hcju/proofwikips huggingface dataset:
Definition:Semi-Inner Product
Let $\C$ be the field of complex numbers. Let $\F$ be a subfield of $\C$. Let $V$ be a vector space over $\F$ A '''semi-inner product''' is a mapping $\innerprod \cdot \cdot: V \times V \to \mathbb F$ that satisfies the following properties: {{begin-axiom}} {{axiom | n = 1         | lc= Conjugate Symmetry         | q = \forall x, y \in V         | m = \quad \innerprod x y = \overline {\innerprod y x} }} {{axiom | n = 2         | lc= Sesquilinearity         | q = \forall x, y, z \in V, \forall a \in \mathbb F         | m = \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z }} {{axiom | n = 3         | lc= Non-Negative Definiteness         | q = \forall x \in V         | m = \quad \innerprod x x \in \R_{\ge 0} }} {{end-axiom}} {{refactor|This section merits a separate page: "Semi-Inner Product in Real Number Field", perhaps.}} If $\mathbb F$ is a subfield of the field of real numbers $\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\overline {\innerprod y x} = \innerprod y x$ for all $x, y \in V$. Then $(1)$ above may be replaced by: {{begin-axiom}} {{axiom | n = 1^\prime         | lc= Symmetry         | q = \forall x, y \in V         | m = \innerprod x y = \innerprod y x }} {{end-axiom}} === Semi-Inner Product Space === {{:Definition:Semi-Inner Product Space}}
Definition:Semi-Inner Product Space
A '''semi-inner product space''' is a vector space together with an associated semi-inner product.
Definition:Inner Product Norm
Let $V$ be an inner product space over a subfield $\Bbb F$ of $\C$. Let $\left \langle{\cdot, \cdot}\right \rangle$ be the inner product of $V$. Then the '''inner product norm''' on $V$ is the mapping $\left\Vert{\cdot}\right\Vert: V \to \R_{\ge 0}$ given by :$\left\Vert{x}\right\Vert := \left\langle{x,x}\right\rangle^{1/2}$.
Definition:Metric Induced by Norm
Let $V$ be a normed vector space. Let $\norm{\,\cdot\,}$ be the norm of $V$. Then the '''induced metric''' or the '''metric induced by $\norm{\,\cdot\,}$''' is the map $d: V \times V \to \R_{\ge 0}$ defined as: :$d \left({x, y}\right) = \left\Vert{x - y}\right\Vert$
Definition:Hilbert Space
Let $V$ be an inner product space over $\Bbb F \in \set {\R, \C}$. Let $d: V \times V \to \R_{\ge 0}$ be the metric induced by the inner product norm $\norm {\,\cdot\,}_V$. If $\struct {V, d}$ is a complete metric space, $V$ is said to be a '''Hilbert space'''.
Definition:Orthogonal (Hilbert Space)
Let $H$ be a Hilbert space. Let $f, g \in H$. Let $\innerprod f g = 0$, where $\innerprod \cdot \cdot$ denotes the inner product. Then $f$ and $g$ are defined as being '''orthogonal''': :$f \perp g$ === Sets === Let $A, B \subseteq H$. Then $A$ and $B$ are defined as '''orthogonal''' : :$\forall a \in A, b \in B: a \perp b$ This is denoted by $A \perp B$.  When $A, B$ are closed linear subspaces of $H$, they are called '''orthogonal subspaces'''. If $A$ consists of only one element $a$, also $a \perp B$ is encountered.
Definition:Convex Set (Vector Space)
Let $V$ be a vector space over $\R$ or $\C$. A subset $A \subseteq V$ is said to be '''convex''' : :$\forall x, y \in A: \forall t \in \closedint 0 1: t x + \paren {1 - t} y \in A$ === Line Segment === {{:Definition:Convex Set (Vector Space)/Line Segment}}
Definition:Distance/Points
=== Metric Space === Let $\struct {X, d}$ be a metric space. {{:Definition:Metric Space/Distance Function}} === Normed Vector Space === {{:Definition:Distance/Points/Normed Vector Space}} === Real Numbers === {{:Definition:Distance/Points/Real Numbers}} === Complex Numbers === {{:Definition:Distance/Points/Complex Numbers}}
Definition:Distance/Sets
=== Real Numbers === {{:Definition:Distance/Sets/Real Numbers}} === Metric Spaces === {{:Definition:Distance/Sets/Metric Spaces}}
Definition:Outer Automorphism
Let $G$ be a group. Let $\phi: G \to G$ be an automorphism which is not an inner automorphism. Then $\phi$ is an '''outer automorphism'''.
