hcju/proofwikips huggingface dataset:
Definition:Derivative/Real Function/With Respect To
Let $f$ be a real function which is differentiable on an open interval $I$. Let $f$ be defined as an equation: $y = \map f x$. Then the '''derivative of $y$ with respect to $x$''' is defined as: :$\displaystyle y^\prime = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h = D_x \, \map f x$ This is frequently abbreviated as '''derivative of $y$''' '''WRT''' or '''w.r.t.''' '''$x$''', and often pronounced something like '''wurt'''. We introduce the quantity $\delta y = \map f {x + \delta x} - \map f x$. This is often referred to as '''the small change in $y$ consequent on the small change in $x$'''. Hence the motivation behind the popular and commonly-seen notation: :$\displaystyle \dfrac {\d y} {\d x} := \lim_{\delta x \mathop \to 0} \dfrac {\map f {x + \delta x} - \map f x} {\delta x} = \lim_{\delta x \mathop \to 0} \dfrac {\delta y} {\delta x}$ Hence the notation $\map {f^\prime} x = \dfrac {\d y} {\d x}$. This notation is useful and powerful, and emphasizes the concept of a derivative as being the limit of a ratio of very small changes. However, it has the disadvantage that the variable $x$ is used ambiguously: both as the point at which the derivative is calculated and as the variable with respect to which the derivation is done. For practical applications, however, this is not usually a problem.
Definition:O Notation/Big-O Notation
'''Big-O notation''' occurs in a variety of contexts. === Sequences === {{:Definition:O Notation/Big-O Notation/Sequence}} === Real Analysis === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Complex Analysis === {{:Definition:O Notation/Big-O Notation/Complex/Infinity}} === General Definition === {{:Definition:O Notation/Big-O Notation/General Definition/Infinity}}
Definition:O Notation/Little-O Notation
'''Little-O notation''' occurs in a variety of contexts. === Sequences === Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers. {{:Definition:O Notation/Little-O Notation/Sequence/Definition 1}} === Real Functions === Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\R$ containing all sufficiently large real numbers. {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} === Point Estimate === {{:Definition:O Notation/Little-O Notation/Real Point}} === Complex Functions === {{:Definition:O Notation/Little-O Notation/Complex Functions}} === Complex Point Estimate === {{:Definition:O Notation/Little-O Notation/Complex Point}} === General Definition for point estimates === {{:Definition:O Notation/Little-O Notation/General Definition/Point}}
Definition:Differential
=== Real Function === {{:Definition:Differential/Real Function/Point}} === Real-Valued Function === {{:Definition:Differential/Real-Valued Function/Point}} === Vector-Valued Function === {{:Definition:Differential/Vector-Valued Function/Point}} === Manifolds === {{:Definition:Differential/Manifolds}} === Functional === {{:Definition:Differential/Functional}}
Definition:Strictly Monotone/Sequence
Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is '''strictly monotone'''  it is either strictly increasing or strictly decreasing.
Definition:Increasing/Mapping
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: S \to T$ be a mapping. Then $\phi$ is '''increasing''' : :$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$ Note that this definition also holds if $S = T$.
Definition:Increasing/Real Function
Let $f$ be a real function. Then $f$ is '''increasing''' : :$x \le y \implies \map f x \le \map f y$
Definition:Strictly Increasing/Mapping
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 mapping. Then $\phi$ is '''strictly increasing''' if: :$\forall x, y \in S: x \prec_1 y \implies \phi \left({x}\right) \prec_2 \phi \left({y}\right)$ Note that this definition also holds if $S = T$.
Definition:Strictly Increasing/Real Function
Let $f$ be a real function. Then $f$ is '''strictly increasing''' : : $x < y \implies f \left({x}\right) < f \left({y}\right)$
Definition:Strictly Decreasing/Mapping
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 mapping. Then $\phi$ is '''strictly decreasing''' : :$\forall x, y \in S: x \prec_1 y \implies \phi \left({y}\right) \prec_2 \phi \left({x}\right)$ Note that this definition also holds if $S = T$.
