hcju/proofwikips huggingface dataset:
Definition:Algorithm/Action
In an algorithm, an '''action''' is part of a step of that algorithm. It is an operation which changes the state of the algorithm in some manner.
Definition:Algorithm/Condition
In an algorithm, a '''condition''' is part of a step of that algorithm. It is an operation which determines the outcome of a decision as to which step to execute next,
Definition:Element is Way Below
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$. Then $x$ '''is way below''' $y$, denoted $x \ll y$,  :for every directed subset $D$ of $S$ ::if $D$ admits a supremum and $y \preceq \sup D$ ::then there exists $d \in D$ such that $x \preceq d$
Definition:Programming Language
A '''programming language''' is a special purpose language used to implement algorithms for a computer to execute.
Definition:Program
A '''program''' is an algorithm which has been written for a computer to execute.
Definition:Algorithm/Formal Specification
An '''algorithm''' can be implemented formally as a computational method $\left({Q, I, \Omega, f}\right)$ as follows: Let $A$ be a finite set of symbols. Let $A^*$ be the set of all collations on $A$: :$\left\{ {x_1 x_2 \cdots x_n: n \ge 0, \forall j: 1 \le j \le n: x_j \in A}\right\}$ The states of the computation are encoded so as to be represented by elements of $A^*$. Let $N \in \Z_{\ge 0}$. Let $Q$ be the set of all ordered pairs $\left({\sigma, j}\right)$ where $\sigma \in A^*, j \in \Z: 0 \le j \le N$. Let $I \subseteq Q$ such that $j = 0$. Let $\Omega \subseteq Q$ such that $j = N$. Let $\theta, \sigma \in A^*$. Then $\theta$ '''occurs in $\sigma$'''  $\sigma$ has the form: :$\alpha \theta \omega$ where $\alpha, \omega \in A^*$. Let $f$ be a mapping of the following type: :$f \left({\left({\sigma, j}\right)}\right) = \begin{cases}\left({\sigma, a_j}\right) : & \sigma_j \text { does not occur in } \sigma \\ \left({\alpha \theta_j \omega, b_j}\right) : & \alpha \text { is the shortest element of $A^*$ such that } \sigma  = \alpha \theta_j \omega\end{cases}$ :$f \left({\left({\sigma, N}\right)}\right) = \left({\sigma, N}\right)$
Definition:Golden Mean/Geometrical Interpretation
Let $\Box ADEB$ be a square. Let $\Box ADFC$ be a rectangle such that: :$AC : AD = AD : BC$ where $AC : AD$ denotes the ratio of $AC$ to $AD$. :200px Then if you remove $\Box ADEB$ from $\Box ADFC$, the sides of the remaining rectangle have the same ratio as the sides of the original one. Thus if $AC = \phi$ and $AD = 1$ we see that this reduces to: :$\phi : 1 = 1 : \phi - 1$ where $\phi$ is the golden mean.
Definition:Golden Mean/Definition 1
Let a line segment $AB$ be divided at $C$ such that: :$AB : AC = AC : BC$ Then the '''golden mean''' $\phi$ is defined as: :$\phi := \dfrac {AB} {AC}$
Definition:Golden Mean/One Minus Golden Mean
The number: :$1 - \phi$ is often denoted $\hat \phi$.
Definition:Golden Mean/Definition 2
The '''golden mean''' is the unique positive real number $\phi$ satisfying: :$\phi = \dfrac {1 + \sqrt 5} 2$
Definition:Golden Mean/Definition 3
The '''golden mean''' is the unique positive real number $\phi$ satisfying: :$\phi = \dfrac 1 {\phi - 1}$
Definition:Basis Expansion/Positive Real Numbers
Let $x \in \R$ be a real number such that $x \ge 0$. Let $b \in \N: b \ge 2$. Let us define the recursive sequence: :$\forall n \in \N: n \ge 1: \sequence {f_n} = \begin {cases} b \paren {x - \floor x} & : n = 1 \\ b \paren {f_{n - 1} - \floor {f_{n - 1} } } & : n > 1 \end{cases}$ Then we define: :$\forall n \in \N: n \ge 1: \sequence {d_n} = \floor {f_n}$ It follows from the method of construction and the definition of the floor function that: :$\forall n: 0 \le f_n < b$ and hence $\forall n: 0 \le d_n \le b - 1$ :$\forall n: f_n = 0 \implies f_{n + 1} = 0$ and hence $d_{n + 1} = 0$. Hence we can express $x = \floor x + \displaystyle \sum_{j \mathop \ge 1} \frac {d_j} {b^j}$ as: :$\sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ where: :$s = \floor x$ :it is not the case that there exists $m \in \N$ such that $d_M = b - 1$ for all $M \ge m$. (That is, the sequence of digits does not end with an infinite sequence of $b - 1$.) This is called the '''expansion of $x$ in base $b$'''. The generic term for such an expansion is a '''basis expansion'''. It follows from the Division Theorem that for a given $b$ and $x$ this expansion is unique.
