hcju/proofwikips huggingface dataset:
Definition:Domain (Set Theory)/Relation/General Definition
Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\displaystyle \mathcal R \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$. The '''domain of $\mathcal R$''' is the set defined as: :$\displaystyle \Dom {\mathcal R} := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$ The concept is usually encountered when $\mathcal R$ is an endorelation on $S$: :$\displaystyle \Dom {\mathcal R} := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$
Definition:Image (Set Theory)/Relation/Relation/General Definition
Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\displaystyle \mathcal R \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$. The '''image of $\mathcal R$''' is the set defined as: :$\Img {\mathcal R} := \set {s_n \in S_n: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in \displaystyle \prod_{i \mathop = 1}^{n - 1} S_i: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$ The concept is usually encountered when $\mathcal R$ is an endorelation on $S$: :$\Img {\mathcal R} := \set {s_n \in S: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$
Definition:Strict Well-Ordering/Definition 1
Let $\prec$ be a strict total ordering on a class $A$. Then $\prec$ is a '''strict well-ordering''' on $A$  $\prec$ is a foundational relation on $A$. That is, expressed symbolically: :${\prec} \mathrel{\operatorname{We}} A \iff \left({\prec \operatorname{Or} A \land {\prec} \mathrel{\operatorname{Fr}} A}\right)$
Definition:Strict Strong Well-Ordering
Let $A$ be a class. Let $\mathcal R$ be a relation on $A$. Then $\mathcal R$ is a '''strict strong well-ordering''' of $A$ : : $\mathcal R$ connects $A$ : $\mathcal R$ is strongly well-founded.  That is, whenever $B$ is a non-empty subclass of $A$, $B$ has an $\mathcal R$-minimal element.
Definition:Mapping/General Definition
Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\displaystyle \RR \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$. Then $\RR$ is a '''mapping''' : :$\displaystyle \forall x := \tuple {x_1, x_2, \ldots, x_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \forall y_1, y_2 \in S_n: \tuple {x, y_1} \in \RR \land \tuple {x, y_2} \in \RR \implies y_1 = y_2$ and :$\displaystyle \forall x := \tuple {x_1, x_2, \ldots, x_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \exists y \in S_n: \tuple {x, y} \in \RR$ Thus, a '''mapping''' is an $n$-ary relation which is: :Many-to-one :Left-total, that is, defined for all elements in the domain.
Definition:Cartesian Product/Family of Sets/Definition 1
Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. The '''Cartesian product of $\family {S_i}_{i \mathop \in I}$''' is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S_i$ for each $i \in I$. This can be denoted $\displaystyle \prod_{i \mathop \in I} S_i$ or, if $I$ is understood, $\displaystyle \prod_i S_i$.
Definition:Cartesian Product/Family of Sets/Definition 2
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. The '''Cartesian product of $\family {S_i}_{i \mathop \in I}$''' is the set: :$\displaystyle \prod_{i \mathop \in I} S_i := \set {f: \paren {f: I \to \bigcup_{i \mathop \in I} S_i} \land \paren {\forall i \in I: \paren {\map f i \in S_i} } }$ where $f$ denotes a mapping. When $S_i = S$ for all $i \in I$, the expression is written: :$\displaystyle S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$ which follows from Union is Idempotent: :$\displaystyle \bigcup_{i \mathop \in I} S = S$
Definition:Order Isomorphism/Wosets
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be well-ordered sets. Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving: :$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$ Then $\phi$ is an '''order isomorphism'''.
Definition:Set-Like Relation
Let $\mathcal R$ be a relation. Let $A$ be a class. Then $\mathcal R$ is '''set-like''' on $A$ : :For all $x \in A$, $\left\{{y \in A: y \mathrel{\mathcal R} x}\right\}$ is a set.
Definition:Boolean Algebra/Definition 2
{{:Definition:Boolean Algebra/Axioms/Definition 2}}
Definition:Strict Well-Ordering/Definition 2
Let $A$ be a set or class. Let $\prec$ be a relation on $A$. Then $\prec$ is a '''strict well-ordering''' of $A$ : : $\prec$ connects $A$ : $\prec$ is well-founded.  That is, whenever $b$ is a non-empty subset of $A$, $b$ has a $\prec$-minimal element.
