hcju/proofwikips huggingface dataset:
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'''  for each partial mapping $f \subseteq S \times T$: : $f \in \mathcal F$  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'''  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}}
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}}
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'''  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 $r$''' : :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 $r$''' : : $(1): \quad$ $G$ 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 $x \mathrel {\mathcal R} y$  $y$ is reachable from $x$. That is, $x \mathrel {\mathcal R} y$  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 $u$  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 $u$  $u \mathrel {\mathcal R} v$.
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 $r$''' : : $(1): \quad$ 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 $G$  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''' : :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''' : : $(1): \quad \left({L, \wedge, \vee, \preceq}\right)$ 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  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  both: :$S$ is a subset of $T$ and :$T$ is a subset of $S$
Definition:Antisymmetric Relation/Definition 1
$\RR$ is '''antisymmetric''' : :$\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$ that is: :$\set {\tuple {x, y}, \tuple {y, x} } \subseteq \RR \implies x = y$
Definition:Antisymmetric Relation/Definition 2
$\mathcal R$ is '''antisymmetric''' : :$\tuple {x, y} \in \mathcal R \land x \ne y \implies \tuple {y, x} \notin \mathcal R$
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.
Definition:Adjacent (Graph Theory)/Vertices
=== 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 $e$'''.
Definition:Incident (Graph Theory)/Digraph
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.
Definition:Adjacent (Graph Theory)/Edges
=== 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'''  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 $E$.
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'''  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'''  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'''  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
 :''{{: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 $z$'''.