Definition:Strictly Decreasing/Real Function
Let $f$ be a real function. Then $f$ is '''strictly decreasing''' : :$x < y \implies \map f y < \map f x$
Definition:Decreasing/Mapping
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a mapping. Then $\phi$ is '''decreasing''' : :$\forall x, y \in S: x \preceq_1 y \implies \map \phi y \preceq_2 \map \phi x$ Note that this definition also holds if $S = T$.
Definition:Decreasing/Real Function
Let $f$ be a real function. Then $f$ is '''decreasing''' : :$x \le y \implies \map f y \le \map f x$.
Definition:Monotone (Order Theory)/Mapping
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 mapping. Then $\phi$ is '''monotone'''  it is either increasing or decreasing. Note that this definition also holds if $S = T$.
Definition:Monotone (Order Theory)/Real Function
This definition continues to hold when $S = T = \R$. Thus, let $f$ be a real function. Then $f$ is '''monotone'''  it is either increasing or decreasing.
Definition:Strictly Monotone/Mapping
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 mapping. Then $\phi$ is '''strictly monotone'''  it is either strictly increasing or strictly decreasing. Note that this definition also holds if $S = T$.
Definition:Strictly Monotone/Real Function
Let $f: S \to \R$ be a real function, where $S \subseteq \R$. Then $f$ is '''strictly monotone'''  it is either strictly increasing or strictly decreasing.
Definition:Analytic Function/Real Numbers
Let $f$ be a real function which is smooth on the open interval $\openint a b$. Let $\xi \in \openint a b$. Let $\openint c d \subseteq \openint a b$ be an open interval such that: :$(1): \quad \xi \in \openint c d$ :$(2): \quad \displaystyle \forall x \in \openint c d: \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } x$ Then $f$ is described as being '''analytic''' at the point $\xi$. That is, a function is analytic at a point if it equals its Taylor series expansion in some interval containing that point.
Definition:Analytic Function/Complex Plane
Let $U \subset \C$ be an open set. Let $f : U \to \C$ be a complex function. Then $f$ is '''analytic''' in $U$  for every $z_0 \in U$ there exists a sequence $(a_n) : \N \to \C$ such that the series: :$\displaystyle \sum_{n\mathop = 0}^\infty a_n(z-z_0)^n$ converges to $f(z)$ in a neighborhood of $z_0$ in $U$.
Definition:Tangent/Geometry
=== Tangent Line === {{:Definition:Tangent/Geometry/Tangent Line}} === Tangent Circles === {{:Definition:Tangent/Geometry/Tangent Circles}}
Definition:Cosecant/Analysis
==== Real Function ==== {{:Definition:Cosecant/Real Function}} ==== Complex Function ==== {{:Definition:Cosecant/Complex Function}}
Definition:Propositional Expansion/Universal Quantifier
Let $\forall$ be the universal quantifier. What $\forall x: \map P x$ means is: :$\mathbf X_1$ has property $P$, and $\mathbf X_2$ has property $P$, and $\mathbf X_3$ has property $P$, and ... This translates into propositional logic as: :$\map P {\mathbf X_1} \land \map P {\mathbf X_2} \land \map P {\mathbf X_3} \land \ldots$ This expression of $\forall x$ as a conjunction is known as the '''propositional expansion''' of $\forall x$.  The '''propositional expansion''' for the universal quantifier can exist in actuality only when the number of objects in the universe is finite. If the universe is infinite, then the '''propositional expansion''' can exist only conceptually, and the universal quantifier cannot be eliminated.
Definition:Propositional Expansion/Existential Quantifier
Let $\exists$ be the existential quantifier. What $\exists x: \map P x$ means is: :At least one of $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ has property $P$. This means: :Either $\mathbf X_1$ has property $P$, or $\mathbf X_2$ has property $P$, or $\mathbf X_3$ has property $P$, or ... This translates into propositional logic as: :$\map P {\mathbf X_1} \lor \map P {\mathbf X_2} \lor \map P {\mathbf X_3} \lor \ldots$ This expression of $\exists x$ as a disjunction is known as the '''propositional expansion''' of $\exists x$. The propositional expansion for the existential quantifier can exist in actuality only when the number of objects in the universe is finite. If the universe is infinite, then the propositional expansion can exist only conceptually, and the existential quantifier cannot be eliminated.
