id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-17100 | Graph is 0-Regular iff Edgeless | Let $G$ be a graph.
Then $G$ is $0$-regular graph {{iff}} $G$ is edgeless. | === Sufficient Condition ===
Let $G$ be an edgeless graph.
By definition of edgeless graph, $G$ has no edges.
That means every vertex of $G$ is incident with no edges.
That is, every vertex of $G$ is of degree $0$.
Hence the result, by definition of $0$-regular graph.
{{qed|lemma}} | Let $G$ be a [[Definition:Graph (Graph Theory)|graph]].
Then $G$ is [[Definition:Regular Graph|$0$-regular graph]] {{iff}} $G$ is [[Definition:Edgeless Graph|edgeless]]. | === Sufficient Condition ===
Let $G$ be an [[Definition:Edgeless Graph|edgeless graph]].
By definition of [[Definition:Edgeless Graph|edgeless graph]], $G$ has no [[Definition:Edge of Graph|edges]].
That means every [[Definition:Vertex of Graph|vertex]] of $G$ is [[Definition:Incident (Undirected Graph)|incident wit... | Graph is 0-Regular iff Edgeless | https://proofwiki.org/wiki/Graph_is_0-Regular_iff_Edgeless | https://proofwiki.org/wiki/Graph_is_0-Regular_iff_Edgeless | [
"Edgeless Graphs",
"Regular Graphs"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Regular Graph",
"Definition:Edgeless Graph"
] | [
"Definition:Edgeless Graph",
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Ver... |
proofwiki-17101 | Edgeless Graph is Bipartite | Let $N_n$ denote the edgeless graph with $n$ vertices.
Then $N_n$ is a bipartite graph. | Because $N_n$ has no edges, it has no cycles longer than $0$.
Thus in particular, it has no odd cycles.
The result follows from Graph is Bipartite iff No Odd Cycles.
{{qed}}
Category:Edgeless Graphs
Category:Bipartite Graphs
onj0q7fygjn9c7y8vyoibcwtir8973h | Let $N_n$ denote the [[Definition:Edgeless Graph|edgeless graph]] with $n$ [[Definition:Vertex of Graph|vertices]].
Then $N_n$ is a [[Definition:Bipartite Graph|bipartite graph]]. | Because $N_n$ has no [[Definition:Edge of Graph|edges]], it has no [[Definition:Cycle (Graph Theory)|cycles]] longer than $0$.
Thus in particular, it has no [[Definition:Odd Cycle (Graph Theory)|odd cycles]].
The result follows from [[Graph is Bipartite iff No Odd Cycles]].
{{qed}}
[[Category:Edgeless Graphs]]
[[Cat... | Edgeless Graph is Bipartite | https://proofwiki.org/wiki/Edgeless_Graph_is_Bipartite | https://proofwiki.org/wiki/Edgeless_Graph_is_Bipartite | [
"Edgeless Graphs",
"Bipartite Graphs"
] | [
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Bipartite Graph"
] | [
"Definition:Graph (Graph Theory)/Edge",
"Definition:Cycle (Graph Theory)",
"Definition:Cycle (Graph Theory)/Odd",
"Graph is Bipartite iff No Odd Cycles",
"Category:Edgeless Graphs",
"Category:Bipartite Graphs"
] |
proofwiki-17102 | Edgeless Graph of Order n has n Components | Let $N_n$ denote the edgeless graph with $n$ vertices.
Then $N_n$ has $n$ components. | Because there are $n$ vertices in $N_n$, it cannot have more than $n$ components.
{{AimForCont}} $N_n$ has fewer than $n$ components.
That would mean that at least $2$ vertices of $N_n$ are connected.
But to be connected, they need to be joined by at least one edge.
But that contradicts the fact that $N_n$ has no edges... | Let $N_n$ denote the [[Definition:Edgeless Graph|edgeless graph]] with $n$ [[Definition:Vertex of Graph|vertices]].
Then $N_n$ has $n$ [[Definition:Component of Graph|components]]. | Because there are $n$ [[Definition:Vertex of Graph|vertices]] in $N_n$, it cannot have more than $n$ [[Definition:Component of Graph|components]].
{{AimForCont}} $N_n$ has fewer than $n$ [[Definition:Component of Graph|components]].
That would mean that at least $2$ [[Definition:Vertex of Graph|vertices]] of $N_n$ ar... | Edgeless Graph of Order n has n Components | https://proofwiki.org/wiki/Edgeless_Graph_of_Order_n_has_n_Components | https://proofwiki.org/wiki/Edgeless_Graph_of_Order_n_has_n_Components | [
"Edgeless Graphs"
] | [
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Component of Graph"
] | [
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Component of Graph",
"Definition:Component of Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Connected (Graph Theory)/Vertices",
"Definition:Connected (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Cont... |
proofwiki-17103 | Complete Graph of Order 1 is Edgeless | The complete graph $K_1$ of order $1$ is the edgeless graph $N_1$. | By definition, $K_1$ has $1$ vertex.
From Complete Graph is Regular, $K_1$ is $0$-regular.
Hence the result from Graph is 0-Regular iff Edgeless.
{{qed}}
Category:Edgeless Graphs
Category:Complete Graphs
oyi3m2zbnw1gysum6opypbdjzd67alx | The [[Definition:Complete Graph|complete graph]] $K_1$ of [[Definition:Order of Graph|order $1$]] is the [[Definition:Edgeless Graph|edgeless graph]] $N_1$. | By definition, $K_1$ has $1$ [[Definition:Vertex of Graph|vertex]].
From [[Complete Graph is Regular]], $K_1$ is [[Definition:Regular Graph|$0$-regular]].
Hence the result from [[Graph is 0-Regular iff Edgeless]].
{{qed}}
[[Category:Edgeless Graphs]]
[[Category:Complete Graphs]]
oyi3m2zbnw1gysum6opypbdjzd67alx | Complete Graph of Order 1 is Edgeless | https://proofwiki.org/wiki/Complete_Graph_of_Order_1_is_Edgeless | https://proofwiki.org/wiki/Complete_Graph_of_Order_1_is_Edgeless | [
"Edgeless Graphs",
"Complete Graphs"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Edgeless Graph"
] | [
"Definition:Graph (Graph Theory)/Vertex",
"Complete Graph is Regular",
"Definition:Regular Graph",
"Graph is 0-Regular iff Edgeless",
"Category:Edgeless Graphs",
"Category:Complete Graphs"
] |
proofwiki-17104 | Complement of Complete Graph is Edgeless Graph | Let $K_n$ denote the complete graph of order $n$.
Then the complement of $K_n$ is the $n$-edgeless graph $N_n$. | By definition of complete graph, each vertex of $K_n$ is adjacent to every other vertex of $K_n$.
Let $\overline {K_n}$ denote the complement of $K_n$.
Let $v$ be a vertex of $\overline {K_n}$.
By definition of complement, $v$ is adjacent to all the vertices of $\overline {K_n}$ that it is not in $K_n$.
But that means ... | Let $K_n$ denote the [[Definition:Complete Graph|complete graph]] of [[Definition:Order of Graph|order]] $n$.
Then the [[Definition:Complement of Simple Graph|complement]] of $K_n$ is the [[Definition:Edgeless Graph|$n$-edgeless graph]] $N_n$. | By definition of [[Definition:Complete Graph|complete graph]], each [[Definition:Vertex of Graph|vertex]] of $K_n$ is [[Definition:Adjacent Vertices (Undirected Graph)|adjacent]] to every other [[Definition:Vertex of Graph|vertex]] of $K_n$.
Let $\overline {K_n}$ denote the [[Definition:Complement of Simple Graph|comp... | Complement of Complete Graph is Edgeless Graph | https://proofwiki.org/wiki/Complement_of_Complete_Graph_is_Edgeless_Graph | https://proofwiki.org/wiki/Complement_of_Complete_Graph_is_Edgeless_Graph | [
"Complete Graphs",
"Edgeless Graphs"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Complement of Graph/Simple Graph",
"Definition:Edgeless Graph"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Complement of Graph/Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Complement of Graph/Simple... |
proofwiki-17105 | Edgeless Graph of Order 1 is Tree | Let $N_1$ denote the edgeless graph with $1$ vertex.
Then $N_1$ is a tree. | By definition, a tree is a simple connected graph with no circuits.
$N_1$ is trivially connected graph.
As $N_1$ is edgeless, it has no edges.
But a circuit is a closed trail with at least one edge.
Hence the result.
{{qed}}
Category:Edgeless Graphs
Category:Tree Theory
nl2aycgpr91he895hzgrfs94keg40mt | Let $N_1$ denote the [[Definition:Edgeless Graph|edgeless graph]] with $1$ [[Definition:Vertex of Graph|vertex]].
Then $N_1$ is a [[Definition:Tree (Graph Theory)|tree]]. | By definition, a [[Definition:Tree (Graph Theory)|tree]] is a [[Definition:Simple Graph|simple]] [[Definition:Connected Graph|connected graph]] with no [[Definition:Circuit (Graph Theory)|circuits]].
$N_1$ is trivially [[Definition:Connected Graph|connected graph]].
As $N_1$ is [[Definition:Edgeless Graph|edgeless]],... | Edgeless Graph of Order 1 is Tree | https://proofwiki.org/wiki/Edgeless_Graph_of_Order_1_is_Tree | https://proofwiki.org/wiki/Edgeless_Graph_of_Order_1_is_Tree | [
"Edgeless Graphs",
"Tree Theory"
] | [
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Tree (Graph Theory)"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Simple Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Circuit (Graph Theory)",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Circuit (Graph Theory)",
"Defin... |
proofwiki-17106 | Edgeless Graph of Order Greater than 1 is Forest | Let $N_n$ denote the edgeless graph with $n$ vertices such that $n > 1$.
Then $N_n$ is a forest. | By definition, a '''forest''' is a simple graph whose components are all trees.
From Edgeless Graph of Order n has n Components $N_n$ has as many components as it has vertices.
