hcju/proofwikips huggingface dataset:
Definition:Symmetric Relation/Definition 2
$\RR$ is '''symmetric'''  it equals its inverse: :$\RR^{-1} = \RR$
Definition:Reflexive Relation/Definition 1
$\RR$ is '''reflexive''' : :$\forall x \in S: \tuple {x, x} \in \RR$
Definition:Reflexive Relation/Definition 2
$\RR$ is '''reflexive'''  it is a superset of the diagonal relation:  :$\Delta_S \subseteq \RR$
Definition:Transitive Relation/Definition 1
$\RR$ is '''transitive''' : :$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$ that is: :$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$
Definition:Transitive Relation/Definition 2
$\RR$ is '''transitive''' : :$\RR \circ \RR \subseteq \RR$ where $\circ$ denotes composite relation.
Definition:Field of Relation
Let $S$ and $T$ be sets. Let $\RR \subseteq S \times T$ be a relation. The '''field''' of $\RR$ is defined as: :$\map {\operatorname {Field} } \RR := \set {x \in S: \exists t \in T: \tuple {x, t} \in \RR} \cup \set {x \in T: \exists s \in S: \tuple {s, x} \in \RR}$ That is, it is the union of the preimage of $\RR$ with its image.
Definition:Coreflexive Relation/Definition 1
$\mathcal R$ is '''coreflexive''' : :$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$
Definition:Coreflexive Relation/Definition 2
$\mathcal R$ is '''coreflexive''' : :$\mathcal R \subseteq \Delta_S$ where $\Delta_S$ is the diagonal relation.
Definition:Finitely Branching
Let $T$ be a tree. Then $T$ is '''finitely branching'''  $T$ is locally finite.
Definition:Preordering/Definition 1
$\RR$ is a '''preordering''' on $S$ : {{begin-axiom}} {{axiom | n = 1         | lc= $\RR$ is reflexive         | q = \forall a \in S         | m = a \mathrel \RR a }} {{axiom | n = 2         | lc= $\RR$ is transitive         | q = \forall a, b, c \in S         | m = a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c }} {{end-axiom}}
Definition:Preordering/Notation
Symbols used to denote a general preordering relation are usually variants on $\lesssim$, $\precsim$ or $\precapprox$. A symbol for a preordering can be reversed, and the sense is likewise inverted: :$a \precsim b \iff b \succsim a$ The notation $a \sim b$ is defined as: :$a \sim b$  $a \precsim b$ and $b \precsim a$ The notation $a \prec b$ is defined as: :$a \prec b$  $a \precsim b$ and $a \not \sim b$
Definition:Preordering/Definition 2
$\mathcal R$ is a '''preordering''' on $S$ : :$(1): \quad \mathcal R \circ \mathcal R = \mathcal R$ :$(2): \quad \Delta_S \subseteq \mathcal R$ where: : $\circ$ denotes relation composition : $\Delta_S$ denotes the diagonal relation on $S$.
Definition:Total Ordering/Definition 1
$\RR$ is a '''total ordering''' on $S$ : :$(1): \quad \RR$ is an ordering on $S$ :$(2): \quad \RR$ is connected That is, $\RR$ is an ordering with no non-comparable pairs: :$\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$
Definition:Total Ordering/Definition 2
$\RR$ is a '''total ordering''' on $S$ : :$(1): \quad \RR \circ \RR = \RR$ :$(2): \quad \RR \cap \RR^{-1} = \Delta_S$ :$(3): \quad \RR \cup \RR^{-1} = S \times S$
Definition:Ordering/Notation
Symbols used to denote a general ordering relation are usually variants on $\preceq$, $\le$ and so on. On , to denote a general ordering relation it is recommended to use $\preceq$ and its variants: :$\preccurlyeq$ :$\curlyeqprec$ To denote the conventional ordering relation in the context of numbers, the symbol $\le$ is to be used, or its variants: : $\leqslant$ : $\leqq$ : $\eqslantless$ The symbol $\subseteq$ is universally reserved for the subset relation. :$a \preceq b$ can be read as: :'''$a$ precedes, or is the same as, $b$'''. Similarly: : $a \preceq b$ can be read as: : '''$b$ succeeds, or is the same as, $a$'''. If, for two elements $a, b \in S$, it is not the case that $a \preceq b$, then the symbols $a \npreceq b$ and $b \nsucceq a$ can be used.