Definition:Group of Outer Automorphisms
Let $G$ be a group. Let $\Aut G$ be the automorphism group of $G$. Let $\Inn G$ be the inner automorphism group of $G$. Let $\dfrac {\Aut G} {\Inn G}$ be the quotient group of $\Aut G$ by $\Inn G$. Then $\dfrac {\Aut G} {\Inn G}$ is called the '''group of outer automorphisms''' of $G$. This group $\dfrac {\Aut G} {\Inn G}$ is often denoted $\Out G$.
Definition:Characteristic Subgroup
Let $G$ be a group. Let $H$ be a subgroup such that: :$\forall \phi \in \Aut G: \phi \sqbrk H = H$ where $\Aut G$ is the automorphism group of $G$. Then $H$ is  '''characteristic (in $G$)''', or '''a characteristic subgroup of $G$'''.
Definition:Projection (Analytic Geometry)
Let $M$ and $N$ be distinct lines through the origin in the plane. The '''projection on $M$ along $N$''' is the mapping $\pr_{M, N}$ such that: :$\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.
Definition:Orthogonal Projection
Let $H$ be a Hilbert space. Let $K$ be a closed linear subspace of $H$. Then the '''orthogonal projection''' on $K$ is the map $P_K: H \to H$ defined by :$k = \map {P_K} h \iff k \in K$ and $\map d {h, k} = \map d {h, K}$ where the latter $d$ signifies distance to a set. {{refactor|Extract the 3 statements below and put them into their own pages}} That $P_K$ is well-defined follows from Unique Point of Minimal Distance. The name orthogonal projection stems from the fact that $\paren {h - \map {P_K} h} \perp K$. This and other properties of $P_K$ are collected in Properties of Orthogonal Projection.
Definition:Normal Series
Let $G$ be a group whose identity is $e$. A '''normal series''' for $G$ is a sequence of (normal) subgroups of $G$: :$\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_n = G$ where $G_{i - 1} \lhd G_i$ denotes that $G_{i - 1}$ is a proper normal subgroup of $G_i$.
Definition:Refinement of Normal Series
Let $G$ be a group whose identity is $e$. Let $\sequence {G_i}_{i \mathop \in \closedint 0 n}$ be a normal series for $G$: :$\sequence {G_i}_{i \mathop \in \closedint 0 n} = \tuple {\set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G}$ Let $\sequence {H_j}_{j \mathop \in \closedint 0 m}$ be another normal series for $G$: :$\sequence {H_j}_{j \mathop \in \closedint 0 m} = \tuple {\set e = H_0 \lhd H_1 \lhd \cdots \lhd H_{m - 1} \lhd H_m = G}$ such that $\sequence {G_i}_{i \mathop \in \closedint 0 n} \subseteq \sequence {H_j}_{j \mathop \in \closedint 0 m}$ Then $\sequence {H_j}_{j \mathop \in \closedint 0 m}$ is a '''refinement''' of $\sequence {G_i}_{i \mathop \in \closedint 0 n}$. That is, a '''refinement''' of a normal series is a normal series which contains all the (normal) subgroups of the original normal series, and may contain more. === Proper Refinement === {{:Definition:Refinement of Normal Series/Proper Refinement}}
Definition:Composition Series
Let $G$ be a finite group. A '''composition series for $G$''' is a normal series for $G$ which has no proper refinement. === Composition Length === {{:Definition:Composition Series/Composition Length}} === Composition Factor === {{:Definition:Composition Series/Composition Factor}}
Definition:Contradictory Statements
Two statements $p$ and $q$ are said to be '''contradictory''' iff: : whenever $p$ is true, $q$ is false. and: : whenever $q$ is true, $p$ is false.
Definition:Contrary Statements
Two statements are said to be '''contrary''' if they can both be false, but they cannot both be true.
Definition:Subcontrary Statements
Two statements are said to be '''subcontrary''' if they can both be true, but they cannot both be false.