Each of those vertices is an instance of $N_1$, the edgeless graph with $1$ vertex.
The result follows from Edgeless Graph of Order 1 is Tree.
... | Let $N_n$ denote the [[Definition:Edgeless Graph|edgeless graph]] with $n$ [[Definition:Vertex of Graph|vertices]] such that $n > 1$.
Then $N_n$ is a [[Definition:Forest|forest]]. | By definition, a '''[[Definition:Forest|forest]]''' is a [[Definition:Simple Graph|simple graph]] whose [[Definition:Component (Graph Theory)|components]] are all [[Definition:Tree (Graph Theory)|trees]].
From [[Edgeless Graph of Order n has n Components]] $N_n$ has as many [[Definition:Component (Graph Theory)|compon... | Edgeless Graph of Order Greater than 1 is Forest | https://proofwiki.org/wiki/Edgeless_Graph_of_Order_Greater_than_1_is_Forest | https://proofwiki.org/wiki/Edgeless_Graph_of_Order_Greater_than_1_is_Forest | [
"Edgeless Graphs",
"Forests"
] | [
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Forest"
] | [
"Definition:Forest",
"Definition:Simple Graph",
"Definition:Component of Graph",
"Definition:Tree (Graph Theory)",
"Edgeless Graph of Order n has n Components",
"Definition:Component of Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Edgeless G... |
proofwiki-17107 | Size of Complete Graph | Let $K_n$ denote the complete graph of order $n$ where $n \ge 0$.
The size of $K_n$ is given by:
:$\size {K_n} = \dfrac {n \paren {n - 1} } 2$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\size {K_n} = \dfrac {n \paren {n - 1} } 2$
First we explore the degenerate case $\map P 0$:
{{begin-eqn}}
{{eqn | l = \size {K_0}
| r = 0
| c = as $K_0$ is the null graph
}}
{{eqn | r = \dfrac {0 \paren {0 - 1... | Let $K_n$ denote the [[Definition:Complete Graph|complete graph]] of [[Definition:Order of Graph|order]] $n$ where $n \ge 0$.
The [[Definition:Size of Graph|size]] of $K_n$ is given by:
:$\size {K_n} = \dfrac {n \paren {n - 1} } 2$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\size {K_n} = \dfrac {n \paren {n - 1} } 2$
First we explore the [[Definition:Degenerate Case|degenerate case]] $\map P 0$:
{{begin-eqn}}
{{eqn | l = \... | Size of Complete Graph | https://proofwiki.org/wiki/Size_of_Complete_Graph | https://proofwiki.org/wiki/Size_of_Complete_Graph | [
"Complete Graphs",
"Proofs by Induction"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Degenerate Case",
"Definition:Null Graph",
"Principle of Mathematical Induction"
] |
proofwiki-17108 | Field Operations of P-adic Numbers | Let $p$ be a prime number.
Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where:
:$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p... | By {{Corollary|Cauchy Sequences form Ring with Unity}}, $\CC$ is a commutative ring of Cauchy sequences with the ring operations defined by:
:$+ : \quad \forall \sequence {x_n}, \sequence {y_n} \in \CC$:
:::$\quad \sequence {x_n} + \sequence {y_n} = \sequence {x_n + y_n}$
:$\circ : \quad \forall \sequence {x_n}, \seque... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|p-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
That is, $\Q_p$ is th... | By {{Corollary|Cauchy Sequences form Ring with Unity}}, $\CC$ is a [[Definition:Ring of Cauchy Sequences|commutative ring of Cauchy sequences]] with the [[Definition:Ring (Abstract Algebra)|ring]] [[Definition:Binary Operation|operations]] defined by:
:$+ : \quad \forall \sequence {x_n}, \sequence {y_n} \in \CC$:
:::$... | Field Operations of P-adic Numbers | https://proofwiki.org/wiki/Field_Operations_of_P-adic_Numbers | https://proofwiki.org/wiki/Field_Operations_of_P-adic_Numbers | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Quotient Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence",
"Definition:Field (Abstract Algebra)",
"Definition:Field ... | [
"Definition:Ring of Cauchy Sequences",
"Definition:Ring (Abstract Algebra)",
"Definition:Operation/Binary Operation",
"Definition:Maximal Ideal",
"Definition:Quotient Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Field (Abstract Algebra)",
"Definition:Operation/Binary Operation",
"Defini... |
proofwiki-17109 | Simple Graph of Maximum Size is Complete Graph | Let $G$ be a simple graph of order $n$ such that $n \ge 1$.
Let $G$ have the largest size of all simple graphs of order $n$.
Then:
:$G$ is the complete graph $K_n$
:its size is $\dfrac {n \paren {n - 1} } 2$. | By definition, $K_n$ is the simple graph of order $n$ such that every vertex of $K_n$ is adjacent to all other vertices.
So, let $G$ have the largest size of all simple graphs of order $n$.
Then by definition of largest size, it is not possible for another edge to be added to $G$.
The only way that could be is if all t... | Let $G$ be a [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$ such that $n \ge 1$.
Let $G$ have the largest [[Definition:Size of Graph|size]] of all [[Definition:Simple Graph|simple graphs]] of [[Definition:Order of Graph|order]] $n$.
Then:
:$G$ is the [[Definition:Complete Graph|c... | By definition, $K_n$ is the [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$ such that every [[Definition:Vertex of Graph|vertex]] of $K_n$ is [[Definition:Adjacent Vertices (Undirected Graph)|adjacent]] to all other [[Definition:Vertex of Graph|vertices]].
So, let $G$ have the large... | Simple Graph of Maximum Size is Complete Graph | https://proofwiki.org/wiki/Simple_Graph_of_Maximum_Size_is_Complete_Graph | https://proofwiki.org/wiki/Simple_Graph_of_Maximum_Size_is_Complete_Graph | [
"Complete Graphs",
"Simple Graphs",
"Simple Graph of Maximum Size is Complete Graph"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Size",
"Definition:Simple Graph",
"Definition:Graph (... |
proofwiki-17110 | Simple Graph where All Vertices and All Edges are Adjacent | Let $G$ be a simple graph in which:
:every vertex is adjacent to every other vertex
and:
:every edge is adjacent to every other edge.
Then $G$ is of order no greater than $3$. | It is seen that examples exist of simple graphs which fulfil the criteria where the order of $G$ is no greater than $3$:
:400px
The cases where the order of $G$ is $1$ or $2$ are trivial.
When the order of $G$ is $3$, the criteria can be verified by inspection.
Let the order of $G = \struct {V, E}$ be $4$ or more.
Let ... | Let $G$ be a [[Definition:Simple Graph|simple graph]] in which:
:every [[Definition:Vertex of Graph|vertex]] is [[Definition:Adjacent Vertices (Undirected Graph)|adjacent]] to every other [[Definition:Vertex of Graph|vertex]]
and:
:every [[Definition:Edge of Graph|edge]] is [[Definition:Adjacent Edges (Undirected Graph... | It is seen that examples exist of [[Definition:Simple Graph|simple graphs]] which fulfil the criteria where the [[Definition:Order of Graph|order]] of $G$ is no greater than $3$:
:[[File:All-Vertices-and-Edges-Adjacent.png|400px]]
The cases where the [[Definition:Order of Graph|order]] of $G$ is $1$ or $2$ are [[Defi... | Simple Graph where All Vertices and All Edges are Adjacent | https://proofwiki.org/wiki/Simple_Graph_where_All_Vertices_and_All_Edges_are_Adjacent | https://proofwiki.org/wiki/Simple_Graph_where_All_Vertices_and_All_Edges_are_Adjacent | [
"Simple Graphs"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Adjacent (Graph Theory)/Edges/Undirected Graph",
"Definition:Graph (Graph The... | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"File:All-Vertices-and-Edges-Adjacent.png",
"Definition:Graph (Graph Theory)/Order",
"Definition:Trivial",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Vertex",
"... |
proofwiki-17111 | Simple Graph whose Vertices all Incident but Edges not Adjacent | Let $G = \struct {V, E}$ be a simple graph such that:
:every vertex is incident with at least one edge
:no two edges are adjacent to each other.
Then $G$ has an even number of vertices. | Suppose there exists a set of $3$ vertices that are connected.
Then at least one of these vertices has at least $2$ edges.
That would mean that at least $2$ edges were incident with the same vertex.
That is, that at least $2$ edges were adjacent to each other.
So, for a simple graph to fulfil the conditions, vertices c... | Let $G = \struct {V, E}$ be a [[Definition:Simple Graph|simple graph]] such that:
:every [[Definition:Vertex of Graph|vertex]] is [[Definition:Incident (Undirected Graph)|incident]] with at least one [[Definition:Edge of Graph|edge]]
:no two [[Definition:Edge of Graph|edges]] are [[Definition:Adjacent Edges (Undirect... | Suppose there exists a [[Definition:Set|set]] of $3$ [[Definition:Vertex of Graph|vertices]] that are [[Definition:Connected Vertices|connected]].
Then at least one of these [[Definition:Vertex of Graph|vertices]] has at least $2$ [[Definition:Edge of Graph|edges]].
That would mean that at least $2$ [[Definition:Edge... | Simple Graph whose Vertices all Incident but Edges not Adjacent | https://proofwiki.org/wiki/Simple_Graph_whose_Vertices_all_Incident_but_Edges_not_Adjacent | https://proofwiki.org/wiki/Simple_Graph_whose_Vertices_all_Incident_but_Edges_not_Adjacent | [
"Simple Graphs"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Adjacent (Graph Theory)/Edges/Undirected Graph",
"Definition:Even Integer",
"Definitio... | [
"Definition:Set",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Connected (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph... |
proofwiki-17112 | Size of Star Graph | Let $S_n$ denote the star graph of order $n$ where $n > 0$.
The size of $S_n$ is given by:
:$\size {S_n} = n - 1$ | By definition of star graph, one vertex is adjacent to all the $n - 1$ other vertices.