Definition:Basis Expansion/Negative Real Numbers
Let $x \in \R: x < 0$. We take the absolute value $y$ of $x$, that is: :$y = \size x$ Then we take the '''expansion of $y$ in base $b$''': :$\size {s . d_1 d_2 d_3 \ldots}_b$ where $s = \floor y$. Finally, the '''expansion of $x$ in base $b$''' is defined as: :$-\sqbrk {s . d_1 d_2 d_3 \ldots}_b$
Definition:Less Than (Real Numbers)
Let $\R_{>0}$ denote the set of  strictly positive real numbers. Let $x, y \in \R$. Then we write $x < y$ : :$ y - x \in \R_{>0}$ and we say that $x$ is '''less than''' $y$.
Definition:Strictly Positive/Real Number/Definition 2
The '''strictly positive real numbers''', written $R_{>0}$, is the subset of $\R$ that satisfies the following: {{begin-axiom}} {{axiom | n = \R_{>0} 1         | lc= Closure under addition         | q = \forall x, y \in \R_{>0}         | m = x + y \in \R_{>0} }} {{axiom | n = \R_{>0} 2         | lc= Closure under multiplication         | q = \forall x, y \in \R_{>0}         | m = xy \in \R_{>0} }} {{axiom | n = \R_{>0} 3         | lc=  Trichotomy         | q = \forall x \in \R         | m = x \in \R_{>0} \lor x = 0 \lor -x \in \R_{>0} }} {{end-axiom}}
Definition:Strictly Positive/Real Number/Definition 1
The '''strictly positive real numbers''' are the set defined as: :$\R_{>0} := \set {x \in \R: x > 0}$ That is, all the real numbers that are strictly greater than zero.
Definition:Decimal Expansion/Decimal Point
The dot that separates the integer part from the fractional part of $x$ is called the '''decimal point'''. That is, it is the radix point when used specifically for a base $10$ representation.
Definition:Number Base/Integers
Let $n \in \Z$ be an integer. Let $b$ be any integer such that $b > 1$. By the Basis Representation Theorem, $n$ can be expressed uniquely in the form: :$\displaystyle n = \sum_{j \mathop = 0}^m r_j b^j$ where: :$m$ is such that $b^m \le n < b^{m + 1}$ :all the $r_j$ are such that $0 \le r_j < b$. <noinclude> {{MissingLinks|The bounds on $n$ are not stated as part of the Basis Representation Theorem. Is there some other link to these bounds?}}</noinclude> <noinclude>{{Improve|The definition is incomplete as the Basis Representation Theorem is only stated for strictly positive integers}}</noinclude> The number $b$ is known as the '''number base''' to which $n$ is represented. $n$ is thus described as being '''(written) in base $b$'''. Thus we can write $\displaystyle n = \sum_{j \mathop = 0}^m {r_j b^j}$ as: :$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$ or, if the context is clear: :${r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$
Definition:Number Base/Real Numbers
Let $x \in \R$ be a real number such that $x \ge 0$. Let $b \in \N: b \ge 2$. See the definition of Basis Expansion for how we can express $x$ in the form: :$x = \sqbrk {s \cdotp d_1 d_2 d_3 \ldots}_b$ Then we express $m$ as for integers, and arrive at: :$x = \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots}_b$ or, if the context is clear: :$r_m r_{m - 1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots_b$
Definition:Number Base/Integer Part
In the basis expansion: :$x = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0 . d_1 d_2 d_3 \ldots}\right]_b$ the part $r_m r_{m-1} \ldots r_2 r_1 r_0$ is known as the '''integer part'''.