Definition:Boolean Algebra/Axioms/Definition 1
A '''Boolean algebra''' is an algebraic system $\struct {S, \vee, \wedge, \neg}$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation. Furthermore, these operations are required to satisfy the following axioms: {{begin-axiom}} {{axiom | n = \text {BA}_1 0         | lc=         | t = $S$ is closed under $\vee$, $\wedge$ and $\neg$ }} {{axiom | n = \text {BA}_1 1         | lc=         | t = Both $\vee$ and $\wedge$ are commutative }} {{axiom | n = \text {BA}_1 2         | lc=         | t = Both $\vee$ and $\wedge$ distribute over the other }} {{axiom | n = \text {BA}_1 3         | lc=         | t = Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively }} {{axiom | n = \text {BA}_1 4         | lc=         | t = $\forall a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$ }} {{end-axiom}}
Definition:Boolean Algebra/Axioms/Definition 2
A '''Boolean algebra''' is an algebraic system $\struct {S, \vee, \wedge, \neg}$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation. Furthermore, these operations are required to satisfy the following axioms: {{begin-axiom}} {{axiom | n = \text {BA}_2 0         | lc= Closure:         | q = \forall a, b \in S         | m = a \vee b \in S }} {{axiom | m = a \wedge b \in S }} {{axiom | m = \neg a \in S }} {{axiom | n = \text {BA}_2 1         | lc= Commutativity:         | q = \forall a, b \in S         | m = a \vee b = b \vee a }} {{axiom | m = a \wedge b = b \wedge a }} {{axiom | n = \text {BA}_2 2         | lc= Associativity:         | q = \forall a, b, c \in S         | m = a \vee \paren {b \vee c} = \paren {a \vee b} \vee c }} {{axiom | m = a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c }} {{axiom | n = \text {BA}_2 3         | lc= Absorption Laws:         | q = \forall a, b \in S         | m = \paren {a \wedge b} \vee b = b }} {{axiom | m = \paren {a \vee b} \wedge b = b }} {{axiom | n = \text {BA}_2 4         | lc= Distributivity:         | q = \forall a, b, c \in S         | m = a \wedge \paren {b \vee c} = \paren {a \wedge b} \vee \paren {a \wedge c} }} {{axiom | m = a \vee \paren {b \wedge c} = \paren {a \vee b} \wedge \paren {a \vee c} }} {{axiom | n = \text {BA}_2 5         | lc= Identity Elements:         | q = \forall a, b \in S         | m = \paren {a \wedge \neg a} \vee b = b }} {{axiom | m = \paren {a \vee \neg a} \wedge b = b }} {{end-axiom}}
Definition:Ordinal/Definition 1
Let $S$ be a set. Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$. Then $S$ is an '''ordinal''' : :$S$ is a transitive set :$\Epsilon \! \restriction_S$ strictly well-orders $S$.
Definition:Ordinal/Definition 3
An '''ordinal''' is a strictly well-ordered set $\struct {S, \prec}$ such that: :$\forall a \in S: S_a = a$ where $S_a$ is the initial segment of $S$ determined by $a$. From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that: :$S_a = \set {x \in S: x \subsetneqq a}$ From Initial Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal.
Definition:Order Isomorphism/Definition 1
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a bijection such that: : $\phi: S \to T$ is order-preserving : $\phi^{-1}: T \to S$ is order-preserving. Then $\phi$ is an '''order isomorphism'''.
Definition:Order Isomorphism/Definition 2
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a surjective order embedding. Then $\phi$ is an '''order isomorphism'''. That is, $\phi$ is an '''order isomorphism''' : :$(1): \quad \phi$ is surjective :$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$
Definition:Order-Reflecting Mapping
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a mapping. Then $\phi$ is an '''order-reflecting mapping''' or '''reflects order''' : : $\forall x, y \in S: \phi \left({x}\right) \preceq_2 \phi\left({y}\right) \implies x \preceq_1 y$
Definition:Probability Theory
'''Probability theory''' is the branch of mathematics which studies probability spaces.
