Definition:Linear Measure/Depth

'''Depth''' is linear measure in a dimension perpendicular to both length and breadth. The choice of '''depth''' is often arbitrary, although in two-dimensional diagrams of three-dimensional figures, '''depth''' is usually imagined as being the dimension perpendicular to the plane the figure is drawn in.


Definition:Set/Implicit Set Definition/Infinite Set

If there is no end to the list of elements in the set, the ellipsis can be left open: :$S = \set {1, 2, 3, \ldots}$ which is taken to mean: :'''$S = $ the set containing $1, 2, 3, $ and so on for ever.'''


Definition:Endorelation/General Definition

An $n$-ary relation $\RR$ on a cartesian space $S^n$ is an '''$n$-ary endorelation on $S$''': :$\RR = \struct {S, S, \ldots, S, R}$ where $R \subseteq S^n$.


Definition:Well-Ordering/Definition 1

The ordering $\preceq$ is a '''well-ordering''' on $S$ {{iff}} ''every'' non-empty subset of $S$ has a smallest element under $\preceq$: :$\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$


Definition:Well-Ordering/Definition 2

The ordering $\preceq$ is a '''well-ordering''' on $S$ {{iff}} $\preceq$ is a well-founded total ordering.


Definition:Domain (Set Theory)/Relation/General Definition

Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\displaystyle \mathcal R \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$. The '''domain of $\mathcal R$''' is the set defined as: :$\displaystyle \Dom {\mathcal R} := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$ The concept is usually encountered when $\mathcal R$ is an endorelation on $S$: :$\displaystyle \Dom {\mathcal R} := \set {\tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \exists s_n \in S_n: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$


Definition:Image (Set Theory)/Relation/Relation/General Definition

Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\displaystyle \mathcal R \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$. The '''image of $\mathcal R$''' is the set defined as: :$\Img {\mathcal R} := \set {s_n \in S_n: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in \displaystyle \prod_{i \mathop = 1}^{n - 1} S_i: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$ The concept is usually encountered when $\mathcal R$ is an endorelation on $S$: :$\Img {\mathcal R} := \set {s_n \in S: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$


Definition:Strict Well-Ordering/Definition 1

Let $\prec$ be a strict total ordering on a class $A$. Then $\prec$ is a '''strict well-ordering''' on $A$ {{iff}} $\prec$ is a foundational relation on $A$. That is, expressed symbolically: :${\prec} \mathrel{\operatorname{We}} A \iff \left({\prec \operatorname{Or} A \land {\prec} \mathrel{\operatorname{Fr}} A}\right)$


Definition:Strict Strong Well-Ordering

Let $A$ be a class. Let $\mathcal R$ be a relation on $A$. Then $\mathcal R$ is a '''strict strong well-ordering''' of $A$ {{iff}}: : $\mathcal R$ connects $A$ : $\mathcal R$ is strongly well-founded.  That is, whenever $B$ is a non-empty subclass of $A$, $B$ has an $\mathcal R$-minimal element.


Definition:Mapping/General Definition

Let $\displaystyle \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$. Let $\displaystyle \RR \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\displaystyle \prod_{i \mathop = 1}^n S_i$. Then $\RR$ is a '''mapping''' {{iff}}: :$\displaystyle \forall x := \tuple {x_1, x_2, \ldots, x_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \forall y_1, y_2 \in S_n: \tuple {x, y_1} \in \RR \land \tuple {x, y_2} \in \RR \implies y_1 = y_2$ and :$\displaystyle \forall x := \tuple {x_1, x_2, \ldots, x_{n - 1} } \in \prod_{i \mathop = 1}^{n - 1} S_i: \exists y \in S_n: \tuple {x, y} \in \RR$ Thus, a '''mapping''' is an $n$-ary relation which is: :Many-to-one :Left-total, that is, defined for all elements in the domain.


Definition:Cartesian Product/Family of Sets/Definition 1

Let $I$ be an indexing set. Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$. The '''Cartesian product of $\family {S_i}_{i \mathop \in I}$''' is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S_i$ for each $i \in I$. This can be denoted $\displaystyle \prod_{i \mathop \in I} S_i$ or, if $I$ is understood, $\displaystyle \prod_i S_i$.


Definition:Cartesian Product/Family of Sets/Definition 2

Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets. The '''Cartesian product of $\family {S_i}_{i \mathop \in I}$''' is the set: :$\displaystyle \prod_{i \mathop \in I} S_i := \set {f: \paren {f: I \to \bigcup_{i \mathop \in I} S_i} \land \paren {\forall i \in I: \paren {\map f i \in S_i} } }$ where $f$ denotes a mapping. When $S_i = S$ for all $i \in I$, the expression is written: :$\displaystyle S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$ which follows from Union is Idempotent: :$\displaystyle \bigcup_{i \mathop \in I} S = S$


Definition:Order Isomorphism/Wosets

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be well-ordered sets. Let $\phi: S \to T$ be a bijection such that $\phi: S \to T$ is order-preserving: :$\forall x, y \in S: x \preceq_1 y \implies \map \phi x \preceq_2 \map \phi y$ Then $\phi$ is an '''order isomorphism'''.


Definition:Set-Like Relation

Let $\mathcal R$ be a relation. Let $A$ be a class. Then $\mathcal R$ is '''set-like''' on $A$ {{iff}}: :For all $x \in A$, $\left\{{y \in A: y \mathrel{\mathcal R} x}\right\}$ is a set.


Definition:Boolean Algebra/Definition 2

{{:Definition:Boolean Algebra/Axioms/Definition 2}}


Definition:Strict Well-Ordering/Definition 2

Let $A$ be a set or class. Let $\prec$ be a relation on $A$. Then $\prec$ is a '''strict well-ordering''' of $A$ {{iff}}: : $\prec$ connects $A$ : $\prec$ is well-founded.  That is, whenever $b$ is a non-empty subset of $A$, $b$ has a $\prec$-minimal element.


Definition:Boolean Algebra/Axioms/Definition 1

A '''Boolean algebra''' is an algebraic system $\struct {S, \vee, \wedge, \neg}$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation. Furthermore, these operations are required to satisfy the following axioms: {{begin-axiom}} {{axiom | n = \text {BA}_1 0         | lc=         | t = $S$ is closed under $\vee$, $\wedge$ and $\neg$ }} {{axiom | n = \text {BA}_1 1         | lc=         | t = Both $\vee$ and $\wedge$ are commutative }} {{axiom | n = \text {BA}_1 2         | lc=         | t = Both $\vee$ and $\wedge$ distribute over the other }} {{axiom | n = \text {BA}_1 3         | lc=         | t = Both $\vee$ and $\wedge$ have identities $\bot$ and $\top$ respectively }} {{axiom | n = \text {BA}_1 4         | lc=         | t = $\forall a \in S: a \vee \neg a = \top, a \wedge \neg a = \bot$ }} {{end-axiom}}


Definition:Boolean Algebra/Axioms/Definition 2

A '''Boolean algebra''' is an algebraic system $\struct {S, \vee, \wedge, \neg}$, where $\vee$ and $\wedge$ are binary, and $\neg$ is a unary operation. Furthermore, these operations are required to satisfy the following axioms: {{begin-axiom}} {{axiom | n = \text {BA}_2 0         | lc= Closure:         | q = \forall a, b \in S         | m = a \vee b \in S }} {{axiom | m = a \wedge b \in S }} {{axiom | m = \neg a \in S }} {{axiom | n = \text {BA}_2 1         | lc= Commutativity:         | q = \forall a, b \in S         | m = a \vee b = b \vee a }} {{axiom | m = a \wedge b = b \wedge a }} {{axiom | n = \text {BA}_2 2         | lc= Associativity:         | q = \forall a, b, c \in S         | m = a \vee \paren {b \vee c} = \paren {a \vee b} \vee c }} {{axiom | m = a \wedge \paren {b \wedge c} = \paren {a \wedge b} \wedge c }} {{axiom | n = \text {BA}_2 3         | lc= Absorption Laws:         | q = \forall a, b \in S         | m = \paren {a \wedge b} \vee b = b }} {{axiom | m = \paren {a \vee b} \wedge b = b }} {{axiom | n = \text {BA}_2 4         | lc= Distributivity:         | q = \forall a, b, c \in S         | m = a \wedge \paren {b \vee c} = \paren {a \wedge b} \vee \paren {a \wedge c} }} {{axiom | m = a \vee \paren {b \wedge c} = \paren {a \vee b} \wedge \paren {a \vee c} }} {{axiom | n = \text {BA}_2 5         | lc= Identity Elements:         | q = \forall a, b \in S         | m = \paren {a \wedge \neg a} \vee b = b }} {{axiom | m = \paren {a \vee \neg a} \wedge b = b }} {{end-axiom}}


Definition:Ordinal/Definition 1

Let $S$ be a set. Let $\Epsilon \! \restriction_S$ be the restriction of the epsilon relation on $S$. Then $S$ is an '''ordinal''' {{iff}}: :$S$ is a transitive set :$\Epsilon \! \restriction_S$ strictly well-orders $S$.


Definition:Ordinal/Definition 3

An '''ordinal''' is a strictly well-ordered set $\struct {S, \prec}$ such that: :$\forall a \in S: S_a = a$ where $S_a$ is the initial segment of $S$ determined by $a$. From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that: :$S_a = \set {x \in S: x \subsetneqq a}$ From Initial Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal.


Definition:Order Isomorphism/Definition 1

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a bijection such that: : $\phi: S \to T$ is order-preserving : $\phi^{-1}: T \to S$ is order-preserving. Then $\phi$ is an '''order isomorphism'''.


Definition:Order Isomorphism/Definition 2

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a surjective order embedding. Then $\phi$ is an '''order isomorphism'''. That is, $\phi$ is an '''order isomorphism''' {{iff}}: :$(1): \quad \phi$ is surjective :$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$


Definition:Order-Reflecting Mapping

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be a mapping. Then $\phi$ is an '''order-reflecting mapping''' or '''reflects order''' {{iff}}: : $\forall x, y \in S: \phi \left({x}\right) \preceq_2 \phi\left({y}\right) \implies x \preceq_1 y$


Definition:Probability Theory

'''Probability theory''' is the branch of mathematics which studies probability spaces.


Definition:Event/Occurrence

Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events, so that $A \subseteq \Omega$ and $B \subseteq \Omega$. Let the outcome of the experiment be $\omega \in \Omega$. Then the following real-world interpretations of the '''occurrence''' of events can be determined: :If $\omega \in A$, then '''$A$ occurs'''. :If $\omega \notin A$, that is $\omega \in \Omega \setminus A$, then '''$A$ does not occur'''.


Definition:Event Space/Discrete

Let $\EE$ be an experiment. Let $\Omega$ be a discrete sample space of $\EE$. Then it is commonplace to take $\Sigma$ to be the power set $\powerset \Omega$ of $\Omega$, that is, the set of all possible subsets of $\Omega$.


Definition:Tangent Map

=== Differentable Mappings === {{:Definition:Tangent Map/Differentable Mappings}} === Affine Transformation === {{:Definition:Tangent Map/Affine Transformation}} Category:Definitions/Affine Geometry Category:Definitions/Differential Geometry pjf1hlk5fpsme5uopj9891idbe7tqn9


Definition:Probability Space/Discrete

Let $\Omega$ be a discrete sample space. Then $\left({\Omega, \Sigma, \Pr}\right)$ is known as a '''discrete probability space'''.


Definition:Probability Space/Continuous

Let $\Omega$ be a continuum. Then $\left({\Omega, \Sigma, \Pr}\right)$ is known as a '''continuous probability space'''.


Definition:Probability Function

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. The probability measure $\Pr$ on $\left({\Omega, \Sigma, \Pr}\right)$ can be considered as a function on elements of $\Omega$ and $\Sigma$.


Definition:Strong Well-Ordering

Let $A$ be a class. Let $\preceq$ be a relation on $A$. Then $\preceq$ is a '''strong well-ordering''' iff: : $\preceq$ is an ordering. : For each non-empty subclass $B \subseteq A$, $B$ has a smallest element with respect to $\preceq$.


Definition:Associate/Integral Domain/Definition 1

'''$x$ is an associate of $y$ (in $D$)''' {{iff}} they are both divisors of each other. That is, $x$ and $y$ are '''associates (in $D$)''' {{iff}} $x \divides y$ and $y \divides x$.


Definition:Associate/Commutative and Unitary Ring

Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $x, y \in R$. Then $x$ and $y$ are '''associates (in $R$)''' {{iff}} there exists a unit $u$ of $\struct {R, +, \circ}$ such that $u \circ x = y$.


