hcju/proofwikips huggingface dataset:
Definition:Bounded Linear Transformation/Bounded Linear Operator
Let $H$ be a Hilbert space. Let $A: H \to H$ be a linear operator. $A$ is a '''bounded linear operator'''  :$\exists c > 0: \forall h \in H: \left\Vert{A h}\right\Vert_H \le c \left\Vert{h}\right\Vert_H$ That is, a '''bounded linear operator''' is a bounded linear transformation from a Hilbert space to itself.
Definition:Arborescence/Root
The vertex $r$ of $G$ is known as the '''root of (the arborescence) $G$'''.
Definition:Arborescence/Also defined as
Some sources, for example {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms||Donald E. Knuth}}, define an arborescence of root $r$ so as to reverse the orientation of $G$, so that the arcs are all directed toward the root rather than away from it.
Definition:Arborescence/Also known as
An arborescence of root $r$ can be referred to as an an '''$r$-arborescence''', or just an '''arborescence'''. Some sources, for example {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms||Donald E. Knuth}}, call an arborescence an '''oriented tree'''.
Definition:Interval/Ordered Set/Half-Open
Let $\struct {S, \preccurlyeq}$ be an ordered set. Let $a, b \in S$. === Left Half-Open Interval === {{:Definition:Interval/Ordered Set/Left Half-Open}} === Right Half-Open Interval === {{:Definition:Interval/Ordered Set/Right Half-Open}}
Definition:Ray (Order Theory)/Open
The following sets are called '''open rays''' or '''open half-lines''': : $\left\{{x \in S: a \prec x}\right\}$ (the strict upper closure of $a$), denoted $a^\succ$ : $\left\{{x \in S: x \prec a}\right\}$ (the strict lower closure of $a$), denoted $a^\prec$.
Definition:Ray (Order Theory)/Closed
The following sets are called '''closed rays''' or '''closed half-lines''': : $\left\{{x \in S: a \preccurlyeq x}\right\}$ (the upper closure of $a$), denoted $a^\succcurlyeq$ : $\left\{{x \in S: x \preccurlyeq a}\right\}$ (the lower closure of $a$), denoted $a^\preccurlyeq$.
Definition:Ray (Order Theory)/Upward-Pointing
An '''upward-pointing ray''' is a ray which is bounded below: : an open ray $a^\succ:= \left\{{x \in S: a \prec x}\right\}$ : a closed ray $a^\succcurlyeq: \left\{{x \in S: a \preccurlyeq x}\right\}$
Definition:Ray (Order Theory)/Downward-Pointing
A '''downward-pointing ray''' is a ray which is bounded above: : an open ray $a^\prec := \left\{{x \in S: x \prec a}\right\}$ : a closed ray $a^\preccurlyeq : \left\{{x \in S: x \preccurlyeq a}\right\}$
Definition:Hereditarily Compact Space/Definition 1
$T$ is ''' hereditarily compact'''  every subspace of $T$ is compact.
Definition:Hereditarily Compact Space/Definition 2
$T$ is ''' hereditarily compact''' : : for each family $\left\langle{U_i}\right\rangle_{i \mathop \in I}$ of open sets of $T$, there exists a finite subset $J \subset I$ such that: :::$\displaystyle \bigcup_{j \mathop \in J} U_j = \bigcup_{i \mathop \in I} U_i$
Definition:Order of Structure/Infinite Structure
Let the underlying set $S$ of $\struct {S, \circ}$ be infinite. Then $\struct {S, \circ}$ is an '''infinite structure'''.
Definition:Order of Structure/Finite Structure
Let the underlying set $S$ of $\struct {S, \circ}$ be finite. Then $\struct {S, \circ}$ a '''finite structure'''.
Definition:Index of Subgroup/Finite
If $G / H$ is a finite set, then $\index G H$ is '''finite''', and $H$ is '''of finite index''' in $G$.
Definition:Index of Subgroup/Infinite
If $G / H$ is an infinite set, then $\index G H$ is '''infinite''', and $H$ is '''of infinite index''' in $G$.