Definition:Closed Linear Span
Let $H$ be a Hilbert space, and let $A \subseteq H$ be a subset. Then the '''closed linear span''' of $A$, denoted $\vee A$, is defined in the following ways: :$(1): \qquad \displaystyle \vee A = \bigcap \Bbb M$, where $\Bbb M$ consists of all closed linear subspace $M$ of $H$ with $A \subseteq M$ :$(2): \qquad \vee A$ is the smallest closed linear subspace $M$ of $H$ with $A \subseteq M$ :$(3): \qquad \vee A = \map \cl {\map {\operatorname {span} } A}$, where $\cl$ denotes closure, and $\operatorname {span}$ denotes linear span.
Definition:Zero Subspace
Let $V$ be a vector space with zero vector $\mathbf 0$. Then the set $(\mathbf 0) := \left\{{\mathbf 0}\right\}$ is called the '''zero subspace''' of $V$. This name is appropriate as $(\mathbf 0)$ is in fact a subspace of $V$, as proved in Zero Subspace is Subspace. {{Explain|I'm not sure about the nomenclature, but this deserves a name. Feel free to adapt if you have sources to back it up.}}
Definition:Length of Group
Let $G$ be a finite group. The '''length''' of $G$ is the length of a composition series for $G$. That is, the '''length''' of $G$ is the number of factors in a composition series for $G$ (not including $G$ itself). The '''length''' of $G$ can be denoted $\map l G$. By the Jordan-Hölder Theorem, all composition series for $G$ have the same length. Therefore, the '''length''' of a finite group $G$ is well-defined.
Definition:Bounded Linear Functional
Let $H$ be a Hilbert space, and let $L$ be a linear functional on $H$. Then $L$ is said to be a '''bounded linear functional''' iff :$\exists c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|$ In view of Continuity of Linear Functionals, a linear functional on a Hilbert space is bounded if and only if it is continuous.
Definition:Orthonormal Subset
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space. Let $S \subseteq V$ be a subset of $V$. Then $S$ is an '''orthonormal subset''' : :$(1): \quad \forall u \in S: \norm u = 1$ where $\norm {\, \cdot \,}$ is the inner product norm. :$(2): \quad S$ is an orthogonal set: ::$\forall u, v \in S: u \ne v \implies \innerprod u v = 0$
Definition:Basis (Hilbert Space)
Let $H$ be a Hilbert space. A '''basis for $H$''' is a maximal orthonormal subset of $H$. Thus, $B$ is a '''basis''' for $H$  for all orthonormal subsets $B'$ of $H$: :$B \subseteq B' \implies B = B'$
Definition:Linear Span
Let $V$ be a vector space over $K$. Let $A \subseteq V$ be a subset of $V$. Then the '''linear span''' of $A$, denoted $\operatorname{span} A$ or $\map {\operatorname{span} } A$, is the set: :$\displaystyle \set {\sum_{k \mathop = 1}^n \alpha_k f_k: n \in \N_{\ge 1}, \alpha_i \in K, f_i \in A}$ The '''linear span''' can be interpreted as the set of all linear combinations (of finite length) of these vectors. === Definition for $\R^n$ === In $\R^n$ (where $n \in \N_{>0}$), above definition translates to: :$\displaystyle \map {\operatorname{span} } {\mathbf v_1, \mathbf v_2, \dotsc, \mathbf v_k} = \set {\sum_{i \mathop = 1}^k \ c_i \ \mathbf v_i: c_i \in \R, \mathbf v_i \in \R^n, 1 \le i \le k}$
Definition:Prime Group
Let $G$ be a group. Then $G$ is a '''prime group'''  the order of $G$ is a prime number.
Definition:Solvable Group
Let $G$ be a finite group. Then $G$ is a '''solvable group'''  it has a composition series in which each factor is a cyclic group.
Definition:Bayesian Probability Model
The '''Bayesian probability model''' is a mathematical model that defines the probability of an event occurring as a degree of belief. That is, that probability is defined as the degree of which it is rational to believe certain statements based on intuition, experience, judgment, or opinion.  {{NamedforDef|Thomas Bayes|cat = Bayes}}
Definition:Amphiboly
An '''amphiboly''' is an ambiguity resulting from poor syntax.