Definition:Associate/Integral Domain/Definition 2

$x$ and $y$ are '''associates (in $D$)''' {{iff}}: :$\ideal x = \ideal y$ where $\ideal x$ and $\ideal y$ denote the ideals generated by $x$ and $y$ respectively.


Definition:Independent Events/Definition 1

The events $A$ and $B$ are defined as '''independent (of each other)''' {{iff}} the occurrence of one of them does not affect the probability of the occurrence of the other one. Formally, $A$ is independent of $B$ {{iff}}: :$\map \Pr {A \mid B} = \map \Pr A$ where $\map \Pr {A \mid B}$ denotes the conditional probability of $A$ given $B$.


Definition:Independent Events/Definition 2

The events $A$ and $B$ are defined as '''independent (of each other)''' {{iff}} the occurrence of both of them together has the same probability as the product of the probabilities of each of them occurring on their own. Formally, $A$ and $B$ are independent {{iff}}: :$\map \Pr {A \cap B} = \map \Pr A \map \Pr B$


Definition:Independent Events/General Definition

Let $\AA = \family {A_i}_{i \mathop \in I}$ be an indexed family of events of $\EE$. Then $\AA$ is '''independent''' {{iff}}, for all finite subsets $J$ of $I$: :$\displaystyle \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$ That is, if the occurrence of any finite collection of $\AA$ has the same probability as the product of each of those sets occurring individually. === Pairwise Independent === {{:Definition:Independent Events/Pairwise Independent}}


Definition:Independent Events/Pairwise Independent

Let $\mathcal A = \family {A_i}_{i \mathop \in I}$ be an indexed family of events of $\EE$. Then $\AA$ is '''pairwise independent''' {{iff}}: :$\forall j, k \in I: \map \Pr {A_j \cap A_k} = \map \Pr {A_j} \, \map \Pr {A_k}$ That is, if every pair of events of $\EE$ are independent of each other. That is, $\AA$ is '''pairwise independent''' {{iff}} the condition for general independence: :$\displaystyle \map \Pr {\bigcap_{i \mathop \in J} A_i} = \prod_{i \mathop \in J} \map \Pr {A_i}$ holds whenever $\card J = 2$.


Definition:Independent Events/Dependent

If $A$ and $B$ are not independent, then they are '''dependent (on each other)''', and vice versa.


Definition:Limit of Sequence of Events/Increasing

Let $\left \langle{A_n}\right \rangle_{n \mathop \in \N}$ be an increasing sequence of events. Then the union: :$\displaystyle A = \bigcup_{i \mathop \in \N} A_i$ of such a sequence is called the '''limit of the sequence $\left \langle{A_n}\right \rangle_{n \in \N}$'''.


Definition:Limit of Sequence of Events/Decreasing

Let $\left \langle{A_n}\right \rangle_{n \in \N}$ be an decreasing sequence of events. Then the intersection: :$\displaystyle A = \bigcap_{i \mathop \in \N} A_i$ of such a sequence is called the '''limit of the sequence $\left \langle{A_n}\right \rangle_{n \in \N}$'''.


Definition:Probability Mass Function/Joint

Let $X: \Omega \to \R$ and $Y: \Omega \to \R$ both be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$. Then the '''joint (probability) mass function''' of $X$ and $Y$ is the (real-valued) function $p_{X, Y}: \R^2 \to \closedint 0 1$ defined as: :$\forall \tuple {x, y} \in \R^2: \map {p_{X, Y} } {x, y} = \begin {cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x \land \map Y \omega = y} } & : x \in \Omega_X \text { and } y \in \Omega_Y \\ 0 & : \text {otherwise} \end {cases}$ That is, $\map {p_{X, Y} } {x, y}$ is the probability that the discrete random variable $X$ takes the value $x$ at the same time that the discrete random variable $Y$ takes the value $y$.


Definition:Geometric Distribution/Shifted

$X$ has the '''shifted geometric distribution with parameter $p$''' {{iff}}: :$\map X \Omega = \set {1, 2, \ldots} = \N_{>0}$ :$\map \Pr {X = k} = p \paren {1 - p}^{k-1}$ where $0 < p < 1$. It is frequently seen as: :$\map \Pr {X = k} = q^{k - 1} p$ where $q = 1 - p$. It is written: :$X \sim \ShiftedGeometric p$


Definition:Negative Binomial Distribution/First Form

$X$ has the '''negative binomial distribution (of the first form) with parameters $n$ and $p$''' if: :$\Img X = \set {0, 1, 2, \ldots}$ :$\map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k \paren {1 - p}^n$ where $0 < p < 1$. It is frequently seen as: :$\map \Pr {X = k} = \dbinom {n + k - 1} {n - 1} p^k q^n$ where $q = 1 - p$.


Definition:Negative Binomial Distribution/Second Form

$X$ has the '''negative binomial distribution (of the second form) with parameters $n$ and $p$''' if: :$\Img X = \set {n, n + 1, n + 2, \dotsc}$ :$\map \Pr {X = k} = \dbinom {k - 1} {n - 1} p^n \paren {1 - p}^{k - n}$ where $0 < p < 1$. It is frequently seen as: :$\map \Pr {X = k} = \dbinom {k - 1} {n - 1} q^{k - n} p^n $ where $q = 1 - p$.


Definition:Variance/Discrete/Definition 1

:$\var X := \expect {\paren {X - \expect X}^2}$ That is: it is the expectation of the squares of the deviations from the expectation.


Definition:Variance/Discrete/Definition 2

:$\displaystyle \var X := \sum_{x \mathop \in \Omega_X} \paren {x - \mu^2} \map \Pr {X = x}$ where: :$\mu := \expect X$ is the expectation of $X$ :$\Omega_X$ is the image of $X$ :$\map \Pr {X = x}$ is the probability mass function of $X$.


Definition:Countable Set/Definition 2

$S$ is '''countable''' {{iff}} it is finite or countably infinite.


Definition:Countable Set/Definition 1

$S$ is '''countable''' {{iff}} there exists an injection: :$f: S \to \N$


Definition:Independent Random Variables/General Definition

Let $X = \tuple {X_1, X_1, \ldots, X_n}$ be an ordered tuple of random variables. Then $X$ is '''independent''' {{iff}}: :$\displaystyle \map \Pr {X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n} = \prod_{k \mathop = 1}^n \map \Pr {X_k = x_k}$ for all $x = \tuple {x_1, x_2, \ldots, x_n} \in \R^n$.


Definition:Independent Random Variables/Pairwise Independent

Let $X = \tuple {X_1, X_1, \ldots, X_n}$ be an ordered tuple of random variables. Then $X$ is '''pairwise independent''' {{iff}} $X_i$ and $X_j$ are independent (of each other) whenever $i \ne j$.


Definition:Independent Random Variables/Dependent

Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Then $X$ and $Y$ are defined as '''dependent (on each other)''' {{iff}} $X$ and $Y$ are not independent (of each other).


Definition:Euler's Number/Limit of Sequence

The sequence $\sequence {x_n}$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$ converges to a limit as $n$ increases without bound. That limit is called '''Euler's Number''' and is denoted $e$.


Definition:Euler's Number/Limit of Series

The series $\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ converges to a limit. This limit is Euler's number $e$.


Definition:Euler's Number/Base of Logarithm

The number $e$ can be defined as the number satisfied by: :$\ln e = 1$ where $\ln e$ denotes the natural logarithm of $e$. That $e$ is unique follows from Logarithm is Strictly Increasing.


Definition:Euler's Number/Exponential Function

The number $e$ can be defined as the number satisfied by: :$e := \exp 1 = e^1$ where $\exp 1$ denotes the exponential function of $1$.


Definition:Euler's Number/Decimal Expansion

The decimal expansion of Euler's number $e$ starts: :$2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$


Definition:Exponential Generating Function

Let $A = \left \langle {a_n}\right \rangle$ be a sequence in $\R$. Then $\displaystyle G_A \left({z}\right) = \sum_{n \mathop \ge 0} \frac {a_n} {n!} z^n$ is called the '''(exponential) generating function''' for the sequence $A$.


Definition:Dismal Addition

'''Dismal addition''' is an operation defined on the natural numbers as follows. Let $a, b \in \N$ be expressed in decimal notation: :$\displaystyle a = \sum_{i \mathop = 0}^n a_i 10^i = \left[{a_n a_{n-1} \ldots a_2 a_1 a_0}\right]_{10}$ :$\displaystyle b = \sum_{j \mathop = 0}^m b_j 10^j = \left[{b_m b_{m-1} \ldots b_2 b_1 b_0}\right]_{10}$ The '''dismal sum''' of $a$ and $b$ is defined as: :$\displaystyle a + b = \sum_{k \mathop = 0}^{\max \left\{{m, n}\right\}} \max \left\{{a_k, b_k}\right\} 10^k$ That is, for each pair of corresponding digits, the maximum is taken.


Definition:Random Variable/Definition 1

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\struct {X, \Sigma'}$ be a measurable space. A '''random variable (on $\struct {\Omega, \Sigma, \Pr}$)''' is a $\Sigma \, / \, \Sigma'$-measurable mapping $f: \Omega \to X$.


Definition:Random Variable/Definition 3

:$\forall x \in \R: X^{-1} \sqbrk {\hointl {-\infty} x} \in \Sigma$ where: :$\hointl {-\infty} x$ denotes the unbounded closed interval $\set {y \in \R: y \le x}$ :$X^{-1} \sqbrk {\hointl {-\infty} x}$ denotes the preimage of $\hointl {-\infty} x$ under $X$.


Definition:Random Variable/Definition 2

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. A '''random variable on $\struct {\Omega, \Sigma, \Pr}$''' is a mapping $X: \Omega \to \R$ such that: :$\forall x \in \R: \set {\omega \in \Omega: \map X \omega \le x} \in \Sigma$


Definition:Tree (Graph Theory)/Node

The vertices of a tree are called its '''nodes'''.


Definition:Tree (Graph Theory)/Infinite

A tree is '''infinite''' if it contains a (countably) infinite number of nodes.


Definition:Tree (Graph Theory)/Finite

A tree is '''finite''' {{iff}} it contains a finite number of nodes.


Definition:Ultrafilter on Set/Definition 4

Let $S$ be a non-empty set. Let $\mathcal F$ be a non-empty set of subsets of $S$. Then $\mathcal F$ is an '''ultrafilter''' on $S$ {{iff}} both of the following hold: : $\mathcal F$ has the finite intersection property : For all $U \subseteq S$, either $U \in \mathcal F$ or $U^c \in \mathcal F$ where $U^c$ is the complement of $U$ in $S$.


Definition:Ultrafilter on Set/Definition 1

Let $S$ be a set. Let $\FF \subseteq \powerset S$ be a filter on $S$. Then $\FF$ is an '''ultrafilter (on $S$)''' {{iff}}: :there is no filter on $S$ which is strictly finer than $\FF$ or equivalently, {{iff}}: :whenever $\GG$ is a filter on $S$ and $\FF \subseteq \mathcal G$ holds, then $\FF = \GG$.


Definition:Finite Character

Let $S$ be a set. Let $\mathcal F$ be a set of subsets of $S$. Then $\mathcal F$ has '''finite character''' {{iff}} for each $A \subseteq S$: : $A \in \mathcal F$ {{iff}} every finite subset of $A$ is in $\mathcal F$.


Definition:Rooted Tree/Ancestor Node/Proper

A '''proper ancestor node''' of $t$ is an ancestor node of $t$ that is not $t$ itself.


Definition:Rooted Tree/Child Node

Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. The '''child nodes''' of $t$ are the elements of the set: :$\left\{{s \in T: \pi \left({s}\right) = t}\right\}$ where $\pi \left({s}\right)$ denotes the parent mapping of $s$. That is, the '''children''' of $t$ are all the nodes of $T$ of which $t$ is the parent. === Grandchild Node === {{:Definition:Rooted Tree/Child Node/Grandchild Node}}


Definition:Rooted Tree/Proper Descendant

A '''proper descendant node''' of $t$ is a descendant of $t$ which is not $t$ itself.


Definition:Rooted Tree/Descendant

Let $T$ be a rooted tree with root $r_T$. Let $t$ be a node of $T$. A '''descendant node''' $s$ of a $t$ is a node such that $t$ is in the path from $s$ to $r_T$. That is, the '''descendant nodes''' of $t$ are all the nodes of $T$ of which $t$ is an ancestor node. === Proper Descendant === {{:Definition:Rooted Tree/Proper Descendant}}


Definition:Rooted Tree/Sibling

Let $T$ be a rooted tree with root $r_T$. Two children of the same node of $T$ are called '''siblings'''. That is, '''siblings''' are nodes which both have the same parent.


