hcju/proofwikips huggingface dataset:
Definition:Marginal Probability Mass Function
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X: \Pr \to \R$ and $Y: \Pr \to \R$ both be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$. Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$. Then the probability mass functions $p_X$ and $p_Y$ are called the '''marginal (probability) mass functions''' of $X$ and $Y$ respectively. The marginal mass function can be obtained from the joint mass function: {{begin-eqn}} {{eqn | l=p_X \left({x}\right)       | r=\Pr \left({X = x}\right)       | c= }} {{eqn | r=\sum_{y \mathop \in \operatorname{Im} \left({Y}\right)} \Pr \left({X = x, Y = y}\right)       | c= }} {{eqn | r=\sum_y p_{X, Y} \left({x, y}\right)       | c= }} {{end-eqn}}
Definition:Moment of Discrete Random Variable
Let $X$ be a discrete random variable. Then the '''$n$th moment of $X$''' is denoted $\mu'_n$ and defined as: :$\mu'_n = \expect {X^n}$ where $\expect {\, \cdot \,}$ denotes the expectation function.
Definition:Diameter of Quadrilateral
Let $ABCD$ be a quadrilateral: :400px The '''diameters''' of $ABCD$ are the lines $AC$ and $BD$ joining their opposite vertices.
Definition:Matrix Exponential
The '''matrix exponential''', denoted $e^{t \mathbf A}$, of the order $n$ constant square matrix $\mathbf A$, is defined to be the unique solution to the initial value problem: :$X' = \mathbf A X, \ \map X {\mathbf 0} = \mathbf I$ where: :$\mathbf I$ is the unit matrix :$X$ is an order $n$ square matrix which is a function of $t$.
Definition:Stability (Differential Equations)
For first-order autonomous systems, define $\phi \left({t, x_0}\right)$ to be the unique solution with initial condition $x \left({0}\right) = x_0$. {{explain|Make this more precise: "for first-order autonomous systems" should be replaced by a statement specifying something like, "Let $S$ be a first-order autonomous system" (or whatever letter is appropriate for such an object in this context), and then "Let $\phi \left({t, x_0}\right)$ be ... of $S$" or whatever.}} Then a solution with initial condition $x_0$ is '''stable''' on $\left[{0 \,.\,.\, \to}\right)$ if: {{explain|Makes it easier to follow if it is introduced as "Let $\mathbf s$ be a solution ... with initial condition $x_0$.}} :given any $\epsilon > 0$, there exists a $\delta > 0$ such that $\left\Vert{x - x_0}\right\Vert < \delta \implies \left\Vert {\phi \left({t, x}\right) - \phi \left({t, x_0}\right)}\right\Vert < \epsilon$ {{explain|Establish links to define the above notation}} {{refactor|Extract these out into separate pages.}} An equilibrium $x_0$ is '''unstable'''  it is not stable. An equilibrium $x_0$ is '''asymptotically stable''' : :For any $x$ in a sufficiently small neighborhood of $x_0$: ::$\displaystyle \lim_{t \to \infty} \phi \left({t, x}\right) = x_0$  {{definition wanted|This page might be better approached by using the machinery of Convergence.}}
Definition:Nonvanishing
A function $f$ is said to be '''nonvanishing''' if it has no zeroes in its domain. That is, $f$ is '''nonvanishing''' : :$\forall x \in \Dom f: \map f x \ne 0$ In this context, $f$ is (usually) either real-valued or complex-valued. In any case, its codomain needs to contain a zero, so at the very least its codomain needs to be a ring.
Definition:Fundamental Matrix
Let $\mathbf x' = A \left({t}\right) \mathbf x$ be a system of $n$ linear first order ODEs. Let $\Phi \left({t}\right)$ be an $n \times n$ matrix function. Then $\Phi \left({t}\right)$ is a '''fundamental matrix''' of the system $\mathbf x' = A \left({t}\right) \mathbf x$ : :it solves the matrix system $\mathbf X'=A(t) \mathbf X$ :$\det \Phi \left({t}\right)$ is nonvanishing
Definition:Lyapunov Function
Let $x_0$ be an equilibrium point of the system $x' = f \left({x}\right)$. Then a function $V$ is a '''Lyapunov function''' of the system on an open set $U$ containing the equilibrium iff: * $V \left({x_0}\right) = 0$ * $V \left({x}\right) > 0$ if $x \in U \setminus \left\{{x_0}\right\}$ * $\nabla V \cdot f \le 0$ for $x \in U$. {{explain|What $\nabla$ means in this context.}} If the inequality is strict except at $x_0$, then $V$ is '''strict'''.
