Definition:Tetration

500pxrightthumb$y = \operatorname{tet}_b \left({x}\right)$ versus $x$ for various $b$ 500pxrightthumb$f = \operatorname{tet}_b \left({x}\right)$ in the $x, b$ plane with levels $f = \text{const}$. === Definition for Integers === For all $x \in \R$, $n \in \Z_{\ge 0}$: :${}^n x := \begin{cases} 1 & : n = 0 \\ x^{ \left({{}^{n - 1} x}\right)} & : n > 0 \\ \end{cases}$ Using Knuth uparrow notation: :$x \uparrow \uparrow n := \begin{cases} 1 & : n = 0 \\ x \uparrow \left({x \uparrow \uparrow \left({n - 1}\right)}\right) & : n > 0 \\ \end{cases}$ === Definition for base $b \ge \exp \left({1 / e}\right)$ === Let $b \in \R$ such that $b \ge \exp \left({\dfrac 1 e}\right)$. Let $L \in \C$ be a fixed point of $\log_b$ such that $\Im \left({L}\right) \ge 0$. Let $C = \C \setminus \left\{ {x \in \R: x \le -2}\right\}$. Let $\operatorname{tet}_b: C \mapsto \C$ be the superfunction of $z \mapsto b^z$ such that: : $\operatorname{tet}_b \left({0}\right) = 1$ : $\forall z \in C: \operatorname{tet}_b \left({z^*}\right) = \operatorname{tet}_b \left({z}\right)^*$ : $\displaystyle \forall x \in \R: \lim_{y \to +\infty} \operatorname{tet}_b \left({x + \mathrm i y}\right) = L$ Then the function $\operatorname{tet}_b$ is called '''tetration to base $b$'''. === Definition for $0 < b < \exp \left({1 / e}\right)$ === Let $b \in \R$ such that $1 < b < \exp \left({\dfrac 1 e}\right)$. Let $L_1, L_2 \in \R: L_1 < L_2$ be the fixed points of $\log_b$. Let $T = \dfrac{2 \pi i} {\ln \left({L_1 \ln \left({b}\right)}\right)}$ Let $C = \C \setminus \left\{ {x + T m, x \in \R: x \le -2, m \in \Z}\right \}$ Let $\operatorname{tet}_b: C \mapsto \C$ be the superfunction of $z \mapsto b^z$ such that: : $\operatorname{tet}_b (0) = 1$ : $\forall z \in C: \operatorname{tet}_b(z^*) = \operatorname{tet}_b(z)^*$ : $\forall z \in C: \operatorname{tet}_b(z) = \operatorname{tet}_b(z+T)$ : $\displaystyle \forall y \in \R: \lim_{x \to -\infty} \operatorname{tet}_b (x + \mathrm i y) = L_2 ~~ $ : $\displaystyle \forall \varepsilon \in \R_{>0}: \exists X \in \R$ such that: :: $\forall x \in \R: x > X: \left|{\operatorname {tet}_b \left({x + i y}\right) - L_1}\right| < \varepsilon$   Then the function $\operatorname{tet}_b$ is called '''tetration to base $b$'''.


Definition:Abundant Number

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $\map A n$ denote the abundance of $n$. === Definition 1 === {{:Definition:Abundant Number/Definition 1}} === Definition 2 === {{:Definition:Abundant Number/Definition 2}} === Definition 3 === {{:Definition:Abundant Number/Definition 3}}


Definition:Deficient Number

Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Deficient Number/Definition 1}} === Definition 2 === {{:Definition:Deficient Number/Definition 2}} === Definition 3 === {{:Definition:Deficient Number/Definition 3}}


Definition:Discrete Uniformity

Let $S$ be a set. The '''discrete uniformity''' on $S$ is the uniformity $\UU$ defined as: :$\UU := \set {u \subseteq S \times S: \Delta_S \subseteq u}$ that is, all subsets of the cartesian product on $S$ which contain the diagonal relation on $S$.


Definition:Quasiuniformity Axioms

{{begin-axiom}} {{axiom | n = \text U 1         | q = \forall u \in \UU         | m = \Delta_S \subseteq u }} {{axiom | n = \text U 2         | q = \forall u, v \in \UU         | m = u \cap v \in \UU }} {{axiom | n = \text U 3         | q = \forall u \in \UU         | m = u \subseteq v \subseteq S \times S \implies v \in \UU }} {{axiom | n = \text U 4         | q = \forall u \in \UU         | m = \exists v \in \UU: v \circ v \subseteq u }} {{end-axiom}} where: :$\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$ :$\circ$ is defined as: ::$u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$


Definition:Uniformity Axioms

{{begin-axiom}} {{axiom | n = \text U 1         | q = \forall u \in \UU         | m = \Delta_S \subseteq u }} {{axiom | n = \text U 2         | q = \forall u, v \in \UU         | m = u \cap v \in \UU }} {{axiom | n = \text U 3         | q = \forall u \in \UU         | m = u \subseteq v \subseteq S \times S \implies v \in \UU }} {{axiom | n = \text U 4         | q = \forall u \in \UU         | m = \exists v \in \UU: v \circ v \subseteq u }} {{axiom | n = \text U 5         | q = \forall u \in \UU         | m = u^{-1} \in \UU }} {{end-axiom}} where: :$\Delta_S$ is the diagonal relation on $S$, that is: $\Delta_S = \set {\tuple {x, x}: x \in S}$ :$\circ$ is defined as: ::$u \circ v := \set {\tuple {x, z}: \exists y \in S: \tuple {x, y} \in v, \tuple {y, z} \in u}$ :$u^{-1}$ is defined as: ::$u^{-1} := \set {\tuple {y, x}: \tuple {x, y} \in u}$ :That is, all elements of $\UU$ are symmetric.


Definition:Comparable Topologies

Let $S$ be a set. Let $\tau_1$ and $\tau_2$ be topologies on $S$. Then $\tau_1$ and $\tau_2$ are '''comparable''' {{iff}} either: :$\tau_1$ is coarser than $\tau_2$ or :$\tau_1$ is finer than $\tau_2$ That is, by definition of coarser and finer, either: :$\tau_1 \subseteq \tau_2$ or :$\tau_1 \supseteq \tau_2$


Definition:Partition Topology

Let $S$ be a set. Let $\PP$ be a partition of $S$. Let $\tau$ be the set of subsets of $S$ defined as: :$a \in \tau \iff a$ is the union of sets of $\PP$ Then $\tau$ is a '''partition topology''' on $S$, and $\struct {S, \tau}$ is a '''partition (topological) space'''.


Definition:Trivial Partition

Let $S$ be a set such that $S \ne \varnothing$. There are two partitions on $S$ which are referred to as '''the trivial partitions on $S$''': === Singleton Partition === {{:Definition:Trivial Partition/Singleton}} === Partition of Singletons === {{:Definition:Trivial Partition/Partition of Singletons}} Category:Definitions/Set Partitions brf8rp4cz616jnoxspgpv6wf5g1lhi0


Definition:Odd-Even Topology

Let $\Z_{>0}$ denote the set of strictly positive integers: :$\Z_{>0} = \set {x \in \Z: x > 0}$ Let $\PP$ be the partition on $\Z_{>0}$ defined as: :$\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$ That is: :$\PP = \set {\set {1, 2}, \set {3, 4}, \set {5, 6}, \ldots}$ Then the topology whose basis is $\PP$ is called the '''odd-even topology'''.


Definition:Deleted Integer Topology

Let $\PP$ be the set: :$\PP = \set {\openint {n - 1} n: n \in \Z_{> 0} }$ that is, the set of all open real intervals of the form: :$\openint 0 1, \openint 1 2, \openint 2 3, \ldots$ Let $S$ be the set defined as: :$S = \displaystyle \bigcup \PP = \R_{\ge 0} \setminus \Z$ that is, the positive real numbers minus the integers. Let $T = \struct {S, \tau}$ be the partition topology whose basis is $\PP$. Then $T$ is called '''the deleted integer topology'''.


Definition:Pseudometrizable Topology

Let $\left({S, d}\right)$ be a pseudometric space. Let $\left({S, \tau_d}\right)$ be the topological space induced by $d$. Then for any topological space which is homeomorphic to such a $\left({S, \tau_d}\right)$, it and its topology are defined as '''pseudometrizable'''.


Definition:Nondeterministic Turing Machine

A '''Nondeterministic''' (or '''Non-deterministic''') '''Turing Machine''' (NTM) is a variation of the classical Turing machine that relaxes the restriction that all the steps in the machine must be definite. That is, given a single internal state and a single character being read on the tape the machine may have more then one possible response. If any sequence of choices puts the machine into a halting state, then the machine stops after $n$ steps, where $n$ is the minimum number of steps needed to put the machine into a halting state.


Definition:Double Pointed Topology

Let $T = \struct {S, \tau_S}$ be a topological space. Let $A = \set {a, b}$ be a doubleton. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on $A$. Let $\struct {T \times D, \tau}$ be the product space of $T$ and $D$. Then $T \times D$ is known as the '''double pointed topology''' on $T$.


Definition:NP Complexity Class

In computation theory, $NP$ is a complexity class in which there exists some nondeterministic Turing machine that will either: : halt within $\map p {\size x}$ steps or: :run forever where $p$ is a polynomial and $\size x$ is the length of the input to the Machine.


Definition:NP-Complete

A problem $L$ is '''NP-complete''' if any problem in the complexity class NP can be reduced in polynomial time and space to $L$. {{NoSources}} Category:Definitions/Mathematical Logic Category:Definitions/Computer Science dom5wx5bpdsj6xzl16vb1capojqru0r


Definition:Particular Point Topology

Let $S$ be a set which is non-empty. Let $p \in S$ be some '''particular point''' of $S$. We define a subset $\tau_p$ of the power set $\powerset S$ as: :$\tau_p = \set {A \subseteq S: p \in A} \cup \set \O$ that is, all the subsets of $S$ which include $p$, along with the empty set. Then $\tau_p$ is a topology called the '''particular point topology on $S$ by $p$''', or just '''a particular point topology'''.


Definition:Finitely Generated Field Extension

Let $E / F$ be a field extension. Then $E$ is said to be '''finitely generated over $F$''' {{iff}}, for some $\alpha_1, \ldots, \alpha_n \in E$: :$E = F \left({\alpha_1, \ldots, \alpha_n}\right)$  where $F \left({\alpha_1, \ldots, \alpha_n}\right)$ is the field in $E$ generated by $F \cup \left\{{\alpha_1, \ldots, \alpha_n}\right\}$.


Definition:Primitive Element of Field Extension

Let $F / K$ be a simple field extension such that $F = K \left({\alpha}\right)$. Then $\alpha$ is a '''primitive element''' of $F$.


Definition:Relative Algebraic Closure

Let $L / K$ be a field extension. The '''relative algebraic closure''' of $K$ contained in $L$ is the set of all elements of $L$ that are algebraic over $K$.


Definition:Algebraically Closed Field

Let $K$ be a field. Then $K$ is '''algebraically closed''' {{iff}}: === Definition 1 === {{:Definition:Algebraically Closed/Definition 1}} === Definition 2 === {{:Definition:Algebraically Closed/Definition 2}} === Definition 3 === {{:Definition:Algebraically Closed/Definition 3}}


Definition:Tower of Fields

A '''tower of fields''': :$F_k / F_{k-1} / \dotsb / F_2 / F_1$ is a finite sequence of field extensions: :$F_{j+1} / F_j$ where $j = 1, \dotsc, k-1$. Category:Definitions/Field Extensions l9bqsmhlpxtxsdmtioa1d46jzyx7lc6


Definition:Formal Derivative of Polynomial

Let $R$ be a ring. Let $R \left[{X}\right]$ be the polynomial ring over $R$. Let $f = a_0 + a_1 X + \cdots + a_n X^n \in R \left[{X}\right]$. The '''formal derivative''' $f'$ of $f$ is the polynomial: :$f' \left({X}\right) = a_1 + 2 a_2 X + \cdots + n a_n X^{n-1}$ Category:Definitions/Polynomial Theory c5nbdn36wiio1alfwfu9ngpdral8jip


Definition:Symmetric Polynomial

Let $K$ be a field. Let $K \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomial forms over $K$. A polynomial $f \in K \sqbrk {X_1, \ldots, X_n}$ is '''symmetric''' {{iff}} for every permutation $\pi$ of $\set {1, 2, \ldots, n}$: :$\map f {X_1, \dotsc, X_n} = \map f {X_{\map \pi 1}, \dotsc, X_{\map \pi n} }$


Definition:Elementary Symmetric Polynomial

Let $K$ be a field. Let $K \left[{X_1, \ldots, X_n}\right]$ be the ring of polynomial forms over $K$. The '''elementary symmetric polynomials in $n$ variables''' are: :$\displaystyle f_r \left({X_1, \ldots, X_n}\right) = \sum_{1 \mathop \le i_1 \mathop < \cdots \mathop < i_r \mathop \le n} x_{i_1} \cdots x_{i_r}: \quad r = 1, \ldots, n$ Category:Definitions/Polynomial Theory 4zvnd1r1knn8xbx8y0ossuy4fdpxqfj


Definition:Splitting Field

=== Of a polynomial === Let $K$ be a field. Let $f$ be a polynomial over $K$. A '''splitting field''' of $f$ over $K$ is a field extension $L / K$ such that: :$f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$ for some $k \in K$, $\alpha_1, \ldots, \alpha_n \in L$. We say that $f$ '''splits''' over $L$. === Of a set of polynomials === Let $K$ be a field. Let $\FF$ be a set of polynomials over $K$. A '''splitting field''' of $\FF$ over $K$ is a field extension $L / K$ such that for any $f \in \FF$: :$f = k \paren {X - \alpha_1} \cdots \paren {X - \alpha_n}$ for some $k \in K$, $\alpha_1, \ldots, \alpha_n \in L$. We say that $\FF$ '''splits''' over $L$.