Definition:Rooted Tree/Branch/Infinite

Let $T$ be a rooted tree with root node $r_T$. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is infinite {{iff}} it has no leaf node at the end.


Definition:Rooted Tree/Branch/Length

Let $T$ be a rooted tree with root node $r_T$. Let $\Gamma$ be a finite branch of $T$. The '''length''' of $\Gamma$ is defined as the number of ancestors of the leaf at the end of that branch.


Definition:Rooted Tree/Branch

Let $T$ be a rooted tree with root node $r_T$. A subset $\Gamma$ of $T$ is a '''branch''' {{iff}} all the following conditions hold: :$(1): \quad$ The root node $r_T$ belongs to $\Gamma$ :$(2): \quad$ The parent of each node in $\Gamma \setminus \left\{{r_T}\right\}$ is in $\Gamma$ :$(3): \quad$ Each node in $\Gamma$ either: ::$(a): \quad$ is a leaf node of $T$ :or: ::$(b): \quad$ has exactly one child node in $\Gamma$.


Definition:Rooted Tree/Root Node

Let $T$ be a rooted tree. The '''root node''' of $T$ is the node of $T$ which is distinguished from the others by being the ancestor node of every node of $T$.


Definition:Finite Join Property

Let $(B, \vee, \wedge, \neg, \bot, \top)$ be a Boolean algebra. Let $J \subseteq B$. Then $J$ has the '''finite join property''' iff for any $x_1, x_2, \dots, x_n \in J$: : $x_1 \vee x_2 \vee \dots \vee x_n \ne \top$ Category:Definitions/Lattice Theory mp6xrnbd4t6fk1u28q618hjklixdisn


Definition:Finite Character/Mappings

Let $S$ and $T$ be sets. Let $\mathcal F$ be a set of mappings from subsets of $S$ to $T$. That is, let $\mathcal F$ be a set of partial mappings from $S$ to $T$. Then $\mathcal F$ has '''finite character''' {{iff}} for each partial mapping $f \subseteq S \times T$: : $f \in \mathcal F$ {{iff}} for each finite subset $K$ of the domain of $f$, the restriction of $f$ to $K$ is in $\mathcal F$.


Definition:Rooted Tree/Branch/Finite

Let $T$ be a rooted tree with root node $r_T$. Let $\Gamma$ be a branch of $T$. Then $\Gamma$ is '''finite''' {{iff}} it has exactly one leaf node.


Definition:Contradictory/Logical Formulas

Let $\Delta$ be a set of propositional formulas. Then $\Delta$ is '''contradictory''' if there exists some propositional formula $P$ such that $P \in \Delta$ and $\neg P \in \Delta$.


Definition:Lattice (Group Theory)

=== Definition 1 === {{:Definition:Lattice_(Group_Theory)/Definition_1}} === Definition 2 === {{:Definition:Lattice_(Group_Theory)/Definition_2}} Category:Definitions/Group Theory giz5748waqza4slf1bjgzp3rvd5si9i


Definition:Finished

=== Finished Set of WFFs of Propositional Logic === {{:Definition:Finished Set of WFFs of Propositional Logic}} === Finished Branch of Propositional Tableau === {{:Definition:Finished Branch of Propositional Tableau}} === Finished Propositional Tableau === {{:Definition:Finished Propositional Tableau}} Category:Definitions/Propositional Tableaus 7fyihzey4x8pypzxbqh54khqnw8ml2n


Definition:Null Sequence (Homological Algebra)

Let $\left({R, +, \cdot}\right)$ be a ring. Let: :$(1): \quad \cdots \longrightarrow M_i \stackrel{d_i}{\longrightarrow} M_{i+1} \stackrel{d_{i+1}}{\longrightarrow} M_{i+2} \stackrel{d_{i+2}}{\longrightarrow} \cdots$ be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$. Then the sequence $(1)$ is '''null''' if $d_i \circ d_{i+1} = 0$ for all $i$.


Definition:Short Exact Sequence of Modules

Let $\left({R, +, \cdot}\right)$ be a ring. Let: :$(1): \quad \cdots \longrightarrow M_i \stackrel{d_i}{\longrightarrow} M_{i+1} \stackrel{d_{i+1}}{\longrightarrow} M_{i+2} \stackrel{d_{i+2}}{\longrightarrow} \cdots$ be a sequence of $R$-modules $M_i$ and $R$-module homomorphisms $d_i$. Then the sequence $(1)$ is a '''short exact sequence''' if it exact, and is finite of the form ::$0 \longrightarrow M_2 \stackrel {d_2} {\longrightarrow} M_3 \stackrel {d_3} {\longrightarrow} M_4 \longrightarrow 0$


Definition:Harmonic Numbers/General Definition

Let $r \in \R_{>0}$. For $n \in \N_{> 0}$ the '''Harmonic numbers order $r$''' are defined as follows: :$\displaystyle H_n^{\paren r} = \sum_{k \mathop = 1}^n \frac 1 {k^r}$


Definition:Homomorphism of Complexes

Let $\left({R, +, \cdot}\right)$ be a ring. Let: :$M: \quad \cdots \longrightarrow M_i \stackrel {d_i} {\longrightarrow} M_{i + 1} \stackrel {d_{i + 1} } {\longrightarrow} M_{i + 2} \stackrel {d_{i + 2} } {\longrightarrow} \cdots$ and :$N: \quad \cdots \longrightarrow N_i \stackrel {d'_i} {\longrightarrow} N_{i + 1} \stackrel {d'_{i + 1} } {\longrightarrow} N_{i + 2} \stackrel {d'_{i + 2} } {\longrightarrow} \cdots$ be two differential complexes of $R$-modules. Let $\phi = \left\{ {\phi_i: i \in \Z}\right\}$ be a family of module homomorphisms $\phi_i: M_i \to N_i$. Then $\phi$ is a '''homomorphism of complexes''' {{iff}} for each $i \in \Z$: :$\phi_{i+1} \circ d_i = \phi_i \circ d'_i$


Definition:Arborescence/Definition 1

$G$ is an '''arborescence of root $r$''' {{iff}}: :For each $v \in V$ there is exactly one directed walk from $r$ to $v$.


Definition:Arborescence/Definition 2

$G$ is an '''arborescence of root $r$''' {{iff}}: : $(1): \quad$ $G$ is an orientation of a tree : $(2): \quad$ For each $v \in V$, $v$ is reachable from $r$.


Definition:Arborescence

Let $G = \left({V, A}\right)$ be a directed graph. Let $r \in V$. === Definition 1 === {{:Definition:Arborescence/Definition 1}} === Definition 2 === {{:Definition:Arborescence/Definition 2}} === Definition 3 === {{:Definition:Arborescence/Definition 3}}


Definition:Reachability Relation/Definition 1

Let $G = \left({V, A}\right)$ be a directed graph. Then the '''reachability relation''' of $G$ is the transitive closure of $A$.


Definition:Reachability Relation/Definition 2

Let $G = \left({V, A}\right)$ be a directed graph. Let $\mathcal R$ be the relation on $V$ defined by letting $x \mathrel {\mathcal R} y$ {{iff}} $y$ is reachable from $x$. That is, $x \mathrel {\mathcal R} y$ {{iff}} there exists a directed walk from $x$ to $y$. Then $\mathcal R$ is the '''reachability relation''' of $G$.


Definition:Reachability Relation

=== Definition 1 === {{:Definition:Reachability Relation/Definition 1}} === Definition 2 === {{:Definition:Reachability Relation/Definition 2}}


Definition:Reachable/Definition 1

Let $G = \left({V, A}\right)$ be a directed graph. Let $u, v \in V$. Then $v$ is '''reachable''' from $u$ {{iff}} there exists a directed walk from $u$ to $v$.


Definition:Reachable/Definition 2

Let $G = \left({V, A}\right)$ be a directed graph. Let $\mathcal R$ be the reachability relation of $G$. That is, $\mathcal R$ is the transitive closure of $A$. Let $u, v \in V$. Then $v$ is '''reachable''' from $u$ {{iff}} $u \mathrel {\mathcal R} v$.


Definition:Reachable

=== Definition 1 === {{:Definition:Reachable/Definition 1}} === Definition 2 === {{:Definition:Reachable/Definition 2}}


Definition:Directed Walk

Let $G = \struct {V, A}$ be a directed graph. A '''directed walk''' in $G$ is a finite or infinite sequence $\sequence {x_k}$ such that: :$\forall k \in \N: k + 1 \in \Dom {\sequence {x_k} }: \tuple {x_k, x_{k + 1} } \in A$


Definition:Arborescence/Definition 3

$G$ is an '''arborescence of root $r$''' {{iff}}: : $(1): \quad$ Each vertex $v \ne r$ is the final vertex of exactly one arc. : $(2): \quad$ $r$ is not the final vertex of any arc. : $(3): \quad$ For each $v \in V$ such that $v \ne r$ there is a directed walk from $r$ to $v$.


Definition:Orientation (Graph Theory)

Let $G = \struct {V, E}$ be a simple graph. Let $H = \struct {V, A}$ be a directed graph. Then $H$ is an '''orientation''' of $G$ {{iff}} both of the following hold: :$G$ is a simple digraph. That is, $A$ is antisymmetric. :$\forall x, y \in V: \paren {\set {x, y} \in E \iff \tuple {x, y} \in A \lor \tuple {y, x} \in A}$


Definition:Pseudocomplement

Let $(L, \wedge, \vee, \preceq)$ be a lattice with smallest element $\bot$. Let $x, x^* \in L$. Then $x^*$ is the '''pseudocomplement''' of $x$ iff : $x^*$ is the greatest element of $L$ such that $x \wedge x^* = \bot$.


Definition:Pseudocomplemented Lattice

Let $(L, \wedge, \vee, \preceq)$ be a lattice with smallest element $\bot$. Then $(L, \wedge, \vee, \preceq)$ is a '''pseudocomplemented lattice''' iff each element $x$ of $L$ has a pseudocomplement. The pseudocomplement of $x$ is denoted $x^*$.


Definition:Relative Pseudocomplement

Let $(L, \wedge, \vee, \preceq)$ be a lattice. Let $x, y \in L$. Then the '''relative pseudocomplement''' of $x$ with respect to $y$ is the greatest element $z \in L$ such that $x \wedge z \preceq y$, if such an element exists. The relative pseudocomplement of $x$ with respect to $y$ is denoted $x \to y$.


Definition:Brouwerian Lattice

Let $\left({L, \wedge, \vee, \preceq}\right)$ be a lattice. Then $\left({L, \wedge, \vee, \preceq}\right)$ is a '''Brouwerian lattice''' {{iff}}: :for each $x, y \in L$: $x$ has a relative pseudocomplement with respect to $y$. This pseudocomplement is denoted $x \to y$.


Definition:Heyting Algebra

Let $\left({L, \wedge, \vee, \preceq}\right)$ be a lattice. Then $\left({L, \wedge, \vee, \preceq}\right)$ is a '''Heyting algebra''' {{iff}}: : $(1): \quad \left({L, \wedge, \vee, \preceq}\right)$ is a Brouwerian lattice : $(2): \quad L$ has a smallest element.


Definition:Conditional/Notational Variants/Sign of Illation

The '''sign of illation''' $-\!\!\!<$ is a notation invented by {{AuthorRef|Charles Sanders Peirce}} to denote the conditional operator. {{AuthorRef|Charles Sanders Peirce|Peirce}} derives $-\!\!\!<$ as a variant of the sign $\le$ for ''less than or equal to'', so as to denote that: : $A \mathop {-\!\!\!<} B$ represents the situation such that whenever a particular statement $A$ is true, then so is statement $B$.


Definition:Infix Notation/Binary Relation

Let $\mathcal R \subseteq S \times T$ be a binary relation. When $\left({s, t}\right) \in \mathcal R$, we can write either: : $\mathcal R \left({s, t}\right)$ or : $s \mathop {\mathcal R} t$ The notation $s \mathop {\mathcal R} t$ is known as '''infix notation'''.