Definition:Inverse Statement
The '''inverse''' of the conditional: : $p \implies q$ is the statement: :$\neg p \implies \neg q$
Definition:Non-reflexive Relation
$\mathcal R$ is '''non-reflexive'''  it is neither reflexive nor antireflexive.
Definition:Non-symmetric Relation
$\RR$ is '''non-symmetric'''  it is neither symmetric nor asymmetric.
Definition:Non-transitive Relation
$\RR$ is '''non-transitive'''  it is neither transitive nor antitransitive.
Definition:Restriction/Operation
Let $\left({S, \circ}\right)$ be an algebraic structure, and let $T \subseteq S$. The '''restriction of $\circ$ to $T \times T$''' is denoted $\circ {\restriction_T}$, and is defined as: :$\forall t_1, t_2 \in T: t_1 \mathbin{\circ {\restriction_T}} t_2 = t_1 \circ t_2$ The notation $\circ {\restriction_T}$ is generally used only if it is necessary to emphasise that $\circ {\restriction_T}$ is strictly different from $\circ$ (through having a different domain). When no confusion is likely to result, $\circ$ is generally used for both. Thus in this context, $\left({T, \circ {\restriction_T}}\right)$ and $\left({T, \circ}\right)$ mean the same thing.
Definition:Restriction/Relation
Let $\RR$ be a relation on $S \times T$. Let $X \subseteq S$, $Y \subseteq T$. The '''restriction of $\RR$ to $X \times Y$''' is the relation on $X \times Y$ defined as: :$\RR {\restriction_{X \times Y} }: = \RR \cap \paren {X \times Y}$ If $Y = T$, then we simply call this the '''restriction of $\RR$ to $X$''', and denote it as $\RR {\restriction_X}$.
Definition:Restriction/Mapping
Let $f: S \to T$ be a mapping. Let $X \subseteq S$. Let $f \sqbrk X \subseteq Y \subseteq T$. The '''restriction of $f$ to $X \times Y$''' is the mapping $f {\restriction_{X \times Y} }: X \to Y$ defined as: :$f {\restriction_{X \times Y} } = f \cap \paren {X \times Y}$ If $Y = T$, then we simply call this the '''restriction of $f$ to $X$''', and denote it as $f {\restriction_X}$.
Definition:Identity (Abstract Algebra)/Left Identity
An element $e_L \in S$ is called a '''left identity''' : :$\forall x \in S: e_L \circ x = x$
Definition:Identity (Abstract Algebra)/Right Identity
An element $e_R \in S$ is called a '''right identity''' : :$\forall x \in S: x \circ e_R = x$
Definition:Identity (Abstract Algebra)/Two-Sided Identity
An element $e \in S$ is called an '''identity (element)'''  it is both a left identity and a right identity: :$\forall x \in S: x \circ e = x = e \circ x$ In Identity is Unique it is established that an identity element, if it exists, is unique within $\struct {S, \circ}$. Thus it is justified to refer to it as '''the''' identity (of a given algebraic structure). This identity is often denoted $e_S$, or $e$ if it is clearly understood what structure is being discussed.
Definition:Inverse (Abstract Algebra)/Left Inverse
An element $x_L \in S$ is called a '''left inverse''' of $x$ : :$x_L \circ x = e_S$
Definition:Inverse (Abstract Algebra)/Right Inverse
An element $x_R \in S$ is called a '''right inverse''' of $x$ : :$x \circ x_R = e_S$
Definition:Inverse (Abstract Algebra)/Inverse
The element $y$ is an '''inverse of $x$''' : :$y \circ x = e_S = x \circ y$ that is,  $y$ is both: :a left inverse of $x$ and: :a right inverse of $x$.
Definition:Idempotence/Mapping
Let $f: S \to S$ be a mapping. Then $f$ is '''idempotent''' : :$\forall x \in S: f \left({f \left({x}\right)}\right) = f \left({x}\right)$ That is,  applying the same mapping a second time to an argument gives the same result as applying it once. And of course, that means the same as applying it as many times as you want. The condition for '''idempotence''' can also be written: :$f \circ f = f$ where $\circ$ denotes composition of mappings.