Definition:Sierpiński Space

The '''Sierpiński space''' is a particular point space with exactly two elements. Its usual presentation is: :$T = \struct {\set {0, 1}, \set {\O, \set 0, \set {0, 1} } }$ that is, as a particular point topology on the set $\set {0, 1}$ where the particular point is $0$. It can also immediately be seen to be an excluded point topology on the set $\set {0, 1}$ where the excluded point is $1$. The '''Sierpiński space''' is considered to be a '''trivial instance''' of both the particular point topology and the excluded point topology.


Definition:Closed Extension Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $p$ be a new element for $S$ such that $S^*_p := S \cup \set p$. Let $\tau^*_p$ be the set defined as: :$\tau^*_p := \set {U \cup \set p: U \in \tau} \cup \set \O$ That is, $\tau^*_p$ is the set of all sets formed by adding $p$ to all the open sets of $\tau$ and including the empty set. Then: :$\tau^*_p$ is the closed extension topology of $\tau$ and: :$T^*_p := \struct {S^*_p, \tau^*_p}$ is the closed extension space of $T = \struct {S, \tau}$.


Definition:Ultraproduct

Let $\mathcal L$ be a first-order language and let $I$ be an infinite set. Let $\mathcal U$ be an ultrafilter on $I$. Let $\mathcal M_i$ be an $\mathcal L$-structure for each $i\in I$ The '''ultraproduct''': :$\mathcal M := \displaystyle \left({\prod_{i \mathop \in I} \mathcal M_i }\right) / \mathcal U$ is an $\mathcal L$-structure defined as follows: :$(1): \quad$ The universe of $\mathcal M$: Let $X$ be the cartesian product: :$\prod_{i \mathop \in I} \mathcal M_i$ Define an equivalence relation $\sim$ on $X$ by: :$\left({a_i}\right)_{i \mathop \in I} \sim \left({b_i}\right)_{i \mathop \in I}$ {{iff}} $\left\{ {i \in I: a_i = b_i}\right \} \in \mathcal U$ The universe of $\mathcal M$ is the set of equivalence classes of $X$ modulo $\sim$. These are essentially sequences taken modulo the equivalence relation above, and are sometimes denoted $\left({m_i}\right)_\mathcal U$. :$(2): \quad$ Interpretation of non-logical symbols of $\mathcal L$ in $\mathcal M$: For each constant symbol $c$, we define $c^\mathcal M$ to be $\left({c^{\mathcal M_i} }\right)_\mathcal U$. For each $n$-ary function symbol $f$, we define $f^\mathcal M$ by setting: :$f^\mathcal M \left({\left({m_{1, i} }\right)_\mathcal U, \dotsc, \left({m_{n, i} }\right)_\mathcal U}\right)$ to be: :$\left({f^{\mathcal M_i} \left({m_{1, i}, \dotsc, m_{n, i} }\right)}\right)_\mathcal U$ For each $n$-ary relation symbol $R$, we define $R^\mathcal M$ to be the set of $n$-tuples: :$\left({\left({m_{1, i} }\right)_\mathcal U, \dots, \left({m_{n, i} }\right)_\mathcal U}\right)$ from $\mathcal M$ such that: :$\left\{ {i \in I: \left({m_{1, i}, \dotsc, m_{n, i} }\right) \in R^\mathcal M_i}\right\} \in \mathcal U$


Definition:Pure Transcendental Extension

Let $K$ be a field. A '''pure transcendental extension''' is a field extension $L / K$ such that: :for some set $\left\{ {X_\alpha: \alpha \in A}\right\}$, $L$ is the field of rational functions in the indeterminates $\left\{ {X_\alpha}\right\}$. Category:Definitions/Field Extensions oolabd8eoks5ushyuqg7tckjyhiloz3


Definition:Algebraically Independent

Let $L / K$ be a field extension. Let $A \subseteq L$ be a subset of $L$. Let $\map K {\set {X_\alpha}_{\alpha \mathop \in A} }$ be the Field of Rational Functions in the indeterminates $\set {X_\alpha: \alpha \mathop \in A}$. Then $A$ is '''algebraically independent''' over $K$ if there exists a homomorphism: :$\phi: \map K {set {X_\alpha}_{\alpha \mathop \in A} } \to L$ such that: :$\map \phi {X_\alpha} = \alpha$


Definition:Transcendence Degree

Let $K$ be a field, and let $L/K$ be a field extension of $K$. The '''transcendence degree''' of $L/K$ is the largest cardinality of an algebraically independent subset $A \subseteq L$. Category:Definitions/Field Extensions l3ehwhvnm2jmnfvlr5laombdga98ywb


Definition:Successor Ordinal

A '''successor ordinal''' is the successor set of an ordinal.


Definition:Limit Ordinal

An ordinal $\lambda$ is a '''limit ordinal''' {{iff}} it is neither the zero ordinal nor a successor ordinal.


Definition:Excluded Point Topology

Let $S$ be a set which is non-empty. Let $p \in S$ be some '''particular point''' of $S$. We define a subset $\tau_{\bar p}$ of the power set $\powerset S$ as: :$\tau_{\bar p} = \set {A \subseteq S: p \notin A} \cup \set S$ That is, all the subsets of $S$ which do not include $p$, along with the set $S$. Then $\tau_{\bar p}$ is a topology called the '''excluded point topology on $S$ by $p$''', or just '''an excluded point topology'''.


Definition:Open Extension Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $p$ be a new element which is not in $S$. Let $S^*_p = S \cup \set p$. Let $\tau^*_p$ be the set defined as: :$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$ That is, $\tau^*_{\bar p}$ is the set of all sets formed by taking all the open sets of $\tau$ and adding to them the set $S^*_p$. Then: :$\tau^*_{\bar p}$ is the open extension topology of $\tau$ and: :$T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ is the open extension space of $T = \struct {S, \tau}$.


Definition:Boubaker Polynomials

The Boubaker polynomials are the components of the following sequence of polynomials: {{begin-eqn}} {{eqn | l = B_0 \left({x}\right)       | r = 1      }} {{eqn | l = B_1 \left({x}\right)       | r = x      }} {{eqn | l = B_2 \left({x}\right)       | r = x^2 + 2      }} {{eqn | l = B_3 \left({x}\right)       | r = x^3 + x      }} {{eqn | l = B_4 \left({x}\right)       | r = x^4 - 2      }} {{eqn | l = B_5 \left({x}\right)       | r = x^5 - x^3 - 3x      }} {{eqn | l = B_6 \left({x}\right)       | r = x^6 - 2x^4 - 3x^2 + 2      }} {{eqn | l = B_7 \left({x}\right)       | r = x^7 - 3x^5 - 2x^3 + 5x      }} {{eqn | l = B_8 \left({x}\right)       | r = x^8 - 4x^6 + 8x^2 - 2      }} {{eqn | l = B_9 \left({x}\right)       | r = x^9 - 5x^7 + 3x^5 + 10x^3 - 7x      }} {{eqn | o = \vdots}} {{end-eqn}}


Definition:Supremum Norm

Let $S$ be a set. Let $\struct {X, \norm {\,\cdot\,} }$ be a normed vector space. Let $\mathcal B$ be the set of bounded mappings $S \to X$. For $f \in \mathcal B$ the '''supremum norm''' of $f$ on $S$ is defined as: :$\norm f_\infty = \sup \set {\norm {\map f x}: x \in S}$ === Supremum Norm on Continuous on Closed Interval Real-Valued Functions === {{:Definition:Supremum Norm/Continuous on Closed Interval Real-Valued Function}}


Definition:Normed Vector Space

Let $\struct {K, +, \circ}$ be a normed division ring. Let $V$ be a vector space over $K$. Let $\norm {\,\cdot\,}$ be a norm on $V$. Then $\struct {V, \norm {\,\cdot\,} }$ is a '''normed vector space'''.


Definition:Banach Space

A '''Banach space''' is a normed vector space where every Cauchy sequence is convergent.


Definition:Contraction Mapping

Let $\struct {X, d_1}$ and $\struct {Y, d_2}$ be metric spaces. Let $f: X \to Y$ be a mapping. Then $f$ is a '''contraction (mapping)''' {{iff}} there exists $\kappa \in \R: 0 \le \kappa < 1$ such that: :$\forall x, y \in X: \map {d_2} {\map f x, \map f y} \le \kappa \, \map {d_1} {x, y}$ That is, $f$ is Lipschitz continuous for a Lipschitz constant less than $1$.


Definition:Type

Let $\MM$ be an $\LL$-structure. Let $A$ be a subset of the universe of $\MM$. Let $\LL_A$ be the language consisting of $\LL$ along with constant symbols for each element of $A$. Viewing $\MM$ as an $\LL_A$-structure by interpreting each new constant as the element for which it is named, let $\map {\operatorname {Th}_A} \MM$ be the set of $\LL_A$-sentences satisfied by $\MM$. An '''$n$-type over $A$''' is a set $p$ of $\LL_A$-formulas in $n$ free variables such that $p \cup \map {\operatorname {Th}_A} \MM$ is satisfiable by some $\LL_A$-structure. {{Disambiguate|Definition:Logical Formula}} === Complete Type === We say that an $n$-type $p$ is '''complete (over $A$)''' {{iff}}: :for every $\LL_A$-formula $\phi$ in $n$ free variables, either $\phi \in p$ or $\phi \notin p$. The set of '''complete $n$-types over $A$''' is often denoted by $\map {S_n^\MM} A$. Given an $n$-tuple $\bar b$ of elements from $\MM$, the '''type of $\bar b$ over $A$''' is the complete $n$-type consisting of those $\LL_A$-formulas $\map \phi {x_1, \dotsc, x_n}$ such that $\MM \models \map \phi {\bar b}$. It is often denoted by $\map {\operatorname {tp}^\MM} {\bar b / A}$. === Realization === Given an $\LL_A$-structure $\NN$, a type $p$ is '''realized''' by an element $\bar b$ of $\NN$ {{iff}}: :$\forall \phi \in p: \NN \models \map \phi {\bar b}$. Such an element $\bar b$ of $\NN$ is a '''realization''' of $p$. === Omission === We say that $\NN$ '''omits''' $p$ if $p$ is not realized in $\NN$. Then $p$ is an '''omission''' from $\NN$.


Definition:Linear Recurrence Relation

A '''linear recurrence relation''' is a recurrence relation which has the form: :$a_n y_{n+k} + a_{n-1} y_{n+k-1} + \cdots + a_0 y_k = b \left({k}\right)$ where $a_0, \ldots, a_n$ are constants. === Homogeneous Linear Recurrence Relation === If in a linear recurrence relation $b \left({k}\right) = 0$, then the equation is '''homogeneous'''. Otherwise it is '''inhomogeneous'''. {{SUBPAGENAME}} {{SUBPAGENAME}} 3nfpbstuhbph1pt4r5euye1w8jnhmok


Definition:Elementary Equivalence

Let $\mathcal{M},\mathcal{N}$ be $\mathcal{L}$-structures. We say that $\mathcal{M}$ and $\mathcal{N}$ are '''elementarily equivalent''' if for all $\mathcal{L}$-sentences $\phi$, we have $\mathcal{M}\models \phi$ if and only if $\mathcal{N}\models \phi$. {{expand}} {{MissingLinks}} {{NoSources}} Category:Definitions/Model Theory s315nv2o7ka885ufob4r45ebb4ooj2p


Definition:Embedded

Let $V$ and $W$ be normed vector spaces. Then $V$ is '''embedded''' in $W$ {{iff}}: : $V \subseteq W$ The inclusion mapping $i_V : V \to W$ is an '''embedding''' of $V$ into $W$. Category:Definitions/Functional Analysis dtnujhyil86qa3mx9up04gs50vebour


Definition:Continuously Embedded

Let $\operatorname{id} : V \to W$ be an embedding of normed vector spaces. Then $V$ is '''continuously embedded''' in $W$ and $\operatorname{id}$ is a '''continuous embedding''' if the identity $\operatorname{id}$ is a continuous mapping. {{explain|Link to the specific context in which continuous mapping is defined}} Category:Definitions/Functional Analysis b9o0pp32jce8l9y8csui48ciz2f9aay


Definition:Excluded Set Topology

Let $S$ be a set which is non-empty. Let $H \subseteq S$ be some subset of $S$. We define a subset $\tau_{\bar H}$ of the power set $\powerset S$ as: :$\tau_{\bar H} = \set {A \subseteq S: A \cap H = \O} \cup \set S$ that is, all the subsets of $S$ which are disjoint from $H$, along with the set $S$. Then $\tau_{\bar H}$ is a topology called the '''excluded set topology on $S$ by $H$''', or just '''an excluded set topology'''.


