Home Metamath Proof ExplorerTheorem List (p. 213 of 325) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-22374) Hilbert Space Explorer (22375-23897) Users' Mathboxes (23898-32447)

Theorem List for Metamath Proof Explorer - 21201-21300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremchpdifbndlem2 21201* Lemma for chpdifbnd 21202. (Contributed by Mario Carneiro, 25-May-2016.)
ψ Λ ψ               ψ ψ

Theoremchpdifbnd 21202* A bound on the difference of nearby ψ values. Theorem 10.5.2 of [Shapiro], p. 427. (Contributed by Mario Carneiro, 25-May-2016.)
ψ ψ

Theoremlogdivbnd 21203* A bound on a sum of logs, used in pntlemk 21253. This is not as precise as logdivsum 21180 in its asymptotic behavior, but it is valid for all and does not require a limit value. (Contributed by Mario Carneiro, 13-Apr-2016.)

Theoremselberg3lem1 21204* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 21198 (written here as Λψ ). Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ        Λ ψ Λ ψ

Theoremselberg3lem2 21205* Lemma for selberg3 21206. Equation 10.4.21 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ Λ ψ

Theoremselberg3 21206* Introduce a log weighting on the summands of ΛΛ, the core of selberg2 21198 (written here as Λψ ). Equation 10.6.7 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
ψ Λ ψ

Theoremselberg4lem1 21207* Lemma for selberg4 21208. Equation 10.4.20 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
Λ ψ        Λ Λ ψ

Theoremselberg4 21208* The Selberg symmetry formula for products of three primes, instead of two. The sum here can also be written in the symmetric form ΛΛΛ; we eliminate one of the nested sums by using the definition of ψ Λ. This statement can thus equivalently be written ψ ΛΛΛ . Equation 10.4.23 of [Shapiro], p. 422. (Contributed by Mario Carneiro, 30-May-2016.)
ψ Λ Λ ψ

Theorempntrval 21209* Define the residual of the second Chebyshev function. The goal is to have , or . (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ        ψ

Theorempntrf 21210 Functionality of the residual. Lemma for pnt 21261. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theorempntrmax 21211* There is a bound on the residual valid for all . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theorempntrsumo1 21212* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd 21213* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 25-May-2016.)
ψ

Theorempntrsumbnd2 21214* A bound on a sum over . Equation 10.1.16 of [Shapiro], p. 403. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ

Theoremselbergr 21215* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ        Λ

Theoremselberg3r 21216* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.8 of [Shapiro], p. 429. (Contributed by Mario Carneiro, 30-May-2016.)
ψ        Λ

Theoremselberg4r 21217* Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.11 of [Shapiro], p. 430. (Contributed by Mario Carneiro, 30-May-2016.)
ψ        Λ Λ

Theoremselberg34r 21218* The sum of selberg3r 21216 and selberg4r 21217. (Contributed by Mario Carneiro, 31-May-2016.)
ψ        Λ Λ Λ

Theorempntsval 21219* Define the "Selberg function", whose asymptotic behavior is the content of selberg 21195. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        Λ ψ

Theorempntsf 21220* Functionality of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theoremselbergs 21221* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theoremselbergsb 21222* Selberg's symmetry formula, using the definition of the Selberg function. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ

Theorempntsval2 21223* The Selberg function can be expressed using the convolution product of the von Mangoldt function with itself. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        Λ Λ Λ

Theorempntrlog2bndlem1 21224* The sum of selberg3r 21216 and selberg4r 21217. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem2 21225* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ               ψ

Theorempntrlog2bndlem3 21226* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem4 21227* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem5 21228* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem6a 21229* Lemma for pntrlog2bndlem6 21230. (Contributed by Mario Carneiro, 7-Jun-2016.)
Λ ψ        ψ

Theorempntrlog2bndlem6 21230* Lemma for pntrlog2bnd 21231. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Λ ψ        ψ

Theorempntrlog2bnd 21231* A bound on . Equation 10.6.15 of [Shapiro], p. 431. (Contributed by Mario Carneiro, 1-Jun-2016.)
ψ

Theorempntpbnd1a 21232* Lemma for pntpbnd 21235. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ

Theorempntpbnd1 21233* Lemma for pntpbnd 21235. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ

Theorempntpbnd2 21234* Lemma for pntpbnd 21235. (Contributed by Mario Carneiro, 11-Apr-2016.)
ψ

Theorempntpbnd 21235* Lemma for pnt 21261. Establish smallness of at a point. Lemma 10.6.1 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntibndlem1 21236 Lemma for pntibnd 21240. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntibndlem2a 21237* Lemma for pntibndlem2 21238. (Contributed by Mario Carneiro, 7-Jun-2016.)
ψ

Theorempntibndlem2 21238* Lemma for pntibnd 21240. The main work, after eliminating all the quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ                                                                              ψ ψ

Theorempntibndlem3 21239* Lemma for pntibnd 21240. Package up pntibndlem2 21238 in quantifiers. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntibnd 21240* Lemma for pnt 21261. Establish smallness of on an interval. Lemma 10.6.2 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 10-Apr-2016.)
ψ