So the size of $S_n$ is at least $n - 1$.
All the other vertices are of degree $1$, and the edges they are incident with are the ones joining them to the distinguished vertex.
Those edges have already been counted.
So the size of $S_... | Let $S_n$ denote the [[Definition:Star Graph|star graph]] of [[Definition:Order of Graph|order]] $n$ where $n > 0$.
The [[Definition:Size of Graph|size]] of $S_n$ is given by:
:$\size {S_n} = n - 1$ | By definition of [[Definition:Star Graph|star graph]], one [[Definition:Vertex of Graph|vertex]] is [[Definition:Adjacent Vertices (Undirected Graph)|adjacent]] to all the $n - 1$ other [[Definition:Vertex of Graph|vertices]].
So the [[Definition:Size of Graph|size]] of $S_n$ is at least $n - 1$.
All the other [[Def... | Size of Star Graph | https://proofwiki.org/wiki/Size_of_Star_Graph | https://proofwiki.org/wiki/Size_of_Star_Graph | [
"Star Graphs",
"Star Graphs"
] | [
"Definition:Star Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Star Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Size",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Grap... |
proofwiki-17113 | Smallest Simple Graph with One Vertex Adjacent to All Others | Let $G = \struct {V, E}$ be a simple graph of order $n$.
Let $G$ be the simple graph with the smallest size such that one vertex is adjacent to all other vertices of $G$.
Then $G$ is the star graph of order $n$ and is of size $n - 1$. | In order for the one vertex in question to be adjacent to all the others, it needs to be incident to $n - 1$ edges.
This is the smallest number of edges required.
Hence $G$ is a star graph.
From Size of Star Graph, the size of $G$ is thus $n - 1$.
{{qed}} | Let $G = \struct {V, E}$ be a [[Definition:Simple Graph|simple graph]] of [[Definition:Order of Graph|order]] $n$.
Let $G$ be the [[Definition:Simple Graph|simple graph]] with the smallest [[Definition:Size of Graph|size]] such that one [[Definition:Vertex of Graph|vertex]] is [[Definition:Adjacent Vertices (Undirecte... | In order for the one [[Definition:Vertex of Graph|vertex]] in question to be [[Definition:Adjacent Vertices (Undirected Graph)|adjacent]] to all the others, it needs to be [[Definition:Incident (Undirected Graphs)|incident]] to $n - 1$ [[Definition:Edge of Graph|edges]].
This is the smallest number of [[Definition:Edg... | Smallest Simple Graph with One Vertex Adjacent to All Others | https://proofwiki.org/wiki/Smallest_Simple_Graph_with_One_Vertex_Adjacent_to_All_Others | https://proofwiki.org/wiki/Smallest_Simple_Graph_with_One_Vertex_Adjacent_to_All_Others | [
"Graph Theory"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Size",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Star Gr... | [
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Incident (Undirected Graphs)",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Star Graph",
"Size of Star Graph",
"Definition:Graph (Graph T... |
proofwiki-17114 | Simple Graph whose Vertices Incident to All Edges | Let $G = \struct {V, E}$ be a simple graph whose vertices are incident to all its edges.
Then $G$ is either:
:the star graph $S_2$, which is also the complete graph $K_2$
:an edgeless graph of any order. | If $G$ has no edges, then all the vertices are incident to all the edges vacuously.
So any of the edgeless graphs $N_n$ for order $n \in \Z_{\ge 0}$ fulfils the criterion.
Suppose $G$ has more than $2$ vertices $v_1, v_2, v_3$ and at least one edge.
{{WLOG}}, let one edge be $v_1 v_2$.
But $v_3$ cannot be incident to e... | Let $G = \struct {V, E}$ be a [[Definition:Simple Graph|simple graph]] whose [[Definition:Vertex of Graph|vertices]] are [[Definition:Incident (Undirected Graph)|incident]] to all its [[Definition:Edge of Graph|edges]].
Then $G$ is either:
:the [[Definition:Star Graph|star graph]] $S_2$, which is also the [[Definition... | If $G$ has no [[Definition:Edge of Graph|edges]], then all the [[Definition:Vertex of Graph|vertices]] are [[Definition:Incident (Undirected Graph)|incident]] to all the [[Definition:Edge of Graph|edges]] [[Definition:Vacuous Truth|vacuously]].
So any of the [[Definition:Edgeless Graph|edgeless graphs]] $N_n$ for [[De... | Simple Graph whose Vertices Incident to All Edges | https://proofwiki.org/wiki/Simple_Graph_whose_Vertices_Incident_to_All_Edges | https://proofwiki.org/wiki/Simple_Graph_whose_Vertices_Incident_to_All_Edges | [
"Graph Theory"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Star Graph",
"Definition:Complete Graph",
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Order"
] | [
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Vacuous Truth",
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Th... |
proofwiki-17115 | Heine-Borel Theorem/Normed Vector Space | Let $\struct {X, \norm {\,\cdot\,} }$ be a finite-dimensional normed vector space.
A subset $K \subseteq X$ is compact {{iff}} $K$ is closed and bounded. | === Necessary Condition ===
{{:Heine-Borel Theorem/Normed Vector Space/Necessary Condition}}{{qed|lemma}} | Let $\struct {X, \norm {\,\cdot\,} }$ be a [[Definition:Finite Dimensional Vector Space|finite-dimensional]] [[Definition:Normed Vector Space|normed vector space]].
A [[Definition:Subset|subset]] $K \subseteq X$ is [[Definition:Compact Subset of Normed Vector Space|compact]] {{iff}} $K$ is [[Definition:Closed Set in ... | === [[Heine-Borel Theorem/Normed Vector Space/Necessary Condition|Necessary Condition]] ===
{{:Heine-Borel Theorem/Normed Vector Space/Necessary Condition}}{{qed|lemma}} | Heine-Borel Theorem/Normed Vector Space | https://proofwiki.org/wiki/Heine-Borel_Theorem/Normed_Vector_Space | https://proofwiki.org/wiki/Heine-Borel_Theorem/Normed_Vector_Space | [
"Closed and Bounded Subset of Normed Vector Space is not necessarily Compact",
"Heine-Borel Theorem",
"Compact Normed Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Dimension of Vector Space/Finite",
"Definition:Normed Vector Space",
"Definition:Subset",
"Definition:Compact Space/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Bounded Subset of Normed Vector Space"
] | [
"Heine-Borel Theorem/Normed Vector Space/Necessary Condition"
] |
proofwiki-17116 | Characteristics of Cycle Graph | Let $G = \struct {V, E}$ be an (undirected) graph whose order is greater than $2$.
Then $G$ is a cycle graph {{iff}}:
:$G$ is connected
:every vertex of $G$ is adjacent to $2$ other vertices
:every edge of $G$ is adjacent to $2$ other edges. | Recall that a cycle is a closed walk with the properties:
:all its edges are distinct
:all its vertices (except for the start and end) are distinct.
From Cycle Graph is Connected, $G$ is connected.
From Cycle Graph is 2-Regular, $G$ is $2$-regular.
Hence every vertex of $G$ is incident with $2$ edges.
{{finish|I hate t... | Let $G = \struct {V, E}$ be an [[Definition:Undirected Graph|(undirected) graph]] whose [[Definition:Order of Graph|order]] is greater than $2$.
Then $G$ is a [[Definition:Cycle Graph|cycle graph]] {{iff}}:
:$G$ is [[Definition:Connected Graph|connected]]
:every [[Definition:Vertex of Graph|vertex]] of $G$ is [[Defini... | Recall that a [[Definition:Cycle (Graph Theory)|cycle]] is a [[Definition:Closed Walk|closed walk]] with the properties:
:all its [[Definition:Edge of Graph|edges]] are [[Definition:Distinct Elements|distinct]]
:all its [[Definition:Vertex of Graph|vertices]] (except for the start and end) are [[Definition:Distinct El... | Characteristics of Cycle Graph | https://proofwiki.org/wiki/Characteristics_of_Cycle_Graph | https://proofwiki.org/wiki/Characteristics_of_Cycle_Graph | [
"Cycle Graphs"
] | [
"Definition:Undirected Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Cycle Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition... | [
"Definition:Cycle (Graph Theory)",
"Definition:Walk (Graph Theory)/Closed",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Distinct/Plural",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Distinct/Plural",
"Cycle Graph is Connected",
"Definition:Connected (Graph Theory)/Graph",
"Cycle Gr... |
proofwiki-17117 | Maximum Number of Arcs in Digraph | Let $D_n$ be a digraph of order $n$ such that $n \ge 1$.
Let $D_n$ have the greatest number of arcs of all digraphs of order $n$.
The number of arcs in $D$ is given by:
:$\size {D_n} = n \paren {n - 1}$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\size {D_n} = n \paren {n - 1}$ | Let $D_n$ be a [[Definition:Digraph|digraph]] of [[Definition:Order of Graph|order]] $n$ such that $n \ge 1$.
Let $D_n$ have the greatest number of [[Definition:Arc of Digraph|arcs]] of all [[Definition:Digraph|digraphs]] of [[Definition:Order of Graph|order]] $n$.
The number of [[Definition:Arc of Digraph|arcs]] in... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\size {D_n} = n \paren {n - 1}$ | Maximum Number of Arcs in Digraph | https://proofwiki.org/wiki/Maximum_Number_of_Arcs_in_Digraph | https://proofwiki.org/wiki/Maximum_Number_of_Arcs_in_Digraph | [
"Digraphs",
"Maximum Number of Arcs in Digraph",
"Proofs by Induction"
] | [
"Definition:Digraph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Digraph/Arc",
"Definition:Digraph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Digraph/Arc"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-17118 | P-adic Norm of p-adic Number is Power of p/Lemma | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.
Let $\sequence {x_n}$ be a Cauchy sequence in $\struct{\Q, \norm {\,\cdot\,}_p}$ such that $\sequence {x_n}$ does not converge to $0$.