Definition:Either-Or Topology

Let $S = \closedint {-1} 1$ be the closed interval on the real number line from $-1$ to $1$. Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$ such that, for any $H \subseteq S$: :$H \in \tau \iff \paren {\set 0 \nsubseteq H \lor \openint {-1} 1 \subseteq H}$ where $\lor$ is the inclusive-or logical connective. Then $\tau$ is the '''either-or topology''', and $T = \struct {S, \tau}$ is the '''either-or space'''


Definition:Type Space

Let $\MM$ be an $\LL$-structure, and let $A$ be a subset of the universe of $\MM$. Let $\map {S_n^\MM} A$ be the set of complete $n$-types over $A$. The '''space of $n$-types over $A$''' is the topological space formed by the set $\map {S_n^\MM} A$ together with the topology arising from the basis which consists of the sets: :$\sqbrk \phi := \set {p \in \map {S_n^\MM} A:\phi \in p}$ for each $\LL_A$-formula $\phi$ with $n$ free variables. Note that each $\sqbrk \phi$ is also closed in this topology, since $\sqbrk \phi$ is the complement of $\sqbrk {\neg \phi}$ in $\map {S_n^\MM} A$. {{LinkWanted|Definition of the notation $\sqbrk \phi$}}


Definition:Isolated Type

Let $T$ be an $\mathcal L$-theory. Let $\phi \left({\bar v}\right)$ be an $\mathcal L$-formula in $n$ free variables $\bar v$ such that $T \cup \phi \left({\bar v}\right)$ is satisfiable. {{Disambiguate|Definition:Logical Formula}} Let $p$ be an $n$-type of $T$. We say that $\phi$ '''isolates''' $p$ if for all $\psi \in p$, we have: :$T \models \forall \bar{v} \left({ \phi \left({\bar v}\right) \rightarrow \psi \left({\bar v}\right) }\right)$ that is, all $\psi$ are semantic consequences of $\phi$. Category:Definitions/Model Theory aufuya5baqc2yul5d7w8w9n3memfcg9


Definition:Witness Property

{{MissingLinks}} An $\LL$-theory $T$ is said to have the '''witness property''' if for every $\LL$-formula $\map \phi v$ with one free variable, there is a constant symbol $c$ in $\LL$ such that: {{Disambiguate|Definition:Logical Formula}} :$T \models \paren {\exists v \map \phi v} \to \map \phi c$ that is, $\paren {\exists v \map \phi v} \to \map \phi c$ is a semantic consequence of $T$. That is, every existential statement satisfied by $T$ is witnessed by a constant. {{NoSources}} Category:Definitions/Model Theory b2x1fd0kttej9lgwie7mff26hbiu8p0


Definition:Theory (Logic)

Let $\mathcal L$ be a logical language. An '''$\mathcal L$-theory''' $T$ is a set of $\mathcal L$-sentences. === Theory of Structure === {{:Definition:Theory (Logic)/Structure}} === Complete Theory === {{:Definition:Theory (Logic)/Complete}} === Maximal Theory === {{:Definition:Theory (Logic)/Maximal}} {{NoSources}} Category:Definitions/Mathematical Logic 6jhezd78oww4rjb0ki3dkmgmgibz9xi


The '''distance to nearest integer function''' $\norm \cdot: \R \to \closedint 0 {\dfrac 1 2}$ is defined in the following ways: === Definition 1 === {{:Definition:Distance to Nearest Integer Function/Definition 1}} === Definition 2 === {{:Definition:Distance to Nearest Integer Function/Definition 2}}


Definition:Stability (Model Theory)

Let $T$ be a complete $\mathcal L$-theory where $\mathcal L$ is countable. Let $\kappa$ be an infinite cardinal. === $\kappa$-Stable === {{:Definition:Stability (Model Theory)/Kappa-Stable Theory}} === Stable === {{:Definition:Stability (Model Theory)/Stable Theory}} === Unstable === {{:Definition:Stability (Model Theory)/Unstable Theory}}


Definition:Finite Complement Topology

Let $S$ be a set whose cardinality is usually specified as being infinite. Let $\tau$ be the set of subsets of $S$ defined as: :$H \in \tau \iff \relcomp S H \text { is finite, or } H = \O$ where $\relcomp S H$ denotes the complement of $H$ relative to $S$. Then $\tau$ is the '''finite complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is a '''finite complement space'''.


Definition:Ramanujan Sum

Let $e: \R \to \C$ be the mapping defined as: :$\forall \alpha \in \R: \map e \alpha := \map \exp {2 \pi i \alpha}$ For $q \in \N_{>0}$, $n \in \N$, the '''Ramanujan sum''' is defined as: :$\displaystyle \map {c_q} n = \sum_{\substack {1 \mathop \le a \mathop \le q \\ \gcd \set {a, q} \mathop = 1} } \map e {\frac {a n} q}$


Definition:Countable Complement Topology

Let $S$ be an infinite set whose cardinality is usually taken to be uncountable. Let $\tau$ be the set of subsets of $S$ defined as: :$H \in \tau \iff \relcomp S H$ is countable, or $H = \O$ where $\relcomp S H$ denotes the complement of $H$ relative to $S$. In this definition, countable is used in its meaning that includes finite. Then $\tau$ is the '''countable complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is a '''countable complement space'''.


Definition:Higher-Aleph Complement Topology

Let $S$ be a set whose cardinality is $\aleph_n$ where $n > 0$. Let $\tau \subseteq \powerset S$ be the set of subsets of $S$ defined as: :$\tau = \set {U \subseteq S: \size {\relcomp S U} = \aleph_m: m < n} \cup \set {U \subseteq S: \relcomp S U \text { is finite} } \cup \O$ That is, $\tau$ is the set of subsets of $S$ whose complements relative to $S$ are of a cardinality strictly less than $S$. Then $\tau$ is an '''$\aleph_m$ complement topology on $S$''', and the topological space $T = \struct {S, \tau}$ is an '''$\aleph_m$ complement space'''. This construction is an extension of the concept of the finite complement topology and the countable complement topology.


Definition:Saturated Model

Let $T$ be an $\mathcal{L}$-theory. Let $\kappa$ be an infinite cardinal. A model $\mathcal{M}$ of $T$ is $\kappa$-'''saturated''' if for every subset $A$ of the  universe of $\mathcal{M}$ of cardinality strictly less than $\kappa$, and for every  $n\in\mathbb N$, every complete $n$-type $p$ over $A$ is realized in $\mathcal{M}$. That is, $\mathcal{M}$ is $\kappa$-saturated if for all $A  \subseteq \mathcal{M}$ with $A<\kappa$, and for all $n\in\mathbb N$, each $p \in S_{n}^{\mathcal{M}} (A)$ is realized in $\mathcal{M}$. We say $\mathcal{M}$ is '''saturated''' if it is $\kappa$-saturated where $\kappa$ is the cardinality of the universe of $\mathcal{M}$. {{SUBPAGENAME}} 1lsodr3oexq9bc7rhx6nzqgdth5h3hq


Definition:Universal Model

Let $T$ be an $\mathcal L$-theory. Let $\kappa$ be an infinite cardinal. A model $\mathcal M$ of $T$ is $\kappa$-'''universal''' {{iff}}: :for every model $\mathcal N$ of $T$ whose universe has cardinality strictly less than $\kappa$, there is an elementary embedding of $\mathcal N$ into $\mathcal M$. That is, $\mathcal M$ is '''$\kappa$-universal''' {{iff}}: :for all models $\mathcal N \models T$ with  cardinality $\left\vert{\mathcal N}\right\vert < \kappa$, there is an elementary embedding $j: \mathcal N \to \mathcal M$. We say $\mathcal M$ is '''universal''' if it is '''$\kappa^+$-universal''' where $\kappa$ is the cardinality of the universe of $\mathcal M$ and $\kappa^+$ is the successor cardinal of $\kappa$. Definitions/Model Theory bklciljojrl7we8ux5qi6os3dtoh5jr


Definition:Categorical (Model Theory)

Let $T$ be an $\mathcal{L}$-theory. Let $\kappa$ be a cardinal. $T$ is $\kappa$-'''categorical''' if whenever $\mathcal{M}$ and $\mathcal{N}$ are models of $T$ of cardinality $\kappa$, then $\mathcal{M}$ and $\mathcal{N}$ are isomorphic.


Definition:Compact Complement Topology

Let $T = \struct {\R, \tau}$ be the real number line with the usual (Euclidean) topology. Let $\tau^*$ be the set defined as: :$\tau^* = \leftset {S \subseteq \R: S = \O \text { or } \relcomp \R S}$ is compact in $\rightset {\struct {\R, \tau} }$ where $\relcomp \R S$ denotes the complement of $S$ in $\R$. Then $\tau^*$ is the '''compact complement topology''' on $\R$, and $T^* = \struct {\R, \tau^*}$ is the '''compact complement space''' on $\R$.


Definition:Graph (Category Theory)

A '''graph''' is an interpretation of a metagraph within set theory. Let $\mathfrak U$ be a class of sets. A metagraph $\mathcal G$ is a '''graph''' if: : $(1): \quad$ The objects form a subset $\operatorname{vert}\mathcal G \subseteq \mathfrak U$ : $(2): \quad$ The morphisms form a subset $\operatorname{edge}\mathcal G \subseteq \mathfrak U$ If the class $\mathfrak U$ is a set, then morphisms are functions, and the domain and codomain in the definition of a morphism are those familiar from set theory. If $\mathfrak U$ is a proper class this is not the case, for example the morphisms of $\mathcal C$ need not be functions. Category:Definitions/Category Theory 145b347zy1dfv7fjzlb3t9u0j2nel1e


Definition:Indiscernible

Let $\mathcal M$ be an $\mathcal L$-structure. Let $I$ be an infinite set. Let $X = \left\{ {x_i \in \mathcal M: i \in I}\right\}$ be an infinite subset of the universe of $\mathcal M$ indexed by $I$. For: : every $n \in \N$ and: : every pair of subsets $\left\{ {i_1, \ldots, i_n}\right\}$ and $\left\{ {j_1, \ldots, j_n}\right\}$ of $I$ each with $n$ distinct elements, let: : $\mathcal M \models \phi \left({x_{i_1}, \ldots, x_{i_n} }\right) \leftrightarrow \phi \left({x_{j_1}, \ldots, x_{j_n} }\right)$ for all $\mathcal L$-formulas $\phi$ with $n$ free variables. Then $X$ is '''(an) indiscernible (set) in $\mathcal M$'''. {{Disambiguate|Definition:Logical Formula}} Informally, $X$ is '''indiscernible (set)''' if $\mathcal M$ cannot distinguish between same-sized ordered tuples over $X$ using $\mathcal L$-formulas.


Definition:Order Indiscernible

Let $\mathcal M$ be an $\mathcal L$-structure. Let $\left({I, \le}\right)$ be an infinite ordered set. Let $X = \left\{ {x_i \in \mathcal M: i \in I}\right\}$ be an infinite subset of the universe of $\mathcal M$ indexed by $I$. Let $A$ be a subset of the universe of $\mathcal M$. $X$ is '''(an) order indiscernible (set) over $A$ in $\mathcal M$''' {{iff}}: : For every $n \in \N$ and every pair of chains $i_1 < \cdots < i_n$ and $j_1 < \cdots < j_n$ in $I$ each with $n$ distinct elements, we have: :: $\mathcal M \models \phi \left({x_{i_1}, \ldots, x_{i_n} }\right) \iff \phi \left({x_{j_1}, \ldots, x_{j_n} }\right)$ : for all $\mathcal L$-formulas $\phi$ with $n$ free variables and parameters from $A$. {{Disambiguate|Definition:Logical Formula}} Informally, $X$ is '''order indiscernible''' {{iff}} $\mathcal M$ cannot distinguish between same-sized increasing ordered tuples over $X$ using $\mathcal L$-formulas.


Definition:Fort Space

Let $S$ be an infinite set. Let $p \in S$ be a particular point of $S$. Let $\tau_p \subseteq \powerset S$ be a subset of the power set of $S$ defined as: :$\tau_p = \leftset {U \subseteq S: p \in \relcomp S U} \text { or } \set {U \subseteq S: \relcomp S U}$ is finite$\rightset{}$ That is, $\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is finite. Then $\tau_p$ is a '''Fort topology''' on $S$, and the topological space $T = \struct {S, \tau_p}$ is a '''Fort space'''.