Definition:Event/Occurrence
Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events, so that $A \subseteq \Omega$ and $B \subseteq \Omega$. Let the outcome of the experiment be $\omega \in \Omega$. Then the following real-world interpretations of the '''occurrence''' of events can be determined: :If $\omega \in A$, then '''$A$ occurs'''. :If $\omega \notin A$, that is $\omega \in \Omega \setminus A$, then '''$A$ does not occur'''.
Definition:Event Space/Discrete
Let $\EE$ be an experiment. Let $\Omega$ be a discrete sample space of $\EE$. Then it is commonplace to take $\Sigma$ to be the power set $\powerset \Omega$ of $\Omega$, that is, the set of all possible subsets of $\Omega$.
Definition:Tangent Map
=== Differentable Mappings === {{:Definition:Tangent Map/Differentable Mappings}} === Affine Transformation === {{:Definition:Tangent Map/Affine Transformation}}
Definition:Probability Space/Discrete
Let $\Omega$ be a discrete sample space. Then $\left({\Omega, \Sigma, \Pr}\right)$ is known as a '''discrete probability space'''.
Definition:Probability Space/Continuous
Let $\Omega$ be a continuum. Then $\left({\Omega, \Sigma, \Pr}\right)$ is known as a '''continuous probability space'''.
Definition:Probability Function
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. The probability measure $\Pr$ on $\left({\Omega, \Sigma, \Pr}\right)$ can be considered as a function on elements of $\Omega$ and $\Sigma$.
Definition:Strong Well-Ordering
Let $A$ be a class. Let $\preceq$ be a relation on $A$. Then $\preceq$ is a '''strong well-ordering''' iff: : $\preceq$ is an ordering. : For each non-empty subclass $B \subseteq A$, $B$ has a smallest element with respect to $\preceq$.
Definition:Associate/Integral Domain/Definition 1
'''$x$ is an associate of $y$ (in $D$)'''  they are both divisors of each other. That is, $x$ and $y$ are '''associates (in $D$)'''  $x \divides y$ and $y \divides x$.
Definition:Associate/Commutative and Unitary Ring
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $x, y \in R$. Then $x$ and $y$ are '''associates (in $R$)'''  there exists a unit $u$ of $\struct {R, +, \circ}$ such that $u \circ x = y$.
Definition:Associate/Integral Domain/Definition 2
$x$ and $y$ are '''associates (in $D$)''' : :$\ideal x = \ideal y$ where $\ideal x$ and $\ideal y$ denote the ideals generated by $x$ and $y$ respectively.
Definition:Independent Events/Definition 1
The events $A$ and $B$ are defined as '''independent (of each other)'''  the occurrence of one of them does not affect the probability of the occurrence of the other one. Formally, $A$ is independent of $B$ : :$\map \Pr {A \mid B} = \map \Pr A$ where $\map \Pr {A \mid B}$ denotes the conditional probability of $A$ given $B$.
Definition:Independent Events/Definition 2
The events $A$ and $B$ are defined as '''independent (of each other)'''  the occurrence of both of them together has the same probability as the product of the probabilities of each of them occurring on their own. Formally, $A$ and $B$ are independent : :$\map \Pr {A \cap B} = \map \Pr A \map \Pr B$
Definition:Independent Events/General Definition
Let $\AA = \family {A_i}_{i \mathop \in I}$ be an indexed family of events of $\EE$. Then $\AA$ is '''independent''' , for all finite subsets $J$ of $I$: :$\displaystyle \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$ That is, if the occurrence of any finite collection of $\AA$ has the same probability as the product of each of those sets occurring individually. === Pairwise Independent === {{:Definition:Independent Events/Pairwise Independent}}
Definition:Independent Events/Pairwise Independent
Let $\mathcal A = \family {A_i}_{i \mathop \in I}$ be an indexed family of events of $\EE$. Then $\AA$ is '''pairwise independent''' : :$\forall j, k \in I: \map \Pr {A_j \cap A_k} = \map \Pr {A_j} \, \map \Pr {A_k}$ That is, if every pair of events of $\EE$ are independent of each other. That is, $\AA$ is '''pairwise independent'''  the condition for general independence: :$\displaystyle \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$ holds whenever $\card J = 2$.