Definition:Ordering/Definition 1
An '''ordering on $S$''' is a relation $\RR$ on $S$ such that: {{begin-axiom}} {{axiom | n = 1         | lc= $\RR$ is reflexive         | q = \forall a \in S         | m = a \mathrel \RR a }} {{axiom | n = 2         | lc= $\RR$ is transitive         | q = \forall a, b, c \in S         | m = a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c }} {{axiom | n = 3         | lc= $\RR$ is antisymmetric         | q = \forall a \in S         | m = a \mathrel \RR b \land b \mathrel \RR a \implies a = b }} {{end-axiom}}
Definition:Ordering/Definition 2
An '''ordering on $S$''' is a relation $\RR$ on $S$ such that: :$(1): \quad \RR \circ \RR = \RR$ :$(2): \quad \RR \cap \RR^{-1} = \Delta_S$ where: :$\circ$ denotes relation composition :$\RR^{-1}$ denotes the inverse of $\RR$ :$\Delta_S$ denotes the diagonal relation on $S$.
Definition:Operation/Binary Operation/Postfix Notation
Let $\circ: S \times T \to U$ be a binary operation. The convention that places the symbol for the operation after the two operands: :$z = \left ({x, y}\right) \circ$ is called '''postfix notation'''.
Definition:Operation/Binary Operation/Prefix Notation
Let $\circ: S \times T \to U$ be a binary operation. The convention that places the symbol for the operation before the two operands: :$z = \circ \left ({x, y}\right)$ is called '''prefix notation'''.
Definition:Surjection/Definition 1
$f: S \to T$ is a '''surjection''' : :$\forall y \in T: \exists x \in \Dom f: \map f x = y$ That is,  $f$ is right-total.
Definition:Surjection/Definition 2
$f: S \to T$ is a '''surjection''' : :$f \sqbrk S = T$ or, in the language and notation of direct image mappings: :$\map {f^\to} S = T$ That is, $f$ is a '''surjection'''  its image equals its codomain: :$\Img f = \Cdm f$
Definition:Ordered Tuple/Ordered Couples and Ordered Pairs
Notice the difference between ordered pairs and ordered couples. By definition, an ordered couple $\tuple {a, b}$ is in fact the set $\set {\tuple {1, a}, \tuple {2, b} }$, where each of $\tuple {1, a}$ and $\tuple {2, b}$ are ordered pairs. It is not possible to use the definition of ordered couple as the definition of ordered pair, as the latter is used to define a mapping, which is then used to define an ordered couple. However, in view of the Equality of Ordered Tuples, it is generally accepted that it is valid to use the notation $\tuple {a, b}$ to mean both an ordered couple and an ordered pair. It is worth bearing this in mind, as there are times when it is important not to confuse them.
Definition:Singleton Relation
Let $S$ and $T$ be sets. Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Then $\mathcal R$ is a singleton relation iff: :$\exists_1 \left({x, y}\right) \in S \times T: \left({x, y}\right) \in \mathcal R$ That is, iff $\mathcal R$ has exactly one element.
Definition:Symmetric Relation/Definition 3
$\mathcal R$ is '''symmetric'''  it is a subset of its inverse: :$\mathcal R \subseteq \mathcal R^{-1}$
Definition:Root of Unity Modulo m
Let $n \in \Z_{>0}$ be a positive integer. Let $m \in \Z_{>1}$ be a positive integer greater than one. Then $a$ is an '''$n^{th}$ root of unity modulo $m$''' : :$a^n \equiv 1 \pmod m$ Solving the following equation over the smallest integers modulo $m$: :$\displaystyle \left({a^n - 1}\right) = \left({a - 1}\right) \left({\sum_{k \mathop = 0}^{n-1} a^k}\right) \equiv 0 \pmod m$ will produce the roots. Any root found will also have $a + k m$ as a solution, where $k \in \Z$ is any integer. Euler's Function $\phi \left({m}\right)$ root of unity modulo $m$ is the set of all positive integers less than $m$. {{explain|The above statement needs tightening: what has been described "$\phi \left({m}\right)$ root of unity modulo $m$" is not clearly defined and (while it possibly may be used to define a set by predicate, is does not appear itself actually to be a set. Difficult to tell.}}
Definition:Subtree (Graph Theory)
Let $T = \left({V, E}\right)$ be a tree. A '''subtree''' of $T$ is a subgraph of $T$ that is also a tree.
Definition:Rooted Subtree
Let $\left({T, r_T}\right)$ be a rooted tree. A '''rooted subtree''' of $T$ is a rooted tree $\left({S, r_S}\right)$ such that: :$S$ is a subtree of $T$ :$r_S = r_T$ Note that the second condition implies that $r_T \in S$.