Definition:Number Base/Fractional Part
In the basis expansion: :$x = \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 . d_1 d_2 d_3 \ldots}_b$ the part $.d_1 d_2 d_3 \ldots$ is known as the '''fractional part'''.
Definition:Number Base/Radix Point
In the basis expansion: :$x = \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0 \cdotp d_1 d_2 d_3 \ldots}_b$ the dot that separates the integer part from the fractional part is called the '''radix point'''.
Definition:Natural Logarithm/Positive Real/Definition 3
Let $x \in \R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$ \displaystyle \ln x := \lim_{n \mathop \to \infty} n \paren {\sqrt [n] x - 1}$
Definition:General Logarithm/Binary
Logarithms base $2$ are becoming increasingly important in computer science. They are often referred to as '''binary logarithms'''.
Definition:Pointwise Equicontinuous
Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces. Let $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ be a family of mappings $f_i: X \to Y$. Then $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ is said to be '''pointwise equicontinuous''' at $x_0 \in A$ : :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall i \in I: \forall x \in A: d \left({x, x_0}\right) < \delta \implies \rho \left({f_i \left({x}\right), f_i \left({x_0}\right)}\right) < \epsilon$
Definition:Uniformly Equicontinuous
Let $X = \left({A, d}\right)$ and $Y = \left({B, \rho}\right)$ be metric spaces. Let $\left\langle{f_i}\right\rangle_{i \mathop \in I} $ be a family of mappings $f_i: X \to Y$. Then $\left\langle{f_i}\right\rangle_{i \mathop \in I}$ is said to be '''uniformly equicontinuous''' on $S \subseteq A$ : :$\forall \epsilon \in \R_{>0} : \exists \delta \in \R_{>0}: \forall i \in I: \forall x, y \in S : d \left({x, y}\right) < \delta \implies \rho \left({f_i \left({x}\right), f_i \left({y}\right) }\right) < \epsilon$
Definition:Discontinuity of the First Kind
Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a real function. Let $f$ be discontinuous at some point $c \in X$. Then $c$ is called a '''discontinuity of the first kind''' of $f$ : :$\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ exist where $\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ denote the limit from the left and limit from the right at $c$ respectively.
Definition:Jump Discontinuity
Let $X$ be an open subset of $\R$. Let $f: X \to Y$ be a real function. Let $f$ be discontinuous at some point $c \in X$. Then $c$ is called a '''jump discontinuity''' of $f$ : :$\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ exist and are not equal where $\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ denote the limit from the left and limit from the right at $c$ respectively. === Jump === {{:Definition:Jump Discontinuity/Jump}}
Definition:Summation/Indexed
The composite is called the '''summation''' of $\tuple {a_1, a_2, \ldots, a_n}$, and is written: :$\displaystyle \sum_{j \mathop = 1}^n a_j = \paren {a_1 + a_2 + \cdots + a_n}$
Definition:Summation/Inequality
The '''summation''' of $\left({a_1, a_2, \ldots, a_n}\right)$ can be written: :$\displaystyle \sum_{1 \mathop \le j \mathop \le n} a_j = \left({a_1 + a_2 + \cdots + a_n}\right)$
Definition:Summation/Propositional Function
Let $R \left({j}\right)$ be a propositional function of $j$. Then we can write the '''summation''' as: :$\displaystyle \sum_{R \left({j}\right)} a_j = \text{ The sum of all $a_j$ such that $R \left({j}\right)$ holds}$. If more than one propositional function is written under the summation sign, they must ''all'' hold.
Definition:Summation/Index Variable
Consider the '''summation''', in either of the three forms: :$\displaystyle \sum_{j \mathop = 1}^n a_j \qquad \sum_{1 \mathop \le j \mathop \le n} a_j \qquad \sum_{\map R j} a_j$ The variable $j$, an example of a bound variable, is known as the '''index variable''' of the summation.