Then:
:$\exists v \in \Z: \ds \lim_{n \mathop \to \infty} \norm{x_n}_p = p^{-v}$ | From P-adic Norm on Rational Numbers is Non-Archimedean Norm, $p$-adic norm is a non-Archimedean norm on the rationals $\Q$.
Since $\sequence {x_n}$ does not converge to $0$, from Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary:
:$\exists N \in \N: \forall n, m > N: \norm {x_n}_p = \norm {x_m}... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime]] $p$.
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence (Normed Division Ring)|Cauchy sequence]] in $\struct{\Q, \norm {\,\cdot\,}_p}$ such tha... | From [[P-adic Norm on Rational Numbers is Non-Archimedean Norm]], [[Definition:P-adic Norm|$p$-adic norm]] is a [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] on the [[Definition:Rational Numbers|rationals $\Q$]].
Since $\sequence {x_n}$ does not [[Definition:Convergent Sequence in Normed Divi... | P-adic Norm of p-adic Number is Power of p/Lemma | https://proofwiki.org/wiki/P-adic_Norm_of_p-adic_Number_is_Power_of_p/Lemma | https://proofwiki.org/wiki/P-adic_Norm_of_p-adic_Number_is_Power_of_p/Lemma | [
"P-adic Norm of p-adic Number is Power of p"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"P-adic Norm forms Non-Archimedean Valued Field/Rational Numbers",
"Definition:P-adic Norm",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Rational Number",
"Definition:Convergent Sequence/Normed Division Ring",
"Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary",
... |
proofwiki-17119 | Loop-Digraph as a Relation | A loop-digraph is the same thing as a relational structure. | A loop-digraph is a graph such that:
:it is not necessarily antireflexive
:it is not necessarily symmetric.
Hence a loop-digraph is an ordered pair:
:$G = \struct {V, E}$
such that:
:$V$ is a set of objects
:$E$ is a relation on $V$.
This defines a relational structure.
{{qed}} | A [[Definition:Loop-Digraph|loop-digraph]] is the same thing as a [[Definition:Relational Structure|relational structure]]. | A [[Definition:Loop-Digraph|loop-digraph]] is a [[Definition:Graph (Graph Theory)|graph]] such that:
:it is not necessarily [[Definition:Antireflexive Relation|antireflexive]]
:it is not necessarily [[Definition:Symmetric Relation|symmetric]].
Hence a [[Definition:Loop-Digraph|loop-digraph]] is an [[Definition:Order... | Loop-Digraph as a Relation | https://proofwiki.org/wiki/Loop-Digraph_as_a_Relation | https://proofwiki.org/wiki/Loop-Digraph_as_a_Relation | [
"Loop-Digraphs",
"Relation Theory"
] | [
"Definition:Loop-Graph/Loop-Digraph",
"Definition:Relational Structure"
] | [
"Definition:Loop-Graph/Loop-Digraph",
"Definition:Graph (Graph Theory)",
"Definition:Antireflexive Relation",
"Definition:Symmetric Relation",
"Definition:Loop-Graph/Loop-Digraph",
"Definition:Ordered Pair",
"Definition:Set",
"Definition:Object",
"Definition:Endorelation",
"Definition:Relational S... |
proofwiki-17120 | Complete Graph is Hamiltonian for Order Greater than 2 | Let $n \in \Z$ be an integer such that $n > 2$.
Let $K_n$ denote the complete graph of order $n$.
Then $K_n$ is Hamiltonian. | First we note that when $n = 2$ there is one edge in $K_n$.
So if you start at one vertex $u$ and travel along that edge to the other vertex $v$, you cannot return to $u$ except by using that same edge.
Consequently $K_2$ is not Hamiltonian.
{{qed|lemma}}
Let $n = 4$.
Let us take two vertices $u, v$ of $K_n$:
From Comp... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n > 2$.
Let $K_n$ denote the [[Definition:Complete Graph|complete graph]] of [[Definition:Order of Graph|order]] $n$.
Then $K_n$ is [[Definition:Hamiltonian Graph|Hamiltonian]]. | First we note that when $n = 2$ there is one [[Definition:Edge of Graph|edge]] in $K_n$.
So if you start at one [[Definition:Vertex of Graph|vertex]] $u$ and travel along that [[Definition:Edge of Graph|edge]] to the other [[Definition:Vertex of Graph|vertex]] $v$, you cannot return to $u$ except by using that same [[... | Complete Graph is Hamiltonian for Order Greater than 2 | https://proofwiki.org/wiki/Complete_Graph_is_Hamiltonian_for_Order_Greater_than_2 | https://proofwiki.org/wiki/Complete_Graph_is_Hamiltonian_for_Order_Greater_than_2 | [
"Complete Graphs",
"Hamiltonian Graphs"
] | [
"Definition:Integer",
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Hamiltonian Graph"
] | [
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Hamiltonian Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Complete Graph is Regul... |
proofwiki-17121 | Cycle Graph is Hamiltonian | Let $n \in \Z$ be an integer such that $n \ge 3$.
Let $C_n$ denote the cycle graph of order $n$.
Then $C_n$ is Hamiltonian. | Recall the definition of a Hamiltonian graph:
:$G$ is Hamiltonian {{iff}} it is an undirected graph that contains a cycle that contains every vertex of $G$ (but not necessarily every edge of $G$).
A cycle graph is a graph which consists of a single cycle.
So all the vertices of $C_n$ are on the same cycle.
The result f... | Let $n \in \Z$ be an [[Definition:Integer|integer]] such that $n \ge 3$.
Let $C_n$ denote the [[Definition:Cycle Graph|cycle graph]] of [[Definition:Order of Graph|order]] $n$.
Then $C_n$ is [[Definition:Hamiltonian Graph|Hamiltonian]]. | Recall the definition of a [[Definition:Hamiltonian Graph|Hamiltonian graph]]:
:$G$ is [[Definition:Hamiltonian Graph|Hamiltonian]] {{iff}} it is an [[Definition:Undirected Graph|undirected graph]] that contains a [[Definition:Cycle (Graph Theory)|cycle]] that contains every [[Definition:Vertex of Graph|vertex]] of $G... | Cycle Graph is Hamiltonian | https://proofwiki.org/wiki/Cycle_Graph_is_Hamiltonian | https://proofwiki.org/wiki/Cycle_Graph_is_Hamiltonian | [
"Cycle Graphs",
"Hamiltonian Graphs"
] | [
"Definition:Integer",
"Definition:Cycle Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Hamiltonian Graph"
] | [
"Definition:Hamiltonian Graph",
"Definition:Hamiltonian Graph",
"Definition:Undirected Graph",
"Definition:Cycle (Graph Theory)",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Cycle Graph",
"Definition:Graph (Graph Theory)",
"Definition:Cycle (Graph Th... |
proofwiki-17122 | Maximum Degree of Vertex in Simple Graph | Let $G = \struct {V, E}$ be a simple graph.
Let $\card V$ denote the order of $G$.
Then no vertex of $G$ has a degree higher than $\card V - 1$. | By definition, a simple graph has no loops or multiple edges.
So a vertex can be incident to only as many edges that will join it to all the other vertices once each.
There are $\card V - 1$ other vertices.
Hence the result.
{{qed}} | Let $G = \struct {V, E}$ be a [[Definition:Simple Graph|simple graph]].
Let $\card V$ denote the [[Definition:Order of Graph|order]] of $G$.
Then no [[Definition:Vertex of Graph|vertex]] of $G$ has a [[Definition:Degree of Vertex|degree]] higher than $\card V - 1$. | By definition, a [[Definition:Simple Graph|simple graph]] has no [[Definition:Loop (Graph Theory)|loops]] or [[Definition:Multiple Edge|multiple edges]].
So a [[Definition:Vertex of Graph|vertex]] can be [[Definition:Incident (Undirected Graph)|incident]] to only as many [[Definition:Edge of Graph|edges]] that will [[... | Maximum Degree of Vertex in Simple Graph | https://proofwiki.org/wiki/Maximum_Degree_of_Vertex_in_Simple_Graph | https://proofwiki.org/wiki/Maximum_Degree_of_Vertex_in_Simple_Graph | [
"Vertices of Graphs",
"Degrees of Vertices",
"Maximum Degree of Vertex in Simple Graph"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex"
] | [
"Definition:Simple Graph",
"Definition:Loop (Graph Theory)",
"Definition:Multigraph/Multiple Edge",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Incident (Graph Theory)/Undirected Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Edge/Join",
"Definition:Graph (G... |
proofwiki-17123 | Degrees of Vertices determine Order and Size of Graph | Let $G = \struct {V, E}$ be a simple graph.
Let the degrees of each of the vertices of $G$ be given.
Then it is possible to determine both the order $\card V$ and size $\card E$ of $G$. | Suppose we are given the degrees of each of the vertices of $G$.
By definition, $\card V$ is simply the count of the vertices.
So if we have been given the degrees of each of the vertices, we must know how many vertices there are.
Then from the Handshake Lemma:
:$\ds \sum_{v \mathop \in V} \map \deg v = 2 \card E$
That... | Let $G = \struct {V, E}$ be a [[Definition:Simple Graph|simple graph]].
Let the [[Definition:Degree of Vertex|degrees]] of each of the [[Definition:Vertex of Graph|vertices]] of $G$ be given.
Then it is possible to determine both the [[Definition:Order of Graph|order]] $\card V$ and [[Definition:Size of Graph|size]]... | Suppose we are given the [[Definition:Degree of Vertex|degrees]] of each of the [[Definition:Vertex of Graph|vertices]] of $G$.
By definition, $\card V$ is simply the count of the [[Definition:Vertex of Graph|vertices]].