Definition:Permutation on n Letters/Two-Row Notation
Let $\pi$ be a permutation on $n$ letters. The '''two-row notation''' for $\pi$ is written as two rows of elements of $\N_n$, as follows: :$\pi = \begin{pmatrix} 1 & 2 & 3 & \ldots & n \\ \map \pi 1 & \map \pi 2 & \map \pi 3 & \ldots & \map \pi n \end{pmatrix}$ The bottom row contains the effect of $\pi$ on the corresponding entries in the top row.
Definition:Inductive Argument
An '''inductive argument''' is a form of argument in which, if all the premises are true, the conclusion is ''probably true'', but might not be.  Such lines of reasoning are ubiquitous in everyday life and in most human endeavors. However, '''inductive arguments''' are only conjectures in the field of mathematics. Such arguments are not truth preserving and therefore they are not proofs.
Definition:Syllogism
A '''syllogism''' is an argument with exactly two premises and one conclusion.
Definition:Membership Relation
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\mathcal R \subseteq S \times \powerset S$ be the relation defined as: :$\tuple {x, A} \in \mathcal R \iff x \in A$ Thus $\mathcal R$ is the relation between elements of $S$ and subsets of $S$ expressing '''membership'''.
Definition:Image (Set Theory)/Relation/Element
Let $s \in S$. The '''image of $s$ by''' (or '''under''') '''$\RR$''' is defined as: :$\map \RR s := \set {t \in T: \tuple {s, t} \in \RR}$ That is, $\map \RR s$ is the set of all elements of the codomain of $\RR$ related to $s$ by $\RR$.
Definition:Image (Set Theory)/Relation
=== Image of a Relation === {{:Definition:Image (Set Theory)/Relation/Relation}} === Image of an Element === {{:Definition:Image (Set Theory)/Relation/Element}} === Image of a Subset === {{:Definition:Image (Set Theory)/Relation/Subset}}
Definition:Image (Set Theory)/Mapping
{{transclude:Definition:Image (Set Theory)/Mapping/Mapping   |section = tc   |title = Image of a Mapping   |link = true   |header = 3   |increase = 1 }} === Image of an Element === {{:Definition:Image (Set Theory)/Mapping/Element}} === Image of a Subset === {{:Definition:Image (Set Theory)/Mapping/Subset}}
Definition:Image (Set Theory)/Mapping/Element
Let $s \in S$. The '''image of $s$ (under $f$)''' is defined as: :$\Img s = \map f s = \displaystyle \bigcup \set {t \in T: \tuple {s, t} \in f}$ That is, $\map f s$ is the element of the codomain of $f$ related to $s$ by $f$.
Definition:Image (Set Theory)/Mapping/Subset
Let $f: S \to T$ be a mapping. Let $X \subseteq S$ be a subset of $S$.
Definition:Tree (Set Theory)
Let $\struct {T, \preceq}$ be an ordered set. Let $\struct {T, \preceq}$ be such that for every $t \in T$, the lower closure of $t$: :$t^\preceq := \set {s \in T: s \preceq t}$  is well-ordered by $\preceq$. Then $\struct {T, \preceq}$ is a '''tree'''. === Branch === A '''branch''' of a tree $T$ is a maximal chain in $T$. === Subtree === A '''subtree''' of a tree $\struct {T, \preceq}$ is an ordered subset $\struct {S, \preceq}$ with the property that: :for every $\forall s \in S: \forall t \in T: t \preceq s \implies t \in S$
Definition:Basic Open Set
Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal B \subseteq \tau$ be a basis for $T$. Let $U \in \mathcal B$. Then $U$ is a '''basic open set''' of $T$. That is, a '''basic open set''' of a topology is an open set of that topology which is an element of a basis for that topology. The basis itself needs to be specified for this definition to make sense.