Definition:Order of Group Element/Infinite
Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Infinite/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Infinite/Definition 2}} === Definition 3 === {{:Definition:Order of Group Element/Infinite/Definition 3}}
Definition:Order of Group Element/Finite
Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$. === Definition 1 === {{:Definition:Order of Group Element/Finite/Definition 1}} === Definition 2 === {{:Definition:Order of Group Element/Finite/Definition 2}}
Definition:Multiplication on 1-Based Natural Numbers
Let $\N_{>0}$ be the $1$-based natural numbers, axiomatized by: {{:Axiom:Axiomatization of 1-Based Natural Numbers}} The operation $\times$ in this axiomatization is called '''multiplication'''.
Definition:Order of Group Element/Definition 1
The '''order of $x$ (in $G$)''', denoted $\order x$, is the smallest $k \in \Z_{> 0}$ such that $x^k = e_G$.
Definition:Order of Group Element/Definition 2
The '''order of $x$ (in $G$)''', denoted $\order x$, is the order of the group generated by $x$: :$\order x := \order {\gen x}$
Definition:Directed Graph/Arc/Endvertex/Final Vertex
The '''final vertex''' of $e$ is the endvertex $v$ which $e$ is incident to.
Definition:Directed Graph/Arc/Endvertex/Initial Vertex
The '''initial vertex''' of $e$ is the endvertex $u$ which $e$ is incident from.
Definition:Integer/Formal Definition/Notation
We have that $\eqclass {\tuple {a, b} } \boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$. As this notation is cumbersome, it is commonplace though technically incorrect to streamline it to $\eqclass {a, b} \boxminus$, or $\eqclass {a, b} {}$. This is generally considered acceptable, as long as it is made explicit as to the precise meaning of $\eqclass {a, b} {}$ at the start of any exposition.
Definition:Injection/Definition 1 a
:$\forall x_1, x_2 \in \Dom f: x_1 \ne x_2 \implies \map f {x_1} \ne \map f {x_2}$
Definition:Injection/Definition 1
A mapping $f$ is '''an injection''', or '''injective''' : :$\forall x_1, x_2 \in \Dom f: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$ That is, an '''injection''' is a mapping such that the output uniquely determines its input.
Definition:Injection/Definition 2
An '''injection''' is a relation which is both one-to-one and left-total.
Definition:Injection/Definition 3
Let $f$ be a mapping. Then $f$ is an injection : :$f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$.
Definition:Injection/Definition 4
Let $f$ be a mapping. $f$ is an '''injection''' : :$\forall y \in \Img f: \card {\map {f^{-1} } y} = \card {\set {f^{-1} \sqbrk {\set y} } } = 1$ where: :$\Img f$ denotes the image set of $f$ :$\card {\, \cdot \,}$ denotes the cardinality of a set :$\map {f^{-1} } y$ is the preimage of $y$ :$f^{-1} \sqbrk {\set y}$ is the preimage of the subset $\set y \subseteq \Img f$.
Definition:Injection/Definition 5
Let $f: S \to T$ be a mapping where $S \ne \O$. Then $f$ is an injection : :$\exists g: T \to S: g \circ f = I_S$ where $g$ is a mapping. That is,  $f$ has a left inverse.
Definition:Injection/Definition 6
Let $f: S \to T$ be a mapping where $S \ne \O$. Then $f$ is an '''injection'''  $f$ is left cancellable: :$\forall X: \forall g_1, g_2: X \to S: f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ where $g_1$ and $g_2$ are arbitrary mappings from an arbitrary set $X$ to the domain $S$ of $f$.
Definition:Associative Operation
Let $S$ be a set. Let $\circ : S \times S \to S$ be a binary operation. Then $\circ$ is '''associative''' : :$\forall x, y, z \in S: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$
Definition:Set/Uniqueness of Elements
A set is '''uniquely determined''' by its elements. This means that the only thing that defines '''what a set is''' is '''what it contains'''. So, how you choose to '''list''' or '''define''' the contents makes '''no difference''' to what the contents actually '''are'''.