Definition:Set Partition/Definition 1

A '''partition''' of $S$ is a set of subsets $\Bbb S$ of $S$ such that: :$(1): \quad$ $\Bbb S$ is pairwise disjoint: $\forall S_1, S_2 \in \Bbb S: S_1 \cap S_2 = \O$ when $S_1 \ne S_2$ :$(2): \quad$ The union of $\Bbb S$ forms the whole set $S$: $\displaystyle \bigcup \Bbb S = S$ :$(3): \quad$ None of the elements of $\Bbb S$ is empty: $\forall T \in \Bbb S: T \ne \O$.


Definition:Set Partition/Definition 2

A '''partition''' of $S$ is a set of non-empty subsets $\Bbb S$ of $S$ such that each element of $S$ lies in exactly one element of $\Bbb S$.


Definition:Cycle Decomposition

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi$ be a permutation on $S_n$. The '''cycle decomposition''' of $\pi$ is the product of the disjoint cycles in which $\pi$ can be expressed.


Definition:Permutation on n Letters/Cycle Notation/Canonical Representation

The permutation: :$\begin{pmatrix}   1 & 2 & 3 & 4 & 5 \\   2 & 1 & 4 & 3 & 5  \end{pmatrix}$ can be expressed in '''cycle notation''' as: :$\begin{pmatrix} 1 & 2 \end{pmatrix} \begin{pmatrix} 3 & 4 \end{pmatrix}$ or as: :$\begin{pmatrix} 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \end{pmatrix} \begin{pmatrix} 1 & 2 \end{pmatrix}$ or as: :$\begin{pmatrix} 4 & 3 \end{pmatrix} \begin{pmatrix} 2 & 1 \end{pmatrix}$ etc. However, only the first is conventional. This is known as the '''canonical representation'''.


Definition:Cantor-Bendixson Rank

Let $\struct {X, \tau}$ be a topological space. Let $S \subseteq X$. For each ordinal $\alpha$, let $S^{\paren \alpha}$ be the $\alpha$th Cantor-Bendixson derivative of $S$. Then the '''Cantor-Bendixson rank''' of $S$ is the least ordinal $\alpha$ such that: :$S^{\paren {\alpha^+} } = S^{\paren \alpha}$ if such an ordinal exists. {{LinkWanted|Definition:Smallest Ordinal, to replace least ordinal}} {{NamedforDef|Georg Cantor|name2 = Ivar Otto Bendixson|cat = Cantor|cat2 = Bendixson}}


Definition:Sign of Permutation on n Letters

Let $n \in \N$ be a natural number. Let $N^*_{\le n}$ denote the set of natural numbers $\left\{ {1, 2, \ldots, n}\right\}$. Let $S_n$ denote the set of permutations on $N^*_{\le n}$. Let $\pi \in S_n$ be a permutation of $N^*_{\le n}$. Let $K$ be the cycle decomposition of $\pi$. Let each cycle of $K$ be factored into transpositions. Let $k$ be the total number of transpositions that compose $K$. The '''sign of $\pi$''' is defined as: :$\forall \pi \in S_n: \sgn \paren \pi = \begin{cases} 1 & : k \text{ even} \\ -1 & : k \text{ odd} \\ \end{cases}$


Definition:Set Equality/Definition 1

$S$ and $T$ are equal {{iff}} they have the same elements: :$S = T \iff \paren {\forall x: x \in S \iff x \in T}$


Definition:Set Equality/Definition 2

$S$ and $T$ are equal {{iff}} both: :$S$ is a subset of $T$ and :$T$ is a subset of $S$


Definition:Antisymmetric Relation/Definition 1

$\RR$ is '''antisymmetric''' {{iff}}: :$\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$ that is: :$\set {\tuple {x, y}, \tuple {y, x} } \subseteq \RR \implies x = y$


Definition:Antisymmetric Relation/Definition 2

$\mathcal R$ is '''antisymmetric''' {{iff}}: :$\tuple {x, y} \in \mathcal R \land x \ne y \implies \tuple {y, x} \notin \mathcal R$


Definition:Direct Proof

A '''direct proof''' is an argument form which establishes the truth of a conclusion by assuming that the premises are true and demonstrating that the conclusion is necessarily true as a consequence.


Definition:Graph (Graph Theory)/Edge/Join

Let $u$ and $v$ be vertices of $G$. Let $e = u v$ be an edge of $G$. Then $e$ '''joins''' the vertices $u$ and $v$.


Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph

Let $G = \struct {V, E}$ be an undirected graph. Two vertices $u, v \in V$ of $G$ are '''adjacent''' if there exists an edge $e = \set {u, v} \in E$ of $G$ to which they are both incident.


Definition:Adjacent (Graph Theory)/Vertices/Digraph

Let $G = \struct {V, E}$ be a digraph. Two vertices $u, v \in V$ of $G$ are '''adjacent''' if there exists an arc $e = \left({u, v}\right) \in E$ of $G$ to which they are both incident.


Definition:Adjacent (Graph Theory)/Vertices/Non-Adjacent

Let $G = \struct {V, E}$ be a graph. Two vertices $u, v \in V$ of $G$ are '''non-adjacent''' if they are not adjacent.


=== Undirected Graph === {{:Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph}} === Digraph === {{:Definition:Adjacent (Graph Theory)/Vertices/Digraph}}


Definition:Incident (Graph Theory)/Undirected Graph

Let $G = \struct {V, E}$ be an undirected graph. Let $u, v \in V$ be vertices of $G$. Let $e = \set {u, v} \in E$ be an edge of $G$: :320px Then: :$u$ and $v$ are each '''incident with $e$''' :$e$ is '''incident with $u$''' and '''incident with $v$'''.


Definition:Incident (Graph Theory)/Digraph/Incident From

: $e$ is '''incident from $u$''' : $v$ is '''incident from $e$'''.


Definition:Incident (Graph Theory)/Digraph/Incident To

: $e$ is '''incident to $v$''' : $u$ is '''incident to $e$'''.


Let $G = \struct {V, E}$ be a digraph. Let $u, v \in V$ be vertices of $G$. Let $e = \tuple {u, v}$ be an arc that is directed from $u$ to $v$: :320px Then the following definitions are used: === Incident From === {{:Definition:Incident (Graph Theory)/Digraph/Incident From}} === Incident To === {{:Definition:Incident (Graph Theory)/Digraph/Incident To}}


Definition:Adjacent (Graph Theory)/Edges/Undirected Graph

Let $G = \struct {V, E}$ be an undirected graph. Two edges $e_1, e_2 \in E$ of $G$ '''adjacent''' if there exists a vertex $v \in V$ to which they are both incident.


Definition:Adjacent (Graph Theory)/Edges/Non-Adjacent

Let $G = \struct {V, E}$ be a graph. Two edges $u, v \in V$ of $G$ are '''non-adjacent''' if they are not adjacent.


Definition:Adjacent (Graph Theory)/Edges/Digraph

Let $G = \struct {V, E}$ be a digraph. Two arcs $e_1, e_2 \in E$ of $G$ '''adjacent''' if there exists a vertex $v \in V$ to which they are both incident.


=== Undirected Graph === {{Definition:Adjacent (Graph Theory)/Edges/Undirected Graph}} === Digraph === {{Definition:Adjacent (Graph Theory)/Edges/Digraph}}


Definition:Adjacent (Graph Theory)/Faces

Let $G = \left({V, E}\right)$ be a planar graph. Two faces of $G$ are '''adjacent''' {{iff}} they are both incident to the same edge (or edges).


Definition:Directed Graph/Symmetric Digraph

Let $D = \struct {V, E}$ be a digraph such that the relation $E$ in $D$ is symmetric. Then $D$ is called a '''symmetric digraph'''.


Definition:Directed Graph/Arc/Endvertex

Let $D = \struct {V, E}$ be a digraph. Let $e = u v$ be an arc of $D$, that is, $e \in E$. The '''endvertices''' of $e$ are the vertices $u$ and $v$.


Definition:Network/Undirected

An '''undirected network''' is a network whose underlying graph is an undirected graph: :450px


Definition:Underlying Graph

Let $N = \left({V, E, f}\right)$ be a network where $f: E \to \R$ is the mapping from the edge set $E$ to the real numbers $\R$. The '''underlying graph''' of $N$ is the graph $G = \left({V, E}\right)$ consisting only of the vertex set $V$ and the edge set $E$. Category:Definitions/Graph Theory 3a2ultitf4nqj9xpxd1sawtxkt6hvuo


Definition:Network/Directed

A '''directed network''' is a network whose underlying graph is a digraph: :320px


Definition:Network/Loop-Network

A '''loop-network''' (directed or undirected) is a loop-graph together with a mapping which maps the edge set into the set $\R$ of real numbers. That is, it is a '''network''' which may have loops.


Definition:Network/Weight

Let $N = \struct {V, E, w}$ be a network with weight function $w: E \to \R$. The values of the elements of $E$ under $w$ are known as the '''weights''' of the edges of $N$. The '''weights''' of a network $N$ can be depicted by writing the appropriate numbers next to the edges of the underlying graph of $N$.


Definition:Network/Weight Function

Let $N = \left({V, E, w}\right)$ be a network. The mapping $w: E \to \R$ is known as the '''weight function''' of $N$.


Definition:Multigraph/Simple Edge

Let $G = \struct {V, E}$ be a multigraph. A '''simple edge''' is an edge $u v$ of $G$ which is the only edge of $G$ which is incident to both $u$ and $v$.


Definition:Multigraph/Multiplicity

The '''multiplicity''' of a multigraph is the maximum multiplicity of its (multiple) edges.


Definition:Loop-Graph/Loop-Multigraph

A '''loop-multigraph''' is a multigraph which is allowed to have loops: :380px


Definition:Graph (Graph Theory)/Notation

Let $G$ be a graph whose order is $p$ and whose size is $q$. Then $G$ can be referred to as a '''$\tuple {p, q}$-graph'''. A wider category: a graph whose order is $n$ can be referred to as an '''$n$-graph'''.


Definition:Path (Graph Theory)/Open

An '''open path''' is a path in which the first and last vertices are distinct.


Definition:Walk (Graph Theory)/Open

An '''open walk''' is a walk whose first vertex and last vertex are distinct. That is, it is a walk which ends on a different vertex from the one where it starts.


Definition:Connected (Graph Theory)/Vertices

Let $G$ be a graph. Two vertices $u, v \in G$ are '''connected''' {{iff}} either: :$(1): \quad u = v$ :$(2): \quad u \ne v$, and there exists a walk between them.


Definition:Connected (Graph Theory)/Graph/Disconnected

Let $G$ be a graph. Then $G$ is '''disconnected''' {{iff}} it is not connected. That is, if there exists (at least) two vertices $u, v \in G$ such that $u$ and $v$ are not connected.


Definition:Connected (Graph Theory)/Graph

Let $G$ be a graph. Then $G$ is '''connected''' {{iff}} every pair of vertices in $G$ is connected.


Definition:Cycle (Graph Theory)/Odd

An '''odd cycle''' is a cycle with odd length, that is, with an odd number of edges.


Definition:Cycle (Graph Theory)/Even

An '''even cycle''' is a cycle with even length, that is, with an even number of edges.


Definition:Edge Deletion

Let $G = \struct {V, E}$ be an (undirected) graph. Let $F \subseteq E$ be a set of edges of $G$. Then the '''graph obtained by deleting $F$ from $G$''', denoted by $G - F$, is the subgraph of $G$ containing the same vertices as $G$ but with all the elements of $F$ removed. That is: :$G - F = \struct {V, E \setminus F}$ Informally, $G - F$ is the graph obtained from $G$ by removing all edges in $F$. If $F$ is a singleton such that $F = \set e$, then $G - F$ may be expressed $G - e$.


Definition:Platonic Graph/Tetrahedron

:400px The graph of the regular tetrahedron is $3$-regular.


Definition:Platonic Graph/Octahedron

:400px The graph of the regular octahedron is $4$-regular.


Definition:Platonic Graph/Icosahedron

:400px The graph of the regular icosahedron is $5$-regular.


Definition:Platonic Graph/Cube

:400px The graph of the cube is $3$-regular.


Definition:Platonic Graph/Dodecahedron

:400px The graph of the regular dodecahedron is $3$-regular.


Definition:Tree (Graph Theory)/Definition 1

A '''tree''' is a simple connected graph with no circuits.


Definition:Tree (Graph Theory)/Definition 2

A '''tree''' is a simple connected graph with no cycles.


Definition:Spanning Tree/Building-Up Method

Start with the edgeless graph $N$ whose vertices correspond with those of $G$. Select edges of $G$ one by one, such that no cycles are created, and add them to $N$. Continue till all vertices are included.