Definition:Sine Integral Function
The '''sine integral function''' is the real function $\Si: \R \to \R$ defined as: :$\map \Si x = \begin{cases} \displaystyle \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t & : x \ne 0 \\ \vphantom x \\ 0 & : x = 0 \\ \end{cases}$
Definition:Greatest
=== Ordered Set === {{:Definition:Greatest/Ordered Set}} === Greatest Set === {{:Definition:Greatest Set by Set Inclusion}} === Mapping === {{:Definition:Greatest/Mapping}}
Definition:Smallest
=== Ordered Set === {{:Definition:Smallest/Ordered Set}} === Smallest Set === {{:Definition:Smallest Set by Set Inclusion}} === Mapping === {{:Definition:Smallest/Mapping}}
Definition:Arc Length
Let $y = \map f x$ be a real function which is: : continuous on the closed interval $\closedint a b$ and: : continuously differentiable on the open interval $\openint a b$. The '''arc length''' $s$ of $f$ between $a$ and $b$ is defined as:  :$s := \displaystyle \int_a^b \sqrt {1 + \paren {\frac {\d y} {\d x} }^2} \rd x$
Definition:Greatest/Ordered Set
Let $\struct {S, \preceq}$ be an ordered set. An element $x \in S$ is '''the greatest element (of $S$)''' : :$\forall y \in S: y \preceq x$ That is, every element of $S$ '''precedes, or is equal to,''' $x$. The Greatest Element is Unique, so calling it '''''the'' greatest element''' is justified. Thus for an element $x$ to be the '''greatest element''', all $y \in S$ must be comparable to $x$.
Definition:Smallest/Ordered Set
Let $\struct {S, \preceq}$ be an ordered set. An element $x \in S$ is '''the smallest element''' : :$\forall y \in S: x \preceq y$ That is, $x$ '''strictly precedes, or is equal to,''' every element of $S$. The Smallest Element is Unique, so calling it '''''the'' smallest element''' is justified. The '''smallest element''' of $S$ is denoted $\min S$. For an element to be the '''smallest element''', all $y \in S$ must be comparable to $x$.
Definition:Radicand
Let $\sqrt[n] P$ denote the $n$th root of an expression. Then $P$ is the '''radicand''' of $\sqrt[n] P$. That is, it is '''the object whose $n$th root is to be taken'''.
Definition:Finite Ring
A finite ring is a ring $\left({R, +, \circ}\right)$ such that $R$ is a finite set.
Definition:Removable Discontinuity
=== Real Function === {{Definition:Removable Discontinuity of Real Function}} {{definition wanted|discuss functions of $n$ variables, complex functions}}
Definition:Generalized Sum
Let $\left({G, +}\right)$ be a commutative topological semigroup. Let $\left({g_i}\right)_{i \in I}$ be an indexed subset of $G$. Consider the set $\mathcal F$ of finite subsets of $I$. Let $\subseteq$ denote the subset relation on $\mathcal F$. By virtue of Finite Subsets form Directed Set, $\left({\mathcal F, \subseteq}\right)$ is a directed set. Define the net: :$\phi: \mathcal F \to G$ by: : $\displaystyle \phi \left({F}\right) = \sum_{i \mathop \in F} g_i$ Then $\phi$ is denoted: : $\displaystyle \sum \left\{{g_i: i \in I}\right\}$ and referred to as a '''generalized sum'''.
Definition:Dimension (Hilbert Space)
Let $H$ be a Hilbert space, and let $E$ be a basis of $H$. Then the '''dimension''' $\dim H$ of $H$ is defined as $\card E$, the cardinality of $E$.
Definition:Successor Set
Let $S$ be a set. The '''successor (set) of $S$''' is defined and denoted: :$S^+ := S \cup \set S$
Definition:Infinite Successor Set
Let $S$ be a set. Then $S$ is an '''infinite successor set''' : :$\O \in S$ :$x \in S \implies x^+ \in S$ where $\O$ is the empty set, and $x^+$ denotes the successor set of $x$. === Axiomatic Set Theory === The concept of '''infinite successor set''' is axiomatised in the Axiom of Infinity in Zermelo-Fraenkel set theory: :$\exists x: \paren {\paren {\exists y: y \in x \land \forall z: \neg \paren {z \in y} } \land \forall u: u \in x \implies \paren {u \cup \set u \in x} }$
Definition:Undefined Term
For a definition to not be circular, the definer must use already defined terms.  However, this process cannot go on indefinitely. If we were to insist on ''everything'' being defined only using previously defined terms, we would enter an infinite regress. Concepts that are not defined in terms of previously defined concepts are called '''undefined terms'''. An undefined term is frequently explained by using an ostensive definition: that is, a statement that ''shows'' what something is, rather than ''explains''.