Definition:Independent Events/Dependent
If $A$ and $B$ are not independent, then they are '''dependent (on each other)''', and vice versa.
Definition:Limit of Sequence of Events/Increasing
Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N}$ be an increasing sequence of events. Then the union: :$\displaystyle A = \bigcup_{i \mathop \in \N} A_i$ of such a sequence is called the '''limit of the sequence $\left \langle{A_n}\right \rangle_{n \in \N}$'''.
Definition:Limit of Sequence of Events/Decreasing
Let $\left \langle{A_n}\right \rangle_{n \in \N}$ be an decreasing sequence of events. Then the intersection: :$\displaystyle A = \bigcap_{i \mathop \in \N} A_i$ of such a sequence is called the '''limit of the sequence $\left \langle{A_n}\right \rangle_{n \in \N}$'''.
Definition:Probability Mass Function/Joint
Let $X: \Omega \to \R$ and $Y: \Omega \to \R$ both be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$. Then the '''joint (probability) mass function''' of $X$ and $Y$ is the (real-valued) function $p_{X, Y}: \R^2 \to \closedint 0 1$ defined as: :$\forall \tuple {x, y} \in \R^2: \map {p_{X, Y} } {x, y} = \begin {cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x \land \map Y \omega = y} } & : x \in \Omega_X \text { and } y \in \Omega_Y \\ 0 & : \text {otherwise} \end {cases}$ That is, $\map {p_{X, Y} } {x, y}$ is the probability that the discrete random variable $X$ takes the value $x$ at the same time that the discrete random variable $Y$ takes the value $y$.
Definition:Geometric Distribution/Shifted
$X$ has the '''shifted geometric distribution with parameter $p$''' : :$\map X \Omega = \set {1, 2, \ldots} = \N_{>0}$ :$\map \Pr {X = k} = p \paren {1 - p}^{k-1}$ where $0 < p < 1$. It is frequently seen as: :$\map \Pr {X = k} = q^{k - 1} p$ where $q = 1 - p$. It is written: :$X \sim \ShiftedGeometric p$
Definition:Negative Binomial Distribution/First Form
$X$ has the '''negative binomial distribution (of the first form) with parameters $n$ and $p$''' if: :$\Img X = \set {0, 1, 2, \ldots}$ :$\map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k \paren {1 - p}^n$ where $0 < p < 1$. It is frequently seen as: :$\map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k q^n$ where $q = 1 - p$.
Definition:Negative Binomial Distribution/Second Form
$X$ has the '''negative binomial distribution (of the second form) with parameters $n$ and $p$''' if: :$\Img X = \set {n, n + 1, n + 2, \dotsc}$ :$\map \Pr {X = k} = \dbinom {k - 1} {n - 1} p^n \paren {1 - p}^{k - n}$ where $0 < p < 1$. It is frequently seen as: :$\map \Pr {X = k} = \dbinom {k - 1} {n - 1} q^{k - n} p^n $ where $q = 1 - p$.
Definition:Variance/Discrete/Definition 1
:$\var X := \expect {\paren {X - \expect X}^2}$ That is: it is the expectation of the squares of the deviations from the expectation.
Definition:Variance/Discrete/Definition 2
:$\displaystyle \var X := \sum_{x \mathop \in \Omega_X} \paren {x - \mu^2} \map \Pr {X = x}$ where: :$\mu := \expect X$ is the expectation of $X$ :$\Omega_X$ is the image of $X$ :$\map \Pr {X = x}$ is the probability mass function of $X$.
Definition:Countable Set/Definition 2
$S$ is '''countable'''  it is finite or countably infinite.