Definition:Spanning Tree/Cutting-Down Method

Start with the graph $G$. Choose any cycle in $G$, and remove any one of its edges. By Condition for Edge to be Bridge, this will not disconnect $G$. Repeat this procedure till no cycles are left in $G$.


Definition:Slope

=== Straight Line === {{:Definition:Slope/Straight Line}} === Curve === {{:Definition:Slope/Curve}}


Definition:Slope/Curve

Let $P$ be a point on a curve $C$. {{explain|Specify the domain -- it is apparent that a curve in a 2-dimensional space is probably to be inferred, but this needs to be made clear.}} The '''slope''' of $C$ at $P$ is defined as the slope of the tangent to $C$ at $P$.


Definition:Slope/Straight Line

Let $L$ be a straight line. The '''slope''' of $L$ is defined as the change in $y$ divided by the change in $x$.


Definition:Octahedron/Regular

A '''regular octahedron''' is an octahedron whose $8$ faces are all congruent equilateral triangles.


Definition:Regular Polyhedron

A '''regular polyhedron''' is a polyhedron: :$(1): \quad$ whose faces are congruent regular polygons :$(2): \quad$ each of whose vertices is the common vertex of the same number of faces.


Definition:Polyhedron

A '''polyhedron''' is a three-dimensional figure that is contained by polygonal faces.


Definition:Platonic Solid

A '''platonic solid''' is a convex polyhedron: :$(1): \quad$ whose faces are congruent regular polygons :$(2): \quad$ each of whose vertices is the common vertex of the same number of faces.


Definition:Prism

:500px {{EuclidDefinition|book = XI|def = 13|name = Prism}}


Definition:Cylinder

A '''cylinder''' is a solid made by rotating a rectangle along one of its sides.


Definition:Kernel of Homomorphism of Differential Complexes

Let $\left({R, +, \cdot}\right)$ be a ring. Let: :$M: \quad \cdots \longrightarrow M_i \stackrel{d_i}{\longrightarrow} M_{i+1} \stackrel{d_{i+1}}{\longrightarrow} M_{i+2} \stackrel{d_{i+2}}{\longrightarrow} \cdots$ and :$N: \quad \cdots \longrightarrow N_i \stackrel{d'_i}{\longrightarrow} N_{i+1} \stackrel{d'_{i+1}}{\longrightarrow} N_{i+2} \stackrel{d'_{i+2}}{\longrightarrow} \cdots$ be two differential complexes of $R$-modules. Let $\phi = \left\{ \phi_i : i \in \Z \right\}$ be a homomorphism $M \to N$. For each $i \in \Z$ let $K_i$ be the kernel of $\phi_i$. For each $i \in \Z$ let $f_i$ be the restriction of $d_i$ to $K_i$. Then the '''kernel''' of $\phi$ is: :$\ker \phi : \quad \cdots \longrightarrow K_i \stackrel{f_i}{\longrightarrow} K_{i+1} \stackrel{f_{i+1}}{\longrightarrow} K_{i+2} \stackrel{f_{i+2}}{\longrightarrow} \cdots$


Definition:Angle/Rectilineal

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/9 - Rectilineal Angle}}'' {{EuclidDefRefNocat|I|9|Rectilineal Angle}} Thus the distinction is made between straight-line angles and curved-line angles. Most of the time the fact that angles are rectilineal is taken for granted.


Definition:Angle/Subtend

Let $AB$ be a line segment and $C$ be a point: :250px The line segment $AB$ is said to '''subtend''' the angle $\angle ACB$.


Definition:Angle/Adjacent

Two angles are '''adjacent''' if they have an intersecting line in common: :250px


Definition:Angle/Unit

The usual units of measurement for angle are as follows: === Degree === {{:Definition:Angle/Unit/Degree}} === Minute === {{:Definition:Angle/Unit/Minute}} === Second === {{:Definition:Angle/Unit/Second}} === Radian === {{:Definition:Angle/Unit/Radian}}


Definition:Addition/Integers

The '''addition operation''' in the domain of integers $\Z$ is written $+$. We have that the set of integers is the Inverse Completion of Natural Numbers. Thus it follows that elements of $\Z$ are the isomorphic images of the elements of equivalence classes of $\N \times \N$ where two tuples are equivalent if the difference between the two elements of each tuple is the same. Thus addition can be formally defined on $\Z$ as the operation induced on those equivalence classes as specified in the definition of integers. That is, the integers being defined as all the difference congruence classes, integer addition can be defined directly as the operation induced by natural number addition on these congruence classes: :$\forall \tuple {a, b}, \tuple {c, d} \in \N \times \N: \eqclass {a, b} \boxminus + \eqclass {c, d} \boxminus = \eqclass {a + c, b + d} \boxminus$


Definition:Addition/Real Modulo Addition

Let $z \in \R$. Let $\R_z$ be the set of residue classes modulo $z$ of $\R$. The addition operation is defined on $\R_z$ as follows: :$\eqclass a z +_z \eqclass b z = \eqclass {a + b} z$ This operation is called '''addition modulo $z$'''.


Definition:Urn

An '''urn''' is a hypothetical container from which one may draw one of a number of objects (usually defined to be balls) of assorted colours. The idea is that one cannot see the ball one is withdrawing until it is out of the urn. Therefore it is not possible to know in advance what colour ball one is taking. The probability that a particular ball being withdrawn has a particular colour depends entirely on the distribution within the urn of the various colours of the balls within. This is used as an abstraction of a randomizing device in probability theory.


Definition:Quadrilateral/Square

A '''square''' is a regular quadrilateral. That is, a regular polygon with $4$ sides. That is, a '''square''' is a plane figure with four sides all the same length and whose angles are all equal. :200px


Definition:Quadrilateral/Oblong

An '''oblong''' is a quadrilateral whose angles are all right angles, but whose sides are ''not'' all the same length: :400px


Definition:Quadrilateral/Rectangle/Containment

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book II/1 - Containment of Rectangle}}'' {{EuclidDefRefNocat|II|1|Containment of Rectangle}}


Definition:Quadrilateral/Rectangle

A '''rectangle''' is a quadrilateral all of whose angles are equal to a right angle, and whose sides ''may or may not'' all be the same length.


Definition:Quadrilateral/Parallelogram/Base

In a given parallelogram, one of the sides is distinguished as being the '''base'''. It is immaterial which is so chosen, but usual practice is that it is one of the two longer sides. In the parallelogram above, line $AB$ is considered to be the '''base'''.


Definition:Quadrilateral/Parallelogram/Altitude

An '''altitude''' of a parallelogram is a line drawn perpendicular to its base, through one of its vertices to the side opposite to the base (which is extended if necessary). In the parallelogram above, line $DE$ is an '''altitude''' of the parallelogram $ABCD$. The term is also used for the length of such a line.


Definition:Quadrilateral/Parallelogram

A '''parallelogram''' is a quadrilateral whose opposite sides are parallel to each other, and whose sides ''may or may not'' all be the same length.   :450px


Definition:Quadrilateral/Rhombus

A '''rhombus''' is a parallelogram whose sides are all the same length. :350px Its angles ''may or may not'' all be equal.


Definition:Quadrilateral/Rhomboid

A '''rhomboid''' is a parallelogram whose sides are ''not'' all the same length. Its angles ''may or may not'' all be equal.


Definition:Quadrilateral/Trapezoid

A '''trapezoid''' is a quadrilateral which has '''exactly one''' pair of sides parallel: :700px


Definition:Quadrilateral/Trapezium

A '''trapezium''' is a quadrilateral with no parallel sides. :300px


Definition:Quadrilateral/Kite

A '''kite''' is an irregular quadrilateral which has both pairs of adjacent sides equal. :300px


Definition:Quadrilateral/Dart

A '''dart''' is an irregular quadrilateral with a reflex angle. :500px


Definition:Boundary (Geometry)/Containment

A geometric figure is said to be '''contained''' by its boundary or boundaries.


Definition:Polygon/Side

The line segments which make up a polygon are known as its '''sides'''. Thus, in the polygon above, the '''sides''' are identified as $a, b, c, d$ and $e$.


Definition:Polygon/Vertex

A corner of a polygon is known as a '''vertex'''. Thus, in the polygon above, the vertices are $A, B, C, D$ and $E$.


Definition:Polygon/Adjacent

=== Adjacent Side to Vertex === {{:Definition:Polygon/Adjacent/Side to Vertex}} === Adjacent Sides === {{:Definition:Polygon/Adjacent/Sides}} === Adjacent Vertex to Side === {{:Definition:Polygon/Adjacent/Vertex to Side}} === Adjacent Vertices === {{:Definition:Polygon/Adjacent/Vertices}}


Definition:Polygon/Opposite

When a polygon has an even number of sides, each side has an '''opposite side''', and each vertex likewise has an '''opposite vertex'''. When a polygon has an odd number of sides, each side has an '''opposite vertex'''. The '''opposite side''' (or '''opposite vertex''') to a given side (or '''vertex''') is that side (or vertex) which has the same number of sides between it and the side (or vertex) in question.


Definition:Polygon/Internal Angle

The '''internal angle''' of a vertex of a polygon is the size of the angle between the sides adjacent to that vertex, as measured ''inside'' the polygon.


Definition:Polygon/External Angle

Contrary to intuition, the '''external angle''' of a vertex of a polygon is ''not'' the size of the angle between the sides forming that vertex, as measured outside the polygon. An '''external angle''' is in fact an angle formed by one side of a polygon and a line produced from an adjacent side. :200px While $\angle AFE$ is the internal angle of vertex $F$, the '''external angle''' of this vertex is $\angle EFG$.


Definition:Polygon/Base

For a given polygon, any one of its sides may be temporarily distinguished from the others, and referred to as the '''base'''. It is immaterial which is so chosen. The usual practice is that the polygon is drawn so that the '''base''' is made horizontal, and at the bottom.


Definition:Polygon/Equilateral

An '''equilateral polygon''' is a polygon in which all the sides are the same length.


Definition:Polygon/Equiangular

An '''equiangular polygon''' is a polygon in which all the vertices have the same angle.


Definition:Polygon/Regular

A '''regular polygon''' is a polygon which is both equilateral and equiangular. That is, in which all the sides are the same length, and all the vertices have the same angle: :300px


Definition:Polygon/Multi-lateral

A '''multi-lateral polygon''' is a term to define a polygon with more than four sides.


Definition:Triangle (Geometry)/Right-Angled/Hypotenuse

In a right-angled triangle, the opposite side to the right angle is known as the '''hypotenuse'''. In the above figure, the side labeled $b$ is the '''hypotenuse'''.


Definition:Triangle (Geometry)/Right-Angled/Adjacent

In a right-angled triangle, for a given non-right angled vertex, the adjacent side which is ''not'' the hypotenuse is referred to as '''the adjacent'''. In the above figure: : the '''adjacent''' to vertex $A$ is side $c$ : the '''adjacent''' to vertex $C$ is side $a$.


Definition:Geometric Figure/Plane Figure

A '''plane figure''' is a geometric figure embedded in the plane.


Definition:Geometric Figure/Three-Dimensional Figure

A '''three-dimensional figure''' is a geometric figure which cannot be embedded in the plane, but which '''can''' be embedded in three-dimensional space.


Definition:Geometric Figure/Diameter

The '''diameter''' of a geometric figure is the greatest length that can be formed between two opposite parallel straight lines that can be drawn tangent to its boundary.


Definition:Triangle (Geometry)/Isosceles/Base

The base of an isosceles triangle is specifically defined to be the side which is a different length from the other two. In the above diagram, $BC$ is the '''base'''.


Definition:Triangle (Geometry)/Isosceles/Base Angles

The two (equal) vertices adjacent to the base of an isosceles triangle are called the '''base angles'''. In the above diagram, $\angle ABC$ and $\angle ACB$ are the '''base angles'''.


Definition:Triangle (Geometry)/Isosceles/Apex

The vertex opposite the base of an isosceles triangle is called the '''apex''' of the triangle. In the above diagram, $A$ is the '''apex'''.


Definition:Triangle (Geometry)/Isosceles/Legs

The sides of an isosceles triangle which are adjacent to the apex are called the '''legs''' of the triangle. In the above diagram, $AB$ and $AC$ are the '''legs'''.


Definition:Triangle (Geometry)/Adjacent

The two sides of a triangle which form a particular vertex are referred to as '''adjacent''' to that angle. Similarly, the two vertices of a triangle to which a particular side contributes are referred to as '''adjacent''' to that side.


Definition:Triangle (Geometry)/Opposite

The side of a triangle which is ''not'' one of the sides adjacent to a particular vertex is referred to as its '''opposite'''. Thus, each vertex has an opposite side, and each side has an opposite vertex.