Definition:Tally Marks
'''Tally marks''' are a system of recording (usually) small numbers. For each unit in a given natural number, a mark is made. Thus, for example, $13$ could be recorded $|||||||||||||$. It is usual for '''tally marks''' to be arranged in groups of $5$, by drawing every fifth line diagonally across the group of $4$ before it. Thus $13$ would then be recorded: :File:Tally5.pngFile:Tally5.pngFile:Tally3.png
Definition:Lebombo Bone
The '''Lebombo bone''' is a baboon's leg bone bearing $29$ notches which were carved into it between $33 \ 000$ B.C.E. and $42 \ 000$ B.C.E. It is the earliest known instance of a system of tally marks. It was found in {{WP|Border_Cave|Border Cave}} in {{WP|South_Africa|South Africa}} in the {{WP|Lebombo_Mountains|Lebombo mountains}}, near the border with {{WP|Swaziland|Swaziland}}. Various untestable hypotheses have been raised concerning this artefact, notably that it was created by an early African woman to count the days between her menses. However, as the bone appears to be broken at one end, it is possible that the count of $29$ is not the complete set of notches that were originally carved.
Definition:Wolf Bone
The '''(Czechoslovakian) wolf bone''' is a wolf bone dating from approximately $30 \ 000$ years ago, bearing $57$ marks which appear to have been made deliberately. It was discovered in $1937$ in {{WP|Czechoslovakia|Czechoslovakia}} at {{WP|Doln%C3%AD_V%C4%9Bstonice_(archaeology)|Dolní Věstonice}} in {{WP|Moravia|Moravia}}, by a team led by {{WP|Karel_Absolon|Karl Absolon}}. The markings are believed to be tally marks, and are arranged in $11$ groups of $5$, with $2$ left over. The untestable hypothesis has been made that the number corresponds to a double lunar period.
Definition:Ishango Bone
The '''Ishango bone''' is a mathematical tool made from a leg bone of a baboon, dating from approximately $25 \ 000$ years ago. It bears a system of markings which may be tally marks, but may in fact be for a more complex purpose. It was found in $1960$ by {{WP|Jean_de_Heinzelin_de_Braucourt|Jean de Heinzelin de Braucourt}} in the {{WP|Belgian_Congo|Belgian Congo}} (now the {{WP|Democratic_Republic_of_the_Congo|Democratic Republic of the Congo}}). It was discovered in the area of {{WP|Ishango|Ishango}} near the {{WP|Semliki_River|Semliki River}}. One row of marks contains the prime numbers between $10$ and $20$, that is, $11$, $13$, $17$ and $19$, which total $60$. Another row contains $9$, $11$, $19$ and $21$, which also total $60$. The third row contains marks which appear to represent a method of multiplication by means of a technique involving doubling and halving. While it is suggested that the markings form a calendar, it is also possible that the markings are coincidental, and perhaps even just put there to make the bone easier to hold.
Definition:Cartesian Product/Cartesian Space/Family of Sets
Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be an family of sets indexed by $I$. Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $S$ be a set such that: :$\forall i \in I: S_i = S$ === Definition 1 === {{:Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1}} === Definition 2 === {{:Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 2}}
Definition:Ordered Tuple/Definition 1
An '''ordered tuple (of length $n$)''' is a finite sequence whose domain is $\N^*_n$.
Definition:Ordered Tuple/Also known as
Some sources refer to an ordered tuple as a '''tuple'''. The term '''ordered $n$-tuple''' can sometimes be seen, particularly for specific instances of $n$.  Instead of writing '''2-tuple''', '''3-tuple''' and '''4-tuple''', the terms '''couple''', '''triple''' and '''quadruple''' are usually used. In the context of abstract algebra, the concept is encountered as '''(associative) word'''.