Definition:Gödel-Bernays Axioms

The '''Gödel-Bernays axioms''' are a conservative extension of the Zermelo-Fraenkel axioms with the axiom of choice (ZFC) that allow comprehension of classes. Although not the standard axioms of set theory, particularly in category theory they spare us any set of all sets-type paradoxes. == Axioms for Sets == The first five axioms are identical to the axioms of the same names from ZFC. The quantified variables range over the universe of sets. === The Axiom of Extension === {{:Axiom:Axiom of Extension/Set Theory}} === The Axiom of Pairing === {{:Axiom:Axiom of Pairing/Set Theory}} === The Axiom of Unions === {{:Axiom:Axiom of Unions/Set Theory}} === The Axiom of Powers === {{:Axiom:Axiom of Powers/Set Theory}} === The Axiom of Infinity === {{:Axiom:Axiom of Infinity/Set Theory}} == Axioms for Classes == In the remaining axioms, the quantified variables range over classes. The first two differ from the ZFC axioms with the same names in this way only. The last two have no analogue among the ZFC axioms. === The Axiom of Extension === {{:Axiom:Axiom of Extension (Classes)}} === The Axiom of Foundation === {{:Axiom:Axiom of Foundation (Classes)}} === Class Comprehension === {{:Axiom:Class Comprehension Schema}} === The Axiom of Limitation of Size === {{:Axiom:Axiom of Limitation of Size}}


Definition:Class (Class Theory)

A '''class''' is a collection of all sets such that a particular condition holds. In class builder notation, this is written as: :$\set {x: \map p x}$ where $\map p x$ is a statement containing $x$ as a free variable.   This is read: :'''All $x$ such that $\map p x$ holds.'''


Definition:Metagraph

A '''metagraph''' $\mathcal G$ consists of: * objects $X, Y, Z, \ldots$ * morphisms $f, g, h, \ldots$ between its objects These are subjected to the following two axioms: {{begin-axiom}} {{axiom | n = 1         | lc= ''Domains''         | t = Every morphism $f$ has associated an object $\operatorname{dom} f$, called the '''domain''' of $f$ }} {{axiom | n = 2         | lc= ''Codomains''         | t = Every morphism $f$ has associated an object $\operatorname{cod} f$, called the '''codomain''' of $f$ }} {{end-axiom}} A '''metagraph''' is purely axiomatic, and does not use set theory. For example, the objects are not "elements of the set of objects", because these axioms are (without further interpretation) unfounded in set theory.


Definition:Path (Category Theory)

Let $\mathcal G$ be a graph. A (non-empty) '''path''' in $\mathcal G$ is a sequence $(i,a_1,\ldots,a_n,j)$ such that: *$n > 1$ *$a_1,\ldots, a_n$ are edges of $\mathcal G$ *$i$ is the source of $a_1$ and $j$ is the destination of $a_n$ *For $1 \leq k < n$ the destination of $a_k$ is the source of $a_{k+1}$ This is usually written in the form: :$\displaystyle i \stackrel{a_1}{\longrightarrow} * \stackrel{a_2}{\longrightarrow} \cdots \stackrel{a_{n-1}}{\longrightarrow} * \stackrel{a_n}{\longrightarrow} j$ For any vertex $i$ the '''empty path''' from $i$ to $i$ is the pair $(i,i)$. Paths are '''composed''' by concatenation: :$\displaystyle (j,b_1,\ldots,b_n,k)(i,a_1,\ldots,a_m,j) = (i,a_1,\ldots,a_m,b_1,\ldots,b_n,k)$ Here we have used the somewhat awkward but more common right-to-left notation for composition. {{SUBPAGENAME}} mtbopkxz4y7tctz5weuq28ngbss9blw


Definition:Minimal Topology

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$. Let $\Theta_S$ be the set of all topologies on $S$: :$\Theta_S = \left\{{\tau \in \mathcal P \left({S}\right): \tau}\right.$ is a topology on $\left.{S}\right\}$ Let $\Phi: \Theta_S \to \left\{{T, F}\right\}$ be a propositional function on $\Theta_S$. Let $\vartheta \in \Theta_S$ have the property that $\Phi \left({\vartheta}\right)$ and: :$\forall \tau \in \Theta_S: \Phi \left({\tau}\right) \implies \vartheta \subseteq \tau$ That is, $\vartheta$ is the coarsest topology on $S$ which satisfies the propositional function $\Phi$. Then $\vartheta$ is the '''minimal topology satisfying $\Phi$'''. {{SUBPAGENAME}} htipkwppnixkuhckesep3patmjqucql


Definition:Minimal (Model Theory)

Let $\mathcal M$ be an $\mathcal L$-structure. Let $M$ be the universe of $\mathcal M$. Let $A$ be a subset of $M$. Let $D \subseteq M^n$ be an infinite $A$-definable set. Let $\phi \left({\bar x, \bar a}\right)$ be an $\mathcal L$-formula with parameters $\bar a$ from $A\subseteq M$ and  free variables $\bar x$ which defines $D$. {{Disambiguate|Definition:Logical Formula}} $D$ is '''minimal''' in $\mathcal M$ {{iff}} every definable subset of $D$ is either finite or cofinite. $\phi$ is '''minimal''' in $\mathcal M$ if $D$ is minimal in $\mathcal M$. $D$ and $\phi$ are '''strongly minimal''' in $\mathcal M$ {{iff}} $\phi$ is minimal in any elementary extension $\mathcal N$ of $\mathcal M$. An $\mathcal L$-theory $T$ is '''strongly minimal''' if for every model $\mathcal N$ of $T$ with universe $N$, the set $N$ is strongly minimal. {{NoSources}} Category:Definitions/Model Theory q7voaf1drllxp3qsiprs092mvfdaebc


Definition:Cofinite Set

Let $S$ be a set. Let $A$ be a subset of $S$. $A$ is '''cofinite in $S$''' {{iff}} the complement of $A$ in $S$ is finite. Category:Definitions/Set Theory m3q8cemkegb05gdyzr23zyx6e9whs5q


Definition:Fortissimo Space

Let $S$ be an uncountably infinite set. Let $p \in S$ be a particular point of $S$. Let $\tau_p \subseteq \powerset S$ be a subset of the power set of $S$ defined as: :$\tau_p = \leftset {U \subseteq S: p \in \relcomp S U}$ or $\set {U \subseteq S: \relcomp S U}$ is countable (either finitely or infinitely)$\rightset {}$ That is, $\tau_p$ is the set of all subsets of $S$ whose complement in $S$ either contains $p$ or is countable. Then $\tau_p$ is a '''Fortissimo topology''' on $S$, and the topological space $T = \struct {S, \tau_p}$ is a '''Fortissimo space'''.


Definition:Divide (Model Theory)

Let $T$ be a complete $\mathcal L$-theory. Let $\mathfrak C$ be a monster model for $T$. Let $A$ be a subset of the universe of $\mathfrak C$. Let $\phi \left({\bar x, \bar b}\right)$ be an $\mathcal L$-formula with free variables $\bar x$ and parameters $\bar b$ from the universe of $\mathfrak C$. {{Disambiguate|Definition:Logical Formula}} Let $\operatorname{tp} \left({\bar b / A}\right)$ denote the type of $\bar b$ over $A$. Then $\phi \left({\bar x, \bar b}\right)$ '''$k$-divides in $\mathfrak C$ over $A$''' {{iff}}: :there exists a sequence $\left\langle{\bar b_i}\right\rangle_{i \mathop \in \N}$ such that $\operatorname{tp} \left({\bar b_i / A}\right) = \operatorname{tp} \left({\bar b / A}\right)$ for each $i \in \N$ and: :for any distinct $k$-many terms $\bar b_{i_1}, \ldots, \bar b_{i_k}$ of the sequence, the set $\left\{{\phi \left({\bar x, \bar b_{i_1} }\right), \ldots, \phi \left({\bar x, \bar b_{i_k} }\right)}\right\}$ is not satisfiable in $\mathfrak C$. Let $\pi \left({\bar x, \bar b}\right)$ be a set of formulas with parameters $\bar b$. $\pi \left({\bar x, \bar b}\right)$ '''$k$-divides''' over $A$ if it implies a formula $\phi \left({x, \bar c}\right)$ which $k$-divides over $A$. Formulas and sets of formulas are said to '''divide''' if they $k$-divide for some $k$.


Definition:Fork

Let $T$ be a complete $\mathcal{L}$-theory. Let $\mathfrak{C}$ be a monster model for $T$. Let $A$ be a subset of the universe of $\mathfrak{C}$. Let  $\pi (\bar{x}, \bar{b})$ be a set of $\mathcal{L}$-formulas with free variables $\bar{x}$ and parameters $\bar{b}$  from the universe of $\mathfrak{C}$. {{Disambiguate|Definition:Logical Formula}} $\pi(\bar{x}, \bar{b})$ '''forks''' over $A$ if it implies some disjunction $\phi_1 (\bar x, \bar c_1) \vee \cdots \vee \phi_n (\bar x, \bar c_n)$ where each $\phi_i (\bar x, \bar c_i)$ divides over $A$. An individual formula $\phi (\bar x, \bar b)$ is said to '''fork''' if the singleton $\{\phi (\bar x, \bar b)\}$ forks.


Definition:Forking Extension

Let $T$ be a complete $\mathcal{L}$-theory. Let $\mathfrak{C}$ be a monster model for $T$. Let $A\subseteq B$ be subsets of the universe of $\mathfrak{C}$. Let $p(\bar x)$ be a complete $n$-type over $B$. Denote by $p\restriction A$ the subset of $p$ consisting of those formulas which involve only parameters from $A$. $p$ is a '''non-forking extension''' of $p\restriction A$ if $p(\bar x)$ does not fork over $A$. {{SUBPAGENAME}} gdkrozd2nkd7dv0v00jxtvtxhtzt815


Definition:Monster Model

Let $T$ be a complete $\mathcal L$-theory. A '''monster model''' of $T$ whose cardinality $\kappa$ is infinite is a model of $T$ which is saturated and homogeneous.


Definition:Finite Set

A set $S$ is defined as '''finite''' {{iff}}: :$\exists n \in \N: S \sim \N_{<n}$ where $\sim$ denotes set equivalence. That is, if there exists an element $n$ of the set of natural numbers $\N$ such that the set of all elements of $\N$ less than $n$ is equivalent to $S$. Equivalently, a '''finite set''' is a set with a count.


Definition:Embedding (Model Theory)

Let $\MM$ and $\NN$ be $\LL$-structures with universes $M$ and $N$ respectively. $j: \MM \to \NN$ is an '''$\LL$-embedding''' {{iff}} it is an injective map $M \to N$ which preserves interpretations of all symbols in $\LL$; that is, such that: :$\map j {\map {f^\MM} {a_1, \dots, a_{n_f} } } = \map {f^\NN} {\map j {a_1}, \ldots, \map j {a_{n_f} } }$ for all function symbols $f$ in $\LL$ and $a_1, \dots, a_{n_f}$ in $M$ :$\tuple {a_1, \ldots, a_{n_R} } \in R^\MM \iff \tuple {\map j {a_1}, \dots, \map j {a_{n_R} } } \in R^\NN$ for all relation symbols $R$ in $\LL$ and $a_1, \dots, a_{n_R}$ in $M$ :$\map j {c^\MM} = c^\NN$ for all constant symbols $c$ in $\LL$. === Partial Embedding === A common method of constructing isomorphisms and elementary embeddings in proofs is to recursively define them a finite number of elements at a time.  For this  purpose, it is useful to have a definition of embeddings using functions which are only defined on a subset of $M$: Let $A \subseteq M$ be a subset of $M$. $j: A \to \NN$ is a '''partial $\LL$-embedding''' {{iff}} it is an injective map $A \to N$ which preserves interpretations of all symbols in $\LL$ applied to elements of $A$; that is, such that: :$\map j {\map {f^\MM} {a_1, \dots, a_{n_f} } } = \map {f^\NN} {\map j {a_1}, \ldots, \map j {a_{n_f} } }$ for  all function symbols $f$ in $\LL$ and $a_1, \dots, a_{n_f}$ in $A$ :$\tuple {a_1, \ldots, a_{n_R} } \in R^\MM \iff \tuple {\map j {a_1}, \dots, \map j {a_{n_R} } } \in R^\NN$ for all relation symbols $R$ in $\LL$ and $a_1, \dots, a_{n_R}$ in $A$ :$\map j {c^\MM} = c^\NN$ for all constant symbols $c$ in $\LL$. === Isomorphism === $j: \MM \to \NN$ is an '''$\LL$-isomorphism''' {{iff}} it is a bijective $\LL$-embedding. === Automorphism === $j: \MM \to \NN$ is an '''$\LL$-automorphism''' {{iff}} it is an $\LL$-isomorphism and $\MM = \NN$. It is often useful to talk about automorphisms which are constant on subsets of $M$.  So, there is a definition and a notation for doing so: Let $A \subseteq M$ be a subset of $M$, and let $b \in M$. An $\LL$-automorphism $j$ is an '''$A$-automorphism''' {{iff}} $\map j a = a$ for all $a\in A$. An $\LL$-automorphism $j$ is an '''$A, b$-automorphism''' {{iff}} it is an $\paren {A \cup \set b}$-automorphism; that is: $\map j a = a$ for all $a \in A$ and also $\map j b = b$. Category:Definitions/Model Theory ph5f7arezx633z7f7ypn6jyva7grsru


Definition:Homogeneous (Model Theory)

Let $T$ be an $\mathcal{L}$-theory. Let $\kappa$ be an infinite cardinal. A model $\mathcal{M}$ of $T$ is $\kappa$-'''homogeneous''' if for every subset $A$ and element $b$ in the universe of $\mathcal{M}$ with the cardinality of $A$ strictly less than $\kappa$, if $f:A\to \mathcal{M}$ is partial elementary, then $f$ extends to an elementary map $f^*: A\cup\{b\} \to \mathcal{M}$. That  is, $\mathcal{M}$ is $\kappa$-homogeneous if for all $A  \subseteq \mathcal{M}$ with  $A<\kappa$ and all $b\in \mathcal{M}$, every elementary $f:A\to \mathcal{M}$ extends to an elementary $f^*:A\cup\{b\}\to\mathcal{M}$. We say $\mathcal{M}$ is  '''homogeneous''' if it is $\kappa$-homogeneous where $\kappa$ is the cardinality of the universe of $\mathcal{M}$.