Definition:Continuously Differentiable
A differentiable function $f$ is '''continuously differentiable'''  $f$ is of differentiability class $C^1$. That is, if the first order derivative of $f$ (and possibly higher) is continuous.
Definition:Homotopy Class
Let $X$ and $Y$ be topological spaces. Let $K \subseteq X$ be any subset. Let $f : X \to Y$ be a continuous mapping. The '''homotopy class''' or '''$K$-homotopy class''' of $f$ is the equivalence class of $f$ under the equivalence relation defined by homotopy relative to $K$. === Homotopy class of path === {{:Definition:Homotopy Class/Path}}
Definition:Cycle (Periodic Solution)
A '''cycle''', or '''periodic solution''', is a solution of a differential equation which is a periodic function.
Definition:Differentiability Class
Let $f: \R \to \R$ be a real function. Then $\map f x$ is of '''differentiability class''' $C^k$  $\dfrac {\d^k} {\d x^k} \map f x$ is continuous. That, $f$ is in '''differentiability class $k$'''  there exists a $k$th derivative of $f$ which is continuous. If $\dfrac {\d^k} {\d x^k} \map f x$ is continuous for all $k \in \N$, then $\map f x$ is of '''differentiability class''' $C^\infty$.
Definition:Direct Image Mapping
Let $S$ and $T$ be sets. Let $\powerset S$ and $\powerset T$ be their power sets. === Relation === {{:Definition:Direct Image Mapping/Relation}} === Mapping === {{:Definition:Direct Image Mapping/Mapping}}
Definition:Intersection (Geometry)
The '''intersection''' of two lines $AB$ and $CD$ is denoted by $AB \cap CD$. The '''intersection''' of two geometric figures is the set of points shared by both figures. Note that this use of $\cap$ is consistent with that of its more usual context of set intersection. When two lines '''intersect''', they are said to '''cut''' each other.
Definition:Generalized Inverse Gaussian Distribution
The '''generalized inverse Gaussian distribution''' ('''GIG''')  is a three-parameter family of continuous probability distributions with probability density function: :$\displaystyle \forall x > 0: f \left({x}\right) = \frac { \left({a/b}\right)^{p/2} } {2 K_p \left({\sqrt{a b} }\right)} x^{\left({p - 1}\right)} e^{-\left({a x + b / x}\right) / 2}$ where: : $K_p$ is a modified Bessel function of the second kind : $a > 0, b > 0, p$ are real.
Definition:Entropy (Probability Theory)
Let $X$ be a discrete random variable that takes on the values of $\set {x_1, x_2, \ldots, x_n}$ and has a probability mass function of $\map p {x_i}$.  Then the '''entropy''' of $X$ is:  :$\ds \map H X := -\sum_{i \mathop = 1}^n \map p {x_i} \log_2 \map p {x_i}$ and is measured in units of '''bits'''.
Definition:Limit of Real Function/Left
Let $\openint a b$ be an open real interval. Let $f: \openint a b \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: b - \delta < x < b \implies \size {\map f x - L} < \epsilon$ where $\R_{>0}$ denotes the set of strictly positive real numbers. That is, for every real strictly positive $\epsilon$ there exists a real strictly positive $\delta$ such that ''every'' real number in the domain of $f$, less than $b$ but within $\delta$ of $b$, has an image within $\epsilon$ of $L$. :400px Then $\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $b$ from the left''', and we write: :$\map f x \to L$ as $x \to b^-$ or :$\displaystyle \lim_{x \mathop \to b^-} \map f x = L$ This is voiced: :'''the limit of $\map f x$ as $x$ tends to $b$ from the left''' and such an $L$ is called: :'''a limit from the left'''.