Definition:Idempotence/Operation
Let $\circ: S \times S \to S$ be a binary operation on $S$. If ''all'' the elements of $S$ are idempotent under $\circ$, then the term can be applied to the operation itself: The binary operation $\circ$ is '''idempotent''' : : $\forall x \in S: x \circ x = x$
Definition:Idempotence/Element
Let $\circ: S \times S \to S$ be a binary operation on $S$. Let $x \in S$ have the property that $x \circ x = x$. Then $x \in S$ is described as '''idempotent under the operation $\circ$'''.
Definition:Unique
Suppose $A$ and $B$ are two objects whose definition is in terms of a given set of properties. If it can be demonstrated that, in order for both $A$ and $B$ to fulfil those properties, it is necessary for $A$ to be equal to $B$, then $A$ (and indeed $B$) is '''unique'''. Equivalently, there is '''one and only one''', or '''exactly one''', such object. Thus, intuitively, an object is '''unique''' if there is precisely one such object. === Unique Existential Quantifier === In the language of predicate logic, '''uniqueness''' can be defined as follows: Let $\map P x$ be a propositional function and let $x$ and $y$ be objects.  {{:Definition:Existential Quantifier/Unique}}
Definition:Paradox
A '''paradox''' is a statement or group of statements that leads to one of the following: * a contradiction * a situation which defies intuition * a result that is merely "puzzling".
Definition:Regular Representations/Left Regular Representation
The mapping $\lambda_a: S \to S$ is defined as: :$\forall x \in S: \map {\lambda_a} x = a \circ x$ This is known as the '''left regular representation of $\struct {S, \circ}$ with respect to $a$'''.
Definition:Regular Representations/Right Regular Representation
The mapping $\rho_a: S \to S$ is defined as: :$\forall x \in S: \map {\rho_a} x = x \circ a$ This is known as the '''right regular representation of $\struct {S, \circ}$ with respect to $a$'''.
Definition:Centralizer/Group Element
Let $\struct {G, \circ}$ be a group. Let $a \in \struct {G, \circ}$. The '''centralizer of $a$ (in $G$)''' is defined as: :$\map {C_G} a = \set {x \in G: x \circ a = a \circ x}$ That is, the centralizer of $a$ is the set of elements of $G$ which commute with $a$.
Definition:Centralizer/Subgroup
Let $\struct {G, \circ}$ be a group. Let $H \le \struct {G, \circ}$. The '''centralizer of $H$ (in $G$)''' is the set of elements of $G$ which commute with all $h \in H$: :$\map {C_G} H = \set {g \in G: \forall h \in H: g \circ h = h \circ g}$
Definition:Centralizer/Ring Subset
Let $S$ be a subset of a ring $\struct {R, +, \circ}$. The '''centralizer of $S$ in $R$''' is defined as: :$\map {C_R} S = \set {x \in R: \forall s \in S: s \circ x = x \circ s}$ That is, the '''centralizer of $S$''' is the set of elements of $R$ which commute with all elements of $S$.
Definition:Center (Abstract Algebra)
=== Group === {{:Definition:Center (Abstract Algebra)/Group}} === Ring === {{:Definition:Center (Abstract Algebra)/Ring}}
Definition:Conjugate (Group Theory)/Element
Let $\struct {G, \circ}$ be a group. === Definition 1 === {{:Definition:Conjugate (Group Theory)/Element/Definition 1}} === Definition 2 === {{:Definition:Conjugate (Group Theory)/Element/Definition 2}}
Definition:Conjugate (Group Theory)/Subset
Let $S \subseteq G, a \in G$. Then the '''$G$-conjugate of $S$ by $a$''' is: :$S^a := \set {y \in G: \exists x \in S: y = a \circ x \circ a^{-1} } = a \circ S \circ a^{-1}$ That is, $S^a$ is the set of all elements of $G$ that are the conjugates of elements of $S$ by $a$. When $G$ is the only group under consideration, we usually just refer to the '''conjugate of $S$ by $a$'''.