Definition:Extension of Rooted Tree
Let $\left({T, r_T}\right)$ and $\left({S, r_S}\right)$ be rooted trees. Then $\left({S, r_S}\right)$ is an '''extension of $T$''' : :$T$ is a subtree of $S$; :$r_S = r_T$. That is,  $T$ is a rooted subtree of $S$. {{expand|Define using parent mapping}}
Definition:Extension of Branch of Rooted Tree
Let $T$ be a rooted tree, and let $\Gamma$ be a branch of $T$. Let $S$ be an extension of $T$, and let $\Gamma'$ be a branch of $S$. Then $\Gamma'$ is an '''extension of $\Gamma$'''  $\Gamma \subseteq \Gamma'$. Informally, a branch may be '''extended''' by successively adding children to its leaf node, one at a time.
Definition:Extension of Propositional Tableau
Let $\left({T, \mathbf H, \Phi}\right)$ be a propositional tableau. === Definition 1 === {{:Definition:Extension of Propositional Tableau/Definition 1}} === Definition 2 === {{:Definition:Extension of Propositional Tableau/Definition 2}}
Definition:Convergent Series/Normed Vector Space
=== Definition 1 === {{:Definition:Convergent Series/Normed Vector Space/Definition 1}} === Definition 2 === {{:Definition:Convergent Series/Normed Vector Space/Definition 2}}
Definition:Convergent Series/Number Field
Let $S$ be one of the standard number fields $\Q, \R, \C$. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$. Let $\sequence {s_N}$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$. It follows that $\sequence {s_N}$ can be treated as a sequence in the metric space $S$. If $s_N \to s$ as $N \to \infty$, the series '''converges to the sum $s$''', and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$. A series is said to be '''convergent'''  it converges to some $s$.
Definition:Extension of Branch of Propositional Tableau
Let $T$ be a propositional tableau, and let $\Gamma$ be a branch of $T$. Let $S$ be an extension of $T$, and let $\Gamma'$ be a branch of $S$. Then $\Gamma'$ is an '''extension of $\Gamma$''' iff $\Gamma \subseteq \Gamma'$. Informally, a branch may be '''extended''' by successively adding children (with WFFs attached to them) to its leaf node, one at a time.
Definition:Geometric Sequence/Common Ratio
The parameter: :$r \in \R: r \ne 0$ is called the '''common ratio''' of $\sequence {x_n}$.
Definition:Geometric Series
Let $\sequence {x_n}$ be a geometric sequence in $\R$: :$x_n = a r^n$ for $n = 0, 1, 2, \ldots$ Then the series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty x_n = a + a r + a r^2 + \cdots + a r^n + \cdots$ is a '''geometric series'''.
Definition:Isolated Point (Metric Space)/Subset
Let $S \subseteq A$ be a subset of $A$. $a \in S$ is an '''isolated point''' of $S$  there exists an open $\epsilon$-ball of $x$ in $M$ containing no points of $S$ other than $a$: :$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a \cap S = \set a$ That is: :$\exists \epsilon \in \R_{>0}: \set {x \in S: \map d {x, a} < \epsilon} = \set a$
Definition:Isolated Point (Metric Space)/Space
$a \in A$ is an '''isolated point''' of $M$  there exists an open $\epsilon$-ball of $x$ containing no points other than $a$: :$\exists \epsilon \in \R_{>0}: \map {B_\epsilon} a = \set a$ That is: :$\exists \epsilon \in \R_{>0}: \set {x \in A: \map d {x, a} < \epsilon} = \set a$
Definition:Natural Logarithm/Positive Real
{{:Definition:Natural Logarithm/Positive Real/Definition 1}}
Definition:Natural Logarithm/Complex
{{:Definition:Natural Logarithm/Complex/Definition 1}}
Definition:General Logarithm/Common
Logarithms base $10$ are often referred to as common logarithms.
Definition:Series/Sequence of Partial Sums
The sequence $\sequence {s_N}$ defined as the indexed summation: :$\displaystyle s_N: = \sum_{n \mathop = 1}^N a_n = a_1 + a_2 + a_3 + \cdots + a_N$ is the '''sequence of partial sums''' of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.
Definition:Series/General/Sequence of Partial Products
The sequence $\left \langle {s_N} \right \rangle$ defined as the indexed iterated operation: :$\displaystyle s_N = \sum_{n \mathop = 1}^N a_n = a_1 \circ a_2 \circ \cdots \circ a_N$ is the '''sequence of partial products''' of the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$.
