hcju/proofwikips huggingface dataset:
Definition:Classes of WFFs
The set of all WFFs of $\mathcal L_1$ formed with relation symbols from $\mathcal P$ and function symbols from $\mathcal F$ can be denoted $WFF \left({\mathcal P, \mathcal F}\right)$. If so desired, the parameters can also be emphasized by writing $WFF \left({\mathcal P, \mathcal F, \mathcal K}\right)$ instead. To specify $\mathcal P$, one speaks of '''WFFs with relation symbols from $\mathcal P$'''. To specify $\mathcal F$, one speaks of '''WFFs with function symbols from $\mathcal F$'''. To specify $\mathcal K$, one speaks of '''WFFs with parameters from $\mathcal K$'''. Of course, combinations of these are possible. Several classes of WFFs are often considered and have special names. === Plain WFF === {{:Definition:Classes of WFFs/Plain WFF}} === Sentence === {{:Definition:Classes of WFFs/Sentence}} === Plain Sentence === {{:Definition:Classes of WFFs/Plain Sentence}}
Definition:Classes of WFFs/Sentence
A WFF is said to be a '''sentence'''  it contains no free variables. To denote particular classes of '''sentences''', $SENT \left({\mathcal P, \mathcal F, \mathcal K}\right)$ and analogues may be used, similar to the notation for classes of WFFs.
Definition:Classes of WFFs/Plain Sentence
A WFF is said to be a '''plain sentence''' iff it is both plain and a sentence. That is, if it contains free variables nor parameters. Thus, '''plain sentences''' are those WFFs which are in $SENT \left({\mathcal P, \mathcal F, \varnothing}\right)$.
Definition:Compound Statement/Ill-Formed
The substatements in a compound statement, which are joined by a connective, may be compound statements themselves.  It is clearly necessary that the interpretation of such a compound statement is unambiguous.  A compound statement is said to be '''ill-formed''' if it is ambiguous as to how its substatements are grouped by the action of the connectives. For example, in natural language: :''I would like some juice or water with ice.'' can mean either: :''I would like some juice, or water with ice.'' or: :''I would like some juice with ice, or water with ice.''
Definition:Set/Implicit Set Definition/Multipart Infinite Set
Let $S$ be a set. Suppose $S$ is to contain: :$(1): \quad$ a never-ending list of elements and :$(2): \quad$ other elements which are unrelated to that list (perhaps another never-ending list). Then a '''semicolon''' is used to separate the various conceptual parts: :$S = \set {1, 3, 5, \ldots; 2, 4, 6, \ldots; \text{red}, \text{orange}, \text{green} }$ Note that ''without'' the semicolon it would appear as though the first list (of odd numbers) '''continued''' as the second list (of even numbers) which in turn '''continued''' as a list of colours, which is absurd.
Definition:Mathematical Logic
'''Mathematical logic''' is a sub-branch of symbolic logic in which the foundations of the assumptions upon which rest mathematics itself are investigated and made rigorous.
Definition:Multi-Value Logic
'''Multi-value logic''' is a branch of logic in which it is admissible for a statements to have a truth value other than just true or false.
Definition:Modal Logic
'''Modal logic''' is a branch of logic in which truth values are more complex than being merely true or false, and which distinguishes between different "modes" of truth.
Definition:Naive Set Theory
'''Naïve set theory''', in contrast with axiomatic set theory, is an approach to set theory which assumes the existence of a universal set, despite the fact that such an assumption leads to paradoxes. A popular alternative (and inaccurate) definition describes this as a : ''non-formalized definition of set theory which describes sets and the relations between them using natural language.'' However, the discipline is founded upon quite as rigid a set of axioms, namely, those of propositional and predicate logic.
Definition:Pure Set Theory
'''Pure set theory''' is a system of set theory in which all elements of sets are themselves sets.
Definition:Set Intersection/Set of Sets
Let $\Bbb S$ be a set of sets The '''intersection of $\Bbb S$''' is: :$\ds \bigcap \Bbb S := \set {x: \forall S \in \Bbb S: x \in S}$ That is, the set of all objects that are elements of all the elements of $\Bbb S$. Thus: :$\ds \bigcap \set {S, T} := S \cap T$
Definition:Set Intersection/Countable Intersection
Let $\mathbb S$ be a set of sets. Let $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$ be a sequence in $\mathbb S$. Let $S$ be the intersection of $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$: :$\displaystyle S = \bigcap_{n \mathop \in \N} S_n$ Then $S$ is a '''countable intersection''' of sets in $\mathbb S$.