Theorempntlemd 21241 Lemma for pnt 21261. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is C^*, is c1, is λ, is c2, and is c3. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;

Theorempntlemc 21242* Lemma for pnt 21261. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is α, is ε, and is K. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;

Theorempntlema 21243* Lemma for pnt 21261. Closure for the constants used in the proof. The mammoth expression is a number large enough to satisfy all the lower bounds needed for . For comparison with Equation 10.6.27 of [Shapiro], p. 434, is x2, is x1, is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemb 21244* Lemma for pnt 21261. Unpack all the lower bounds contained in , in the form they will be used. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is x. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;               ;

Theorempntlemg 21245* Lemma for pnt 21261. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, is j^* and is ĵ. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemh 21246* Lemma for pnt 21261. Bounds on the subintervals in the induction. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemn 21247* Lemma for pnt 21261. The "naive" base bound, which we will slightly improve. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemq 21248* Lemma for pntlemj 21250. (Contributed by Mario Carneiro, 7-Jun-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemr 21249* Lemma for pntlemj 21250. (Contributed by Mario Carneiro, 7-Jun-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemj 21250* Lemma for pnt 21261. The induction step. Using pntibnd 21240, we find an interval in which is sufficiently large and has a much smaller value, (instead of our original bound ). (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                                                ..^

Theorempntlemi 21251* Lemma for pnt 21261. Eliminate some assumptions from pntlemj 21250. (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                                  ..^

Theorempntlemf 21252* Lemma for pnt 21261. Add up the pieces in pntlemi 21251 to get an estimate slightly better than the naive lower bound . (Contributed by Mario Carneiro, 13-Apr-2016.)
ψ                                    ;                                                         ;                                           ;

Theorempntlemk 21253* Lemma for pnt 21261. Evaluate the naive part of the estimate. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlemo 21254* Lemma for pnt 21261. Combine all the estimates to establish a smaller eventual bound on . (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntleme 21255* Lemma for pnt 21261. Package up pntlemo 21254 in quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                    ;                                                         ;

Theorempntlem3 21256* Lemma for pnt 21261. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ                                           ψ

Theorempntlemp 21257* Lemma for pnt 21261. Wrapping up more quantifiers. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                           ;

Theorempntleml 21258* Lemma for pnt 21261. Equation 10.6.35 in [Shapiro], p. 436. (Contributed by Mario Carneiro, 14-Apr-2016.)
ψ                                           ;               ψ

Theorempnt3 21259 The Prime Number Theorem, version 3: the second Chebyshev function tends asymptotically to . (Contributed by Mario Carneiro, 1-Jun-2016.)
ψ

Theorempnt2 21260 The Prime Number Theorem, version 2: the first Chebyshev function tends asymptotically to . (Contributed by Mario Carneiro, 1-Jun-2016.)

Theorempnt 21261 The Prime Number Theorem: the number of prime numbers less than tends asymptotically to as goes to infinity. (Contributed by Mario Carneiro, 1-Jun-2016.)
π

13.4.13  Ostrowski's theorem

Theoremabvcxp 21262* Raising an absolute value to a power less than one yields another absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)
AbsVal

Theorempadicfval 21263* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theorempadicval 21264* Value of the p-adic absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.)

Theoremostth2lem1 21265* Lemma for ostth2 21284, although it is just a simple statement about exponentials which does not involve any specifics of ostth2 21284. If a power is upper bounded by a linear term, the exponent must be less than one. Or in big-O notation, for any . (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremqrngbas 21266 The base set of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqdrng 21267 The rationals form a division ring. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng0 21268 The zero element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrng1 21269 The unit element of the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngneg 21270 The additive inverse in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds

Theoremqrngdiv 21271 The division operation in the field of rationals. (Contributed by Mario Carneiro, 8-Sep-2014.)
flds        /r

Theoremqabvle 21272 By using induction on , we show a long-range inequality coming from the triangle inequality. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremqabvexp 21273 Induct the product rule abvmul 15872 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostthlem1 21274* Lemma for ostth 21286. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theoremostthlem2 21275* Lemma for ostth 21286. Refine ostthlem1 21274 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)
flds        AbsVal

Theoremqabsabv 21276 The regular absolute value function on the rationals is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabv 21277* The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvf 21278* The p-adic absolute value is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theorempadicabvcxp 21279* All positive powers of the p-adic absolute value are absolute values. (Contributed by Mario Carneiro, 9-Sep-2014.)
flds        AbsVal

Theoremostth1 21280* - Lemma for ostth 21286: trivial case. (Not that the proof is trivial, but that we are proving that the function is trivial.) If is equal to on the primes, then by complete induction and the multiplicative property abvmul 15872 of the absolute value, is equal to on all the integers, and ostthlem1 21274 extends this to the other rational numbers. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem2 21281* Lemma for ostth2 21284. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem3 21282* Lemma for ostth2 21284. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2lem4 21283* Lemma for ostth2 21284. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth2 21284* - Lemma for ostth 21286: regular case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth3 21285* - Lemma for ostth 21286: p-adic case. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