So if we have been given the [[Definition:Degree of Vertex|degrees]] of each of the [[Definition:... | Degrees of Vertices determine Order and Size of Graph | https://proofwiki.org/wiki/Degrees_of_Vertices_determine_Order_and_Size_of_Graph | https://proofwiki.org/wiki/Degrees_of_Vertices_determine_Order_and_Size_of_Graph | [
"Degrees of Vertices"
] | [
"Definition:Simple Graph",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size"
] | [
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Handshake Lemma",
"Definition:Addition/Integers",
"Definition:Degree ... |
proofwiki-17124 | Order and Size of Graph do not determine Degrees of Vertices | Let $G = \struct {V, E}$ be a simple graph.
Let both the order $\card V$ and size $\card E$ of $G$ be given.
Then it is not always possible to determine the degrees of each of the vertices of $G$. | The following $2$ simple graphs both have order $4$ and size $4$:
:400px
But:
:the graph on the left has vertices with degrees $1, 2, 2, 3$
:the graph on the right has vertices with degrees $2, 2, 2, 2$.
{{qed}} | Let $G = \struct {V, E}$ be a [[Definition:Simple Graph|simple graph]].
Let both the [[Definition:Order of Graph|order]] $\card V$ and [[Definition:Size of Graph|size]] $\card E$ of $G$ be given.
Then it is not always possible to determine the [[Definition:Degree of Vertex|degrees]] of each of the [[Definition:Verte... | The following $2$ [[Definition:Simple Graph|simple graphs]] both have [[Definition:Order of Graph|order]] $4$ and [[Definition:Size of Graph|size]] $4$:
:[[File:Chartrand-exercise-2-1-6.png|400px]]
But:
:the [[Definition:Simple Graph|graph]] on the left has [[Definition:Vertex of Graph|vertices]] with [[Definition:De... | Order and Size of Graph do not determine Degrees of Vertices | https://proofwiki.org/wiki/Order_and_Size_of_Graph_do_not_determine_Degrees_of_Vertices | https://proofwiki.org/wiki/Order_and_Size_of_Graph_do_not_determine_Degrees_of_Vertices | [
"Degrees of Vertices"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"File:Chartrand-exercise-2-1-6.png",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Simple Graph",
"Definition:Graph (Graph The... |
proofwiki-17125 | Size of Regular Graph in terms of Degree and Order | Let $G = \struct {V, E}$ be an $r$-regular graph of order $p$.
Let $q$ denote the size of $G$.
Then:
:$q = \dfrac {p r} 2$
when such an $r$-regular graph exists.
If an $r$-regular graph of order $p$ does exist, then $p r$ is an even integer. | Let $G$ be of order $p$, size $q$ and $r$-regular.
Then by definition $G$ has:
:$p$ vertices each of which is of degree $r$.
:$q$ edges.
Thus:
{{begin-eqn}}
{{eqn | l = q
| r = \dfrac 1 2 \sum_{v \mathop \in V} \map \deg v
| c = Handshake Lemma
}}
{{eqn | r = \dfrac 1 2 \sum_{v \mathop \in V} r
| c = ... | Let $G = \struct {V, E}$ be an [[Definition:Regular Graph|$r$-regular graph]] of [[Definition:Order of Graph|order]] $p$.
Let $q$ denote the [[Definition:Size of Graph|size]] of $G$.
Then:
:$q = \dfrac {p r} 2$
when such an [[Definition:Regular Graph|$r$-regular graph]] exists.
If an [[Definition:Regular Graph|$r... | Let $G$ be of [[Definition:Order of Graph|order]] $p$, [[Definition:Size of Graph|size]] $q$ and [[Definition:Regular Graph|$r$-regular]].
Then by definition $G$ has:
:$p$ [[Definition:Vertex of Graph|vertices]] each of which is of [[Definition:Degree of Vertex|degree]] $r$.
:$q$ [[Definition:Edge of Graph|edges]].
... | Size of Regular Graph in terms of Degree and Order | https://proofwiki.org/wiki/Size_of_Regular_Graph_in_terms_of_Degree_and_Order | https://proofwiki.org/wiki/Size_of_Regular_Graph_in_terms_of_Degree_and_Order | [
"Regular Graphs"
] | [
"Definition:Regular Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Regular Graph",
"Definition:Regular Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Even Integer"
] | [
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Regular Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Handshake Lemma",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Defi... |
proofwiki-17126 | Same Degrees of Vertices does not imply Graph Isomorphism | Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be graphs such that:
:$\card {\map V G} = \card {\map V H}$
where $\card {\map V G}$ denotes the order of $G$.
Let $\phi: G \to H$ be a mapping which preserves the degrees of the vertices:
:$\forall v \in \map V G: \map {\deg_H} {\map \phi v}... | Proof by Counterexample:
:400px
Consider a bijection $\phi: \map V {G_1} \to \map V {G_2}$, where $G_1$ is the graph on the left and $G_2$ is the graph on the right.
The vertices $v_1$, $v_2$ and $v_5$ of $G_2$ are each adjacent to both of the others.
Because $\phi$ is a bijection, it must map $3$ vertices of $G_1$ to ... | Let $G = \struct {\map V G, \map E G}$ and $H = \struct {\map V H, \map E H}$ be [[Definition:Graph (Graph Theory)|graphs]] such that:
:$\card {\map V G} = \card {\map V H}$
where $\card {\map V G}$ denotes the [[Definition:Order of Graph|order]] of $G$.
Let $\phi: G \to H$ be a [[Definition:Mapping|mapping]] which p... | [[Proof by Counterexample]]:
:[[File:Same-degree-non-isomorphic-graphs.png|400px]]
Consider a [[Definition:Bijection|bijection]] $\phi: \map V {G_1} \to \map V {G_2}$, where $G_1$ is the [[Definition:Graph (Graph Theory)|graph]] on the left and $G_2$ is the [[Definition:Graph (Graph Theory)|graph]] on the right.
The... | Same Degrees of Vertices does not imply Graph Isomorphism | https://proofwiki.org/wiki/Same_Degrees_of_Vertices_does_not_imply_Graph_Isomorphism | https://proofwiki.org/wiki/Same_Degrees_of_Vertices_does_not_imply_Graph_Isomorphism | [
"Degrees of Vertices",
"Graph Isomorphisms"
] | [
"Definition:Graph (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Mapping",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Isomorphism (Graph Theory)"
] | [
"Proof by Counterexample",
"File:Same-degree-non-isomorphic-graphs.png",
"Definition:Bijection",
"Definition:Graph (Graph Theory)",
"Definition:Graph (Graph Theory)",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Bijection",
"Definition:Graph (G... |
proofwiki-17127 | Order 1 Simple Graph is Unique up to Isomorphism | Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be simple graphs of order $1$.
Then $G_1$ and $G_2$ are isomorphic. | There is only one bijection from $\map V {G_1}$ to $\map V {G_2}$.
There are no vertices adjacent to the sole vertex in $\map V {G_1}$
There are no vertices adjacent to the sole vertex in $\map V {G_2}$.
Hence the result, vacuously.
{{qed}} | Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be [[Definition:Simple Graph|simple graphs]] of [[Definition:Order of Graph|order]] $1$.
Then $G_1$ and $G_2$ are [[Definition:Graph Isomorphism|isomorphic]]. | There is only one [[Definition:Bijection|bijection]] from $\map V {G_1}$ to $\map V {G_2}$.
There are no [[Definition:Vertex of Graph|vertices]] [[Definition:Adjacent Vertices of Graph|adjacent]] to the sole [[Definition:Vertex of Graph|vertex]] in $\map V {G_1}$
There are no [[Definition:Vertex of Graph|vertices]] [... | Order 1 Simple Graph is Unique up to Isomorphism | https://proofwiki.org/wiki/Order_1_Simple_Graph_is_Unique_up_to_Isomorphism | https://proofwiki.org/wiki/Order_1_Simple_Graph_is_Unique_up_to_Isomorphism | [
"Graph Isomorphisms"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Isomorphism (Graph Theory)"
] | [
"Definition:Bijection",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Vacu... |
proofwiki-17128 | Edgeless Graphs of Order n are Isomorphic | Let $n \in \Z_{>0}$ be a positive integer.
Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be edgeless graphs of order $n$.
Then $G_1$ and $G_2$ are isomorphic. | Let $\phi$ be a bijection from $\map V {G_1}$ to $\map V {G_2}$.
This is possible because $\card {\map V {G_1} } = \card {\map V {G_2} } = n$.
Let $v_a \in \map V {G_1}$.
By definition, $v_a$ is adjacent to no other vertices of $G_1$.
Similarly by definition, $\map \phi {v_a}$ is adjacent to no other vertices of $G_2$.... | Let $n \in \Z_{>0}$ be a [[Definition:Positive Integer|positive integer]].
Let $G_1 = \struct {\map V {G_1}, \map E {G_1} }$ and $G_2 = \struct {\map V {G_2}, \map E {G_2} }$ be [[Definition:Edgeless Graph|edgeless graphs]] of [[Definition:Order of Graph|order]] $n$.
Then $G_1$ and $G_2$ are [[Definition:Graph Isomor... | Let $\phi$ be a [[Definition:Bijection|bijection]] from $\map V {G_1}$ to $\map V {G_2}$.
This is possible because $\card {\map V {G_1} } = \card {\map V {G_2} } = n$.
Let $v_a \in \map V {G_1}$.
