:Minimal Infinite Successor Set/Definition 2
The '''minimal infinite successor set''' $\omega$ is defined as the set of all finite ordinals: :$\omega := \set {\alpha : \text{$\alpha$ is a finite ordinal} }$
Definition:Root of Unity/Complex/First
The root $e^{2 i \pi / n}$ is known as the '''first (complex) $n$th root of unity'''.
Definition:Finite Algebraic Structure
A '''finite algebraic structure''' is an algebraic structure whose underlying set is finite.
Definition:Euclidean Relation/Left-Euclidean
$\mathcal R$ is '''left-Euclidean''' : :$\tuple {x, z} \in \mathcal R \land \tuple {y, z} \in \mathcal R \implies \tuple {x, y} \in \mathcal R$
Definition:Hasse Diagram/Examples
These are examples of Hasse diagrams : :350px $\qquad$ 350px The diagram on the left illustrates the "Divisor" ordering on the set $S = \left\{{1, 2, 3, 4, 6, 8, 12, 24}\right\}$ where $S$ is the set of all elements of $\N_{>0}$ which divide $24$. The diagram on the right illustrates the "Subset" relation on the power set $\mathcal P \left({S}\right)$ where $S = \left\{{1, 2, 3}\right\}$.
Definition:Smallest/Ordered Set/Subset
Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. An element $x \in T$ is '''the smallest element of $T$''' : :$\forall y \in T: x \preceq \restriction_T y$ where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.
Definition:Inverse Relation/Ordering Notation
To denote the inverse of an ordering, the conventional technique is to reverse the symbol. Thus: :$\succeq$ denotes $\preceq^{-1}$ :$\succcurlyeq$ denotes $\preccurlyeq^{-1}$ :$\curlyeqsucc$ denotes $\curlyeqprec^{-1}$ and so: :$a \preceq b \iff b \succeq a$ :$a \preccurlyeq b \iff b \succcurlyeq a$ :$a \curlyeqprec b \iff b \curlyeqsucc a$ Similarly for the standard symbols used to denote an ordering on numbers: :$\ge$ denotes $\le^{-1}$ :$\geqslant$ denotes $\leqslant^{-1}$ :$\eqslantgtr$ denotes $\eqslantless^{-1}$ and so on.