Theoremostth 21286* Ostrowski's theorem, which classifies all absolute values on . Any such absolute value must either be the trivial absolute value , a constant exponent times the regular absolute value, or a positive exponent times the p-adic absolute value. (Contributed by Mario Carneiro, 10-Sep-2014.)
flds        AbsVal

PART 14  GRAPH THEORY

To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to Definitions in [Bollobas] p. 1-8. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic kinds of graphs:

TermReferenceDefinitionRemarks
(Undirected) Hypergraph df-uhgra 21288 an ordered pair of a set and a function into the powerset of ( ).
An element of is called "vertex", an element of is called "edge", the function is called the "edge-function" .
In this most general definition of a graph, an "edge" may connect three or more vertices with each other, compare with the definition in Section I.1 in [Bollobas] p. 7.
Undirected multigraph df-umgra 21301 a graph such that is a function into the set of (proper or not proper) unordered pairs of .A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 3940).
According to the definition in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed".
Undirected simple graph with loops df-uslgra 21319 a graph such that is a one-to-one function into the set of (proper or not proper) unordered pairs of .This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph without loops (in short "simple graph") df-usgra 21320 a graph such that is a one-to-one function into the set of (proper) unordered pairs of .An ordered pair of two distinct sets and (the "usual" definition of a "graph", see, for example, the definition in Section I.1 in [Bollobas] p. 1) can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see ausisusgra 21333.
Finite graph---a graph with finite sets and .In simple graphs, is finite if is finite, see usgrafis 21382. The number of edges is limited by (or " choose 2") with , see usgramaxsize 21449. Analogously, the number of edges of an undirected simple graph with loops is limited by . In multigraphs, however, can be infinite although is finite.
Graph of finite size---a graph with finite set , i.e. with a finite number of edges.A graph can be of finite size although is infinite.

Terms and properties of graphs:
TermReferenceDefinitionRemarks
Edge joining (two) vertices --- An edge "joins" the vertices v1, v2, ... vn ( ) if = { v1, v2, ... vn }. If , = { v1 } is a "loop", if , = { v1 , v2 } is an egde as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge see definition in Section I.1 in [Bollobas] p. 1. If an edge joins the vertices v1, v2, ... vn ( ), then the vertices v1, v2, ... vn are called the "endvertices" of the edge .
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v1, v2, ... vn ( ) are "adjacent" if there is an edge e = { v1, v2, ... vn } joining these vertices. In this case, the vertices are "incident" with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or "connected" by the edge e.
(Two) Adjacent edges The edges e0, e1, ... en ( ) are "adjacent" if they have exactly one common endvertex. Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph see definition in Section I.1 in [Bollobas] p. 3 the "order" of a graph is the number of vertices in the graph ().
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 the "size" of a graph is the number of edges in the graph ().
Neighborhood of a vertex df-nbgra 21386 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex by an edge is called a "neighbor" of the vertex . The set of neighbors of a vertex is called the "neighborhood" (or "open neighborhood") of the vertex . The "closed neighborhood" is the union of the (open) neighborhood of the vertex with .
Degree of a vertex df-vdgr 21618 The "degree" of a vertex is the number of the edges having this vertex as endvertex. In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex usgravd0nedg 21636 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 21388 A vertex is called "universal" if it is connected with every other vertex of the graph by an edge, thus having degree .

Special kinds of graphs:
TermReferenceDefinitionRemarks
Complete graph df-cusgra 21387 A graph is called "complete" if each pair of vertices is connected by an edge. The size of a complete undirected simple graph of order is (or " choose 2"), see cusgrasize 21440.
Empty graph umgra0 21313 and usgra0 21343 A graph is called "empty" if it has no edges.
Null graph usgra0v 21344 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 21362 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 21610 resp. definition in Section I.1 in [Bollobas] p. 6 A graph is called "connected" if for each pair of vertices there is a path between these vertices.

For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.

14.1  Undirected graphs - basics

14.1.1  Undirected hypergraphs

Syntaxcuhg 21287 Extend class notation with undirected hypergraphs.
UHGrph

Definitiondf-uhgra 21288* Define the class of all undirected hypergraphs. An undirected hypergraph is a pair of a set and a function into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremreluhgra 21289 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgrav 21290 The classes of vertices and edges of an undirected hypergraph are sets. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremisuhgra 21291 The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgraf 21292 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgrafun 21293 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgrass 21294 An edge is a subset of vertices, analogous to umgrass 21307. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph

Theoremuhgraeq12d 21295 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph UHGrph

Theoremuhgrares 21296 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 21312. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UHGrph UHGrph

Theoremuhgra0 21297 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 21313. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UHGrph

Theoremuhgra0v 21298 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UHGrph

Theoremuhgraun 21299 If and are hypergraphs, then is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting in two edges between two vertices), analogous to umgraun 21316. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
UHGrph        UHGrph        UHGrph

14.1.2  Undirected multigraphs

Syntaxcumg 21300 Extend class notation with undirected multigraphs.
UMGrph

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306