Definition:Triangle (Geometry)/Base

For a given triangle, one of the sides can be distinguished as being the '''base'''. It is immaterial which is so chosen. The usual practice is that the triangle is drawn so that the '''base''' is made horizontal, and at the bottom. In the above diagram, it would be conventional for the side $AC$ to be identified as the '''base'''.


Definition:Triangle (Geometry)/Apex

Having selected one side of a triangle to be the base, the opposite vertex to that base is called the '''apex'''. In the above diagram, if $AC$ is taken to be the base of $\triangle ABC$, then $B$ is the '''apex'''.


Definition:Triangle (Geometry)/Height

The '''height''' of a triangle is the length of a perpendicular from the apex to whichever side has been chosen as its base. That is, the length of the '''altitude''' so defined.


Definition:Triangle (Geometry)/Nomenclature

:300px The vertices of a triangle are conventionally labeled $A, B, C$ (or with other uppercase letters), and the sides with lowercase letters corresponding to the opposite vertex, as above. In order to emphasize that a particular vertex being referred to is in fact a vertex, the symbol $\angle$ is often placed by the letter corresponding to that vertex. Thus, for example: : $\angle A$ is adjacent to sides $b$ and $c$ : Side $a$ is adjacent to $\angle B$ and $\angle C$ : $\angle A$ is opposite side $a$ : Side $a$ is opposite $\angle A$.


Definition:Triangle (Geometry)/Equilateral

An '''equilateral triangle''' is a triangle in which all three sides are the same length: That is, a regular polygon with $3$ sides. :300px


Definition:Triangle (Geometry)/Scalene

A '''scalene triangle''' is a triangle in which all three sides are of different lengths. :500px


Definition:Triangle (Geometry)/Acute

An '''acute triangle''' is a triangle in which all three of the vertices are acute angles.


Definition:Triangle (Geometry)/Obtuse

An '''obtuse triangle''' is a triangle in which one of the vertices is an obtuse angle.


Definition:Quadrilateral/Trapezoid/Height

The '''height''' of a '''trapezoid''' is defined as the length of a line perpendicular to the bases. In the above diagram, the '''heights''' of the given trapezoids are indicated by the letter $h$.


Definition:Icosahedron/Regular

A '''regular icosahedron''' is an icosahedron whose $20$ faces are all congruent equilateral triangles.


Definition:Dodecahedron/Regular

A '''regular dodecahedron''' is a dodecahedron whose $12$ faces are all congruent regular pentagons.


Definition:Tetrahedron/Regular

A '''regular tetrahedron''' is a tetrahedron whose $4$ faces are all congruent equilateral triangles.


Definition:Plane Number

A '''plane number''' is the product of two (natural) numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VII/16 - Plane Number}}'' {{EuclidDefRefNocat|VII|16|Plane Number}}


Definition:Solid Number

A '''solid number''' is the product of three (natural) numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book VII/17 - Solid Number}}'' {{EuclidDefRefNocat|VII|17|Solid Number}}


Definition:Incommensurable

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. $a$ and $b$ are '''incommensurable''' {{iff}} $\dfrac a b$ is irrational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}


Definition:Commensurable

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. $a$ and $b$ are '''commensurable''' iff $\dfrac a b$ is rational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/1 - Commensurable}}'' {{EuclidDefRefNocat|X|1|Commensurable}}


Definition:Inclination

=== Straight Line to Plane === {{Definition:Inclination/Straight Line to Plane}} === Plane to Plane === {{Definition:Inclination/Plane to Plane}}


Definition:Similar Inclination

{{EuclidDefinition|book=XI|def=7|name=Similarly Inclined}} Category:Definitions/Angles Category:Definitions/Solid Geometry o1c4yyqx26loroltm7v20o0ja4d48u9


Definition:Sphere/Geometry/Axis

By definition, a sphere is made by turning a semicircle around a straight line. That straight line is called the '''axis of the sphere'''. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/15 - Axis of Sphere}}'' {{EuclidDefRefNocat|XI|15|Axis of Sphere}}


Definition:Sphere/Geometry/Diameter

The '''diameter of a sphere''' is the length of any straight line drawn from a point on the surface to another point on the surface through the center.


Definition:Cylinder/Axis

{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/22 - Axis of Cylinder}}'' {{EuclidDefRefNocat|XI|22|Axis of Cylinder}} In the above diagram, the '''axis''' of the cylinder $ACBEFD$ is the straight line $GH$.


Definition:Cylinder/Base

{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/23 - Base of Cylinder}}'' {{EuclidDefRefNocat|XI|23|Base of Cylinder}} In the above diagram, the '''bases''' of the cylinder $ACBEDF$ are the faces $ABC$ and $DEF$.


Definition:Cylinder/Similar Cylinders

Let $h_1$ and $h_2$ be the heights of two cylinders. Let $d_1$ and $d_2$ be the diameters of the bases of the two cylinders. Then the two cylinders are '''similar''' {{iff}}: :$\dfrac {h_1} {h_2} = \dfrac {d_1} {d_2}$ {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders}}'' {{EuclidDefRefNocat|XI|24|Similar Cones and Cylinders}}


Definition:Segment of Circle/Base

The '''base''' of a segment of a circle is the straight line forming one of the boundaries of the seqment. In the above diagram, $AB$ is the '''base''' of the highlighted segment.


Definition:Segment of Circle/Angle of Segment

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/7 - Angle of Segment}}'' {{EuclidDefRefNocat|III|7|Angle of Segment}} That is, it is the angle the base makes with the circumference where they meet. It can also be defined as the angle between the base and the tangent to the circle at the end of the base: :300px


Definition:Sector/Angle

The '''angle of a sector''' is the angle between the two radii which delimit the sector. In the above diagram, the '''angle of the sector''' $BAC$ is $\theta$. The conjugate angle of $\theta$ also forms a '''sector''', denoted $CAB$ (see above).


Definition:Inclination/Straight Line to Plane

{{EuclidDefinition|book = XI|def = 5|name = Inclination of Straight Line}}


Definition:Inclination/Plane to Plane

{{EuclidDefinition|book = XI|def = 6|name = Inclination of Plane}}


Definition:Pyramid

:400px Let $P$ be a polygon. Let $Q$ be a point not in the plane of $P$. From each vertex of $P$, let lines be drawn to $Q$. The polyhedron which is contained by $P$ and the triangles formed by the sides of $P$ and the lines to $Q$ is a '''pyramid'''.


Definition:Right Circular Cone/Axis

Let $K$ be a right circular cone. Let point $A$ be the apex of $K$. Let point $O$ be the center of the base of $K$. Then the line $AO$ is the '''axis''' of $K$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/19 - Axis of Cone}}'' {{EuclidDefRefNocat|XI|19|Axis of Cone}}


Definition:Right Circular Cone/Base

Let $\triangle AOB$ be a right-angled triangle such that $\angle AOB$ is the right angle. Let $K$ be the right circular cone formed by the rotation of $\triangle AOB$ around $OB$. Let $BC$ be the circle described by $B$. The '''base''' of $K$ is the plane surface enclosed by the circle $BC$. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/20 - Base of Cone}}'' {{EuclidDefRefNocat|XI|20|Base of Cone}}


Definition:Right Circular Cone/Acute-Angled

Let $K$ be a right circular cone. Then $K$ is '''acute-angled''' {{iff}} the opening angle of $K$ is an acute angle.


Definition:Right Circular Cone/Right-Angled

Let $K$ be a right circular cone. Then $K$ is '''right-angled''' {{iff}} the opening angle of $K$ is a right angle.


Definition:Right Circular Cone/Obtuse-Angled

Let $K$ be a right circular cone. Then $K$ is '''obtuse-angled''' {{iff}} the opening angle of $K$ is an obtuse angle.


Definition:Circle/Center

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/16 - Center of Circle}}'' {{EuclidDefRefNocat|I|16|Center of Circle}} In the above diagram, the '''center''' is the point $A$.


Definition:Right Circular Cone/Similar Cones

Let $h_1$ and $h_2$ be the lengths of the axes of two right circular cones. Let $d_1$ and $d_2$ be the lengths of the diameters of the bases of the two right circular cones. Then the two right circular cones are '''similar''' {{iff}}: :$\dfrac {h_1} {h_2} = \dfrac {d_1} {d_2}$ {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book XI/24 - Similar Cones and Cylinders}}'' {{EuclidDefRefNocat|XI|24|Similar Cones and Cylinders}}


Definition:Commensurable in Square

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''commensurable in square''' iff $\left ({\dfrac a b}\right)^2$ is rational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}}


Definition:Incommensurable in Square

Let $a, b \in \R_{>0}$ be (strictly) positive real numbers. Then $a$ and $b$ are '''incommensurable in square''' iff $\left ({\dfrac a b}\right)^2$ is irrational. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X/2 - Commensurable in Square}}'' {{EuclidDefRefNocat|X|2|Commensurable in Square}}


Definition:Binomial (Euclidean)

Let $a$ and $b$ be two (strictly) positive real numbers such that: : $(1): \quad \dfrac a b \notin \Q$ : $(2): \quad \left({\dfrac a b}\right)^2 \in \Q$ where $\Q$ denotes the set of rational numbers. Then $a + b$ is a '''binomial'''. {{:Euclid:Proposition/X/36}} === First Binomial === {{Definition:Binomial (Euclidean)/First Binomial}} === Second Binomial === {{Definition:Binomial (Euclidean)/Second Binomial}} === Third Binomial === {{Definition:Binomial (Euclidean)/Third Binomial}} === Fourth Binomial === {{Definition:Binomial (Euclidean)/Fourth Binomial}} === Fifth Binomial === {{Definition:Binomial (Euclidean)/Fifth Binomial}} === Sixth Binomial === {{Definition:Binomial (Euclidean)/Sixth Binomial}}


Definition:Binomial (Euclidean)/First Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''first binomial''' {{iff}}: : $(1): \quad a \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2} } a \in \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/1 - First Binomial}}'' {{EuclidDefRefNocat|X (II)|1|First Binomial}}


Definition:Circle/Circumference

The '''circumference''' of a circle is the line that forms its boundary.


Definition:Binomial (Euclidean)/Second Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''second binomial''' {{iff}}: : $(1): \quad b \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/2 - Second Binomial}}'' {{EuclidDefRefNocat|X (II)|2|Second Binomial}}


Definition:Binomial (Euclidean)/Third Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''third binomial''' {{iff}}: : $(1): \quad a \notin \Q$ : $(2): \quad b \notin \Q$ : $(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/3 - Third Binomial}}'' {{EuclidDefRefNocat|X (II)|3|Third Binomial}}


Definition:Binomial (Euclidean)/Fourth Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''fourth binomial''' {{iff}}: : $(1): \quad a \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/4 - Fourth Binomial}}'' {{EuclidDefRefNocat|X (II)|4|Fourth Binomial}}


Definition:Binomial (Euclidean)/Fifth Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''fifth binomial''' {{iff}}: : $(1): \quad b \in \Q$ : $(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/5 - Fifth Binomial}}'' {{EuclidDefRefNocat|X (II)|5|Fifth Binomial}}


Definition:Binomial (Euclidean)/Sixth Binomial

Let $a$ and $b$ be two (strictly) positive real numbers such that $a + b$ is a binomial. Then $a + b$ is a '''sixth binomial''' {{iff}}: : $(1): \quad: a \notin \Q$ : $(2): \quad: b \notin \Q$ : $(3): \quad: \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (II)/6 - Sixth Binomial}}'' {{EuclidDefRefNocat|X (II)|6|Sixth Binomial}}


Definition:Circle/Diameter

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/17 - Diameter of Circle}}'' {{EuclidDefRefNocat|I|17|Diameter of Circle}} In the above diagram, the line $CD$ is a '''diameter'''.


Definition:Circle/Radius

A '''radius''' of a circle is a straight line segment whose endpoints are the center and the circumference of the circle. In the above diagram, the line $AB$ is a '''radius'''.


Definition:Circle/Arc

An '''arc''' of a circle is any part of its circumference.


Definition:Circle/Semicircle

{{EuclidDefinition|book=I|def=18|name=Semicircle}}


Definition:Circle/Chord

A '''chord''' of a circle is a straight line segment whose endpoints are on the circumference of the circle. In the diagram above, the lines $CD$ and $EF$ are both '''chords'''.