Definition:Summation/Infinite
Let an infinite number of values of $j$ satisfy the propositional function $R \left({j}\right)$. Then the precise meaning of $\displaystyle \sum_{R \left({j}\right)} a_j$ is: :$\displaystyle \sum_{R \left({j}\right)} a_j = \left({\lim_{n \mathop \to \infty} \sum_{\substack {R \left({j}\right) \\ -n \mathop \le j \mathop < 0}} a_j}\right) + \left({\lim_{n \mathop \to \infty} \sum_{\substack {R \left({j}\right) \\ 0 \mathop \le j \mathop \le n} } a_j}\right)$ provided that both limits exist. If either limit ''does'' fail to exist, then the '''infinite summation''' does not exist.
Definition:Real Interval/Bounded
Let $I$ be an interval. Let $I$ be either open, half-open or closed. Then $I$ is said to be a '''bounded (real) interval'''.
Definition:Fiber of Truth
Let $P: X \to \set {\mathrm T, \mathrm F}$ be a propositional function defined on a domain $X$. The '''fiber of truth (under $P$)''' is the preimage, or '''fiber''', of $\mathrm T$ under $P$: :$\set {x \in X: \map P x = \mathrm T}$ That is, the elements of $X$ whose image under $P$ is $\mathrm T$.
Definition:Way Below Closure
Let $\struct {S, \preceq}$ be an ordered set. Let $x \in S$. The '''way below closure''' of $x$, denoted by $x^\ll$, is defined by :$x^\ll := \set {y \in S: y \ll x}$ where $y \ll x$ denotes that $y$ is way below $x$.
Definition:Axiom of Approximation
Let $\left({S, \preceq}\right)$ be an ordered set. '''Axiom of approximation''' says  :$\forall x \in S: x = \sup \left({x^\ll}\right)$ where $x^\ll$ denotes the way below closure of $x$.
Definition:Continuous Ordered Set
Let $\left({S, \preceq}\right)$ be an ordered set. {{explain|$x^\ll$}} Then $\left({S, \preceq}\right)$ is '''continuous'''  :(for all elements $x$ of $S$: $x^\ll$ is directed) and :$\left({S, \preceq}\right)$ is up-complete and satisfies axiom of approximation.
Definition:Euler's Number/Base of Exponential
There is a number $x \in \R$ such that: :$\displaystyle \lim_{h \to 0} \frac{ x^{h} - 1 }{ h } = 1$ This number is called '''Euler's Number''' and is denoted $e$.
Definition:Midpoint-Convex
Let $f$ be a real function defined on a real interval $I$. $f$ is '''midpoint-convex''' : :$\forall x, y \in I: f \left({\dfrac {x + y} 2}\right) \le \dfrac {f \left({x}\right) + f \left({y}\right)} 2$
Definition:Midpoint-Concave
Let $f$ be a real function defined on a real interval $I$. $f$ is '''midpoint-concave''' : :$\forall x, y \in I: f \left({\dfrac {x + y} 2}\right) \ge \dfrac {f \left({x}\right) + f \left({y}\right)} 2$
Definition:Strictly Midpoint-Convex
Let $f$ be a real function defined on a real interval $I$. $f$ is '''strictly midpoint-convex''' : :$\forall x, y \in I : f \left({\dfrac {x + y} 2}\right) < \dfrac {f \left({x}\right) + f \left({y}\right)} 2$
Definition:Strictly Midpoint-Concave
Let $f$ be a real function defined on a real interval $I$. $f$ is '''strictly midpoint-concave''' : :$\forall x, y \in I : f \left({\dfrac {x + y} 2}\right) > \dfrac {f \left({x}\right) + f \left({y}\right)} 2$
Definition:Summation/Propositional Function/Iverson's Convention
Let $\displaystyle \sum_{R \left({j}\right)} a_j$ be the summation over all $a_j$ such that $j$ satisfies $R$. This can also be expressed: :$\displaystyle \sum_{j \mathop \in \Z} a_j \left[{R \left({j}\right)}\right]$ where $\left[{R \left({j}\right)}\right]$ is Iverson's convention.