Definition:Arens-Fort Space

Let $S$ be the set $\Z_{\ge 0} \times \Z_{\ge 0}$ be the Cartesian product of the set of positive integers $\Z_{\ge 0}$: :$S = \set {0, 1, 2, \ldots} \times \set {0, 1, 2, \ldots}$ Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$ such that: :$(1): \quad \forall H \subseteq S: \tuple {0, 0} \notin H \implies H \in \tau$ :$(2): \quad H \subseteq S: \tuple {0, 0} \in H$ and, for all but a finite number of $m \in \Z_{\ge 0}$, the sets $S_m$ defined as: ::::$S_m = \set {n: \tuple {m, n} \notin H}$ :::are finite. That is, $H$ is allowed to be in $\tau$ if, considering $S = \Z_{\ge 0} \times \Z_{\ge 0}$ as the lattice points of the first quadrant of a Cartesian plane: Either: :$H$ does not contain $\tuple {0, 0}$ :$H$ contains $\tuple {0, 0}$, and only a finite number of the columns of $S$ are allowed to omit an infinite number of points in $H$. Then $\tau$ is the '''Arens-Fort topology''' on $S = \Z_{\ge 0} \times \Z_{\ge 0}$, and the topological space $T = \struct {S, \tau}$ is the '''Arens-Fort space'''.


Definition:Elementary Embedding

Let $\mathcal{M}$ and $\mathcal{N}$ be $\mathcal{L}$-structures with universes $M$ and $N$ respectively. An $\mathcal{L}$-embedding $j:\mathcal{M}\to \mathcal{N}$ is an '''elementary embedding''' {{iff}} it preserves truth; that is: :$\mathcal{M} \models \phi(a_1,\dots, a_n) \iff \mathcal{N} \models \phi(j(a_1),\dots, j(a_n))$ holds for all $n\in\mathbb N$, all $\mathcal{L}$-formulas $\phi$ with $n$ free variables, and for all $a_1,\dots,a_n \in M$. === Partial Elementary Embedding === {{:Definition:Elementary Embedding/Partial Elementary Embedding}} {{NoSources}} Category:Definitions/Model Theory 8i39p8d5esovaf6leawjz7ytpscjl32


Definition:Modified Fort Space

Let $N$ be an infinite set. Let $\set a$ and $\set b$ be singleton sets such that $a \ne b$ and $a, b \notin N$. Let $S = N \cup \set a \cup \set b$. Let $\tau_{a, b}$ be the set of subsets of $S$ defined as: :$\tau_{a, b} = \set {H \subseteq N} \cup \set {H \subseteq S: \paren {a \in H \lor b \in H} \land N \setminus H \text { is finite} }$ That is, a subset $H$ of $S$ is in $\tau_{a, b}$ {{iff}} either: :$(1): \quad H$ is any subset of $N$ or: :$(2): \quad$ if $a$ or $b$ or both are in $H$, then $H$ is in $S$ only if it is cofinite in $S$, that is, that it contains all but a finite number of points of $S$ (or $N$, equivalently). Then $\tau_{a, b}$ is a '''modified Fort topology''' on $a$ and $b$, and the topological space $T = \struct {S, \tau_{a, b} }$ is a '''modified Fort space'''.


Definition:Algebraic (Model Theory)

Let $\mathcal{M}$ be an $\mathcal{L}$-structure with universe $M$. Let $A$ be a subset of $M$. and let $\bar b$ be an ordered $n$-tuple of elements from $M$. Let $\mathcal{L}_A$ be the language formed by adding constant symbols to $\mathcal{L}$ for each element of $A$. $\bar b$ is '''algebraic''' over $A$ if there is an $\mathcal{L}_A$-formula $\phi(\bar x)$ with $n$ free variables such that $\mathcal{M}\models \phi(\bar b)$ and the set $\{\bar m \in M^n : \mathcal{M}\models \phi(\bar m)\}$ has only finitely many elements. {{Disambiguate|Definition:Logical Formula}}


Definition:Open Point

Let $T = \left({S, \tau}\right)$ be a topological space. Let $p \in S$ such that $\left\{{p}\right\} \in \tau$. That is, let $\left\{{p}\right\}$ be an open set of $T$. Then $p$ is an open point of $T$. {{SUBPAGENAME}} rs9ypa7mfatx7czrg2prm3peck65rh5


Definition:Age (Model Theory)

Let $\MM$ be an $\LL$-structure. An '''age''' of $\MM$ is a class $K$ of $\LL$-structures such that: * if $\AA$ is a finitely generated $\LL$-structure such that there is an $\LL$-embedding $\AA \to \MM$, then $\AA$ is isomorphic to some structure in $K$, * no two structures in $K$ are isomorphic, and * $K$ does not contain any structures which are not finitely generated or do not embed into $\MM$. That is, $K$ is an age of $\MM$ if it contains exactly one representative from each isomorphism type of the finitely-generated structures that embed into $\MM$. {{MissingLinks|finitely generated structure}} Category:Definitions/Model Theory 5hf61yvyjc9bw34ihvj4d6x21weg2x6


Let $M = \left({A, d}\right)$ be a metric space. === Isolated Point in Subset === {{:Definition:Isolated Point (Metric Space)/Subset}} === Isolated Point in Space === When $S = A$ this reduces to: {{:Definition:Isolated Point (Metric Space)/Space}}


Definition:Isolated Point

=== Topology === {{:Definition:Isolated Point (Topology)/Space}} === Metric Space === {{:Definition:Isolated Point (Metric Space)/Space}} === Complex Analysis === {{:Definition:Isolated Point (Complex Analysis)}} === Real Analysis === {{:Definition:Isolated Point (Real Analysis)}} Category:Definitions/Topology Category:Definitions/Isolated Points ij9snn0nz9qu0psws311i158x6bnib9


Definition:Isolated Point (Real Analysis)

Let $S \subseteq \R$ be a subset of the set of real numbers. Let $\alpha \in S$. Then $\alpha$ is an '''isolated point of $S$''' {{iff}} there exists an open interval of $\R$ whose midpoint is $\alpha$ which contains no points of $S$ except $\alpha$: :$\exists \epsilon \in \R_{>0}: \openint {\alpha - \epsilon} {\alpha + \epsilon} \cap S = \set \alpha$


Definition:Limit Point/Topology/Set

Let $A \subseteq S$. ==== Definition from Open Neighborhood ==== {{:Definition:Limit Point/Topology/Set/Definition 1}} ==== Definition from Closure ==== {{:Definition:Limit Point/Topology/Set/Definition 2}} ==== Definition from Adherent Point ==== {{:Definition:Limit Point/Topology/Set/Definition 3}} ==== Definition from Relative Complement ==== {{:Definition:Limit Point/Topology/Set/Definition 4}}


Definition:Limit Point/Complex Analysis

Let $S \subseteq \C$ be a subset of the set of complex numbers. Let $z_0 \in \C$.  Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$ for a given $\epsilon \in \R$ such that $\epsilon > 0$. Then $z_0$ is a '''limit point of $S$''' {{iff}} ''every'' deleted $\epsilon$-neighborhood $\map {N_\epsilon} {z_0} \setminus \set {z_0}$ of $z_0$ contains a point in $S$: :$\forall \epsilon \in \R_{>0}: \paren {\map {N_\epsilon} {z_0} \setminus \set {z_0} } \cap S \ne \O$ that is: :$\forall \epsilon \in \R_{>0}: \set {z \in S: 0 < \cmod {z - z_0} < \epsilon} \ne \O$


Definition:Limit Point/Metric Space

Let $M = \struct {S, d}$ be a metric space. Let $A \subseteq S$ be a subset of $S$. Let $\alpha \in S$.  Then $\alpha$ is a '''limit point of $A$''' {{iff}} ''every'' deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $A$: :$\forall \epsilon \in \R_{>0}: \map {B_\epsilon} \alpha \setminus \set \alpha \cap A \ne \O$ that is: :$\forall \epsilon \in \R_{>0}: \set {x \in A: 0 < \map d {x, \alpha} < \epsilon} \ne \O$ Note that $\alpha$ does not have to be an element of $A$ to be a '''limit point'''.


Definition:Deleted Neighborhood/Topology

Let $T = \left({S, \tau}\right)$ be a topological space. Let $x \in S$. Let $V \subseteq S$ be a neighborhood of $x$. Then $V \setminus \left\{{x}\right\}$ is called a '''deleted neighborhood of $x$'''. That is, it is a neighborhood of $x$ with $x$ itself ''removed''.


Definition:Deleted Neighborhood/Metric Space

Let $M = \left({A, d}\right)$ be a metric space. Let $x \in A$. Let $B_\epsilon \left({x}\right)$ be the open $\epsilon$-ball neighborhood of $x$. Then the '''deleted $\epsilon$-neighborhood''' of $x$ is defined as $B_\epsilon \left({x}\right) \setminus \left\{{x}\right\}$. That is, it is the open $\epsilon$-ball neighborhood of $x$ with $x$ itself ''removed''. It can also be defined as: : $\left\{{y \in A: 0 < d \left({x, y}\right) < \epsilon}\right\}$


Definition:Deleted Neighborhood/Complex Analysis

Let $z_0 \in \C$ be a point in the complex plane. Let $\map {N_\epsilon} {z_0}$ be the $\epsilon$-neighborhood of $z_0$. Then the '''deleted $\epsilon$-neighborhood''' of $z_0$ is defined as $\map {N_\epsilon} {z_0} \setminus \set {z_0}$. That is, it is the $\epsilon$-neighborhood of $z_0$ with $z_0$ itself ''removed''. It can also be defined as: :$\map {N_\epsilon} {z_0} \setminus \set {z_0} : = \set {z \in A: 0 < \cmod {z_0 - z} < \epsilon}$ from the definition of $\epsilon$-neighborhood.


Definition:Deleted Neighborhood/Real Analysis

Let $\alpha \in \R$ be a real number. Let $N_\epsilon \left({\alpha}\right)$ be the $\epsilon$-neighborhood of $\alpha$: :$N_\epsilon \left({\alpha}\right) := \left({\alpha - \epsilon \,.\,.\, \alpha + \epsilon}\right)$ Then the '''deleted $\epsilon$-neighborhood''' of $\alpha$ is defined as $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\}$. That is, it is the $\epsilon$-neighborhood of $\alpha$ with $\alpha$ itself ''removed''. It can also be defined as: : $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} : = \left\{{x \in \R: 0 < \left \vert{\alpha - x}\right \vert < \epsilon}\right\}$ or : $N_\epsilon \left({\alpha}\right) \setminus \left\{{\alpha}\right\} : = \left({\alpha - \epsilon \,.\,.\, \alpha}\right) \cup \left({\alpha \,.\,.\, \alpha + \epsilon}\right)$ from the definition of $\epsilon$-neighborhood.