Definition:Algebra (Abstract Algebra)
In the context of abstract algebra, in particular ring theory and linear algebra, the following varieties of '''algebra''' exist: * Definition:Boolean Algebra * Definition:Algebra over Ring: an $R$-module $G_R$ over a commutative ring $R$ with a bilinear mapping $\oplus: G^2 \to G$. * Definition:Algebra over Field: a vector space $G_F$ over a field $F$ with a bilinear mapping $\oplus: G^2 \to G$. * Definition:Real Algebra: an algebra over a field where the field in question is the field of real numbers $\R$. * Definition:Division Algebra: an algebra over a field $\left({A_F, \oplus}\right)$ such that $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$. * Definition:Associative Algebra: an algebra over a ring in which the bilinear mapping $\oplus$ is associative.  * Definition:Unitary Algebra, also known as a Unital Algebra: an algebra over a ring $\left({A_R, \oplus}\right)$ in which there exists an identity element, that is, a '''unit''', usually denoted $1$, for $\oplus$. * Definition:Unitary Division Algebra: a division algebra $\left({A_F, \oplus}\right)$ in which there exists an identity element, that is, a '''unit''', usually denoted $1$, for $\oplus$. * Definition:Graded Algebra: an algebra over a ring where the ring has a gradation, that is, is a graded ring. * Definition:Filtered Algebra: an algebra over a field which has a sequence of subalgebras which constitute a gradation. * Definition:Quadratic Algebra: a filtered algebra whose generator consists of degree one elements, with defining relations of degree 2. * Definition:Lie Algebra
Definition:Ideal of Ring/Left Ideal
$J$ is a '''left ideal of $R$''' : :$\forall j \in J: \forall r \in R: r \circ j \in J$ that is, : :$\forall r \in R: r \circ J \subseteq J$
Definition:Ideal of Ring/Right Ideal
$J$ is a '''right ideal of $R$''' : :$\forall j \in J: \forall r \in R: j \circ r \in J$ that is, : :$\forall r \in R: J \circ r \subseteq J$
Definition:Divisor (Algebra)/Ring with Unity
Let $\struct {R, +, \circ}$ be an ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \in D$. We define the term '''$x$ divides $y$ in $R$''' as follows: :$x \mathrel {\divides_R} y \iff \exists t \in R: y = t \circ x$ When no ambiguity results, the subscript is usually dropped, and '''$x$ divides $y$ in $R$''' is just written $x \divides y$.
Definition:Divisor (Algebra)/Integer
Let $\struct {\Z, +, \times}$ be the ring of integers. Let $x, y \in \Z$. Then '''$x$ divides $y$''' is defined as: :$x \divides y \iff \exists t \in \Z: y = t \times x$
Definition:Divisor (Algebra)/Factorization
Let $x, y \in D$ where $\struct {D, +, \times}$ is an integral domain. Let $x$ be a divisor of $y$. Then by definition it is possible to find some $t \in D$ such that $y = t \times x$. The act of breaking down such a $y$ into the product $t \circ x$ is called '''factorization'''.
Definition:Divisor (Algebra)/Integer/Aliquot Part
An '''aliquot part''' of an integer $n$ is a divisor of $n$ which is strictly less than $n$.
Definition:Divisor (Algebra)/Integer/Aliquant Part
An '''aliquant part''' of an integer $n$ is a positive integer which is less than $n$ but is not a divisor of $n$.
Definition:Common Divisor/Integral Domain
Let $\struct {D, +, \times}$ be an integral domain. Let $S \subseteq D$ be a finite subset of $D$. Let $c \in D$ such that $c$ divides all the elements of $S$, that is: :$\forall x \in S: c \divides x$ Then $c$ is a '''common divisor''' (or '''common factor''') of all the elements in $S$.
Definition:Common Divisor/Integers
Let $S$ be a finite set of integers, that is: :$S = \set {x_1, x_2, \ldots, x_n: \forall k \in \N^*_n: x_k \in \Z}$ Let $c \in \Z$ such that $c$ divides all the elements of $S$, that is: :$\forall x \in S: c \divides x$ Then $c$ is a '''common divisor''' (or '''common factor''') of all the elements in $S$.