Definition:Circle/Equality

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/1 - Equal Circles}}'' {{EuclidDefRefNocat|III|1|Equal Circles}}


Definition:Line/Endpoint

Each of the points at either end of a line segment is called an '''endpoint''' of that line segment. Similarly, the point at which an infinite half-line terminates is called '''the endpoint''' of that line. {{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/3 - Line Extremities}}'' {{EuclidDefRefNocat|I|3|Line Extremities}}


Definition:Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a > b$. Then $a - b$ is an '''apotome''' {{iff}}: :$(1): \quad \dfrac a b \notin \Q$ :$(2): \quad \paren {\dfrac a b}^2 \in \Q$ where $\Q$ denotes the set of rational numbers. {{:Euclid:Proposition/X/73}}


Definition:Apotome/First Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''first apotome''' {{iff}}: :$(1): \quad a \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/1 - First Apotome}}'' {{EuclidDefRefNocat|X (III)|1|First Apotome}}


Definition:Apotome/Second Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''second apotome''' {{iff}}: :$(1): \quad b \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/2 - Second Apotome}}'' {{EuclidDefRefNocat|X (III)|2|Second Apotome}}


Definition:Apotome/Third Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''third apotome''' {{iff}}: :$(1): \quad a \notin \Q$ :$(2): \quad b \notin \Q$ :$(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \in \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/3 - Third Apotome}}'' {{EuclidDefRefNocat|X (III)|3|Third Apotome}}


Definition:Line/Straight Line

{{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book I/4 - Straight Line}}'' {{EuclidDefRefNocat|I|4|Straight Line}}


Definition:Line/Curve

A '''curve''' is a line which may or may not be straight.


Definition:Curve/Arc

An '''arc''' is a specific part of a curve.


Definition:Apotome/Fourth Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''fourth apotome''' {{iff}}: :$(1): \quad a \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/4 - Fourth Apotome}}'' {{EuclidDefRefNocat|X (III)|4|Fourth Apotome}}


Definition:Apotome/Fifth Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''fifth apotome''' {{iff}}: :$(1): \quad b \in \Q$ :$(2): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/5 - Fifth Apotome}}'' {{EuclidDefRefNocat|X (III)|5|Fifth Apotome}}


Definition:Apotome/Sixth Apotome

Let $a, b \in \set {x \in \R_{>0} : x^2 \in \Q}$ be two rationally expressible numbers such that $a - b$ is an apotome. Then $a - b$ is a '''sixth apotome''' {{iff}}: :$(1): \quad a \notin \Q$ :$(2): \quad b \notin \Q$ :$(3): \quad \dfrac {\sqrt {a^2 - b^2}} a \notin \Q$ where $\Q$ denotes the set of rational numbers. {{EuclidSaid}} :''{{Definition:Euclid's Definitions - Book X (III)/6 - Sixth Apotome}}'' {{EuclidDefRefNocat|X (III)|6|Sixth Apotome}}


Definition:Line/Straight Line Segment

A '''straight line segment''' is a line segment which is straight. {{EuclidSaid}} :''A straight line segment can be drawn joining any two points.'' ({{BookLink|The Elements|Euclid}}: Postulates: Euclid's Second Postulate)


Definition:Line/Segment

A '''line segment''' is any line (straight or not) which terminates at two points. === Straight Line Segment === {{:Definition:Line/Straight Line Segment}} === Endpoint === {{:Definition:Line/Endpoint}} === Midpoint === {{:Definition:Line/Midpoint}}


Definition:Line/Infinite Half-Line

An '''infinite half-line''' is a line which terminates at an endpoint at one end, but has no such endpoint at the other.


Definition:Line/Equality

Two line segments are '''equal''' {{iff}} they have the same length.


Definition:Incircle

Let $P$ be a polygon. Let $C$ be a circle which is inscribed within $P$. :300px Then $C$ is an '''incircle''' of $P$. === Incenter === {{:Definition:Incircle/Incenter}} === Inradius === {{:Definition:Incircle/Inradius}}


Definition:Circumcircle

Let $P$ be a polygon. Let $C$ be a circle which is circumscribed around $P$. :300px Then $C$ is a '''circumcircle''' of $P$.


Definition:Incircle of Triangle/Incenter

The center of an incircle of a triangle is called an '''incenter of the triangle'''.


Definition:Incircle of Triangle/Inradius

A radius of an incircle of a triangle is called an '''inradius of the triangle'''. In the above diagram, $r$ is an inradius.


Definition:Circumcircle of Triangle/Circumcenter

The center of a circumcircle of a triangle is called a '''circumcenter of a triangle'''.


Definition:Circumcircle of Triangle/Circumradius

A radius of a circumcircle of a triangle is called a '''circumradius of a triangle'''. In the above diagram, $R$ is a circumradius.


Definition:Circumcenter

Let $P$ be a polygon. Let $P$ have a circumcircle $C$. :300px Then the center of $C$ is the '''circumcenter''' of $P$.


Definition:Excircle of Triangle/Exradius

A radius of an excircle of a triangle is called an '''exradius of the triangle'''. In the above diagram, $r$ is an exradius.


Definition:Excircle of Triangle/Excenter

The center of an excircle of a triangle is called an '''excenter of the triangle'''.


Definition:Tangent/Geometry/Tangent Line

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/2 - Tangent to Circle}}'' {{EuclidDefRefNocat|III|2|Tangent to Circle}} :300px In the above diagram, the line is '''tangent''' to the circle at the point $C$.


Definition:Tangent/Geometry/Tangent Circles

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book III/3 - Tangent Circles}}'' {{EuclidDefRefNocat|III|3|Tangent Circles}} :400px In the above diagram, the two circles are '''tangent''' to each other at the point $C$.


Definition:Piecewise Continuous Function

Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. $f$ is '''piecewise continuous''' {{iff}}: :there exists a finite subdivision $\left\{ {x_0, x_1, \ldots, x_n}\right\}$ of $\left[{a \,.\,.\, b}\right]$, where $x_0 = a$ and $x_n = b$, such that: ::for all $i \in \left\{ {1, 2, \ldots, n}\right\}$, $f$ is continuous on $\left({x_{i − 1} \,.\,.\, x_i}\right)$.


Let $f$ be a real function defined on a closed interval $\closedint a b$. === Definition 1 === {{:Definition:Piecewise Continuously Differentiable Function/Definition 1}} === Definition 2 === {{:Definition:Piecewise Continuously Differentiable Function/Definition 2}}


Definition:Cone/Double Napped Cone

A '''double napped cone''' is a cone where the lines joining the apex to the circumference of the base extend indefinitely in either dimension: :300px


Definition:Cone/Double Napped Cone/Nappe

Each of the halves of a double napped cone is a '''nappe'''.


Definition:Plane Surface

{{EuclidDefinition|book = I|def = 7|name = Plane Surface}}


Definition:Plane Surface/The Plane

'''The plane''' is the term used for the general plane surface which is infinite in all directions.


Definition:Cone/Apex

In the above diagram, the point $A$ is known as the '''apex''' of the cone.


Definition:Cone/Base

The plane figure $PQR$ is called the '''base''' of the cone.


Definition:Right Circular Cone/Opening Angle

Let $K$ be a right circular cone. Let point $A$ be the apex of $K$. Let $B$ and $C$ be the endpoints of a diameter of the base of $K$. Then the angle $\angle BAC$ is the '''opening angle''' of $K$. In the above diagram, $\phi$ is the '''opening angle''' of the right circular cone depicted.


Definition:Conic Section

=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property}}


Definition:Ellipse

=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Ellipse}} === Focus-Directrix Property === {{:Definition:Ellipse/Focus-Directrix}} === Equidistance Property === {{:Definition:Ellipse/Equidistance}}


Definition:Parallel (Geometry)/Lines

{{EuclidSaid}} :''{{:Definition:Euclid's Definitions - Book I/23 - Parallel Lines}}'' {{EuclidDefRefNocat|I|23|Parallel Lines}} The contemporary interpretation of the concept of parallelism declares that a straight line is parallel to itself.


Definition:Parallel (Geometry)/Planes

Two planes are '''parallel''' {{iff}}, when produced indefinitely, do not intersect at any point. {{EuclidDefinition|book = XI|def = 8|name = Parallel Planes}} The contemporary interpretation of the concept of parallelism declares that a plane is parallel to itself.


Definition:Right Circular Cone/Directrix

Let $K$ be a right circular cone. Let $B$ be the base of $K$. The circumference of $B$ is the '''directrix''' of $K$.


Definition:Right Circular Cone/Generatrix

Let $K$ be a right circular cone. Let $A$ be the apex of $K$. Let $B$ be the base of $K$. Then a line joining the apex of $K$ to its directrix is a '''generatrix of $K$'''.


Definition:Generatrix

A '''generatrix''' is an element of a set of straight lines which constitute a given surface.


Definition:Conic Section/Intersection with Cone/Parabola

:400px Let $\phi = \theta$. Then $K$ is a parabola.


Definition:Conic Section/Intersection with Cone/Circle

:400px Let $\phi = \dfrac \pi 2 - \theta$, thereby making $D$ perpendicular to the axis of $C$. Then $D$ and $B$ are parallel, and so $K$ is a circle.


Definition:Conic Section/Intersection with Cone/Ellipse

:400px Let $\theta < \phi < \dfrac \pi 2 - \theta$. That is, the angle between $D$ and the axis of $C$ is between that for which $K$ is a circle and that which $K$ is a parabola. Then $K$ is an ellipse.


Definition:Conic Section/Intersection with Cone/Hyperbola

:400px Let $\phi < \theta$. Then $K$ is a hyperbola. Note that in this case $D$ intersects $C$ in two places: one for each nappe of $C$.


Definition:Ellipse/Focus-Directrix

400px Let $D$ be a straight line. Let $F$ be a point. Let $\epsilon \in \R: 0 < \epsilon < 1$. Let $K$ be the locus of points $b$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition: :$\epsilon \, p = q$ Then $K$ is an '''ellipse'''.


Definition:Ellipse/Equidistance

:400px Let $F_1$ and $F_2$ be two points in the plane. Let $d$ be a length greater than the distance between $F_1$ and $F_2$. Let $K$ be the locus of points $P$ which are subject to the condition: :$d_1 + d_2 = d$ where: :$d_1$ is the distance from $P$ to $F_1$ :$d_2$ is the distance from $P$ to $F_2$. Then $K$ is an '''ellipse'''. This property is known as the '''equidistance property'''.


Definition:Line/Midpoint

Let $L = AB$ be a line segment whose endpoints are $A$ and $B$. Let $M$ be a point on $L$ such that the line segment $AM$ is equal to the line segment $MB$. Then $M$ is the '''midpoint''' of $L$.


Definition:Ellipse/Major Axis

The '''major axis''' of $K$ is the line segment passing through both $F_1$ and $F_2$ whose endpoints are where it intersects $K$.


Definition:Ellipse/Minor Axis

The '''minor axis''' of $K$ is the line segment through the center of $K$ perpendicular to the major axis of $K$ such that its endpoints are the points of intersection with $K$.


Definition:Ellipse/Minor Axis/Semi-Minor Axis

A '''semi-minor axis''' of $K$ is either half of the minor axis of $K$ from its midpoint to its endpoint.


Definition:Ellipse/Major Axis/Semi-Major Axis

A '''semi-major axis''' of $K$ is either half of the major axis of $K$ from its midpoint to its endpoint.


Definition:Ellipse/Covertex

A '''covertex''' of $K$ is either one of the endpoints of the minor axis of $K$.


Definition:Ellipse/Vertex

A '''vertex''' of $K$ is either one of the endpoints of the major axis of $K$.


Definition:Cartesian Plane

324pxrightthumbA general point $Q = \tuple {x, y}$ in the Cartesian plane The '''Cartesian plane''' is a Cartesian coordinate system of $2$ dimensions. Every point on the plane can be identified uniquely by means of an ordered pair of real coordinates $\tuple {x, y}$, as follows: Identify one distinct point on the plane as the origin $O$. Select a point $P$ on the plane different from $O$. Construct an infinite straight line through $O$ and $P$ and call it the $x$-axis. Identify the $x$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P$ The orientation of the $x$-axis is determined by the relative positions of $O$ and $P$. It is conventional to locate $P$ to the right of $O$, so as to arrange that: :to the right of the origin, the numbers on the $x$-axis are positive :to the left of the origin, the numbers on the $x$-axis are negative. Construct an infinite straight line through $O$ perpendicular to the $x$-axis and call it the $y$-axis. Identify the point $P'$ on the $y$-axis such that $OP' = OP$. Identify the $y$-axis with the real number line such that: :$0$ is identified with the origin $O$ :$1$ is identified with the point $P'$ The orientation of the $y$-axis is determined by the position of $P'$ relative to $O$. It is conventional to locate $P'$ such that, if one were to imagine being positioned at $O$ and facing along the $x$-axis towards $P$, then $P'$ is on the left.  Hence with the conventional orientation of the $x$-axis as horizontal and increasing to the right: :going vertically "up" the page or screen from the origin, the numbers on the $y$-axis are positive :going vertically "down" the page or screen from the origin, the numbers on the $y$-axis are negative.