By definition, $v_a$ is [[Definition:Adjacent Vertices of Graph|adjacent]] to no other [[Definition:Vertex of Graph|verti... | Edgeless Graphs of Order n are Isomorphic | https://proofwiki.org/wiki/Edgeless_Graphs_of_Order_n_are_Isomorphic | https://proofwiki.org/wiki/Edgeless_Graphs_of_Order_n_are_Isomorphic | [
"Graph Isomorphisms"
] | [
"Definition:Positive/Integer",
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Isomorphism (Graph Theory)"
] | [
"Definition:Bijection",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Vacuous Truth",
"Definition:Adjacent (Graph Theory)/Vertices",
"Definition:Adjacent (Gra... |
proofwiki-17129 | Isomorphism Classes for Order 2 Simple Graphs | There are $2$ equivalence classes for simple graphs of order $2$ under graph isomorphism:
:the edgeless graph of order $2$
and
:the complete graph of order $2$. | {{ProofWanted|Too tedious to contemplate.}} | There are $2$ [[Definition:Equivalence Class|equivalence classes]] for [[Definition:Simple Graph|simple graphs]] of [[Definition:Order of Graph|order]] $2$ under [[Definition:Graph Isomorphism|graph isomorphism]]:
:the [[Definition:Edgeless Graph|edgeless graph]] of [[Definition:Order of Graph|order]] $2$
and
:the [[D... | {{ProofWanted|Too tedious to contemplate.}} | Isomorphism Classes for Order 2 Simple Graphs | https://proofwiki.org/wiki/Isomorphism_Classes_for_Order_2_Simple_Graphs | https://proofwiki.org/wiki/Isomorphism_Classes_for_Order_2_Simple_Graphs | [
"Graph Isomorphisms"
] | [
"Definition:Equivalence Class",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Isomorphism (Graph Theory)",
"Definition:Edgeless Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Order"
] | [] |
proofwiki-17130 | Isomorphism Classes for Order 4 Size 3 Simple Graphs | There are $3$ equivalence classes for simple graphs of order $4$ and size $3$ under isomorphism:
:400px | {{tidy}}
{{MissingLinks}}
The fact that the $3$ graphs given are not isomorphic follows from Vertex Condition for Isomorphic Graphs.
The vertices have degrees as follows:
:Graph $1$: $2, 2, 1, 1$
:Graph $2$: $3, 1, 1, 1$
:Graph $3$: $2, 2, 2, 0$
The fact that there are no more isomorphism classes of such graphs can be ... | There are $3$ [[Definition:Equivalence Class|equivalence classes]] for [[Definition:Simple Graph|simple graphs]] of [[Definition:Order of Graph|order]] $4$ and [[Definition:Size of Graph|size]] $3$ under [[Definition:Graph Isomorphism|isomorphism]]:
:[[File:Isomorphism-Classes-Order4-Size3-Simple.png|400px]] | {{tidy}}
{{MissingLinks}}
The fact that the $3$ [[Definition:Simple Graph|graphs]] given are not [[Definition:Graph Isomorphism|isomorphic]] follows from [[Vertex Condition for Isomorphic Graphs]].
The [[Definition:Vertex of Graph|vertices]] have [[Definition:Degree of Vertex|degrees]] as follows:
:Graph $1$: $2, 2, ... | Isomorphism Classes for Order 4 Size 3 Simple Graphs | https://proofwiki.org/wiki/Isomorphism_Classes_for_Order_4_Size_3_Simple_Graphs | https://proofwiki.org/wiki/Isomorphism_Classes_for_Order_4_Size_3_Simple_Graphs | [
"Graph Isomorphisms"
] | [
"Definition:Equivalence Class",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Order",
"Definition:Graph (Graph Theory)/Size",
"Definition:Isomorphism (Graph Theory)",
"File:Isomorphism-Classes-Order4-Size3-Simple.png"
] | [
"Definition:Simple Graph",
"Definition:Isomorphism (Graph Theory)",
"Vertex Condition for Isomorphic Graphs",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Proof by Cases"
] |
proofwiki-17131 | Differential Equation governing First-Order Reaction | Let a substance decompose spontaneously in a '''first-order reaction'''.
The differential equation which governs this reaction is given by:
:$-\dfrac {\d x} {\d t} = k x$
where:
:$x$ determines the quantity of substance at time $t$.
:$k \in \R_{>0}$. | From the definition of a first-order reaction, the rate of change of the quantity of the substance is proportional to the quantity of the substance present at any time.
As the rate of change is a decrease, this rate will be negative.
Thus the differential equation governing this reaction is given by:
:$-\dfrac {\d x} {... | Let a [[Definition:Substance|substance]] decompose spontaneously in a '''[[Definition:First-Order Reaction|first-order reaction]]'''.
The [[Definition:Differential Equation|differential equation]] which governs this reaction is given by:
:$-\dfrac {\d x} {\d t} = k x$
where:
:$x$ determines the quantity of [[Definiti... | From the definition of a [[Definition:First-Order Reaction|first-order reaction]], the rate of change of the quantity of the [[Definition:Substance|substance]] is [[Definition:Direct Proportion|proportional]] to the quantity of the [[Definition:Substance|substance]] present at any time.
As the rate of change is a decr... | Differential Equation governing First-Order Reaction | https://proofwiki.org/wiki/Differential_Equation_governing_First-Order_Reaction | https://proofwiki.org/wiki/Differential_Equation_governing_First-Order_Reaction | [
"First-Order Reactions"
] | [
"Definition:Substance",
"Definition:First-Order Reaction",
"Definition:Differential Equation",
"Definition:Substance",
"Definition:Time"
] | [
"Definition:First-Order Reaction",
"Definition:Substance",
"Definition:Proportion",
"Definition:Substance",
"Definition:Differential Equation"
] |
proofwiki-17132 | Formula for Radiocarbon Dating | Let $Q$ be a quantity of a sample of dead organic material (usually wood) whose time of death is to be determined.
Let $t$ years be the age of $Q$ which is to be determined.
Let $r$ denote the ratio of the quantity of carbon-14 remaining in $Q$ after time $t$ to the quantity of carbon-14 in $Q$ at the time of its death... | Let $x_0$ denote the ratio of carbon-14 to carbon-12 in $Q$ at the time of its death.
Let $x$ denote the ratio of carbon-14 to carbon-12 in $Q$ after time $t$.
Thus:
:$r = \dfrac x {x_0}$
It is assumed that the rate of decay of carbon-14 is a first-order reaction.
Hence we use:
{{begin-eqn}}
{{eqn | l = x
| r = x... | Let $Q$ be a quantity of a sample of dead organic material (usually wood) whose time of death is to be determined.
Let $t$ [[Definition:Year|years]] be the age of $Q$ which is to be determined.
Let $r$ denote the [[Definition:Ratio|ratio]] of the quantity of [[Definition:Carbon/14|carbon-14]] remaining in $Q$ after ... | Let $x_0$ denote the [[Definition:Ratio|ratio]] of [[Definition:Carbon/14|carbon-14]] to [[Definition:Carbon/12|carbon-12]] in $Q$ at the time of its death.
Let $x$ denote the [[Definition:Ratio|ratio]] of [[Definition:Carbon/14|carbon-14]] to [[Definition:Carbon/12|carbon-12]] in $Q$ after [[Definition:Time|time]] $t... | Formula for Radiocarbon Dating | https://proofwiki.org/wiki/Formula_for_Radiocarbon_Dating | https://proofwiki.org/wiki/Formula_for_Radiocarbon_Dating | [
"Radiometric Dating"
] | [
"Definition:Time/Unit/Year",
"Definition:Ratio",
"Definition:Carbon/14",
"Definition:Carbon/14",
"Definition:Time/Unit/Year"
] | [
"Definition:Ratio",
"Definition:Carbon/14",
"Definition:Carbon/12",
"Definition:Ratio",
"Definition:Carbon/14",
"Definition:Carbon/12",
"Definition:Time",
"Definition:Carbon/14",
"Definition:First-Order Reaction",
"First-Order Reaction",
"Definition:Time/Unit/Year",
"Definition:Carbon/14",
"... |
proofwiki-17133 | Period of Reciprocal of Prime | Consider the decimal expansion of the reciprocal $\dfrac 1 p$ of a prime $p$.
If $p \nmid a$, the decimal expansion of $\dfrac 1 p$ is periodic in base $a$ and its period of recurrence is the order of $a$ modulo $p$.
If $p \divides a$, the decimal expansion of $\dfrac 1 p$ in base $a$ terminates. | === Case $1$: $p \divides a$ ===
Let $q = \dfrac a p$.
Then $\dfrac 1 p = \dfrac q a$.
So the decimal expansion of $\dfrac 1 p$ in base $a$ is $0.q$ and terminates.
{{qed|lemma}} | Consider the [[Definition:Decimal Expansion|decimal expansion]] of the [[Definition:Reciprocal|reciprocal]] $\dfrac 1 p$ of a [[Definition:Prime Number|prime]] $p$.
If $p \nmid a$, the [[Definition:Decimal Expansion|decimal expansion]] of $\dfrac 1 p$ is [[Definition:Period of Recurrence|periodic]] in base $a$ and its... | === Case $1$: $p \divides a$ ===
Let $q = \dfrac a p$.
Then $\dfrac 1 p = \dfrac q a$.
So the [[Definition:Decimal Expansion|decimal expansion]] of $\dfrac 1 p$ in base $a$ is $0.q$ and [[Definition:Termination of Basis Expansion|terminates]].
{{qed|lemma}} | Period of Reciprocal of Prime | https://proofwiki.org/wiki/Period_of_Reciprocal_of_Prime | https://proofwiki.org/wiki/Period_of_Reciprocal_of_Prime | [
"Reciprocals",
"Prime Numbers"
] | [
"Definition:Decimal Expansion",
"Definition:Reciprocal",
"Definition:Prime Number",
"Definition:Decimal Expansion",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Multiplicative Order of Integer",
"Definition:Decimal Expansion",
"Definitio... | [
"Definition:Decimal Expansion",
"Definition:Basis Expansion/Termination",
"Definition:Decimal Expansion"
] |
proofwiki-17134 | Maximum Period of Reciprocal of Prime | Let $p$ be a prime number such that $p$ is not a divisor of $10$.
The maximum period of recurrence of the reciprocal of $p$ when expressed in decimal notation is $p - 1$. | First we dispose of the case where $p$ ''is'' a divisor of $10$:
When $p \divides 10$, $\dfrac 1 p$ expressed in decimal notation is a terminating decimal:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2
| r = 0 \cdotp 5
}}
{{eqn | l = \dfrac 1 5
| r = 0 \cdotp 2
}}
{{end-eqn}}
{{qed|lemma}}
So, let $p$ be such that $... | Let $p$ be a [[Definition:Prime Number|prime number]] such that $p$ is not a [[Definition:Divisor of Integer|divisor]] of $10$.