Definition:Limit of Real Function/Right
Let $\Bbb I = \openint a b$ be an open real interval. Let $f: \Bbb I \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \Bbb I: a < x < a + \delta \implies \size {\map f x - L} < \epsilon$ where $\R_{>0}$ denotes the set of strictly positive real numbers. That is, for every real strictly positive $\epsilon$ there exists a real strictly positive $\delta$ such that ''every'' real number in the domain of $f$, greater than $a$ but within $\delta$ of $a$, has an image within $\epsilon$ of $L$. :400px Then $\map f x$ is said to '''tend to the limit $L$ as $x$ tends to $a$ from the right''', and we write: :$\map f x \to L$ as $x \to a^+$ or :$\displaystyle \lim_{x \mathop \to a^+} \map f x = L$ This is voiced : '''the limit of $\map f x$ as $x$ tends to $a$ from the right''' and such an $L$ is called: : '''a limit from the right'''.
Definition:Limit of Sequence
=== Topological Space === {{:Definition:Limit Point/Topology/Sequence}} === Metric Space === {{:Definition:Limit of Sequence/Metric Space}} === Normed Division Ring === {{:Definition:Limit of Sequence/Normed Division Ring}} === Normed Vector Space === {{:Definition:Limit of Sequence/Normed Vector Space}}
Definition:Bounded Sequence
A special case of a bounded mapping is a '''bounded sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Then $\sequence {x_n}$ is '''bounded'''  $\exists m, M \in T$ such that $\forall i \in \N$: :$(1): \quad m \preceq x_i$ :$(2): \quad x_i \preceq M$ That is,  it is bounded above and bounded below.
Definition:Bézout Numbers
Let $a, b \in \Z$ such that $a \ne 0$ or $b \ne 0$. Let $d$ be the greatest common divisor of $a$ and $b$. By Bézout's Lemma: :$\exists x, y \in \Z: a x + b y = d$ The numbers $x$ and $y$ are known as '''Bézout numbers''' of $a$ and $b$.
Definition:Parity of Integer
Let $z \in \Z$ be an integer. The '''parity''' of $z$ is whether it is even or odd. That is: :an integer of the form $z = 2 n$, where $n$ is an integer, is of '''even parity'''; :an integer of the form $z = 2 n + 1$, where $n$ is an integer, is of '''odd parity'''. :If $z_1$ and $z_2$ are either both even or both odd, $z_1$ and $z_2$ have '''the same parity'''.  :If $z_1$ is even and $z_2$ is odd, then $z_1$ and $z_2$ have '''opposite parity'''.
Definition:Parity (Permutation)
Let $n \in \N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$, that is, let $\rho$ be a permutation of $S_n$. The '''parity''' of $\rho$ is defined as follows: === Even Permutation === {{:Definition:Even Permutation}} === Odd Permutation === {{:Definition:Odd Permutation}} where $\map \sgn \rho$ denotes the sign of $\rho$.
Definition:Linear Operator
A '''linear operator''' is a linear transformation from a module into itself.
Definition:Codomain (Set Theory)
=== Relation === {{:Definition:Codomain (Set Theory)/Relation}} === Mapping === The term '''codomain''' is usually seen when the relation in question is actually a mapping: {{:Definition:Codomain (Set Theory)/Mapping}}
Definition:Kernel of Group Homomorphism
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism. The '''kernel''' of $\phi$ is the subset of the domain of $\phi$ defined as: :$\map \ker \phi := \phi \sqbrk {e_G} = \set {x \in G: \map \phi x = e_H}$ where $e_H$ is the identity of $H$. That is, $\map \ker \phi$ is the subset of $G$ that maps to the identity of $H$.
Definition:Kernel of Ring Homomorphism
Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings. Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. The '''kernel''' of $\phi$ is the subset of the domain of $\phi$ defined as: :$\map \ker \phi = \set {x \in R_1: \map \phi x = 0_{R_2} }$ where $0_{R_2}$ is the zero of $R_2$. That is, $\map \ker \phi$ is the subset of $R_1$ that maps to the zero of $R_2$. From Ring Homomorphism Preserves Zero it follows that $0_{R_1} \in \map \ker \phi$ where $0_{R_1}$ is the zero of $R_1$.