Definition:Ordered Tuple/Notation
Notation for an ordered tuple varies throughout the literature. There are also specialised instances of an ordered tuple where the convention is to use angle brackets. However, it is common for an ordered tuple to be denoted: :$\tuple {a_1, a_2, \ldots, a_n}$ extending the notation for an ordered pair. For example: $\tuple {6, 3, 3}$ is the ordered triple $f$ defined as: :$\map f 1 = 6, \map f 2 = 3, \map f 3 = 3$ The notation: :$\sequence {a_1, a_2, \ldots, a_n}$ is recommended when use of round brackets would be ambiguous. Other notations which may be encountered are: :$\sqbrk {a_1, a_2, \ldots, a_n}$ :$\set {a_1, a_2, \ldots, a_n}$ but both of these are strongly discouraged: the square bracket format because there are rendering problems on this site, the latter because it is too easily confused with set notation. In order to further streamline notation, it is common to use the more compact $\sequence {a_n}$ for $\sequence {a_k}_{1 \mathop \le k \mathop \le n}$. Some sources, particularly in such fields as communication theory, where the elements of the domain of the ordered tuple is a specific set of symbols, use the notation $x_1 x_2 \cdots x_n$ for $\tuple {x_1, x_2, \dotsc, x_n}$.
Definition:Ordered Tuple/Definition 2
Let $\family {S_i}_{i \mathop \in \N_n}$ be a family of sets indexed by $\N_n$. Let $\displaystyle \prod_{i \mathop \in \N_n} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in \N_n}$. An '''ordered tuple of length $n$''' of $\family {S_i}$ is an element of $\displaystyle \prod_{i \mathop \in \N_n} S_i$.
Definition:Direct Image Mapping/Mapping
Let $f \subseteq S \times T$ be a mapping from $S$ to $T$. The '''direct image mapping''' of $f$ is the mapping $f^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq S$ to its image under $f$: :$\forall X \in \powerset S: \map {f^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \map f s = t} & : X \ne \O \\ \O & : X = \O \end {cases}$
Definition:Direct Image Mapping/Relation
Let $\RR \subseteq S \times T$ be a relation on $S \times T$. The '''direct image mapping''' of $\RR$ is the mapping $\RR^\to: \powerset S \to \powerset T$ that sends a subset $X \subseteq T$ to its image under $\RR$: :$\forall X \in \powerset S: \map {\RR^\to} X = \begin {cases} \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR} & : X \ne \O \\ \O & : X = \O \end {cases}$
Definition:Inverse Image Mapping
Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. === Relation === Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. {{Definition:Inverse Image Mapping/Relation/Definition 1}} === Mapping === Let $f: S \to T$ be a mapping. {{Definition:Inverse Image Mapping/Mapping/Definition 1}}
Definition:Inverse Image Mapping/Relation
Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. === Definition 1 === {{:Definition:Inverse Image Mapping/Relation/Definition 1}} === Definition 2 === {{:Definition:Inverse Image Mapping/Relation/Definition 2}}
Definition:Inverse Image Mapping/Mapping
Let $f: S \to T$ be a mapping. === Definition 1 === {{:Definition:Inverse Image Mapping/Mapping/Definition 1}} === Definition 2 === {{:Definition:Inverse Image Mapping/Mapping/Definition 2}}
Definition:Equivalence Relation/Also denoted as
When discussing equivalence relations, various notations are used for $\tuple {x, y} \in \RR$. Examples are: :$x \mathrel \RR y$ :$x \equiv \map y \RR$ :$x \equiv y \pmod \RR$ and so on. Specialised equivalence relations generally have their own symbols, which can be defined as they are needed. Such symbols include: :$\cong$, $\equiv$, $\sim$, $\simeq$, $\approx$
Definition:Equivalence Relation/Definition 2
$\RR$ is an equivalence relation : :$\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$ where: :$\Delta_S$ denotes the diagonal relation on $S$ :$\RR^{-1}$ denotes the inverse relation :$\circ$ denotes composition of relations
Definition:Equivalence Relation/Definition 1
Let $\RR$ be: :$(1): \quad$ reflexive :$(2): \quad$ symmetric :$(3): \quad$ transitive Then $\RR$ is an '''equivalence relation''' on $S$.