Definition:Power (Algebra)/Natural Number
Let $\N$ denote the natural numbers. For each $m \in \N$, recursively define $e_m: \N \to \N$ to be the mapping: :$e_m \left({n}\right) = \begin{cases} 1 & : n = 0 \\ m \times e_m \left({x}\right) & : n = x + 1 \end{cases}$ where: : $+$ denotes natural number addition. : $\times$ denotes natural number multiplication. $e_m \left({n}\right)$ is then expressed as a binary operation in the form: :$m^n := e_m \left({n}\right)$ and is called '''$m$ to the power of $n$'''.
Definition:Topological Semigroup
Let $\left({S, \circ}\right)$ be a semigroup. On that same underlying set $S$, let $\left({S, \tau}\right)$ be a topological space. Then $\left({S, \circ, \tau}\right)$ is said to be a '''topological semigroup''' if: :$\circ: \left({S, \tau}\right) \times \left({S, \tau}\right) \to \left({S, \tau}\right)$ is a continuous mapping where $\left({S, \tau}\right) \times \left({S, \tau}\right)$ is considered as $S \times S$ with the product topology.
Definition:Isomorphism (Hilbert Spaces)
Let $H, K$ be Hilbert spaces. Denote by $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ their respective inner products. An '''isomorphism''' between $H$ and $K$ is a map $U: H \to K$, such that: :$(1): \quad U$ is a linear map :$(2): \quad U$ is surjective :$(3): \quad \forall g, h \in H: \innerprod g h_H = \innerprod {U g} {U h}_K$ These three requirements may be summarized by stating that $U$ be a surjective isometry. Furthermore, Surjection that Preserves Inner Product is Linear shows that requirement $(1)$ is superfluous. If such an '''isomorphism''' $U$ exists, $H$ and $K$ are said to be '''isomorphic'''. As the name '''isomorphism''' suggests, Hilbert Space Isomorphism is Equivalence Relation.
Definition:Isometry (Hilbert Spaces)
Let $H, K$ be Hilbert spaces, and denote by $\innerprod \cdot \cdot_H$ and $\innerprod \cdot \cdot_K$ their respective inner products. A linear map $U: H \to K$ is called an '''isometry''' : :$\forall g,h \in H: \innerprod g h_H = \innerprod {U g} {U h}_K$ {{explain|Notation $U g, U h$ may need to be amended so as to make it clear that it means the application of the mapping $U$ on both of $g$ and $h$. If this is indeed what it means (as is probable) then suggest $\map U g$ and $\map U h$.}}
Definition:Unitary Operator
A '''unitary operator''' is an isomorphism from a Hilbert space $H$ to itself.
Definition:Riemann Sum
Let $f$ be a real function defined on the closed interval $\mathbb I = \closedint a b$. Let $\Delta$ be a subdivision of $\mathbb I$. For $1 \le i \le n$: :let $\Delta x_i = x_i - x_{i - 1}$ :let $c_i \in \closedint {x_{i - 1} } {x_i}$. The summation: :$\displaystyle \sum_{i \mathop = 1}^n \map f {c_i} \Delta x_i$ is called a '''Riemann sum''' of $f$ for the subdivision $\Delta$. === Geometric Interpretation ===
Definition:Total Relation
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. Then $\mathcal R$ is defined as '''total''' : :$\forall a, b \in S: \tuple {a, b} \in \mathcal R \lor \tuple {b, a} \in \mathcal R$ That is,  every pair of elements is related (either or both ways round).
Definition:Signed Area
Consider the plane region described by a graph in the $xy$-plane. The '''signed area''' of the region is the area such that: : The area of the graph on or above the $x$-axis is defined as positive : The area of the graph below the $x$-axis is defined as negative. {{explain|As this is a definition which requires the use of spatial relationships, might be good to have a diagram of this.}}
Definition:Initial Segment
Let $\struct {S, \preceq}$ be a well-ordered set. Let $a \in S$. The '''initial segment (of $S$) determined by $a$''' is defined as: :$S_a := \set {b \in S: b \preceq a \land b \ne a}$ which can also be rendered as: :$S_a := \set {b \in S: b \prec a}$ That is, $S_a$ is the set of all elements of $S$ that strictly precede $a$. That is, $S_a$ is the strict lower closure of $a$ (in $S$). By extension, $S_a$ is described as '''an initial segment (of $S$)'''.
Definition:Preordering/Preordered Set
Let $S$ be a set. Let $\precsim$ be a preordering on $S$. Then the relational structure $\left({S, \precsim}\right)$ is called a '''preordered set'''.