Definition:Product Notation (Algebra)/Index
The composite is called the '''product''' of $\tuple {a_1, a_2, \ldots, a_n}$, and is written: :$\displaystyle \prod_{j \mathop = 1}^n a_j = \paren {a_1 \times a_2 \times \cdots \times a_n}$
Definition:Product Notation (Algebra)/Inequality
The '''product''' of $\left({a_1, a_2, \ldots, a_n}\right)$ can be written: :$\displaystyle \prod_{1 \mathop \le j \mathop \le n} a_j = \left({a_1 \times a_2 \times \cdots \times a_n}\right)$
Definition:Product Notation (Algebra)/Propositional Function
Let $R \left({j}\right)$ be a propositional function of $j$. Then we can write: :$\displaystyle \prod_{R \left({j}\right)} a_j = \text{ The product of all } a_j \text{ such that } R \left({j}\right) \text{ holds}$. If more than one propositional function is written under the product sign, they must ''all'' hold.
Definition:Product Notation (Algebra)/Infinite
Let an infinite number of values of $j$ satisfy the propositional function $\map R j$. Then the precise meaning of $\displaystyle \prod_{\map R j} a_j$ is: :$\displaystyle \prod_{\map R j} a_j = \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ -n \mathop \le j \mathop < 0} } a_j} \times \paren {\lim_{n \mathop \to \infty} \prod_{\substack {\map R j \\ 0 \mathop \le j \mathop \le n} } a_j}$ provided that both limits exist. If either limit ''does'' fail to exist, then the '''infinite product''' does not exist.
Definition:Product Notation (Algebra)/Index Variable
Consider the '''product''', in either of the three forms: :$\displaystyle \prod_{j \mathop = 1}^n a_j \qquad \prod_{1 \mathop \le j \mathop \le n} a_j \qquad \prod_{R \left({j}\right)} a_j$ The variable $j$, an example of a bound variable, is known as the '''index variable''' of the product.
Definition:Summation/Inequality/Multiple Indices
Let $\displaystyle \sum_{0 \mathop \le j \mathop \le n} a_j$ denote the '''summation''' of $\left({a_0, a_1, a_2, \ldots, a_n}\right)$. Summands with multiple indices can be denoted by propositional functions in several variables, for example: :$\displaystyle \sum_{0 \mathop \le i \mathop \le n} \left({\sum_{0 \mathop \le j \mathop \le n} a_{i j} }\right) = \sum_{0 \mathop \le i, j \mathop \le n} a_{i j}$ :$\displaystyle \sum_{0 \mathop \le i \mathop \le n} \left({\sum_{0 \mathop \le j \mathop \le i} a_{i j} }\right) = \sum_{0 \mathop \le j \mathop \le i \mathop \le n} a_{i j}$
Definition:Telescoping Series
A '''telescoping series''' is a series whose partial sums eventually only have a fixed number of terms after cancellation through algebraic manipulation.
Definition:Norm of Subdivision
Let $\closedint a b$ be a closed interval of the set of real numbers $\R$. Let $\Delta = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a finite subdivision of $\closedint a b$. Then the '''norm''' of $\Delta$ is defined as: :$\max \set {x_1 - x_0, x_2 - x_1, \ldots, x_n - x_{n - 1} }$ and is denoted $\norm \Delta$.