Definition:Minimal/Ordered Set/Definition 1
An element $x \in T$ is a '''minimal element of $T$''' : :$\forall y \in T: y \preceq x \implies x = y$ That is, the only element of $T$ that $x$ '''succeeds or is equal to''' is itself.
Definition:Minimal/Ordered Set/Definition 2
An element $x \in T$ is a '''minimal element of $T$''' : :$\neg \exists y \in T: y \prec x$ where $y \prec x$ denotes that $y \preceq x \land y \ne x$. That is,  $x$ has '''no strict predecessor'''.
Definition:Minimal/Ordered Set/Comparison with Smallest Element
Compare the definition of minimal element with that of a smallest element. An element $x \in T$ is '''the''' smallest element of $T$ : :$\forall y \in T: x \preceq y$ That is, $x$ is comparable to, and precedes, or is equal to, every $y \in T$. An element $x \in T$ is '''a''' minimal element of $T$ : :$y \preceq x \implies x = y$ That is, $x$ precedes, or is equal to, every $y \in T$ which ''is'' comparable to $x$. If ''all'' elements are comparable to $x$, then such a minimal element is indeed '''the smallest element'''. Note that when an ordered set is in fact a totally ordered set, the terms '''minimal element''' and '''smallest element''' are equivalent.
Definition:Maximal/Ordered Set/Definition 1
An element $x \in T$ is a '''maximal element of $T$''' : :$x \preceq y \implies x = y$ That is, the only element of $S$ that $x$ '''precedes or is equal to''' is itself.
Definition:Maximal/Ordered Set/Definition 2
An element $x \in T$ is a '''maximal element of $T$''' : :$\neg \exists y \in T: x \prec y$ where $x \prec y$ denotes that $x \preceq y \land x \ne y$. That is,  $x$ has '''no strict successor'''.
Definition:Maximal/Ordered Set/Comparison with Greatest Element
Compare the definition of maximal element with that of a greatest element. An element $x \in T$ is '''the''' greatest element of $T$ : :$\forall y \in T: y \preceq x$ That is, $x$ is comparable to, and succeeds, or is equal to, every $y \in S$. An element $x \in S$ is '''a''' maximal element of $T$ : :$x \preceq y \implies x = y$ That is, $x$ succeeds, or is equal to, every $y \in S$ which ''is'' comparable to $x$. If ''all'' elements are comparable to $x$, then such a maximal element is indeed '''the greatest element'''. Note that when an ordered set is in fact a totally ordered set, the terms '''maximal element''' and '''greatest element''' are equivalent.
Definition:Choice Function/Use of Axiom of Choice
The Axiom of Choice (abbreviated '''AoC''' or '''AC''') is the following statement: :''All $\mathbb S$ as above have a choice function.'' It can be shown that the AoC it does not follow from the other usual axioms of set theory, and that it is relative consistent to these axioms (i.e., that AoC does not make the axiom system inconsistent, provided it was consistent without AoC). Note that for any given set $S \in \mathbb S$, one can select an element from it (without using AoC). AoC guarantees that there is a choice function, i.e., a function that "simultaneously" picks elements of all $S \in \mathbb S$.   AoC is needed to prove statements such as "all countable unions of finite sets are countable" (for many specific such unions this  can be shown without AoC), and AoC is equivalent to many other mathematical statements such as "every vector space has a basis".
Definition:Bessel Function/First Kind
A '''Bessel function of the first kind of order $n$''' is a Bessel function which is non-singular at the origin It is usually denoted $\map {J_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's equation''' to which $\map {J_n} x$ forms a solution.
Definition:Bessel Function/Second Kind
A '''Bessel function of the second kind of order $n$''' is a Bessel function which is singular at the origin. It is usually denoted $\map {Y_n} x$, where $x$ is the dependent variable of the instance of '''Bessel's equation''' to which $\map {Y_n} x$ forms a solution.