Definition:Set Union/Set of Sets
Let $\mathbb S$ be a set of sets. The '''union of $\mathbb S$''' is: :$\ds \bigcup \mathbb S := \set {x: \exists X \in \mathbb S: x \in X}$ That is, the set of all elements of all elements of $\mathbb S$. Thus the general union of two sets can be defined as: :$\ds \bigcup \set {S, T} = S \cup T$
Definition:Set Union/Countable Union
Let $\mathbb S$ be a set of sets. Let $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$ be a sequence in $\mathbb S$. Let $S$ be the union of $\left\langle{S_n}\right\rangle_{n \mathop \in \N}$: :$\displaystyle S = \bigcup_{n \mathop \in \N} S_n$ Then $S$ is a '''countable union''' of sets in $\mathbb S$.
Definition:Set Union/Finite Union
Let $S = S_1 \cup S_2 \cup \ldots \cup S_n$. Then: :$\displaystyle S = \bigcup_{i \mathop \in \N^*_n} S_i = \set {x: \exists i \in \N^*_n: x \in S_i}$ where $\N^*_n = \set {1, 2, 3, \ldots, n}$. If it is clear from the context that $i \in \N^*_n$, we can also write $\displaystyle \bigcup_{\N^*_n} S_i$.
Definition:Proof System
A '''proof system''' $\mathscr P$ for $\mathcal L$ comprises: * '''Axioms''' and/or '''axiom schemata'''; * '''Rules of inference''' for deriving theorems. It is usual that a '''proof system''' does this by declaring certain arguments concerning $\mathcal L$ to be valid. Informally, a '''proof system''' amounts to a precise account of what constitutes a '''(formal) proof'''.
Definition:Set Intersection/Finite Intersection
Let $S = S_1 \cap S_2 \cap \ldots \cap S_n$. Then: :$\displaystyle S = \bigcap_{i \mathop \in \N^*_n} S_i := \set {x: \forall i \in \N^*_n: x \in S_i}$ where $\N^*_n = \set {1, 2, 3, \ldots, n}$. If it is clear from the context that $i \in \N^*_n$, we can also write $\displaystyle \bigcap_{\N^*_n} S_i$.
Definition:Set Union/Family of Sets
Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Then the '''union''' of $\family {S_i}$ is defined as: :$\ds \bigcup_{i \mathop \in I} S_i := \set {x: \exists i \in I: x \in S_i}$
Definition:Set Intersection/Family of Sets
Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. Then the '''intersection''' of $\family {S_i}$ is defined as: :$\displaystyle \bigcap_{i \mathop \in I} S_i := \set {x: \forall i \in I: x \in S_i}$
Definition:Zero (Ordinal)
The '''zero ordinal''', denoted $0$, is the empty set $\O$.
Definition:Generalized Momentum
The '''generalized momentum of analytical (Lagrangian, Hamiltonian) formulations of classical mechanics''' is defined as the partial derivative of the Lagrangian with regards to the time derivative of generalized coordinates: :$p_i = \dfrac {\partial\mathcal L} {\partial \dot q_i}$ where: : $p_i$ is the $i$th coordinate of the generalized momenta : $\mathcal L$ is the Lagrangian : $\dot q_i$ is the time derivative of the generalized coordinates $q_i$.
Definition:Action Applied by System
The '''action applied by a system''' from state $1$ to state $2$ is defined as the definite integral of the Lagrangian over time from state $1$ to state $2$: :$\displaystyle S_{12} = \int_{t_1}^{t_2} \mathcal L \ \mathrm d t$ where: : $S_{12}$ is the '''action''' from $1$ to $2$ : $t$ is time : $\mathcal L$ is the Lagrangian.
Definition:Set Product
Let $S$ and $T$ be sets. Let $P$ be a set and let $\phi_1: P \to S$ and $\phi_2: P \to T$ be mappings such that: :For all sets $X$ and all mappings $f_1: X \to S$ and $f_2: X \to T$ there exists a unique mapping $h: X \to P$ such that: ::$\phi_1 \circ h = f_1$ ::$\phi_2 \circ h = f_2$ :that is, such that: ::$\begin{xy}\xymatrix@+1em@L+3px{ &  X   \ar[ld]_*+{f_1}   \ar@{-->}[d]^*+{h}   \ar[rd]^*+{f_2} \\  S &  P   \ar[l]^*+{\phi_1}   \ar[r]_*+{\phi_2} &  T }\end{xy}$ :is a commutative diagram. Then $P$, together with the mappings $\phi_1$ and $\phi_2$, is called '''a product of $S$ and $T$'''.