Definition:Cartesian Coordinate System/X Coordinate

Let $x$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $x$-axis. Then $x$ is known as the '''$x$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $x$-axis, then $x$ is positive. If $Q$ is in the negative direction along the real number line that is the $x$-axis, then $x$ is negative.


Definition:Cartesian Coordinate System/Y Coordinate

Let $y$ be the length of the line segment from the origin $O$ to the foot of the perpendicular from $Q$ to the $y$-axis. Then $y$ is known as the '''$y$ coordinate'''. If $Q$ is in the positive direction along the real number line that is the $y$-axis, then $y$ is positive. If $Q$ is in the negative direction along the real number line that is the $y$-axis, then $y$ is negative.


Definition:Cartesian Plane/Quadrants

For ease of reference, the cartesian plane is often divided into four quadrants by the axes: === First Quadrant === {{:Definition:First Quadrant}} === Second Quadrant === {{:Definition:Second Quadrant}} === Third Quadrant === {{:Definition:Third Quadrant}} === Fourth Quadrant === {{:Definition:Fourth Quadrant}} Note that the axes themselves are generally not considered to belong to any quadrant.


Definition:Axis/Coordinate Axes

Consider a coordinate system. One of the reference lines of such a system is called an '''axis'''.


Definition:Axis/X-Axis

In a cartesian coordinate system, the '''$x$-axis''' is the one usually depicted and visualised as going from left to right. It consists of all the points in the real vector space in question (usually either $\R^2$ or $\R^3$) at which all the elements of its coordinates but $x$ are zero.


Definition:Axis/Y-Axis

In a cartesian coordinate system, the '''$y$-axis''' is the one usually depicted and visualised as going from "bottom" to "top" of the paper (or screen). It consists of all the points in the real vector space in question (usually either $\R^2$ or $\R^3$) at which all the elements of its coordinates but $y$ are zero.


Definition:Axis/Z-Axis/Right-Hand Rule

The usual convention for the orientation of the $z$-axis is that of the right-hand rule: Let the coordinate axes be oriented as follows: :Let the $x$-axis increase from '''West''' to '''East'''. :Let the $y$-axis increase from '''South''' to '''North'''. Then the $z$-axis increases from '''below''' to '''above'''. If the $x$-axis and $y$-axis are aligned with a piece of paper or a screen aligned perpendicular to the line of sight, this translates into the following orientation: :Let the $x$-axis increase from '''left''' to '''right'''. :Let the $y$-axis increase from '''bottom''' to '''top'''. Then the $z$-axis increases from '''behind''' to '''in front''' (that is, from '''further away''' to '''closer in''').


Definition:Axis/Z-Axis

In a cartesian coordinate system, the '''$z$-axis''' is the axis passing through $x = 0, y = 0$ which is perpendicular to both the $x$-axis and the $y$-axis. It consists of all the points in the real vector space in question (usually $\R^3$) at which all the elements of its coordinates but $z$ are zero.


Definition:Formal Language

A '''formal language''' is a structure $\mathcal L$ which comprises: : A set of symbols $\mathcal A$ called the alphabet of $\mathcal L$ : A collation system with the unique readability property for $\mathcal A$ : A formal grammar that determines which collations belong to the formal language and which do not.


Definition:Coordinate System/Coordinate/Element of Ordered Pair

Let $\tuple {a, b}$ be an ordered pair. The following terminology is used: :$a$ is called the '''first coordinate''' :$b$ is called the '''second coordinate'''. This definition is compatible with the equivalent definition in the context of Cartesian coordinate systems.


Definition:Polar Coordinates

'''Polar coordinates''' are a technique for unique identification of points on the plane. A distinct point $O$ is identified. === Pole === {{:Definition:Polar Coordinates/Pole}} === Polar Axis === {{:Definition:Polar Coordinates/Polar Axis}}


Definition:Polar Coordinates/Radial Coordinate

The length of $OP$ is called the '''radial coordinate''' of $P$, and usually labelled $r$.


Definition:Polar Coordinates/Pole

The point $O$ is referred to as the '''pole''' of the polar coordinate plane.


Definition:Polar Coordinates/Polar Axis

A ray is drawn from $O$, usually to the right, and referred to as the '''polar axis'''.


Definition:Polar Coordinates/Angular Coordinate

The angle measured anticlockwise from the polar axis to $OP$ is called the '''angular coordinate''' of $P$, and usually labelled $\theta$. If the angle is measured clockwise from the polar axis to $OP$, its value is considered negative.


Definition:Meaningful Product

Let $\left({S, \circ}\right)$ be a semigroup. Let $a_1, \ldots, a_n$ be a sequence of elements of $S$. Then we define a '''meaningful product''' of $a_1, \ldots, a_n$ inductively as follows: If $n = 1$ then the only meaningful product is $a_1$. If $n > 1$ then a meaningful product is defined to be any product of the form: :$\left({a_1 \ldots a_m}\right)\left({a_{m+1} \ldots a_n}\right)$ where $m < n$ and $\left({a_1 \ldots a_m}\right)$ and  $\left({a_{m+1} \ldots a_n}\right)$ are meaningful products of $m$ and $n - m$ elements respectively.


Definition:Standard n Product

Let $\left({S, \circ}\right)$ be a semigroup. Let $a_1, \ldots, a_n$ be a sequence of elements over $S$. Then we denote the '''standard n product''' of $a_1, \ldots, a_n$ as: :$\displaystyle \prod_{i \mathop = 1}^n a_i$ We define it inductively as follows: If $n = 1$ then: :$\displaystyle \prod_{i \mathop = 1}^1 a_i = a_1$ If $n > 1$ then: :$\displaystyle \prod_{i \mathop = 1}^n a_i = \left({\displaystyle \prod_{i \mathop = 1}^{n-1} a_i}\right) a_n$


Definition:Cartesian Plane/Quadrants/First

:400px Quadrant $\text{I}: \quad$ The area above the $x$-axis and to the right of the $y$-axis is called '''the first quadrant'''. That is, '''the first quadrant''' is where both the $x$ coordinate and the $y$ coordinate of a point are positive.


Definition:Cartesian Plane/Quadrants/Third

:400px Quadrant $\text{III}: \quad$ The area below the $x$-axis and to the left of the $y$-axis is called '''the third quadrant'''. That is, '''the third quadrant''' is where both the $x$ coordinate and the $y$ coordinate of a point are negative.


Definition:Cartesian Plane/Quadrants/Fourth

:400px Quadrant $\text{IV}: \quad$ The area below the $x$-axis and to the right of the $y$-axis is called '''the fourth quadrant'''. That is, '''the fourth quadrant''' is where the $x$ coordinate of a point is positive and the $y$ coordinate of a point is negative.


Definition:Cartesian Plane/Quadrants/Second

:400px Quadrant $\text{II}: \quad$ The area above the $x$-axis and to the left of the $y$-axis is called '''the second quadrant'''. That is, '''the second quadrant''' is where the $x$ coordinate of a point is negative and the $y$ coordinate of a point is positive.


Definition:Canonical Epimorphism

Let $m \in \Z$. Let $f:\Z \to \Z_m$ be a mapping such that: :$\forall n \in \Z: \map f n = \eqclass n m$ where: ::$\Z_m$ denotes the integers modulo $m$. ::$\eqclass n m$ denotes the residue class of $n$ modulo $m$. Then $f$ is referred to as the '''canonical epimorphism''' ( '''from $\Z$ to $\Z_m$'''). That this is an epimorphism is proved in Canonical Epimorphism is Epimorphism. {{expand|Note that this definition is unnecessarily specific - it is an instance of the concept as applied to the integers modulo $m$. It may be appropriate to link to the fuller definition as given in Definition:Quotient Epimorphism, or depending on the context in Hungerford it may be better to rewrite this as a proof that it is such a specific instance. OTOH that ought already to have been documented somewhere (search around) as this area of group theory has already been covered in considerable detail.<br>In fact I've found it: Ring Epimorphism from Integers to Integers Modulo m.}}


Definition:Tangent Function

=== Definition from Triangle === {{:Definition:Tangent Function/Definition from Triangle}} === Definition from Circle === {{:Definition:Tangent Function/Definition from Circle}} === Real Function === {{:Definition:Tangent Function/Real}} === Complex Function === {{:Definition:Tangent Function/Complex}}


Definition:Tangent Function/Real

Let $x \in \R$ be a real number. The real function $\tan x$ is defined as: :$\tan x = \dfrac {\sin x} {\cos x}$ where: : $\sin x$ is the sine of $x$ : $\cos x$ is the cosine of $x$. The definition is valid for all $x \in \R$ such that $\cos x \ne 0$.


Definition:Tangent Function/Complex

Let $z \in \C$ be a complex number. The complex function $\tan z$ is defined as: :$\tan z = \dfrac {\sin z} {\cos z}$ where: : $\sin z$ is the sine of $z$ : $\cos z$ is the cosine of $z$. The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.


Definition:Cotangent/Real Function

Let $x \in \R$ be a real number. The real function $\cot x$ is defined as: :$\cot x = \dfrac {\cos x} {\sin x} = \dfrac 1 {\tan x}$ where: : $\sin x$ is the sine of $x$ : $\cos x$ is the cosine of $x$ : $\tan x$ is the tangent of $x$ The definition is valid for all $x \in \R$ such that $\sin x \ne 0$.


Definition:Cotangent/Complex Function

Let $z \in \C$ be a complex number. The complex function $\cot z$ is defined as: :$\cot z = \dfrac {\cos z} {\sin z} = \dfrac 1 {\tan z}$ where: : $\sin z$ is the sine of $z$ : $\cos z$ is the cosine of $z$ : $\tan z$ is the tangent of $z$ The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.


Definition:Secant Function/Real

Let $x \in \R$ be a real number. The real function $\sec x$ is defined as: :$\sec x = \dfrac 1 {\cos x}$ where $\cos x$ is the cosine of $x$. The definition is valid for all $x \in \R$ such that $\cos x \ne 0$.


Definition:Secant Function/Complex

Let $z \in \C$ be a complex number. The complex function $\sec z$ is defined as: :$\sec z = \dfrac 1 {\cos z}$ where $\cos z$ is the cosine of $z$. The definition is valid for all $z \in \C$ such that $\cos z \ne 0$.


Definition:Cosecant/Real Function

Let $x \in \C$ be a real number. The real function $\csc x$ is defined as: :$\csc x = \dfrac 1 {\sin x}$ where $\sin x$ is the sine of $x$. The definition is valid for all $x \in \R$ such that $\sin x \ne 0$.


Definition:Cosecant/Complex Function

Let $z \in \C$ be a complex number. The complex function $\csc z$ is defined as: :$\csc z = \dfrac 1 {\sin z}$ where $\sin z$ is the sine of $z$. The definition is valid for all $z \in \C$ such that $\sin z \ne 0$.


Definition:Sine/Definition from Triangle

:400px In the above right triangle, we are concerned about the angle $\theta$. The '''sine''' of $\angle \theta$ is defined as being $\dfrac {\text{Opposite}} {\text{Hypotenuse}}$.


Definition:Sine/Definition from Circle/First Quadrant

Consider a unit circle $C$ whose center is at the origin of a cartesian plane. :500px Let $P = \tuple {x, y}$ be the point on $C$ in the first quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.


Definition:Sine/Definition from Circle/Second Quadrant

:500px Let $P = \tuple {x, y}$ be the point on $C$ in the second quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.


Definition:Sine/Definition from Circle/Third Quadrant

:500px Let $P = \tuple {x, y}$ be the point on $C$ in the third quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.


Definition:Sine/Definition from Circle/Fourth Quadrant

:500px Let $P = \tuple {x, y}$ be the point on $C$ in the fourth quadrant such that $\theta$ is the angle made by $OP$ with the $x$-axis. Let $AP$ be the perpendicular from $P$ to the $x$-axis. Then the '''sine''' of $\theta$ is defined as the length of $AP$.


Definition:Sine/Definition from Circle

{{:Definition:Sine/Definition from Circle/First Quadrant}}