Definition:Range of Sequence
Let $\sequence {x_n}_{n \mathop \in A}$ be a sequence. The '''range of $\sequence {x_n}$''' is the set: :$\set {x_n: n \mathop \in A}$
Definition:Real Sequence
A '''real sequence''' is a sequence (usually infinite) whose codomain is the set of real numbers $\R$.
Definition:Increasing/Sequence
Let $\struct {S, \preceq}$ be a totally ordered set. Let $A$ be a subset of the natural numbers $\N$. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''increasing''' : :$\forall j, k \in A: j < k \implies a_j \preceq a_k$
Definition:Decreasing/Sequence
Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''decreasing''' : :$\forall j, k \in A: j < k \implies a_k \preceq a_j$
Definition:Strictly Increasing/Sequence
Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''strictly increasing''' : :$\forall j, k \in A: j < k \implies a_j \prec a_k$
Definition:Strictly Decreasing/Sequence
Let $\struct {S, \preceq}$ be a totally ordered set. Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is '''strictly decreasing''' : :$\forall j, k \in A: j < k \implies a_k \prec a_j$
Definition:Monotone (Order Theory)/Sequence
Let $\struct {S, \preceq}$ be a totally ordered set. A sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is '''monotone'''  it is either increasing or decreasing.
Definition:Rational Sequence
A '''rational sequence''' is a sequence (usually infinite) whose codomain is the set of rational numbers $\Q$.
Definition:Self-Map
Let $S$ be a set. A '''self-map on $S$''' is a mapping from $S$ to itself: :$f: S \to S$
Definition:Renaming Mapping
Let $f: S \to T$ be a mapping. The '''renaming mapping''' $r: S / \RR_f \to \Img f$ is defined as: :$r: S / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = \map f x$ where: :$\RR_f$ is the equivalence induced by the mapping $f$ :$S / \RR_f$ is the quotient set of $S$ determined by $\RR_f$ :$\eqclass x {\RR_f}$ is the equivalence class of $x$ under $\RR_f$.
Definition:Fixed Element
=== Fixed Point of General Mapping === {{:Definition:Fixed Point}} === Fixed Element of Permutation === The concept is particularly important when studying permutations in the context of group theory: {{:Definition:Fixed Element of Permutation}}
Definition:Peano Structure
A '''Peano structure''' $\struct {P, 0, s}$ comprises a set $P$ with a successor mapping $s: P \to P$ and a non-successor element $0$. These three together are required to satisfy Peano's axioms.
Definition:Ordered Set
An '''ordered set''' is a relational structure $\struct {S, \preceq}$ such that the relation $\preceq$ is an ordering. Such a structure may be: :A partially ordered set (poset) :A totally ordered set (toset) :A well-ordered set (woset) depending on whether the ordering $\preceq$ is: :A partial ordering :A total ordering :A well-ordering.
Definition:Hasse Diagram
Let $\struct {S, \preceq}$ be an ordered set. A '''Hasse diagram''' is a method of representing $\struct {S, \preceq}$ as a graph $G$, in which: :$(1):\quad$ The vertices of $G$ represent the elements of $S$ :$(2):\quad$ The edges of $G$ represent the elements of $\preceq$ :$(3):\quad$ If $x, y \in S: x \preceq y$ then the edge representing $x \preceq y$ is drawn so that $x$ is lower down the page than $y$. ::::That is, the edge ascends (usually obliquely) from $x$ to $y$ :$(4):\quad$ If $x \preceq y$ and $y \preceq z$, then as an ordering is transitive it follows that $x \preceq z$. ::::But in a '''Hasse diagram''', the relation $x \preceq z$ is not shown. ::::Transitivity is implicitly expressed by the fact that $z$ is higher up than $x$, and can be reached by tracing a path from $x$ to $z$ completely through ascending edges.
Definition:Conditional/Antecedent
In a conditional $p \implies q$, the statement $p$ is the '''antecedent'''.