Definition:Integral of Integrable Function over Measurable Set
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $E \in \Sigma$. Let $f: X \to \overline \R$ be a $\mu$-integrable function. Then the '''$\mu$-integral of $f$ over $E$''' is defined by: :$\displaystyle \int_E f \rd \mu := \int \chi_E \cdot f \rd \mu$ where: :$\chi_E$ is the characteristic function of $E$ :$\chi_E \cdot f$ is the pointwise product of $\chi_E$ and $f$ :the integral sign on the  denotes $\mu$-integration of the function $\chi_E \cdot f$.
Definition:Measure with Density
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \overline{\R}_{\ge 0}$ be a positive $\mu$-measurable function. Then the '''measure with density $f$ with respect to $\mu$''', denoted $f \mu$, is defined by: :$f \mu \left({E}\right) := \displaystyle \int_E f \, \mathrm d \mu$ where $\displaystyle \int_E f \, \mathrm d \mu$ is the $\mu$-integral of $f$ over $E$.
Definition:Indexing Set/Family of Sets
Let $\mathcal S$ be a set of sets. Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of elements of $\mathcal S$ indexed by $I$. Then $\family {S_i}_{i \mathop \in I}$ is referred to as an '''indexed family of sets'''.
Definition:Set Partition/Finite Expansion
Let $S$ be a set. Let $\Bbb S = \set {S_1, S_2, \ldots, S_n}$ form a partition of $S$. Then the representation by such a partition $\displaystyle \bigcup_{k \mathop = 1}^n S_k = S$ is also called a '''finite expansion''' of $S$. The notations: :$S = S_1 \mid S_2 \mid \cdots \mid S_n$ or: :$\Bbb S = \set {S_1 \mid S_2 \mid \cdots \mid S_n}$ are sometimes seen.
Definition:Partitioning
Let $S$ be a set. Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$ such that: :$(1): \quad \forall i \in I: S_i \ne \O$, that is, none of $S_i$ is empty :$(2): \quad \ds S = \bigcup_{i \mathop \in I} S_i$, that is, $S_i$ is the union of $\family {S_i}_{i \mathop \in I}$ :$(3): \quad \forall i, j \in I: i \ne j \implies S_i \cap S_j = \O$, that is, the elements of $\family {S_i}_{i \mathop \in I}$ are pairwise disjoint. Then $\family {S_i}_{i \in I}$ is a '''partitioning''' of $S$. The image of this '''partitioning''' is the set $\set {S_i: i \in I}$ and is called a partition of $S$. Note the difference between: :the '''partitioning''', which is an indexing function (that is a mapping) and :the partition, which is the effect (that is, the image) of that mapping.
Definition:Set Partition/Component
The elements $S_1, S_2, \ldots \in \mathbb S$ are known as the '''components''' of the partition.
Definition:P-Sequence Space
Let $p \in \R$, $p \ge 1$. The '''$p$-sequence space''', denoted $\ell^p$ or $\ell^p \left({\N}\right)$, is defined as: :$\displaystyle \ell^p := \left\{{\left\langle{z_n}\right\rangle_{n \in \N} \in \C^\N: \sum_{n \mathop = 0}^\infty \left\vert{z_n}\right\vert^p < \infty}\right\}$ As such, $\ell^p$ is a subspace of $\C^\N$, the space of all complex sequences.
Definition:Set Product/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. Let $P$ be a set. Let $\family {\phi_i}_{i \mathop \in I}$ be an indexed family of mappings $\phi_i: P \to S_i$ for all $i \in I$ such that: :For all sets $X$ and all indexed families $\family {f_i}_{i \mathop \in I}$ of mappings $f_i: X \to S_i$ there exists a unique mapping $h: X \to P$ such that: ::$\forall i \in I: \phi_i \circ h = f_i$ :that is, such that for all $i \in I$: ::$\begin{xy}\xymatrix@+1em@L+3px{  X   \ar@{-->}[d]_*+{h}   \ar[dr]^*+{f_i} \\  P   \ar[r]_*{\phi_i} &  S_i }\end{xy}$ :is a commutative diagram. Then $P$, together with the family of mappings $\family {\phi_i}_{i \mathop \in I}$, is called '''a product of (the family) $\family {S_i}_{i \mathop \in I}$'''. This '''product of $\family {S_i}_{i \mathop \in I}$''' can be denoted $\struct {P, \family {\phi_i}_{i \mathop \in I} }$.