Definition:Positive Series
Let $\displaystyle s = \sum_{n \mathop = 1}^\infty a_n$ be a series in the real numbers $\R$. The series $s$ is a '''positive series'''  either: :$\forall n \in \N: a_n > 0$ or: :$\forall n \in \N: a_n < 0$ That is, if all terms of $\sequence {a_n}$ are either all (strictly) positive or (strictly) negative.
Definition:Absolutely Convergent Series/General
Let $V$ be a normed vector space with norm $\norm {\, \cdot \,}$. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ in $V$ is '''absolutely convergent'''  $\displaystyle \sum_{n \mathop = 1}^\infty \norm {a_n}$ is a convergent series in $\R$.
Definition:Absolutely Convergent Series/Real Numbers
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in the real number field $\R$. Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is '''absolutely convergent''' : :$\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}$ is convergent where $\size {a_n}$ denotes the absolute value of $a_n$.
Definition:Alternating Series
Let $\displaystyle s = \sum_{n \mathop = 1}^\infty a_n$ be a series in the real numbers $\R$. The series $s$ is an '''alternating series'''  the terms of $\sequence {a_n}$ alternate between positive and negative.
Definition:Absolutely Convergent Series/Complex Numbers
Let $S = \displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in the complex number field $\C$. Then $S$ is '''absolutely convergent''' : :$\displaystyle \sum_{n \mathop = 1}^\infty \cmod {a_n}$ is convergent where $\cmod {a_n}$ denotes the complex modulus of $a_n$.
Definition:Harmonic Series
The series defined as: :$\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 n = 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \cdots$ is known as '''the harmonic series'''.
Definition:Mercator's Constant
'''Mercator's constant''' is the real number: {{begin-eqn}} {{eqn | l = \ln 2       | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n       | c =  }} {{eqn | r = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \dotsb       | c =  }} {{eqn | r = 0 \cdotp 69314 \, 71805 \, 59945 \, 30941 \, 72321 \, 21458 \, 17656 \, 80755 \, 00134 \, 360 \ldots \ldots       | c =  }} {{end-eqn}}
Definition:Cauchy Product
Let $A := \displaystyle \sum_{n \mathop = 0}^\infty a_n$ and $B := \displaystyle \sum_{n \mathop = 0}^\infty b_n$ be two  series. The '''Cauchy product''' $C$ of $A$ and $B$ is defined as: :$\displaystyle C := \sum_{n \mathop = 0}^\infty c_n = \sum_{n \mathop = 0}^\infty a_n \cdot \sum_{n \mathop = 0}^\infty b_n$ where: :$\displaystyle \forall n \in \N: c_n = \sum_{k \mathop = 0}^n a_k b_{n - k} = \sum_{k \mathop = 0}^n a_{n - k} b_k$ {{NamedforDef|Augustin Louis Cauchy|cat=Cauchy}}
Definition:Power Series/Coefficient
The terms: :$a_0, a_1, \ldots, a_n, \ldots$ are the '''coefficients''' of $\displaystyle \sum_{n \mathop = 0}^\infty a_n \paren {x - \xi}^n$.
Definition:Maclaurin Series
Let $f$ be a real function which is smooth on the open interval $\openint a b$. Then the '''Maclaurin series expansion of $f$''' is: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$ It is ''not'' necessarily the case that this power series is convergent with sum $\map f x$. {{NamedforDef|Colin Maclaurin|cat = Maclaurin}}
Definition:Determinant/Matrix/Order 1
:$\begin {vmatrix} a_{1 1} \end {vmatrix} = a_{1 1}$ Thus the determinant of an order $1$ matrix is that element itself.