Definition:Conditional/Consequent
In a conditional $p \implies q$, the statement $q$ is the '''consequent'''.
Definition:Cardinal
Let $S$ be a set. Associated with $S$ there exists a set $\map \Card S$ called the '''cardinal of $S$'''. It has the properties: : $(1): \quad \map \Card S \sim S$ that is, $\map \Card S$ is (set) equivalent to $S$ : $(2): \quad S \sim T \iff \map \Card S = \map \Card T$
Definition:Philosophical Position
A '''philosophical position''' is, broadly speaking, a belief that a particular statement is either true or false. In the context of mathematics the term is usually reserved for particular fundamental questions whose truth values have not been explicitly determined in any chosen frame of reference or axiom schema. Hence, in order to progress down a particular logical path, one needs to make a decision as to which it is. Usually for such questions, its answer is of far-reaching importance. It having been necessary to take such a position, one then finds that diverging schools of thought arise: one which '''accepts''' and one which '''rejects''' the truth of the question. The answer to the question itself is added to the particular axiom schema for that school to form an expanded schema.
Definition:Set of Sets
A '''set of sets''' is a set, whose elements are themselves all sets. Those elements can themselves be assumed to be subsets of some particular fixed set which is frequently referred to as the universe.
Definition:Well-Orderable Set
Let $S$ be a set. If it is possible to construct an ordering $\preceq$ on $S$ such that $\preceq$ is a well-ordering, then $S$ is defined as being '''well-orderable'''.
Definition:Choice Function
Let $\mathbb S$ be a set of sets such that: :$\forall S \in \mathbb S: S \ne \O$ that is, none of the sets in $\mathbb S$ may be empty. A '''choice function on $\mathbb S$''' is a mapping $f: \mathbb S \to \bigcup \mathbb S$ satisfying: :$\forall S \in \mathbb S: \map f S \in S$. That is, for any set in $\mathbb S$, a '''choice function''' selects an element from that set.   The domain of $f$ is $\mathbb S$.
Definition:Distributive Operation/Right
The operation $\circ$ is '''right distributive''' over the operation $*$ : :$\forall a, b, c \in S: \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$
Definition:Distributive Operation/Left
The operation $\circ$ is '''left distributive''' over the operation $*$ : :$\forall a, b, c \in S: a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$
Definition:Relation Induced by Partition
Let $S$ be a set. Let $\Bbb S$ be a partition of a set $S$. Let $\mathcal R \subseteq S \times S$ be the relation defined as: :$\forall \tuple {x, y} \in S \times S: \tuple {x, y} \in \mathcal R \iff \exists T \in \Bbb S: \set {x, y} \subseteq T$ Then $\mathcal R$ is the '''(equivalence) relation induced by (the partition) $\Bbb S$'''.
Definition:Definition
A '''definition''' lays down the meaning of a concept. It is a statement which tells the reader '''what something is'''. It can be understood as an equation in (usually) natural language.
Definition:Stipulative Definition
A '''stipulative definition''' is a definition which defines how to interpret the meaning of a symbol. It '''stipulates''', or lays down, the meaning of a symbol in terms of previously defined symbols or concepts. The symbol used for a stipulative definition is: :$\text {(the symbol being defined)} := \text {(the meaning of that symbol)}$ This can be written the other way round: :$\text {(a concept being assigned a symbol)} =: \text {(the symbol for it)}$ when it is necessary to emphasise that the symbol has been crafted to abbreviate the notation for the concept.
Definition:Ostensive Definition
An '''ostensive definition''' is a definition which ''shows'' what a symbol is, rather than use words to ''explain'' what it is or what it does. As an example of an '''ostensive definition''', we offer up: :The symbol used for a stipulative definition is $:=$, as in: ::$\text {(the symbol being defined)} := \text {(the meaning of that symbol)}$
Definition:Formal Grammar/Bottom-Up/Extremal Clause
Let $\mathcal F$ be a formal language. Let the rules of formation of $\mathcal F$ be defined in a bottom-up manner. <section notitle="true" name="tc"> The '''extremal clause''' of a bottom-up grammar is the final rule which excludes all collations other than those specified in the formation rules from being well-formed formulas. </section>
Definition:Occurrence (Formal Systems)
Let $\mathcal F$ be a formal language. Let $S, T$ be collations in the alphabet of $\mathcal F$. Each place where $S$ appears in $T$ is called an '''occurrence of $S$ in $T$'''.