Definition:Continuous Extension/Real Function
Let $A$, $B \subseteq \R$ be subsets of the real numbers such that $A \subseteq B$. Let $f: A \to \R$ and $g: B \to \R$ be continuous real functions. Then $g$ is a '''continuous extension''' of $f$ : :$\forall x \in A : f \left({x}\right) = g \left({x}\right)$
Definition:Definite Integral/Riemann
Let $\Delta$ be a finite subdivision of $\closedint a b$, $\Delta = \set {x_0, \ldots, x_n}$, $x_0 = a$ and $x_n = b$. Let there for $\Delta$ be a corresponding sequence $C$ of sample points $c_i$, $C = \tuple {c_1, \ldots, c_n}$, where $c_i \in \closedint {x_{i - 1} } {x_i}$ for every $i \in \set {1, \ldots, n}$. Let $\map S {f; \Delta, C}$ denote the Riemann sum of $f$ for the subdivision $\Delta$ and the sample point sequence $C$. Then $f$ is said to be '''(properly) Riemann integrable''' on $\closedint a b$ : :$\exists L \in \R: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall$ finite subdivisions $\Delta$ of $\closedint a b: \forall$ sample point sequences $C$ of $\Delta: \norm \Delta < \delta \implies \size {\map S {f; \Delta, C} - L} < \epsilon$ where $\norm \Delta$ denotes the norm of $\Delta$. The real number $L$ is called the '''Riemann integral''' of $f$ over $\closedint a b$ and is denoted: :$\displaystyle \int_a^b \map f x \rd x$
Definition:Definite Integral/Darboux
Let $f$ be bounded on $\closedint a b$. Suppose that: :$\displaystyle \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$ where $\displaystyle \underline {\int_a^b}$ and $\displaystyle \overline {\int_a^b}$ denote the lower integral and upper integral, respectively. Then the '''definite (Darboux) integral of $f$ over $\closedint a b$''' is defined as: :$\displaystyle \int_a^b \map f x \rd x = \underline {\int_a^b} \map f x \rd x = \overline {\int_a^b} \map f x \rd x$ $f$ is formally defined as '''(properly) integrable over $\closedint a b$ in the sense of Darboux''', or '''(properly) Darboux integrable over $\closedint a b$'''. More usually (and informally), we say: :'''$f$ is (Darboux) integrable over $\closedint a b$.'''
Definition:Auxiliary Relation
Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. Let $\mathcal R \subseteq S \times S$ be a relation on $S$. Then $\mathcal R$ is an '''auxiliary relation'''  {{begin-axiom}} {{axiom | n = 1         | q = \forall x, y \in S         | m = \left({x, y}\right) \in \mathcal R \implies x \preceq y }} {{axiom | n = 2         | q = \forall x, y, z, u \in S         | m = x \preceq y \land \left({y, z}\right) \in \mathcal R \land z \preceq u \implies \left({x, u}\right) \in \mathcal R }} {{axiom | n = 3         | q = \forall x, y, z \in S         | m = \left({x, z}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x \vee y, z}\right) \in \mathcal R }} {{axiom | n = 4         | q = \forall x \in S         | m = \left({\bot, x}\right) \in \mathcal R }} {{end-axiom}}
Definition:Sign of Ordered Tuple
Let $n \in \N$ be a natural number such that $n > 1$. Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers. Let $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ be the product of differences of $\tuple {x_1, x_2, \ldots, x_n}$: :$\displaystyle \map {\Delta_n} {x_1, x_2, \ldots, x_n} = \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$ The '''sign of $\tuple {x_1, x_2, \ldots, x_n}$''' is defined and denoted as: :$\map \epsilon {x_1, x_2, \ldots, x_n} := \map \sgn {\Delta_n}$ where $\sgn$ denotes the signum function. That is: :$\displaystyle \map \epsilon {x_1, x_2, \ldots, x_n} := \map \sgn {\prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j} }$ where: :$\map \sgn \pi := \sqbrk {x > 0} - \sqbrk {x < 0}$ :$\sqbrk {x > 0}$ etc. is Iverson's convention.
Definition:Floor Function/Definition 1
The '''floor function of $x$''' is defined as the supremum of the set of integers no greater than $x$: :$\floor x := \sup \set {m \in \Z: m \le x}$ where $\le$ is the usual ordering on the real numbers.
Definition:Floor Function/Notation
Before around $1970$, the usual symbol for the '''floor function''' was $\sqbrk x$. The notation $\floor x$ for the '''floor function''' is a relatively recent development. Compare the notation for the corresponding ceiling function, $\ceiling x$, which in the context of discrete mathematics is used almost as much. Some sources use $\map {\mathrm {fl} } x$ for the '''floor function''' of $x$. However, this notation is clumsy, and will not be used on .
Definition:Floor Function/Definition 3
The '''floor function of $x$''' is the unique integer $\floor x$ such that: :$\floor x \le x < \floor x + 1$
Definition:Ceiling Function/Notation
The notation $\ceiling x$ for the '''ceiling function''' is a relatively recent development. Compare the notation $\floor x$ for the corresponding  floor function.