Definition:Isolated Point (Complex Analysis)

Let $S \subseteq \C$ be a subset of the set of real numbers. Let $z \in S$. Then $z$ is an '''isolated point of $S$''' {{iff}} there exists a neighborhood of $z$ in $\C$ which contains no points of $S$ except $z$: :$\exists \epsilon \in \R_{>0}: \map {N_\epsilon} z \cap S = \set z$


Definition:Lipschitz Quaternion

A '''Lipschitz quaternion''' is a quaternion whose components are all integers. The set $\Bbb L$ of all '''Lipschitz quaternions''' can therefore be defined as: :$\Bbb L = \left\{{a + b \mathbf i + c \mathbf j + d \mathbf k \in \Bbb H: a, b, c, d \in \Z}\right\}$ {{NamedforDef|Rudolf Lipschitz|cat=Lipschitz}}


Definition:Hurwitz Quaternion

A '''Hurwitz quaternion''' is a quaternion whose components are all either: :integers or: :integers plus a half, that is, halves of odd integers. The set $H$ of all '''Hurwitz quaternions''' can therefore be defined as: :$H = \left\{{a + b \mathbf i + c \mathbf j + d \mathbf k \in \Bbb H: \left({a, b, c, d \in \Z}\right) \text { or } \left({a, b, c, d \in \Z + \dfrac 1 2}\right)}\right\}$


Definition:Multiple Pointed Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $A$ be a finite set whose cardinality is greater than $1$. Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on $A$. Let $T \times D$ be the product space of $T$ and $D$. Then $T \times D$ is known as the '''multiple pointed topology''' on $T$. It is seen that $T \times D$ is conceptually equivalent to taking the space $T$ and replacing each point with a finite set of topologically indistinguishable points.


Definition:Box Topology

Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be an $I$-indexed family of topological spaces. Let $X$ be the cartesian product of $\family {X_i}_{i \mathop \in I}$, that is: :$\displaystyle X := \prod_{i \mathop \in I} X_i$ Define: :$\displaystyle \BB := \set {\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}$ Then $\BB$ is a synthetic basis on $X$, as shown on Basis for Box Topology. The '''box topology''' on $X$ is defined as the topology $\tau$ generated by the synthetic basis $\BB$.


Definition:Big Model

Let $\mathcal{M}$ be an $\mathcal{L}$-structure with universe $M$. Let $\kappa$ be a cardinal. $\mathcal{M}$ is '''$\kappa$-big''' if for every subset $A\subset M$ with cardinality $|A| < \kappa$, the following holds: :if $\mathcal{L}_A$ is the language obtained from $\mathcal{L}$ by adding new constant symbols for each $a\in A$, then :for every language $\mathcal{L}_A^*$ obtained by adding a new relation symbol $R$ to $\mathcal{L}_A$, and :for every $\mathcal{L}_A^*$-structure $\mathcal{N}$ such that $\mathcal{M}$ and $\mathcal{N}$ are elementary equivalent as $\mathcal{L}_A$-structures, :there is a relation $R^\mathcal{M}$ on $M$ such that $(\mathcal{M},R^\mathcal{M})$ is elementary equivalent to $\mathcal{N}$ as an $\mathcal{L}_A^*$-structure.


Definition:Word Metric

Let $\struct {G, \circ}$ be a group. Let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \in S \iff x \in S$). The '''word metric''' on $G$ with respect to $S$ is the metric $d_S$ defined as follows: :For any $g, h \in G$, let $\map {d_S} {g, h}$ be the minimum length among the finite sequences $\tuple {x_1, \dots, x_n}$ with each $x_i \in S$ such that $g \circ x_1 \circ \cdots \circ x_n = h$. Informally, $\map {d_S} {g, h}$ is the smallest number of elements from $S$ that one needs to multiply by to get from $g$ to $h$.


Definition:Continuous Extension

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Let $A, B \subseteq S_1$ be subsets of $S_1$ such that $A \subseteq B$. Let $f: A \to S_2$ and $g: B \to S_2$ be continuous mappings. Then $g$ is a '''continuous extension''' of $f$ {{iff}}: :$\forall s \in A: f \left({s}\right) = g \left({s}\right)$ That is, a '''continuous extension''' of $f$ is a continuous mapping on a superset which agrees with $f$ on the domain of $f$. Simply, it is a continuous mapping which is an extension. === Real Function === {{:Definition:Continuous Extension/Real Function}} Category:Definitions/Continuity s7yn9e1314qrkrgqebxi7io2vrl0bb7


Definition:Diagonal Mapping

Let $S$ be a set. Let $S \times S$ be the Cartesian product of $S$ with itself. Then the '''diagonal mapping on $S$''' is defined as $\Delta: S \to S \times S$: :$\forall x \in S: \Delta \left({x}\right) = \left({x, x}\right)$ Clearly $\Delta$ is an injection, and is not a surjection unless $S$ is a singleton. {{SUBPAGENAME}} tgmqf7v43mpneequ5iqkxn1v1zkho6g


Definition:Minimal Arithmetic

'''Minimal arithmetic''' is the set $Q$ of theorems of the recursive set of sentences in the language of arithmetic containing exactly: {{begin-axiom}} {{axiom | n = \text M 1         | q = \forall x         | m = \map s x \ne 0 }} {{axiom | n = \text M 2         | q = \forall x, y         | m = \map s x = \map s y \implies x = y }} {{axiom | n = \text M 3         | q = \forall x         | m = x + 0 = x }} {{axiom | n = \text M 4         | q = \forall x, y         | m = x + \map s y = \map s {x + y} }} {{axiom | n = \text M 5         | q = \forall x         | m = x \cdot 0 = 0 }} {{axiom | n = \text M 6         | q = \forall x, y         | m = x \cdot \map s y = \paren {x \cdot y} + x }} {{axiom | n = \text M 7         | q = \forall x         | m = \neg x < 0 }} {{axiom | n = \text M 8         | q = \forall x, y         | m = x < \map s y \iff \paren {x < y \lor x = y} }} {{axiom | n = \text M 9         | q = \forall x         | m = 0 < x \iff x \ne 0 }} {{axiom | n = \text M 10         | q = \forall x, y         | m = \map s x < y \iff \paren {x < y \land y \ne \map s x} }} {{end-axiom}}


Definition:Language of Arithmetic

A '''language of arithmetic''' is a signature for predicate logic consisting of: * the binary operation symbols: $+$ and $\cdot$ * the unary function symbol: $s$ * the binary relation symbol: $<$ * the constant symbol: $0$


Definition:Euclid's Definitions - Book I/2 - Line

A '''line''' is breadthless length.


Definition:Euclid's Definitions - Book I/4 - Straight Line

A '''straight line''' is a line which lies evenly with the points on itself.


Definition:Euclid's Definitions - Book I/3 - Line Extremities

The extremities of a line are points.


Definition:Euclid's Definitions - Book XI/2 - Extremity of Solid

An extremity of a solid is a surface.


Definition:Euclid's Definitions - Book I/7 - Plane Surface

A '''plane surface''' is a surface which lies evenly with the straight lines on itself.


Definition:Euclid's Definitions - Book I/6 - Surface Extremities

The extremities of a surface are lines.


Definition:Limit Point/Real Analysis

Let $S \subseteq \R$ be a subset of the real numbers. Let $\xi \in \R$ and let $S_\xi$ be the set defined as: :$S_\xi := \left\{{x: x \in S, x \ne \xi}\right\}$ Then $\xi$ is a '''limit point''' of $S$ {{iff}} $\xi$ is at zero distance from $S_\xi$.


Definition:Euclid's Definitions - Book I/1 - Point

A '''point''' is that which has no part.


Definition:Euclid's Definitions - Book I/5 - Surface

A '''surface''' is that which has length and breadth only.


Definition:Euclid's Definitions - Book I/8 - Plane Angle

A '''plane angle''' is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.


Definition:Euclid's Definitions - Book I/9 - Rectilineal Angle

And when the lines containing the angle are straight, the angle is called '''rectilineal'''.


Definition:Euclid's Definitions - Book I/10 - Right Angle

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is '''right''', and the straight line standing on the other is called a '''perpendicular''' to that on which it stands.


Definition:Euclid's Definitions - Book I/11 - Obtuse Angle

An '''obtuse angle''' is an angle greater than a right angle.


Definition:Euclid's Definitions - Book I/12 - Acute Angle

An '''acute angle''' is an angle less than a right angle.


Definition:Euclid's Definitions - Book I/13 - Boundary

A '''boundary''' is that which is an extremity of anything.


Definition:Euclid's Definitions - Book I/14 - Geometric Figure

A '''figure''' is that which is contained by any boundary or boundaries.


Definition:Euclid's Definitions - Book I/15 - Circle

A '''circle''' is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another;


Definition:Euclid's Definitions - Book I/16 - Center of Circle

And the point is called the '''center of the circle'''.


Definition:Euclid's Definitions - Book I/17 - Diameter of Circle

A '''diameter of the circle''' is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the center.


Definition:Euclid's Definitions - Book I/18 - Semicircle

A '''semicircle''' is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.


Definition:Euclid's Definitions - Book I/19 - Rectilineal Figure

'''Rectilineal figures''' are those which are contained by straight lines, '''trilateral''' figures being those contained by three, '''quadrilateral''' those contained by four, and '''multi-lateral''' those contained by more than four straight lines.


Definition:Euclid's Definitions - Book I/22 - Quadrilaterals

Of quadrilateral figures, a '''square''' is that which is both equilateral and right-angled; an '''oblong''' that which is right-angled but not equilateral; a '''rhombus''' that which is equilateral but not right-angled; and a '''rhomboid''' that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called '''trapezia'''.


Definition:Euclid's Definitions - Book I/20 - Triangles by Sides

Of trilateral figures, an '''equilateral triangle''' is that which has its three sides equal, an '''isosceles triangle''' that which has two of its sides alone equal, and a '''scalene triangle''' that which has its three sides unequal.


Definition:Euclid's Definitions - Book I/21 - Triangles by Angles

Further, of trilateral figures, a '''right-angled triangle''' is that which has a right angle, an '''obtuse-angled triangle''' that which has an obtuse angle, and an '''acute-angled triangle''' that which has its three angles acute.


Definition:Euclid's Definitions - Book I/23 - Parallel Lines

'''Parallel''' straight lines are straight lines which, being in the same plane and being produced indefinitely in either direction, do not meet one another in either direction.


Definition:Euclid's Definitions - Book II/1 - Containment of Rectangle

Any rectangular parallelogram is said to be '''contained''' by the two straight lines containing the right angle.


Definition:Euclid's Definitions - Book II/2 - Gnomon

And in any parallelogrammic area let any one whatever of the parallelograms about its diameter with the two complements be called a '''gnomon'''.


Definition:Euclid's Definitions - Book III/1 - Equal Circles

'''Equal circles''' are those the diameters of which are equal, or the radii of which are equal.


Definition:Euclid's Definitions - Book III/2 - Tangent to Circle

A straight line is said to '''touch a circle''' which, meeting the circle and being produced, does not cut the circle.


Definition:Euclid's Definitions - Book III/3 - Tangent Circles

Circles are said to '''touch one another''' which, meeting one another, do not cut one another.


Definition:Euclid's Definitions - Book III/4 - Equally Distant in Circle

In a circle straight lines are said to be '''equally distant from the center''' when the perpendiculars drawn to them from the center are equal.


Definition:Euclid's Definitions - Book III/5 - Greater Distance in Circle

And that straight line is said to be '''at a greater distance''' on which the greater perpendicular falls.


Definition:Euclid's Definitions - Book III/6 - Segment of Circle

A '''segment of a circle''' is the figure contained by a straight line and a circumference of a circle.


Definition:Euclid's Definitions - Book III/7 - Angle of Segment

An '''angle of a segment''' is that contained by a straight line and a circumference of a circle.


Definition:Euclid's Definitions - Book III/8 - Angle in Segment

An '''angle in a segment''' is the angle which, when a point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, is contained by the straight lines so joined.


Definition:Euclid's Definitions - Book III/9 - Stand on Circumference

And, when the straight lines containing the angle cut off a circumference, the angle is said to '''stand upon''' that circumference.


Definition:Euclid's Definitions - Book III/10 - Sector of Circle

A '''sector of a circle''' is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.


Definition:Euclid's Definitions - Book III/11 - Similar Segments

'''Similar segments of circles''' are those which admit equal angles, or in which the angles are equal to one another.


Definition:Euclid's Definitions - Book IV/1 - Inscribe

A rectilineal figure is said to be '''inscribed in a rectilineal figure''' when the respective angles of the inscribed figure lie on the respective sides of that in which it is inscribed.


Definition:Euclid's Definitions - Book IV/2 - Circumscribe

Similarly a figure is said to be '''circumscribed about a figure''' when the respective sides of the circumscribed figure pass through the respective angles of that about which it is circumscribed.


Definition:Euclid's Definitions - Book IV/3 - Inscribed in Circle

A rectilineal figure is said to be '''inscribed in a circle''' when each angle of the inscribed figure lies on the circumference of the circle.


Definition:Euclid's Definitions - Book IV/4 - Circumscribed about Circle

A rectilineal figure is said to be '''circumscribed about a circle''', when each side of the circumscribed figure touches the circumference of the circle.