Definition:Countable Set/Definition 1
$S$ is '''countable'''  there exists an injection: :$f: S \to \N$
Definition:Independent Random Variables/General Definition
Let $X = \tuple {X_1, X_1, \ldots, X_n}$ be an ordered tuple of random variables. Then $X$ is '''independent''' : :$\displaystyle \map \Pr {X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n} = \prod_{k \mathop = 1}^n \map \Pr {X_k = x_k}$ for all $x = \tuple {x_1, x_2, \ldots, x_n} \in \R^n$.
Definition:Independent Random Variables/Pairwise Independent
Let $X = \tuple {X_1, X_1, \ldots, X_n}$ be an ordered tuple of random variables. Then $X$ is '''pairwise independent'''  $X_i$ and $X_j$ are independent (of each other) whenever $i \ne j$.
Definition:Independent Random Variables/Dependent
Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Then $X$ and $Y$ are defined as '''dependent (on each other)'''  $X$ and $Y$ are not independent (of each other).
Definition:Euler's Number/Limit of Sequence
The sequence $\sequence {x_n}$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$ converges to a limit as $n$ increases without bound. That limit is called '''Euler's Number''' and is denoted $e$.
Definition:Euler's Number/Limit of Series
The series $\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ converges to a limit. This limit is Euler's number $e$.
Definition:Euler's Number/Base of Logarithm
The number $e$ can be defined as the number satisfied by: :$\ln e = 1$ where $\ln e$ denotes the natural logarithm of $e$. That $e$ is unique follows from Logarithm is Strictly Increasing.
Definition:Euler's Number/Exponential Function
The number $e$ can be defined as the number satisfied by: :$e := \exp 1 = e^1$ where $\exp 1$ denotes the exponential function of $1$.
Definition:Euler's Number/Decimal Expansion
The decimal expansion of Euler's number $e$ starts: :$2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$
Definition:Exponential Generating Function
Let $A = \left \langle {a_n}\right \rangle$ be a sequence in $\R$. Then $\displaystyle G_A \left({z}\right) = \sum_{n \mathop \ge 0} \frac {a_n} {n!} z^n$ is called the '''(exponential) generating function''' for the sequence $A$.
Definition:Dismal Addition
'''Dismal addition''' is an operation defined on the natural numbers as follows. Let $a, b \in \N$ be expressed in decimal notation: :$\displaystyle a = \sum_{i \mathop = 0}^n a_i 10^i = \left[{a_n a_{n-1} \ldots a_2 a_1 a_0}\right]_{10}$ :$\displaystyle b = \sum_{j \mathop = 0}^m b_j 10^j = \left[{b_m b_{m-1} \ldots b_2 b_1 b_0}\right]_{10}$ The '''dismal sum''' of $a$ and $b$ is defined as: :$\displaystyle a + b = \sum_{k \mathop = 0}^{\max \left\{{m, n}\right\}} \max \left\{{a_k, b_k}\right\} 10^k$ That is, for each pair of corresponding digits, the maximum is taken.
Definition:Random Variable/Definition 1
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\struct {X, \Sigma'}$ be a measurable space. A '''random variable (on $\struct {\Omega, \Sigma, \Pr}$)''' is a $\Sigma \, / \, \Sigma'$-measurable mapping $f: \Omega \to X$.
Definition:Random Variable/Definition 3
:$\forall x \in \R: X^{-1} \sqbrk {\hointl {-\infty} x} \in \Sigma$ where: :$\hointl {-\infty} x$ denotes the unbounded closed interval $\set {y \in \R: y \le x}$ :$X^{-1} \sqbrk {\hointl {-\infty} x}$ denotes the preimage of $\hointl {-\infty} x$ under $X$.
Definition:Random Variable/Definition 2
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. A '''random variable on $\struct {\Omega, \Sigma, \Pr}$''' is a mapping $X: \Omega \to \R$ such that: :$\forall x \in \R: \set {\omega \in \Omega: \map X \omega \le x} \in \Sigma$
Definition:Tree (Graph Theory)/Node
The vertices of a tree are called its '''nodes'''.
Definition:Tree (Graph Theory)/Infinite
A tree is '''infinite''' if it contains a (countably) infinite number of nodes.