The [[Definition:Maximum|maximum]] [[Definition:Period of Recurrence|period of recurrence]] of the [[Definition:Reciprocal|reciprocal]] of $p$ when expressed in [[Definition:Decimal Notation|... | First we dispose of the case where $p$ ''is'' a [[Definition:Divisor of Integer|divisor]] of $10$:
When $p \divides 10$, $\dfrac 1 p$ expressed in [[Definition:Decimal Notation|decimal notation]] is a [[Definition:Terminating Decimal|terminating decimal]]:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2
| r = 0 \cdotp 5
}... | Maximum Period of Reciprocal of Prime | https://proofwiki.org/wiki/Maximum_Period_of_Reciprocal_of_Prime | https://proofwiki.org/wiki/Maximum_Period_of_Reciprocal_of_Prime | [
"Reciprocals",
"Prime Numbers"
] | [
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Maximum Value of Real Function",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Reciprocal",
"Definition:Decimal Notation"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Decimal Notation",
"Definition:Terminating Decimal",
"Period of Reciprocal of Prime",
"Definition:Basis Expansion/Recurrence/Period",
"Definition:Multiplicative Order of Integer",
"Definition:Integer",
"Fermat's Little Theorem",
"Definition:Basis E... |
proofwiki-17135 | Maximum Abscissa for Loop of Folium of Descartes | Consider the folium of Descartes defined in parametric form as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
:500px
The point on the loop at which the $x$ value is at a maximum occurs when $t = \sqrt [3] {\dfrac 1 2}$, corresponding to the point $P$ defined as:
:$P = \tu... | We calculate the derivative of $x$ {{WRT|Differentiation}} $t$:
{{begin-eqn}}
{{eqn | l = \dfrac {\d x} {\d t}
| r = \map {\dfrac \d {\d t} } {\dfrac {3 a t} {1 + t^3} }
| c =
}}
{{eqn | r = \dfrac {\paren {1 + t^3} \times 3 a - 3 a t \paren {3 t^2} } {\paren {1 + t^3}^2}
| c = Quotient Rule for Deri... | Consider the [[Definition:Folium of Descartes|folium of Descartes]] defined in [[Definition:Folium of Descartes/Parametric Form|parametric form]] as:
:$\begin {cases} x = \dfrac {3 a t} {1 + t^3} \\ y = \dfrac {3 a t^2} {1 + t^3} \end {cases}$
:[[File:FoliumOfDescartes.png|500px]]
The [[Definition:Point|point]] on... | We calculate the [[Definition:Derivative|derivative]] of $x$ {{WRT|Differentiation}} $t$:
{{begin-eqn}}
{{eqn | l = \dfrac {\d x} {\d t}
| r = \map {\dfrac \d {\d t} } {\dfrac {3 a t} {1 + t^3} }
| c =
}}
{{eqn | r = \dfrac {\paren {1 + t^3} \times 3 a - 3 a t \paren {3 t^2} } {\paren {1 + t^3}^2}
|... | Maximum Abscissa for Loop of Folium of Descartes | https://proofwiki.org/wiki/Maximum_Abscissa_for_Loop_of_Folium_of_Descartes | https://proofwiki.org/wiki/Maximum_Abscissa_for_Loop_of_Folium_of_Descartes | [
"Folium of Descartes"
] | [
"Definition:Folium of Descartes",
"Definition:Folium of Descartes/Parametric Form",
"File:FoliumOfDescartes.png",
"Definition:Point"
] | [
"Definition:Derivative",
"Quotient Rule for Derivatives",
"Definition:Stationary Point",
"Behaviour of Parametric Equations for Folium of Descartes according to Parameter",
"Definition:Maximum Value of Real Function/Local",
"Category:Folium of Descartes"
] |
proofwiki-17136 | Third Derivative of Natural Logarithm Function | Let $\ln x$ be the natural logarithm function.
Then:
:$\map {D^3_x} {\ln x} = \dfrac 2 {x^3}$ | {{begin-eqn}}
{{eqn | l = \map {D^3_x} {\ln x}
| r = \map {D_x} {\map {D^2_x} {\ln x} }
| c =
}}
{{eqn | r = \map {D_x} {-\dfrac 1 {x^2} }
| c = Second Derivative of Natural Logarithm Function
}}
{{eqn | r = -\paren {-2 \paren {\dfrac 1 {x^3} } }
| c = Power Rule for Derivatives: Integer Index
... | Let $\ln x$ be the [[Definition:Natural Logarithm|natural logarithm function]].
Then:
:$\map {D^3_x} {\ln x} = \dfrac 2 {x^3}$ | {{begin-eqn}}
{{eqn | l = \map {D^3_x} {\ln x}
| r = \map {D_x} {\map {D^2_x} {\ln x} }
| c =
}}
{{eqn | r = \map {D_x} {-\dfrac 1 {x^2} }
| c = [[Second Derivative of Natural Logarithm Function]]
}}
{{eqn | r = -\paren {-2 \paren {\dfrac 1 {x^3} } }
| c = [[Power Rule for Derivatives/Integer I... | Third Derivative of Natural Logarithm Function | https://proofwiki.org/wiki/Third_Derivative_of_Natural_Logarithm_Function | https://proofwiki.org/wiki/Third_Derivative_of_Natural_Logarithm_Function | [
"Derivatives involving Logarithm Functions",
"Natural Logarithms"
] | [
"Definition:Natural Logarithm"
] | [
"Second Derivative of Natural Logarithm Function",
"Power Rule for Derivatives/Integer Index"
] |
proofwiki-17137 | Condition for Denesting of Square Root/Lemma | Let $a, b, c, d \in \Q_{\ge 0}$.
Suppose $\sqrt b \notin \Q$.
Then:
:$\sqrt {a + \sqrt b} = \sqrt {c + \sqrt d} \implies a = c, b = d$ | {{begin-eqn}}
{{eqn | l = \sqrt {a + \sqrt b}
| r = \sqrt {c + \sqrt d}
}}
{{eqn | ll= \leadsto
| l = a + \sqrt b
| r = c + \sqrt d
}}
{{eqn | ll= \leadsto
| l = a - c
| r = \sqrt d - \sqrt b
}}
{{eqn | ll= \leadsto
| l = \sqrt d - \sqrt b
| o = \in
| r = \Q
| c = R... | Let $a, b, c, d \in \Q_{\ge 0}$.
Suppose $\sqrt b \notin \Q$.
Then:
:$\sqrt {a + \sqrt b} = \sqrt {c + \sqrt d} \implies a = c, b = d$ | {{begin-eqn}}
{{eqn | l = \sqrt {a + \sqrt b}
| r = \sqrt {c + \sqrt d}
}}
{{eqn | ll= \leadsto
| l = a + \sqrt b
| r = c + \sqrt d
}}
{{eqn | ll= \leadsto
| l = a - c
| r = \sqrt d - \sqrt b
}}
{{eqn | ll= \leadsto
| l = \sqrt d - \sqrt b
| o = \in
| r = \Q
| c = [... | Condition for Denesting of Square Root/Lemma | https://proofwiki.org/wiki/Condition_for_Denesting_of_Square_Root/Lemma | https://proofwiki.org/wiki/Condition_for_Denesting_of_Square_Root/Lemma | [
"Condition for Denesting of Square Root"
] | [] | [
"Rational Subtraction is Closed",
"Difference of Two Squares",
"Rational Division is Closed",
"Rational Subtraction is Closed",
"Rational Division is Closed",
"Definition:Contradiction",
"Category:Condition for Denesting of Square Root"
] |
proofwiki-17138 | First Order ODE/y' + y = 0 | The first order ODE:
:$\dfrac {\d y} {\d x} + y = 0$
has the general solution:
:$y = C e^{-x}$
where $C$ is an arbitrary constant. | This first order ODE is in the form:
:$\dfrac {\d y} {\d x} + k y = 0$
where $k = 1$.
From First Order ODE: $\d y = k y \rd x$, this has the solution:
:$y = C e^{-x}$ | The [[Definition:First Order ODE|first order ODE]]:
:$\dfrac {\d y} {\d x} + y = 0$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C e^{-x}$
where $C$ is an [[Definition:Arbitrary Constant|arbitrary constant]]. | This [[Definition:First Order ODE|first order ODE]] is in the form:
:$\dfrac {\d y} {\d x} + k y = 0$
where $k = 1$.