Definition:Greatest Common Divisor/Integral Domain
Let $\struct {D, +, \times}$ be an integral domain whose zero is $0$. Let $a, b \in D: a \ne 0 \lor b \ne 0$. Let $d \divides a$ denote that $d$ is a divisor of $a$. Let $d \in D$ have the following properties: : $(1): \quad d \divides a \land d \divides b$ : $(2): \quad c \divides a \land c \divides b \implies c \divides d$ Then $d$ is called '''a greatest common divisor of $a$ and $b$''' (abbreviated '''GCD''' or '''gcd''') and denoted $\gcd \set {a, b}$. That is, in the integral domain $D$, $d$ is the GCD of $a$ and $b$ : : $d$ is a common divisor of $a$ and $b$ : Any other common divisor of $a$ and $b$ also divides $d$. We see that, trivially: :$\gcd \set {a, b} = \gcd \set {b, a}$ so the set notation is justified.
Definition:Greatest Common Divisor/Integers
Let $a, b \in \Z: a \ne 0 \lor b \ne 0$. === Definition 1 === {{:Definition:Greatest Common Divisor/Integers/Definition 1}} === Definition 2 === {{:Definition:Greatest Common Divisor/Integers/Definition 2}} This is denoted $\gcd \set {a, b}$.
Definition:Compact Space/Topology
=== Definition 1 === {{:Definition:Compact Space/Topology/Definition 1}} === Definition 2 === {{:Definition:Compact Space/Topology/Definition 2}} === Definition 3 === {{:Definition:Compact Space/Topology/Definition 3}} === Definition 4 === {{:Definition:Compact Space/Topology/Definition 4}} === Definition 5 === {{:Definition:Compact Space/Topology/Definition 5}}
Definition:Compact Space/Real Analysis
Let $\R$ be the real number space considered as a topological space under the Euclidean topology. Let $H \subseteq \R$. Then $H$ is '''compact in $\R$'''  $H$ is closed and bounded.
Definition:Ordered Tree
Let $T$ be a rooted tree whose root is $v$. Let $O$ be an orchard whose trees are subtrees of $v$. {{explain|Cannot find a reference to an "orchard" in this context -- probably means "ordered forest".}} Then $T$ is an '''ordered tree'''.
Definition:Closed Ball
{{:Definition:Closed Ball/Metric Space}}
Definition:Average Value of Function
Let $f$ be an integrable function on some closed interval $\closedint a b$. The '''average value of $f$''' (or '''mean value of $f$''') on $\closedint a b$ is defined as: :$\displaystyle \frac 1 {b - a} \int_a^b \map f x \rd x$
Definition:Unit Circle
The '''unit circle''' is a circle with radius equal to $1$.
Definition:Cantor Set/Limit of Intersections
Define, for $n \in \N$, subsequently: :$\map k n := \dfrac {3^n - 1} 2$ :$\displaystyle A_n := \bigcup_{i \mathop = 1}^{\map k n} \openint {\frac {2 i - 1} {3^n} } {\frac {2 i} {3^n} }$ Since $3^n$ is always odd, $\map k n$ is always an integer, and hence the union will always be perfectly defined. Consider the closed interval $\closedint 0 1 \subset \R$. Define: :$\CC_n := \closedint 0 1 \setminus A_n$ The '''Cantor set''' $\CC$ is defined as: :$\displaystyle \CC = \bigcap_{n \mathop = 1}^\infty \CC_n$
Definition:Ternary Notation
'''Ternary notation''' is the technique of expressing numbers in base $3$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 3^j$ where: :$\forall j \in \Z: r_j \in \set {0, 1, 2}$
Definition:Cantor Set/Ternary Representation
Consider the closed interval $\closedint 0 1 \subset \R$. The '''Cantor set''' $\CC$ consists of all the points in $\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$. From Representation of Ternary Expansions, if any number has two different ternary representations, for example: :$\dfrac 1 3 = 0.10000 \ldots = 0.02222$ then at most one of these can be written without any $1$'s in it. Therefore this representation of points of $\CC$ is unique.