Definition:Pythagoreans
The '''Pythagoreans''' were a semi-mystical cult which dated from around $550$ B.C.E., founded by {{AuthorRef|Pythagoras of Samos}}. Can claim to be the world's first university. It is feasible to suggest that their initial work in the field of geometry may have formed the basis of at least the first two books of . Attendees of the school were divided into two classes: :the '''Probationers''' (or '''listeners''') :the '''Pythagoreans'''. A student was a '''listener''' for $3$ years, after which he was allowed to be initiated into the class of '''Pythagoreans''', who were allowed to learn what was considered to be the deeper secrets. '''Pythagoreans''' were a closely-knit brotherhood, who held all their worldly goods in common, and were bound by oath never to reveal the secrets of the Founder. There exists a legend that one of the '''Pythagoreans''' was thrown overboard to drown after having revealed the existence of the regular dodecahedron. For some considerable time they dominated the political life of {{WP|Crotone|Croton}}, where they were based, but in $501$ B.C.E. there was a popular revolt in which a number of the leaders of the school were murdered. {{AuthorRef|Pythagoras of Samos|Pythagoras}} himself was murdered soon after. Some sources state that the reasons for this were based on the fact that their puritanical philosophy was at odds with the contemporary way of thinking. Others suggest that there was a reaction against their autocratic rule. Whatever the truth of the matter, between $501$ and about $460$ B.C.E. the political influence of the cult was destroyed. Its survivors scattered, many of them fleeing to {{WP|Thebes,_Egypt|Thebes}} in {{WP|Upper_Egypt|Upper Egypt}}, where they continued to exist more as a philosophical and mathematical society for another couple of centuries, secretive and ascetic to the end, publishing nothing, ascribing all their discoveries to the Master, {{AuthorRef|Pythagoras of Samos|Pythagoras}} himself. === Quadrivium === {{:Definition:Pythagoreans/Quadrivium}} === Trivium === {{:Definition:Pythagoreans/Trivium}}
Definition:Method of Exhaustion
The '''method of exhaustion''' is a technique for calculating an approximation to the physical extent of a geometric figure whose extremities are curved lines. To measure the area of a circle, for example, this is done by: : Circumscribing a polygon around the circle : Inscribing a polygon inside the circle : Measuring the areas of those polygons : Noting that the area of the circle is between those two : Increasing the number of sides on those polygons to make them closer and closer to the circle and each other. As the number of sides of the polygons increases, their areas become ever closer to each other.
Definition:Markov Chain
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a sequence of random variables in a countable set $S$. Let $\map \Pr X$ denote the probability of the random variable $X$. Let $\sequence {X_n}_{n \mathop \ge 0}$ satisfy the Markov property: :$\map \Pr {X_{n + 1} = i_{n + 1} \mid X_0 = i_0, X_1 = i_1, \ldots, X_n = i_n} = \map \Pr {X_{n + 1} = i_{n + 1} \mid X_n = i_n}$ for all $n \ge 0$ and all $i_0, i_1, \ldots, i_{n + 1} \in S$. Then $\sequence {X_n}_{n \mathop \ge 0}$ is a '''Markov chain'''.
Definition:Markov Chain/State Space
The set $S$ is called the '''state space''' of the Markov chain.
Definition:Tool
A '''tool''' is a physical device used as an aid to a process.
Definition:Straightedge
A '''straightedge''' is an ideal tool for constructing straight lines. A '''straightedge''' is of unlimited length, but has no markings on it, so it cannot be used for measurement.
Definition:Compass
A '''compass''' is an ideal tool for drawing circles. Hence it can be used according to Euclid's third postulate to construct a circle using any two given points: :$(1): \quad$ the center and: :$(2): \quad$ an arbitrary point on the circumference.
Definition:Compass and Straightedge Construction
A '''compass and straightedge construction''' is a technique of drawing geometric figures using only a straightedge and a compass. The operations available are: :using the straightedge to draw a straight line determined by two given points :using the compass to draw a circle whose center is at a given point and whose radius is the distance between two given points :finding the points of intersection between straight lines and circles.
Definition:Pólya's Urn
'''Pólya's urn''' is an implementation of an urn such that, when a ball of a particular colour is taken out and examined, that ball is replaced and another of the same colour is added. {{NamedforDef|George Pólya|cat = Pólya}}