Definition:Determinant/Matrix/Order 2
{{begin-eqn}} {{eqn | l = \begin {vmatrix} a_{1 1} & a_{1 2} \\ a_{2 1} & a_{2 2} \end{vmatrix}       | r = \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1}       | c =  }} {{eqn | r = a_{1 1} a_{2 2} - a_{1 2} a_{2 1}       | c =  }} {{end-eqn}}
Definition:Determinant/Matrix/Order 3
:$\map \det {\mathbf A} = \begin {vmatrix} a_{1 1} & a_{1 2} & a_{1 3} \\ a_{2 1} & a_{2 2} & a_{2 3} \\ a_{3 1} & a_{3 2} & a_{3 3} \end {vmatrix}$ Then: {{begin-eqn}} {{eqn | l = \map \det {\mathbf A}       | r = a_{1 1} \begin {vmatrix} a_{2 2} & a_{2 3} \\ a_{3 2} & a_{3 3} \end {vmatrix} - a_{1 2} \begin {vmatrix} a_{2 1} & a_{2 3} \\ a_{3 1} & a_{3 3} \end {vmatrix} + a_{1 3} \begin {vmatrix} a_{2 1} & a_{2 2} \\ a_{3 1} & a_{3 2} \end{vmatrix}       | c =  }} {{eqn | r = \map \sgn {1, 2, 3} a_{1 1} a_{2 2}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {1, 3, 2} a_{1 1} a_{2 3}  a_{3 2}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {2, 1, 3} a_{1 2} a_{2 1}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {2, 3, 1} a_{1 2} a_{2 3}  a_{3 1}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {3, 1, 2} a_{1 3} a_{2 1}  a_{3 2}       | c =  }} {{eqn | o =        | ro= +       | r = \map \sgn {3, 2, 1} a_{1 3} a_{2 2}  a_{3 1}       | c =  }} {{eqn | r = a_{1 1} a_{2 2}  a_{3 3}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 1} a_{2 3}  a_{3 2}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 2} a_{2 1}  a_{3 3}       | c =  }} {{eqn | o =        | ro= +       | r = a_{1 2} a_{2 3}  a_{3 1}       | c =  }} {{eqn | o =        | ro= +       | r = a_{1 3} a_{2 1}  a_{3 2}       | c =  }} {{eqn | o =        | ro= -       | r = a_{1 3} a_{2 2}  a_{3 1}       | c =  }} {{end-eqn}} and thence in a single expression as: :$\ds \map \det {\mathbf A} = \frac 1 6 \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^3 \sum_{k \mathop = 1}^3 \sum_{r \mathop = 1}^3 \sum_{s \mathop = 1}^3 \sum_{t \mathop = 1}^3 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$ where $\map \sgn {i, j, k}$ is the sign of the permutation $\tuple {i, j, k}$ of the set $\set {1, 2, 3}$. The values of the various instances of $\map \sgn {\lambda_1, \lambda_2, \lambda_3}$ are obtained by applications of Parity of K-Cycle.
Definition:Extension of Propositional Tableau/Definition 2
An '''extension of $T$''' is a propositional tableau $\left({S, \mathbf H', \Phi'}\right)$ such that: :$S$ is an extension of $T$; :$\mathbf H = \mathbf H'$; :$\Phi'$ is an extension of $\Phi$.
Definition:Extension of Propositional Tableau/Definition 1
A tableau $T'$ is an '''extension of $T$''' if $T'$ can be obtained from $T$ by repeatedly adding nodes to the leaf nodes of $T$ (by means of the tableau extension rules).
Definition:Ordered Statistic
Let $X_1, X_2, X_3, \ldots, X_N$ be a sequence of $N$ independent random variables chosen from an identical probability space. Let $X_{(1)},X_{(2)},X_{(3)},\ldots,X_{(N)}$ be these measures re-arranged in increasing order. Then $X_{(i)}$ is said to be the '''$i$th ordered statistic'''.
Definition:Exterior (Topology)/Notation
The exterior of $H$ can be denoted: :$\map {\mathrm {Ext} } H$ :$H^e$ The first is regarded by some as cumbersome, but has the advantage of being clear. $H^e$ is neat and compact, but has the disadvantage of being relatively obscure. On , the notation of choice is $H^e$.
Definition:Exterior (Topology)/Definition 1
The '''exterior''' of $H$ is the complement of the closure of $H$ in $T$.
Definition:Exterior (Topology)/Definition 2
The '''exterior''' of $H$ is the interior of the complement of $H$ in $T$.
Definition:Enumeration/Finite
Let $X$ be a finite set of cardinality $n \in \N$. An '''enumeration''' of $X$ is a bijection $x: \N_n \to X$, where $\N_n = \set {1, \ldots, n}$.
Definition:Enumeration/Countably Infinite
Let $X$ be a countably infinite set. An '''enumeration''' of $X$ is a bijection $x: \N \to X$.