Definition:Tree (Graph Theory)/Finite
A tree is '''finite'''  it contains a finite number of nodes.
Definition:Ultrafilter on Set/Definition 4
Let $S$ be a non-empty set. Let $\mathcal F$ be a non-empty set of subsets of $S$. Then $\mathcal F$ is an '''ultrafilter''' on $S$  both of the following hold: : $\mathcal F$ has the finite intersection property : For all $U \subseteq S$, either $U \in \mathcal F$ or $U^c \in \mathcal F$ where $U^c$ is the complement of $U$ in $S$.
Definition:Ultrafilter on Set/Definition 1
Let $S$ be a set. Let $\FF \subseteq \powerset S$ be a filter on $S$. Then $\FF$ is an '''ultrafilter (on $S$)''' : :there is no filter on $S$ which is strictly finer than $\FF$ or equivalently, : :whenever $\GG$ is a filter on $S$ and $\FF \subseteq \mathcal G$ holds, then $\FF = \GG$.
Definition:Finite Character
Let $S$ be a set. Let $\mathcal F$ be a set of subsets of $S$. Then $\mathcal F$ has '''finite character'''  for each $A \subseteq S$: : $A \in \mathcal F$  every finite subset of $A$ is in $\mathcal F$.
Definition:Rooted Tree/Ancestor Node/Proper
A '''proper ancestor node''' of $t$ is an ancestor node of $t$ that is not $t$ itself.
Definition:Rooted Tree/Child Node
Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. The '''child nodes''' of $t$ are the elements of the set: :$\left\{{s \in T: \pi \left({s}\right) = t}\right\}$ where $\pi \left({s}\right)$ denotes the parent mapping of $s$. That is, the '''children''' of $t$ are all the nodes of $T$ of which $t$ is the parent. === Grandchild Node === {{:Definition:Rooted Tree/Child Node/Grandchild Node}}
Definition:Rooted Tree/Proper Descendant
A '''proper descendant node''' of $t$ is a descendant of $t$ which is not $t$ itself.
Definition:Rooted Tree/Descendant
Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. A '''descendant node''' $s$ of a $t$ is a node such that $t$ is in the path from $s$ to $r_T$. That is, the '''descendant nodes''' of $t$ are all the nodes of $T$ of which $t$ is an ancestor node. === Proper Descendant === {{:Definition:Rooted Tree/Proper Descendant}}
Definition:Rooted Tree/Sibling
Let $T$ be a rooted tree with root $r_T$. Two children of the same node of $T$ are called '''siblings'''. That is, '''siblings''' are nodes which both have the same parent.
Definition:Rooted Tree/Branch/Infinite
Let $T$ be a rooted tree with root node $r_T$. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is infinite  it has no leaf node at the end.
Definition:Rooted Tree/Branch/Length
Let $T$ be a rooted tree with root node $r_T$. Let $\Gamma$ be a finite branch of $T$. The '''length''' of $\Gamma$ is defined as the number of ancestors of the leaf at the end of that branch.
Definition:Rooted Tree/Branch
Let $T$ be a rooted tree with root node $r_T$. A subset $\Gamma$ of $T$ is a '''branch'''  all the following conditions hold: :$(1): \quad$ The root node $r_T$ belongs to $\Gamma$ :$(2): \quad$ The parent of each node in $\Gamma \setminus \left\{{r_T}\right\}$ is in $\Gamma$ :$(3): \quad$ Each node in $\Gamma$ either: ::$(a): \quad$ is a leaf node of $T$ :or: ::$(b): \quad$ has exactly one child node in $\Gamma$.
Definition:Rooted Tree/Root Node
Let $T$ be a rooted tree. The '''root node''' of $T$ is the node of $T$ which is distinguished from the others by being the ancestor node of every node of $T$.
Definition:Finite Join Property
Let $(B, \vee, \wedge, \neg, \bot, \top)$ be a Boolean algebra. Let $J \subseteq B$. Then $J$ has the '''finite join property''' iff for any $x_1, x_2, \dots, x_n \in J$: : $x_1 \vee x_2 \vee \dots \vee x_n \ne \top$