Definition:Euclid's Definitions - Book IV/5 - Circle Inscribed

Similarly a circle is said to be '''inscribed in a figure''' when the circumference of the circle touches each side of the figure in which it is inscribed.


Definition:Euclid's Definitions - Book IV/6 - Circle Circumscribed

A circle is said to be '''circumscribed about a figure''' when the circumference of the circle passes through each angle of the figure about which it is circumscribed.


Definition:Euclid's Definitions - Book IV/7 - Fitted Into Circle

A straight line is said to be '''fitted into a circle''' when its extremities are on the circumference of the circle.


Definition:Measure (Geometry)

A geometric quantity $A$ is said to '''measure''' another quantity $B$ when the size of $A$ is a divisor of the size of $B$. Category:Definitions/Geometry 7pv73327m91o2y8shdhh3f65fao156f


Definition:Part

=== Of Integer === {{:Definition:Divisor (Algebra)/Integer/Aliquot Part}} === Of Real Number === {{:Definition:Divisor (Algebra)/Real Number/Part}}


Definition:Euclid's Definitions - Book V/1 - Part

A magnitude is a '''part''' of a magnitude, the less of the greater, when it measures the greater.


Definition:Euclid's Definitions - Book V/2 - Multiple

The greater is a '''multiple''' of the less when it is measured by the less.


Definition:Euclid's Definitions - Book V/3 - Ratio

A '''ratio''' is a sort of relation in respect of size between two magnitudes of the same kind.


Definition:Euclid's Definitions - Book V/4 - Existence of Ratio

Magnitudes are said to '''have a ratio''' to one another which are capable, when multiplied, of exceeding one another.


Definition:Euclid's Definitions - Book V/5 - Equality of Ratios

Magnitudes are said to '''be in the same ratio''', the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.


Definition:Euclid's Definitions - Book V/6 - Proportion

Let magnitudes which have the same ratio be called '''proportional'''.


Definition:Euclid's Definitions - Book V/7 - Greater Ratio

When, of the equimultiples, the multiple of the first magnitude exceeds the multiple of the second, but the multiple of the third does not exceed the multiple of the fourth, then the first is said to '''have a greater ratio''' to the second than the third has to the fourth.


Definition:Euclid's Definitions - Book V/8 - Proportion in Three Terms

A proportion in three terms is the least possible.


Definition:Euclid's Definitions - Book V/9 - Duplicate Ratio

When three magnitudes are proportional, the first is said to have to the third the '''duplicate ratio''' of that which it has to the second.


Definition:Euclid's Definitions - Book V/10 - Triplicate Ratio

When four magnitudes are $<$ continuously $>$ proportional, the first is said to have to the fourth the '''triplicate ratio''' of that which it has to the second, and so on continually, whatever be the proportion.


Definition:Term (Algebra)

A '''term''' is either a variable or a constant. Let $a \circ b$ be an expression. Then each of $a$ and $b$ are known as the '''terms''' of the expression. The word '''term''' is usually used when the operation $\circ$ is addition, that is $+$.


Definition:Term (Natural Language)

A '''term''' is a noun which is assigned a specified definition within mathematics. In a more specialized context, the word '''term''' is used for an element of an expression. Category:Definitions/Language Definitions 13knjhv833n7iyw3wt2kfec2jupvprb


Definition:Euclid's Definitions - Book V/11 - Corresponding Magnitudes

The term '''corresponding magnitudes''' is used of antecedents in relation to antecedents, and of consequents in relation to consequents.


Definition:Euclid's Definitions - Book V/12 - Alternate Ratio

'''Alternate ratio''' means taking the antecedent in relation to the antecedent and the consequent in relation to the consequent.


Definition:Euclid's Definitions - Book V/13 - Inverse Ratio

'''Inverse ratio''' means taking the consequent as antecedent in relation to the antecedent as consequent.


Definition:Euclid's Definitions - Book V/14 - Composition of Ratio

'''Composition of a ratio''' means taking the antecedent together with the consequent as one in relation to the consequent by itself.


Definition:Euclid's Definitions - Book V/15 - Separation of Ratio

'''Separation of a ratio''' means taking the excess by which the antecedent exceeds the consequent in relation to the consequent by itself.


Definition:Euclid's Definitions - Book V/17 - Ratio Ex Aequali

A '''ratio ex aequali''' arises when, there being several magnitudes and another set equal to them in multitude which makes two and two are in the same proportion, as the first is to the last among the first magnitudes, so is the first to the last among the second magnitudes;<br/>Or, in other words, it means taking the extreme terms by virtue of the removal of the intermediate terms.


Definition:Euclid's Definitions - Book V/16 - Conversion of Ratio

'''Conversion of a ratio''' means taking the antecedent in relation to the excess by which the antecedent exceeds the consequent.


Definition:Euclid's Definitions - Book V/18 - Perturbed Proportion

A '''perturbed proportion''' arises when, there being three magnitudes and another set equal to them in multitude, as antecedent is to consequent among the first magnitudes, so is antecedent to consequent among the second magnitudes, while, as the consequent is to a third among the first magnitudes, so is a third to the antecedent among the second magnitudes.


Definition:Euclid's Definitions - Book VI/1 - Similar Rectilineal Figures

'''Similar rectilineal figures''' are such as have their angles severally equal and the sides about the equal angles proportional.


Definition:Euclid's Definitions - Book VI/2 - Reciprocally Related Figures

Two figures are '''reciprocally related''' when there are in each of the two figures antecedent and consequent ratios.


Definition:Euclid's Definitions - Book VI/3 - Extreme and Mean Ratio

A straight line is said to have been '''cut in extreme and mean ratio''' when, as the whole line is to the greater segment, so is the greater to the less.


Definition:Euclid's Definitions - Book VI/4 - Height

The '''height''' of any figure is the perpendicular drawn from the vertex to the base.


Definition:Euclid's Definitions - Book VII/1 - Unit

An '''unit''' is that of which each of the things that exist is called one.


Definition:Euclid's Definitions - Book VII/2 - Number

A '''number''' is a multitude composed of units.


Definition:Euclid's Definitions - Book VII/3 - Part

A number is '''a part''' of a number, the less of the greater, when it measures the greater;


Definition:Euclid's Definitions - Book VII/4 - Parts

but '''parts''' when it does not measure it.


Definition:Euclid's Definitions - Book VII/5 - Multiple

The greater number is a '''multiple''' of the less when it is measured by the less.


Definition:Euclid's Definitions - Book VII/6 - Even Number

An '''even number''' is that which is divisible into two equal parts.


Definition:Euclid's Definitions - Book VII/7 - Odd Number

An '''odd number''' is that which is not divisible into two equal parts, or that which differs by an unit from an even number.


Definition:Euclid's Definitions - Book VII/8 - Even Times Even

An '''even-times even number''' is that which is measured by an even number according to an even number.


Definition:Euclid's Definitions - Book VII/9 - Even Times Odd

An '''even-times odd number''' is that which is measured by an even number according to an odd number.


Definition:Euclid's Definitions - Book VII/10 - Odd Times Odd

An '''odd-times odd number''' is that which is measured by an odd number according to an odd number.


Definition:Euclid's Definitions - Book VII/11 - Prime Number

A '''prime number''' is that which is measured by an unit alone.


Definition:Euclid's Definitions - Book VII/12 - Relatively Prime

Numbers '''prime to one another''' are those which are measured by an unit alone as a common measure.


Definition:Euclid's Definitions - Book VII/13 - Composite Number

A '''composite number''' is that which is measured by some number.


Definition:Euclid's Definitions - Book VII/14 - Relatively Composite

Numbers '''composite to one another''' are those which are measured by some number as a common measure.


Definition:Euclid's Definitions - Book VII/15 - Multiply

A number is said to '''multiply''' a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.


Definition:Euclid's Definitions - Book VII/16 - Plane Number

And, when two numbers having multiplied one another make some number, the number so produced is called '''plane''', and its '''sides''' are the numbers which have multiplied one another.


Definition:Euclid's Definitions - Book VII/17 - Solid Number

And, when three numbers having multiplied one another make some number, the number so produced is '''solid''', and its '''sides''' are the numbers which have multiplied one another.


Definition:Euclid's Definitions - Book VII/18 - Square Number

A '''square number''' is equal multiplied by equal, or a number which is contained by two equal numbers.


Definition:Euclid's Definitions - Book VII/19 - Cube Number

And a '''cube''' is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.


Definition:Euclid's Definitions - Book VII/20 - Proportional

Numbers are '''proportional''' when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.


Definition:Euclid's Definitions - Book VII/21 - Similar Numbers

'''Similar plane''' and '''solid''' numbers are those which have their sides proportional.


Definition:Euclid's Definitions - Book VII/22 - Perfect Number

A '''perfect number''' is that which is equal to its own parts.


Definition:Euclid's Definitions - Book X/1 - Commensurable

Those magnitudes are said to be '''commensurable''' which are measured by the same same measure, and those '''incommensurable''' which cannot have any common measure.


Definition:Euclid's Definitions - Book X/2 - Commensurable in Square

Straight lines are '''commensurable in square''' when the squares on them are measured by the same area, and '''incommensurable in square''' when the squares on them cannot possibly have any area as a common measure.


Definition:Euclid's Definitions - Book X/3 - Rational Line Segment

With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called '''rational''', and those straight lines which are commensurable with it, whether in length and in square or square only, '''rational''', but those which are incommensurable with it '''irrational'''.


Definition:Euclid's Definitions - Book X/4 - Rational Area

And let the square on the assigned straight line be called '''rational''' and those areas which are commensurable with it '''rational''', but those which are incommensurable with it '''irrational''', and the straight lines which produce them '''irrational''', that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.


Definition:Euclid's Definitions - Book X (II)/1 - First Binomial

Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a '''first binomial''' straight line;


Definition:Euclid's Definitions - Book X (II)/2 - Second Binomial

but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a '''second binomial''';


Definition:Euclid's Definitions - Book X (II)/3 - Third Binomial

and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a '''third binomial'''.


Definition:Euclid's Definitions - Book X (II)/4 - Fourth Binomial

Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a '''fourth binomial''';


Definition:Euclid's Definitions - Book X (II)/5 - Fifth Binomial

if the lesser, a '''fifth binomial''';


Definition:Euclid's Definitions - Book X (II)/6 - Sixth Binomial

and if neither, a '''sixth binomial'''.


Definition:Euclid's Definitions - Book X (III)/1 - First Apotome

Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole be commensurable in length with the rational straight line set out, let the apotome be called a '''first apotome'''.


Definition:Euclid's Definitions - Book X (III)/2 - Second Apotome

But if the annex be commensurable in length with the rational straight line set out, and the square on the whole be greater than that on the annex by the square on a straight line commensurable in length with the whole, let the apotome be called a '''second apotome'''.


Definition:Euclid's Definitions - Book X (III)/3 - Third Apotome

But if neither be commensurable in length with the rational straight line set out, and the square on the whole be greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a '''third apotome'''.


Definition:Euclid's Definitions - Book X (III)/4 - Fourth Apotome

Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in length with the rational straight line set out, let the apotome be called a '''fourth apotome''';


Definition:Euclid's Definitions - Book X (III)/5 - Fifth Apotome

if the annex be so commensurable, a '''fifth''';


Definition:Euclid's Definitions - Book X (III)/6 - Sixth Apotome

and if neither, a '''sixth'''.


Definition:Euclid's Definitions - Book XI/1 - Solid

A '''solid''' is that which has length, breadth, and depth.


Definition:Euclid's Definitions - Book XI/3 - Line at Right Angles to Plane

A '''straight line''' is '''at right angles to a plane''' when it makes right angles with all the straight lines which meet it and are in the plane.


Definition:Euclid's Definitions - Book XI/4 - Plane at Right Angles to Plane

A '''plane''' is '''at right angles to a plane''' when the straight lines drawn, in one of the planes, at right angles to the common section of the planes are at right angles to the remaining plane.


Definition:Euclid's Definitions - Book XI/5 - Inclination of Straight Line

The '''inclination of a straight line to a plane''' is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.


Definition:Euclid's Definitions - Book XI/6 - Inclination of Plane

The '''inclination of a plane to a plane''' is the acute angle contained by the straight lines drawn at right angles to the common section at the same point, one in each of the planes.


Definition:Euclid's Definitions - Book XI/7 - Similarly Inclined

A plane is said to be '''similarly inclined''' to a plane as another is to another when the said angles of the inclinations are equal to one another.


Definition:Euclid's Definitions - Book XI/8 - Parallel Planes

'''Parallel planes''' are those which do not meet.


Definition:Euclid's Definitions - Book XI/9 - Similar Solid Figures

'''Similar solid figures''' are those contained by similar planes equal in multitude.


Definition:Euclid's Definitions - Book XI/10 - Similar Equal Solid Figures

'''Equal and similar solid figures''' are those contained by similar planes equal in multitude and in magnitude.