Definition:Euclidean Plus Metric
Let $\R$ be the set of real numbers. Let $\set {r_i}$ be an enumeration of the rational numbers $\Q$. The '''Euclidean plus metric''' $d: \R \times \R \to \R$ is the metric defined as: :$\map d {x, y} := \set {x - y} + \displaystyle \sum_{i \mathop = 1}^\infty 2^{-i} \map \inf {1, \size {\max_{j \mathop \le i} \frac 1 {\size {x - r_j} } - \max_{j \mathop \le i} \frac 1 {\size {y - r_j} } } }$ Thus $d$ adds to the Euclidean metric between $x$ and $y$ a contribution which measures the relative distances of $x$ and $y$ from $\Q$.
Definition:Tautology/Formal Semantics
Let $\mathcal L$ be a logical language. Let $\mathscr M$ be a formal semantics for $\mathcal L$. A logical formula $\phi$ of $\mathcal L$ is a '''tautology for $\mathscr M$''' : :$\phi$ is valid in every structure $\mathcal M$ of $\mathscr M$ That $\phi$ is a '''tautology for $\mathscr M$''' can be denoted as: :$\models_{\mathscr M} \phi$
Definition:Tautology/Formal Semantics/Boolean Interpretations
Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is called a '''tautology (for boolean interpretations)''' : :$\map v {\mathbf A} = T$ for every boolean interpretation $v$ of $\mathbf A$. That $\mathbf A$ is a '''tautology''' may be denoted as: :$\models_{\mathrm {BI} } \mathbf A$
Definition:Contradiction
A '''contradiction''' is a statement which is ''always false'', independently of any relevant circumstances that could theoretically influence its truth value. This has the form: :$p \land \neg p$ or, equivalently: :$\neg p \land p$ that is: :'''$p$ is true and, at the same time, $p$ is not true.'''
Definition:Rational Number Space
Let $\Q$ be the set of rational numbers. Let $d: \Q \times \Q \to \R$ be the Euclidean metric on $\Q$. Let $\tau_d$ be the topology on $\Q$ induced by $d$. Then $\struct {\Q, \tau_d}$ is the '''rational number space'''.
Definition:Irrational Number Space
Let $\mathbb I := \R \setminus \Q$ be the set of irrational numbers. Let $d: \mathbb I \times \mathbb I \to \R$ be the Euclidean metric on $\mathbb I$. Let $\tau_d$ be the topology on $\mathbb I$ induced by $d$. Then $\struct {\mathbb I, \tau_d}$ is the '''irrational number space'''.
Definition:Unsatisfiable/Boolean Interpretations
Let $\mathbf A$ be a WFF of propositional logic. $\mathbf A$ is called '''unsatisfiable (for boolean interpretations)''' : :$\map v {\mathbf A} = F$ for every boolean interpretation $v$ for $\mathbf A$. In terms of validity, this can be rendered: :$v \not \models_{\mathrm {BI} } \mathbf A$ that is, $\mathbf A$ is invalid in every boolean interpretation of $\mathbf A$.
Definition:Boolean Interpretation/Formal Semantics/Invalid
$\phi$ is declared '''($\mathrm{BI}$-)invalid''' in a boolean interpretation $v$ iff: :$v \left({\phi}\right) = F$ Symbolically, this can be expressed as: :$v \not\models_{\mathrm{BI}} \phi$
Definition:Scattered Space/Definition 1
A topological space $T = \struct {S, \tau}$ is '''scattered'''  it contains no non-empty subset which is dense-in-itself. That is, $T = \struct {S, \tau}$ is '''scattered'''  every non-empty subset $H$ of $S$ contains at least one point which is isolated in $H$.
Definition:Scattered Space/Also defined as
According to {{BookReference|Counterexamples in Topology|1978|Lynn Arthur Steen|author2 = J. Arthur Seebach, Jr.|ed = 2nd|edpage = Second Edition}}, a topological space $T$ is defined as '''scattered''': : ''... if it contains no non-empty dense-in-itself subsets; ...'' and it is immaterial whether those subsets are closed or not. On the other hand, [http://planetmath.org/encyclopedia/ScatteredSpace.html PlanetMath's definition] specifically requires that in order for a space to be classified as '''scattered''', only its closed subsets are required to contain one or more isolated points. There are few other reliable definitions to be found (the concept can be found neither on Wikipedia nor even MathWorld), and when the concept is used at all, the definitions go either way. However, it is apparent that the two definitions are equivalent, and so ultimately it does not matter which definition is used.
Definition:Scattered Space/Definition 2
A topological space $T = \left({S, \tau}\right)$ is '''scattered'''  it contains no non-empty closed set which is dense-in-itself. That is, $T = \left({S, \tau}\right)$ is '''scattered'''  every non-empty closed set $H$ of $S$ contains at least one point which is isolated in $H$.