Definition:Cartesian Product/Family of Sets
=== Definition 1 === {{:Definition:Cartesian Product/Family of Sets/Definition 1}} === Definition 2 === {{:Definition:Cartesian Product/Family of Sets/Definition 2}} === Uncountable Cartesian Product === Using this notation, it is then possible to define the Cartesian product of an uncountable family: {{:Definition:Cartesian Product/Uncountable}}
Definition:Cartesian Product/Cartesian Space
Let $S$ be a set. The '''cartesian $n$th power of $S$''', or '''$S$ to the power of $n$''', is defined as: :$\displaystyle S^n = \prod_{k \mathop = 1}^n S = \set {\tuple {x_1, x_2, \ldots, x_n}: \forall k \in \N^*_n: x_k \in S}$ Thus $S^n = \underbrace {S \times S \times \cdots \times S}_{\text{$n$ times} }$ Alternatively it can be defined recursively: :$S^n = \begin{cases} S: & n = 1 \\ S \times S^{n - 1} & n > 1 \end{cases}$ The set $S^n$ called a '''cartesian space'''. An element $x_j$ of an ordered tuple $\tuple {x_1, x_2, \ldots, x_n}$ of a '''cartesian space''' $S^n$ is known as a basis element of $S^n$.
Definition:Projection (Mapping Theory)/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets. Let $\displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. For each $j \in I$, the '''$j$th projection on $\displaystyle S = \prod_{i \mathop \in I} S_i$''' is the mapping $\pr_j: S \to S_j$ defined by: :$\map {\pr_j} {\family {s_i}_{i \mathop \in I} } = s_j$ where $\family {s_i}_{i \mathop \in I}$ is an arbitrary element of $\displaystyle \prod_{i \mathop \in I} S_i$.
Definition:Seminorm
Let $\struct{K, +, \circ}$ be a division ring with norm $\norm{\,\cdot\,}_K$. Let $V$ be a vector space over $\struct{K, \norm{\,\cdot\,}_K}$, with zero $0_V$. A '''seminorm''' on $V$ is a map from $V$ to the positive reals $\norm{\cdot}: V \to \R_{\ge 0}$ satisfying the following properties (for all $x,y \in V$ and $\lambda \in K$): {| | align="left" | '''N2:''' | align="left" | Positive homogeneity: | align="left" | $\norm{\lambda x} = \norm{\lambda}_K \times \norm{x}$ |- | align="left" | '''N3:''' | align="left" | Triangle inequality: | align="left" | $\norm {x + y} \leq \norm{x} + \norm{y}$ |} {{refactor|Use Axiom template}} These may be referred to as the '''seminorm axioms'''. The '''N2''' and '''N3''' markings originate from the fact that these axioms are also used in defining norms.
Definition:P-Seminorm
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $p \in \R$, $p \ge 1$. Let $\mathcal{L}^p \left({\mu}\right)$ be Lebesgue $p$-space for $\mu$. The '''$p$-seminorm''' on $\mathcal{L}^p \left({\mu}\right)$ is the mapping $\left\Vert{\cdot}\right\Vert_p : \mathcal{L}^p \left({\mu}\right) \to \R_{\ge 0}$ defined by: :$\displaystyle \forall f \in \mathcal{L}^p \left({\mu}\right): \left\Vert{f}\right\Vert_p := \left({\int \left\vert{f}\right\vert^p \, \mathrm d \mu}\right)^{1/p}$ That the '''$p$-seminorm''' is in fact a seminorm is proved on $p$-Seminorm is Seminorm.
Definition:Friedman Number
A '''Friedman number (base $n$)''' is a (positive) integer which is the result of an expression in base $n$ arithmetic which contains exactly its digits. The expression is subject to the following constraints: :$(1): \quad$ The arithmetic operators $+$, $-$, $\times$, $\div$ and exponentiation are the only operators which are allowed. :$(2): \quad$ Parentheses are allowed, but only in order to override the default operator precedence, otherwise every number would trivially be Friedman by $n = (n)$. :$(3): \quad$ Leading zeroes are not allowed, otherwise other numbers would trivially be Friedman by, for example, $011 = 10 + 1$.