Definition:Euclid's Definitions - Book XI/11 - Solid Angle

A '''solid angle''' is the inclination constituted by more than two lines which meet one another and are not in the same surface, towards all the lines.<br/>Otherwise: A '''solid angle''' is that which is contained by more than two plane angles which are not in the same plane and are constructed to one point.


Definition:Euclid's Definitions - Book XI/12 - Pyramid

A '''pyramid''' is a solid figure, contained by planes, which is constructed from one plane to one point.


Definition:Euclid's Definitions - Book XI/13 - Prism

A '''prism''' is a solid figure contained by planes of which, namely those which are opposite, are equal, similar and parallel, while the rest are parallelograms.


Definition:Euclid's Definitions - Book XI/14 - Sphere

When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''sphere'''.


Definition:Euclid's Definitions - Book XI/15 - Axis of Sphere

The '''axis of the sphere''' is the straight line which remains fixed about which the semicircle is turned.


Definition:Euclid's Definitions - Book XI/16 - Center of Sphere

The '''center of the sphere''' is the same as that of the semicircle.


Definition:Euclid's Definitions - Book XI/17 - Diameter of Sphere

A '''diameter of the sphere''' is any straight line drawn through the centre and terminated in both directions by the surface of the sphere.


Definition:Euclid's Definitions - Book XI/18 - Cone

When, one side of those about the right angle in a right-angled triangle remaining fixed, the triangle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''cone'''.<br>And, if the straight line which remains fixed be equal to the remaining side about the right angle which is carried round, the cone will be '''right-angled'''; if less, '''obtuse-angled'''; and if greater, '''acute-angled'''.


Definition:Euclid's Definitions - Book XI/19 - Axis of Cone

The '''axis of the cone''' is the straight line which remains fixed and about which the triangle is turned.


Definition:Euclid's Definitions - Book XI/20 - Base of Cone

And the '''base''' is the circle described by the straight line which is carried round.


Definition:Euclid's Definitions - Book XI/21 - Cylinder

When, one side of those about the right angle in a rectangular parallelogram remaining fixed, the parallelogram is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a '''cylinder'''.


Definition:Euclid's Definitions - Book XI/22 - Axis of Cylinder

The '''axis of the cylinder''' is the straight line which remains fixed and about which the parallelogram is turned.


Definition:Euclid's Definitions - Book XI/23 - Base of Cylinder

And the '''bases''' are the circles described by the two sides opposite to one another which are carried round.


Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders

'''Similar cones and cylinders''' are those in which the axes and the diameters of the bases are proportional.


Definition:Euclid's Definitions - Book XI/25 - Cube

A '''cube''' is a solid figure contained by six equal squares.


Definition:Euclid's Definitions - Book XI/26 - Octahedron

An '''octahedron''' is a solid figure contained by eight equal and equilateral triangles.


Definition:Euclid's Definitions - Book XI/27 - Icosahedron

An '''icosahedron''' is a solid figure contained by twenty equal and equilateral triangles.


Definition:Euclid's Definitions - Book XI/28 - Dodecahedron

A '''dodecahedron''' is a solid figure contained by twelve equal, equilateral, and equiangular pentagons.


Definition:Conjugate of Quadratic Irrational

Let $\alpha = r + s \sqrt n$ be a quadratic irrational. Then its '''conjugate''' is defined as: :$\tilde \alpha = r - s \sqrt n$ Thus $\alpha$ and $\tilde \alpha$ are known as '''conjugate quadratic irrationals'''. Notation may vary.


Definition:Pentagon

A '''pentagon''' is a polygon with exactly $5$ sides: :300px


Definition:Hexagon

A '''hexagon''' is a polygon with exactly $6$ sides. :300px


Definition:Equimultiples

Let $a$ and $b$ be numbers. Then '''equimultiples''' of $a$ and $b$ are numbers of the form $c a$ and $c b$ where $c$ is also a number. That is, they are products arising from the multiplications of two numbers by the same common number.


Definition:Ratio

Let $x$ and $y$ be quantities which have the same dimensions. Let $\dfrac x y = \dfrac a b$ for two numbers $a$ and $b$. Then the '''ratio of $x$ to $y$''' is defined as: :$x : y = a : b$ It explicitly specifies how many times the first number contains the second.


Definition:Archimedean Property

Let $\struct  {S, \circ}$ be a closed algebraic structure on which there exists either an ordering or a norm. Let $\cdot: \Z_{>0} \times S \to S$ be the operation defined as: :$m \cdot a = \begin{cases} a & : m = 1 \\ a \circ \paren {\paren {m - 1} \cdot a} & : m > 1 \end {cases}$ === Archimedean Property on Norm === {{:Definition:Archimedean Property/Norm}} === Archimedean Property on Ordering === {{:Definition:Archimedean Property/Ordering}}


Definition:Alternate Ratio

Let $a : b$ and $c : d$ where $a : b$ denotes the ratio of $a$ to $b$. Then the '''alternate ratio''' is defined as $a : c$ and $b : d$. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/12 - Alternate Ratio}}'' {{EuclidDefRefNocat|V|12|Alternate Ratio}} Category:Definitions/Euclidean Algebra piabn5rg6ig22rgmo41i3zxjb9hcgt9


Definition:Ratio Ex Aequali

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/17 - Ratio Ex Aequali}}'' {{EuclidDefRefNocat|V|17|Ratio Ex Aequali}} Category:Definitions/Euclidean Algebra 94lau22sv5d0itmfa7ufokv79qlyzlv


Definition:Proportion/Perturbed

Let $a, b, c$ and $A, B, C$ be magnitudes. $a, b, c$ are '''in perturbed proportion''' to $A, B, C$ {{iff}}: :$a : b = B : C$ :$b : c = A : B$ {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book V/18 - Perturbed Proportion}}'' {{EuclidDefRefNocat|V|18|Perturbed Proportion}}


Definition:Duplicate Ratio

Let $a, b, c$ be magnitudes such that: :$a : b = b : c$ Then $a$ has the '''duplicate ratio''' to $c$ of the ratio it has to $b$. That is: :$a : c$ is the '''duplicate ratio''' of $a : b$. From the definition of ratio: {{begin-eqn}} {{eqn | l=a : b       | r=b : c       | c= }} {{eqn | ll=\implies       | l=\dfrac a b       | r=\dfrac b c       | c= }} {{eqn | ll=\implies       | l=\dfrac a c       | r=\dfrac a b \dfrac b c       | c= }} {{eqn | ll=\implies       | l=\dfrac a c       | r=\left({\dfrac a b}\right)^2       | c= }} {{end-eqn}} That is: : $a : c = \left({a : b}\right)^2$ {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/9 - Duplicate Ratio}} {{EuclidDefRefNocat|V|9|Duplicate Ratio}} Category:Definitions/Euclidean Algebra 746epqd5j2n81g9dzz4z7nsjb2hwbr5


Definition:Triplicate Ratio

Let $a, b, c, d$ be magnitudes such that: :$a : b = b : c = c : d$ Then $a$ has the '''triplicate ratio''' to $d$ of the ratio it has to $b$. That is: :$a : d$ is the '''triplicate ratio''' of $a : b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/10 - Triplicate Ratio}} {{EuclidDefRefNocat|V|10|Triplicate Ratio}} Category:Definitions/Euclidean Algebra dhtr67om9nqt1ghro0y9v2eg2faj15o


Definition:Corresponding Magnitudes

Let $R_1 = a : b$ and $R_2 = c : d$ be ratios. Then the '''corresponding magnitudes''' of $a$ and $b$ in $R_2$ are $c$ and $d$ respectively. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/11 - Corresponding Magnitudes}} {{EuclidDefRefNocat|V|11|Corresponding Magnitudes}} Category:Definitions/Euclidean Algebra o82jjujsnjdiaptw7gincdz3rsfbct3


Definition:Inverse Ratio

Let $R = a : b$ be a ratio. Then the '''inverse ratio''' to $R$ is $b : a$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/13 - Inverse Ratio}} {{EuclidDefRefNocat|V|13|Inverse Ratio}}


Definition:Composition of Ratio

Let $R = a : b$ be a ratio. Then the '''composition of $R$''' is the ratio $a + b : b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/14 - Composition of Ratio}} {{EuclidDefRefNocat|V|14|Composition of Ratio}} Category:Definitions/Euclidean Algebra 2cu4arsolrxnoen6gc1bp5i2iut11o9


Definition:Separation of Ratio

Let $R = a : b$ be a ratio. Then the '''separation of $R$''' is the ratio $a - b : b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/15 - Separation of Ratio}} {{EuclidDefRefNocat|V|15|Separation of Ratio}} Category:Definitions/Euclidean Algebra bugz6tpgo3owyiqguubjszvwcneh1qm


Definition:Conversion of Ratio

Let $R = a : b$ be a ratio. Then the '''conversion of $R$''' is the ratio $a : a - b$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book V/16 - Conversion of Ratio}} {{EuclidDefRefNocat|V|16|Conversion of Ratio}} Category:Definitions/Euclidean Algebra p48tjawh10chh0rk1xgurfmzezaevut


Definition:Section of Line by Line

Let a geometrical line $A$ cross over (or intersect) another line $B$. The point where they cross is called the '''section''' of the $B$ by $A$ (or equivalently, of $A$ by $B$).


Definition:Similar Figures

Two rectilineal figures are '''similar''' {{iff}}: :They have corresponding angles, all of which are equal :They have corresponding sides, all of which are proportional.


Definition:Reciprocal Relation

{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VI/2 - Reciprocally Related Figures}} {{EuclidDefRefNocat|VI|2|Reciprocally Related Figures}}


Definition:Similar Triangles

Similar triangles are triangles whose corresponding angles are the same, but whose corresponding sides may be of different lengths. :300px Thus $\triangle ABC$ is similar to $\triangle DEF$: :$\angle ABC = \angle EFD$ :$\angle BCA = \angle EDF$ :$\angle CAB = \angle DEF$


Definition:Rectification

'''Rectification''' is an archaic term meaning '''the process of finding length''', specifically in the context of arcs.


Definition:Reciprocal Proportion

Let $P$ and $Q$ be geometric figures of the same type (that is, having the same number and configuration of sides). Let $A$ and $B$ be sides of $P$, and let $C$ and $D$ be sides of $Q$, such that $A$ and $C$ are corresponding sides, and $B$ and $D$ also be corresponding sides. Then $P$ and $Q$ have sides which are '''in reciprocal proportion''', or are '''reciprocally proportional''', if: :$A : D = B : C$ where $A : D$ is the ratio of the lengths of $A$ and $D$.


Definition:Compound Ratio

Let $R_1 = a : b$ and $R_2 = c : d$ be ratios. The '''compound ratio''' of $R_1$ and $R_2$ is defined as: :$R_1 R_2 = a c : b d$ That is, it is the product of the two ratios.


Definition:Minimal Polynomial

Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. === Definition 1 === {{Definition:Minimal Polynomial/Definition 1}} === Definition 2 === {{Definition:Minimal Polynomial/Definition 2}} === Definition 3 === {{Definition:Minimal Polynomial/Definition 3}}


Definition:Algebraic Closure

Let $K$ be a field. An '''algebraic closure''' of $K$ is an algebraically closed algebraic field extension of $K$. An '''algebraic closure''' of $K$ can be denoted $\overline K$.


Definition:Normal Extension

=== Definition 1 === {{:Definition:Normal Extension/Definition 1}} === Definition 2 === {{:Definition:Normal Extension/Definition 2}}


Definition:Separable Extension

Let $K$ be a field. Let $L/K$ be an algebraic field extension. Then $L/K$ is a '''separable extension''' {{iff}} every $\alpha\in L$ is separable over $K$.


Definition:Galois Extension/Finite

Let $L/K$ be a finite field extension. === Definition 1 === {{:Definition:Galois Extension/Finite/Definition 1}} === Definition 2 === {{:Definition:Galois Extension/Finite/Definition 2}} === Definition 3 === {{:Definition:Galois Extension/Finite/Definition 3}}


Definition:Intermediate Field

Let $L / K$ be a field extension. Let $F$ be a field such that: : $K \subset F \subset L$ Then $F$ is an '''intermediate field'''. Category:Definitions/Field Extensions dgtlfaiy3osbe2h6rjje26ygd3hsfy4


Definition:Embedding (Galois Theory)

Let $K$ and $L$ be fields. A (field) monomorphism $\phi: K \to L$ is called an '''embedding''' of $K$ in $L$. Category:Definitions/Field Theory 0qbhx2myjtxif74p84yor3alybnkzrx


Definition:Inclusion-Reversing Mapping

Let $A, B$ be sets of sets and $\phi: A \to B$ be a mapping. Then $\phi$ is '''inclusion-reversing''' {{iff}}: : for every pair of sets $a_1, a_2 \in A$ such that $a_1 \subseteq a_2$: :: $\phi \left({a_2}\right) \subseteq \phi \left({a_1}\right)$