Definition:Null String
A '''null string''' (or '''empty string''') is a string with no symbols in it. In particular, the '''null string''' is a word. A null string has a length of $0$.
Definition:Substring
Let $\mathcal L$ be a formal language with alphabet $\mathcal A$. Let $S$ be a string in $\mathcal A$. Let $T$ be a string in $\mathcal A$ such that: :$S = S_1 T S_2$ where: * $S_1$ and $S_2$ are strings in $\mathcal A$ (possibly null); * $S_1 T S_2$ is the concatenation of $S_1$, $T$ and $S_2$. Then $T$ is called a '''substring of $S$'''. It follows from this definition that $S$ is a substring of itself (by considering $S_1$ and $S_2$ as both null).
Definition:Subordinate
Let $\mathbf A$ be a WFF of propositional logic. Let $\circ$ and $\ast$ be connectives. Then $\circ$ is '''subordinate to $\ast$ (in $\mathbf A$)''' iff the scope of $\circ$ is a well-formed part of the scope of $\ast$.
Definition:Venn Diagram
A '''Venn diagram''' is a technique for the graphic depiction of the interrelationship between a small number (usually $3$ or fewer) of sets. The following diagram illustrates the various operations between three sets. :500px The circles represent the sets $S_1$, $S_2$ and $S_3$. The '''white''' surrounding box represents the universal set $\mathbb U$. Each of the areas inside the various circle represents an intersection between the various sets and their complements, as follows: :The '''gray''' area represents $S_1 \cap S_2 \cap S_3$. :The '''purple''' area represents $S_1 \cap S_2 \cap \overline {S_3}$. :The '''orange''' area represents $S_1 \cap \overline {S_2} \cap S_3$. :The '''green''' area represents $\overline {S_1} \cap S_2 \cap S_3$. :The '''red''' area represents $S_1 \cap \overline {S_2} \cap \overline {S_3}$. :The '''blue''' area represents $\overline {S_1} \cap S_2 \cap \overline {S_3}$. :The '''yellow''' area represents $\overline {S_1} \cap \overline {S_2} \cap S_3$. :The surrounding '''white''' area represents $\overline {S_1} \cap \overline {S_2} \cap \overline {S_3}$. The notation $\overline {S_1}$ denotes set complement. If it is required to show on a diagram that a particular intersection is empty, then it is generally shaded '''black'''.
Definition:Order Type
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be totally ordered sets. Then $S$ and $T$ '''have the same (order) type'''  they are order isomorphic.
Definition:Order Type of Natural Numbers
The order type of $\struct {\N, \le}$ is denoted $\omega$ ('''omega''').
Definition:Ordered Sum
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be tosets. Let: * the order type of $\left({S, \preceq_1}\right)$ be $\theta_1$ * the order type of $\left({T, \preceq_2}\right)$ be $\theta_2$. Let $S \cup T$ be the union of $S$ and $T$. We define the ordering $\preceq$ on $S$ and $T$ as: :$\forall s \in S, t \in T: a \preceq b \iff \begin{cases} a \preceq_1 b & : a \in S \land b \in S \\ a \preceq_2 b & : \neg \left({a \in S \land b \in S}\right) \land \left({a \in T \land b \in T}\right) \\ & : a \in S, b \in T \end{cases}$ That is: : If $a$ and $b$ are both in $S$, they are ordered as they are in $S$. : If $a$ and $b$ are not both in $S$, but they ''are'' both in $T$, they are ordered as they are in $T$. : Otherwise, that is if $a$ and $b$ are in ''both'' sets, their ordering in $S$ takes priority over that in $T$. The ordered set $\left({S \cup T, \preceq}\right)$ is called the '''ordered sum''' of $S$ and $T$, and is denoted $S + T$.