Definition:Lebesgue Space/L-Infinity
The '''Lebesgue $\infty$-space for $\mu$''', denoted $\mathcal{L}^\infty \left({\mu}\right)$, is defined as: :$\displaystyle \mathcal{L}^\infty \left({\mu}\right) := \left\{{f \in \mathcal M \left({\Sigma}\right): \text{$f$ is a.e. bounded}}\right\}$ and so consists of all $\Sigma$-measurable $f: X \to \R$ that are almost everywhere bounded, that is, subject to: :$\exists c \in \R: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0$ {{refactor|The following can be split off into separate pages, as they need proving in their own right.}} $\mathcal{L}^\infty \left({\mu}\right)$ can be endowed with the supremum seminorm $\left\Vert{\cdot}\right\Vert_\infty$ by: :$\displaystyle \forall f \in \mathcal{L}^\infty \left({\mu}\right): \left\Vert{f}\right\Vert_\infty := \inf \, \left\{{c \ge 0: \mu \left({\left\{{\left\vert{f}\right\vert > c}\right\}}\right) = 0}\right\}$ If, subsequently, we introduce the equivalence $\sim$ by: :$f \sim g \iff \left\Vert{f - g}\right\Vert_\infty = 0$ we obtain the quotient space $L^\infty \left({\mu}\right) := \mathcal{L}^\infty \left({\mu}\right) / \sim$, which is also called '''Lebesgue $\infty$-space for $\mu$'''.
Definition:Product Sigma-Algebra
Let $\struct {X, \Sigma_1}$ and $\struct {Y, \Sigma_2}$ be measurable spaces. The '''product $\sigma$-algebra''' of $\Sigma_1$ and $\Sigma_2$ is denoted $\Sigma_1 \otimes \Sigma_2$, and defined as: :$\Sigma_1 \otimes \Sigma_2 := \map \sigma {\set {S_1 \times S_2: S_1 \in \Sigma_1 \land S_2 \in \Sigma_2} }$ where: :$\sigma$ denotes generated $\sigma$-algebra :$\times$ denotes Cartesian product. This '''product $\sigma$-algebra''' $\Sigma_1 \otimes \Sigma_2$ is a $\sigma$-algebra on $X \times Y$. === Product of Measurable Spaces === {{:Definition:Product of Measurable Spaces}}
Definition:Product of Measurable Spaces
The '''product of $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$''' is the measurable space: :$\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$
Definition:Product Measure
Let $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ be $\sigma$-finite measure spaces. Let $\left({X \times Y, \Sigma_1 \otimes \Sigma_2}\right)$ be the product measurable space of $\left({X, \Sigma_1}\right)$ and $\left({Y, \Sigma_2}\right)$. The '''product measure of $\mu$ and $\nu$''', denoted $\mu \times \nu$, is the measure defined by: :$\forall E_1 \in \Sigma_1, E_2 \in \Sigma_2: \mu \times \nu \left({E_1 \times E_2}\right) = \mu \left({E_1}\right) \nu \left({E_2}\right)$ That this uniquely defines a measure on $\Sigma_1 \otimes \Sigma_2$ is shown on Uniqueness of Product Measures and Existence of Product Measures. === Product Measure Space === {{:Definition:Product Measure Space}}
Definition:Product Measure Space
The '''product of $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$''' is the measure space: :$\left({X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}\right)$
Definition:Survival Function
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \overline{\R}$ be a positive $\Sigma$-measurable function. The '''survival function of $f$''' is the mapping $F: \R \to \R$ defined by: :$\forall t \in \R: F \left({t}\right) := \mu \left({\left\{{f \ge t}\right\}}\right)$ where $\left\{{f > t}\right\}$ is the set $\left\{{x \in X: f \left({x}\right) \ge t}\right\}$.
Definition:Characteristic Function (Set Theory)/Set
Let $E \subseteq S$. The '''characteristic function of $E$''' is the function $\chi_E: S \to \set {0, 1}$ defined as: :$\map {\chi_E} x = \begin {cases} 1 & : x \in E  \\  0 & : x \notin E \end {cases}$ That is: :$\map {\chi_E} x = \begin {cases} 1 & : x \in E  \\  0 & : x \in \relcomp S E \end {cases}$ where $\relcomp S E$ denotes the complement of $E$ relative to $S$.
Definition:Characteristic Function (Set Theory)/Relation
The concept of a characteristic function of a subset carries over directly to relations. Let $\RR \subseteq S \times T$ be a relation. The '''characteristic function of $\RR$''' is the function $\chi_\RR: S \times T \to \set {0, 1}$ defined as: :$\map {\chi_\RR} {x, y} = \begin {cases} 1 & : \tuple {x, y} \in \RR  \\  0 & : \tuple {x, y} \notin \RR \end{cases}$ It can be expressed in Iverson bracket notation as: :$\map {\chi_\RR} {x, y} = \sqbrk {\tuple {x, y} \in \RR}$ More generally, let $\displaystyle \mathbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$. Let $\RR \subseteq \mathbb S$ be an $n$-ary relation on $\mathbb S$. The '''characteristic function of $\RR$''' is the function $\chi_\RR: \mathbb S \to \set {0, 1}$ defined as: :$\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \begin {cases} 1 & : \tuple {s_1, s_2, \ldots, s_n} \in \RR  \\  0 & : \tuple {s_1, s_2, \ldots, s_n} \notin \RR \end {cases}$ It can be expressed in Iverson bracket notation as: :$\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \sqbrk {\tuple {s_1, s_2, \ldots, s_n} \in \RR}$
Definition:Zero (Cardinal)
The cardinal associated with the empty set $\O$ is called '''zero''', and is denoted $0$. More informally, this means that '''zero''' is defined as being '''the number of elements in the empty set'''.
Definition:One (Cardinal)
The cardinal associated with the singleton $\left\{{\varnothing}\right\}$ is called '''one''', and is denoted $1$. More informally, this means that '''one''' is defined as being '''the number of elements in a singleton set'''.
Definition:Product of Cardinals
Let $A$ and $B$ be sets. Let $\mathbf a$ and $\mathbf b$ be the cardinals associated respectively with $A$ and $B$. Then the '''product''' of $\mathbf a$ and $\mathbf b$ is defined as: :$\mathbf a \mathbf b := \operatorname{Card} \left({A \times B}\right)$ where: :$A \times B$ denotes the Cartesian product of $A$ and $B$ :$\operatorname{Card} \left({A \times B}\right)$ denotes the cardinal associated with $A \times B$.
Definition:Quasigroup/Right Quasigroup
$\left({S, \circ}\right)$ is a '''right quasigroup''' : :for all $a \in S$, the right regular representation $\rho_a$ is a permutations on $S$. That is: :$\forall a, b \in S: \exists ! x: x \circ a = b$
Definition:(1-3) Parastrophe
$\struct {S, *}$ is a '''$(1-3)$ parastrophe of $\struct {S, \circ}$''' : :$\forall x_1, x_2, x_3 \in S: x_1 \circ x_2 = x_3 \iff x_3 * x_2 = x_1$
Definition:Quasigroup/Left Quasigroup
$\left({S, \circ}\right)$ is a '''left quasigroup''' : :for all $a \in S$, the left regular representation $\lambda_a$ is a permutations on $S$. That is: :$\forall a, b \in S: \exists ! x: a \circ x = b$
Definition:(2-3) Parastrophe
$\struct {S, *}$ is a '''$(2-3)$ parastrophe of $\struct {S, \circ}$''' : :$\forall x_1, x_2, x_3 \in S: x_1 \circ x_2 = x_3 \iff x_1 * x_3 = x_2$
Definition:B-Algebra
Let $\struct {X, \circ}$ be an algebraic structure. Then $\struct {X, \circ}$ is a '''$B$-algebra''' : {{begin-axiom}} {{axiom | n = \text {AC}         | q = \forall x, y \in X         | m = x \circ y \in X }} {{axiom | n = \text A 0         | m = \exists 0 \in X }} {{axiom | n = \text A 1         | q = \forall x \in X         | m = x \circ x = 0 }} {{axiom | n = \text A 2         | q = \forall x \in X         | m = x \circ 0 = x }} {{axiom | n = \text A 3         | q = \forall x, y, z \in X         | m = \paren {x \circ y} \circ z = x \circ \paren {z \circ \paren {0 \circ y} } }} {{end-axiom}}
Definition:Power (B-Algebra)
Let $\struct {X, \circ}$ be a $B$-algebra. For any $x \in X$ and $n \in \N$, define the '''$n$th power of $x$''', denoted $x^n$, inductively: :$x^n = \begin{cases} 0 & \text {if $n = 0$} \\ x^{n - 1} \circ \paren {0 \circ x} & \text {if $n \ge 1$} \end{cases}$
Definition:Commutative B-Algebra
Let $\struct {X, \circ}$ be a $B$-algebra. Then $\struct {X, \circ}$ is said to be '''$0$-commutative''' (or just '''commutative''') : :$\forall x, y \in X: x \circ (0 \circ y) = y \circ (0 \circ x)$