From [[First Order ODE/dy = k y dx|First Order ODE: $\d y = k y \rd x$]], this has the solution:
:$y = C e^{-x}$ | First Order ODE/y' + y = 0 | https://proofwiki.org/wiki/First_Order_ODE/y'_+_y_=_0 | https://proofwiki.org/wiki/First_Order_ODE/y'_+_y_=_0 | [
"First Order ODE/dy = k y dx"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Definition:First Order Ordinary Differential Equation",
"First Order ODE/dy = k y dx"
] |
proofwiki-17139 | Linear Second Order ODE/y'' = 1 over 1 - x^2 | The second order ODE:
:$(1): \quad y'' = \dfrac 1 {1 - x^2}$
has the general solution:
:$y = x \tanh^{-1} x + \map \ln {1 - x^2} + C x + D$ | {{begin-eqn}}
{{eqn | l = y''
| r = \dfrac 1 {1 - x^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d^2 y} {\d x^2} \rd x
| r = \int \frac 1 {1 - x^2} \rd x
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d x}
| r = \tanh^{-1} x + C
| c = Primitive of $\dfrac 1... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' = \dfrac 1 {1 - x^2}$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = x \tanh^{-1} x + \map \ln {1 - x^2} + C x + D$ | {{begin-eqn}}
{{eqn | l = y''
| r = \dfrac 1 {1 - x^2}
| c =
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d^2 y} {\d x^2} \rd x
| r = \int \frac 1 {1 - x^2} \rd x
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d y} {\d x}
| r = \tanh^{-1} x + C
| c = [[Primitive of Recipro... | Linear Second Order ODE/y'' = 1 over 1 - x^2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_=_1_over_1_-_x^2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_=_1_over_1_-_x^2 | [
"Examples of Constant Coefficient LSOODEs"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Primitive of Inverse Hyperbolic Tangent of x over a"
] |
proofwiki-17140 | First Order ODE/y' = e^x | The first order ODE:
:$y' = e^x$
has the general solution:
:$y = e^x + C$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = y'
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \d y
| r = \int e^x \rd x
| c = Solution to Separable Differential Equation
}}
{{eqn | ll= \leadsto
| l = y
| r = e^x + C
| c = Primitive of Constant, Primitive of Exponential Function
}}
{{e... | The [[Definition:First Order ODE|first order ODE]]:
:$y' = e^x$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = e^x + C$
where $C$ is an [[Definition:Arbitrary Constant|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = y'
| r = e^x
| c =
}}
{{eqn | ll= \leadsto
| l = \int \d y
| r = \int e^x \rd x
| c = [[Solution to Separable Differential Equation]]
}}
{{eqn | ll= \leadsto
| l = y
| r = e^x + C
| c = [[Primitive of Constant]], [[Primitive of Exponential Funct... | First Order ODE/y' = e^x | https://proofwiki.org/wiki/First_Order_ODE/y'_=_e^x | https://proofwiki.org/wiki/First_Order_ODE/y'_=_e^x | [
"First Order ODE/dy = k y dx"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Solution to Separable Differential Equation",
"Primitive of Constant",
"Primitive of Exponential Function"
] |
proofwiki-17141 | Semilattice Homomorphism is Order-Preserving | Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices.
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a semilattice homomorphism.
Let $\preceq_1$ be the ordering on $S$ defined by:
:$a \preceq_1 b \iff \paren {a \circ b} = b$
Let $\preceq_2$ be the ordering on $T$ defined by:
:$x \preceq_2 y \iff \paren {... | {{begin-eqn}}
{{eqn | l = a \preceq_1 b
| o = \leadstoandfrom
| r = a \circ b = b
| c = Definition of the ordering $\preceq_1$
}}
{{eqn | o = \leadsto
| r = \map \phi { a \circ b} = \map \phi b
}}
{{eqn | o = \leadstoandfrom
| r = \map \phi a * \map \phi b = \map \phi b
| c = {{Defof... | Let $\struct {S, \circ}$ and $\struct {T, *}$ be [[Definition:Semilattice|semilattices]].
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a [[Definition:Semilattice Homomorphism|semilattice homomorphism]].
Let $\preceq_1$ be the [[Definition:Ordering|ordering]] on $S$ defined by:
:$a \preceq_1 b \iff \paren {a \... | {{begin-eqn}}
{{eqn | l = a \preceq_1 b
| o = \leadstoandfrom
| r = a \circ b = b
| c = Definition of the [[Definition:Ordering|ordering]] $\preceq_1$
}}
{{eqn | o = \leadsto
| r = \map \phi { a \circ b} = \map \phi b
}}
{{eqn | o = \leadstoandfrom
| r = \map \phi a * \map \phi b = \map \p... | Semilattice Homomorphism is Order-Preserving | https://proofwiki.org/wiki/Semilattice_Homomorphism_is_Order-Preserving | https://proofwiki.org/wiki/Semilattice_Homomorphism_is_Order-Preserving | [
"Semilattice Homomorphisms",
"Increasing Mappings"
] | [
"Definition:Semilattice",
"Definition:Semilattice Homomorphism",
"Definition:Ordering",
"Definition:Ordering",
"Definition:Increasing/Mapping"
] | [
"Definition:Ordering",
"Definition:Ordering"
] |
proofwiki-17142 | Order-Preserving Mapping Not Always Semilattice Homomorphism | Let $\struct {S, \circ}$ and $\struct {T, *}$ be semilattices.
Let $\preceq_1$ be the ordering on $S$ defined by:
:$a \preceq_1 b \iff \paren {a \circ b} = b$
Let $\preceq_2$ be the ordering on $T$ defined by:
:$x \preceq_2 y \iff \paren {x * y} = y$
Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be an o... | Let $S = \set {a, b, c}$.
Let $\circ : S \times S \to S$ be defined by:
:$a \circ b = b \circ a = c$
:$a \circ c = c \circ a = c$
:$b \circ c = c \circ b = c$
:$a \circ a = a$
:$b \circ b = b$
:$c \circ c = c$
Then a manual check shows that $\circ$ is:
:closed
:associative
:commutative
:idempotent
So $\struct {S, \circ... | Let $\struct {S, \circ}$ and $\struct {T, *}$ be [[Definition:Semilattice|semilattices]].
Let $\preceq_1$ be the [[Definition:Ordering|ordering]] on $S$ defined by:
:$a \preceq_1 b \iff \paren {a \circ b} = b$
Let $\preceq_2$ be the [[Definition:Ordering|ordering]] on $T$ defined by:
:$x \preceq_2 y \iff \paren {x * ... | Let $S = \set {a, b, c}$.
Let $\circ : S \times S \to S$ be defined by:
:$a \circ b = b \circ a = c$
:$a \circ c = c \circ a = c$
:$b \circ c = c \circ b = c$
:$a \circ a = a$
:$b \circ b = b$
:$c \circ c = c$
Then a manual check shows that $\circ$ is:
:[[Definition:Closed Algebraic Structure|closed]]
:[[Definition:A... | Order-Preserving Mapping Not Always Semilattice Homomorphism | https://proofwiki.org/wiki/Order-Preserving_Mapping_Not_Always_Semilattice_Homomorphism | https://proofwiki.org/wiki/Order-Preserving_Mapping_Not_Always_Semilattice_Homomorphism | [
"Semilattice Homomorphisms",
"Increasing Mappings"
] | [
"Definition:Semilattice",
"Definition:Ordering",
"Definition:Ordering",
"Definition:Increasing/Mapping",
"Definition:Semilattice Homomorphism"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Associative Operation",
"Definition:Commutative/Operation",
"Definition:Idempotence/Operation",
"Definition:Semilattice",
"Definition:Ordering",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Associative... |
proofwiki-17143 | Schwarz's Lemma/Lemma | Let $D$ be the unit disk centred at $0$.
Let $g : D \to \C$ be a complex function such that:
:$\map g z = \begin {cases} \dfrac {\map f z} z & z \ne 0 \\ \\ \map {f'} 0 & z = 0\end {cases}$
Then $g$ is holomorphic on $D$. | By Differentiable Function is Continuous, $f$ is continuous.
So by Quotient Rule for Continuous Complex Functions:
:$g$ is continuous on $D \setminus \set 0$.
We aim to show that $f$ is continuous on $D$.
Note that since $f$ is holomorphic on $D$ and $0 \in D$ we have, by the definition of the complex derivative:
:$\... | Let $D$ be the [[Definition:Unit Disk|unit disk]] centred at $0$.
Let $g : D \to \C$ be a [[Definition:Complex Function|complex function]] such that:
:$\map g z = \begin {cases} \dfrac {\map f z} z & z \ne 0 \\ \\ \map {f'} 0 & z = 0\end {cases}$
Then $g$ is [[Definition:Holomorphic Function|holomorphic]] on $D$. | By [[Differentiable Function is Continuous]], $f$ is [[Definition:Continuous Complex Function|continuous]].
So by [[Quotient Rule for Continuous Complex Functions]]:
:$g$ is [[Definition:Continuous Complex Function|continuous]] on $D \setminus \set 0$.
We aim to show that $f$ is [[Definition:Continuous Complex Funct... | Schwarz's Lemma/Lemma | https://proofwiki.org/wiki/Schwarz's_Lemma/Lemma | https://proofwiki.org/wiki/Schwarz's_Lemma/Lemma | [
"Complex Analysis",
"Schwarz's Lemma"
] | [
"Definition:Unit Disk",
"Definition:Complex Function",
"Definition:Holomorphic Function"
] | [
"Differentiable Function is Continuous",
"Definition:Continuous Complex Function",
"Combination Theorem for Continuous Functions/Complex/Quotient Rule",
"Definition:Continuous Complex Function",
"Definition:Continuous Complex Function",
"Definition:Holomorphic Function",
"Definition:Derivative/Complex F... |
proofwiki-17144 | Factorial as Product of Consecutive Factorials/Lemma 2 | Let $n \in \N$.
Then $\paren {2 n - 2}! \, \paren {2 n - 1}! > \paren {3 n - 1}!$ for all $n \ge 7$. | We prove the result by induction on $n$. | Let $n \in \N$.
Then $\paren {2 n - 2}! \, \paren {2 n - 1}! > \paren {3 n - 1}!$ for all $n \ge 7$. | We prove the result by [[Principle of Mathematical Induction|induction]] on $n$. | Factorial as Product of Consecutive Factorials/Lemma 2 | https://proofwiki.org/wiki/Factorial_as_Product_of_Consecutive_Factorials/Lemma_2 | https://proofwiki.org/wiki/Factorial_as_Product_of_Consecutive_Factorials/Lemma_2 | [
"Factorials",
"Factorial as Product of Consecutive Factorials"
] | [] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-17145 | Factorial as Product of Consecutive Factorials/Lemma 1 | Let $n \in \N$.
Then $\paren {2 n - 1}! \, \paren {2 n}! > \paren {3 n - 1}!$ for all $n > 1$. | Let $n, k \in \N_{> 0}$.
Suppose $n > 1$ and $n > k$.
We show that $\paren {k + 1} \paren {2 n - k} > 2 n + k$.
For $k = 1$:
:$2 \paren {2 n - 1} = 4 n - 2 \ge 2 n + 2 > 2 n + 1$
For $k > 1$:
{{begin-eqn}}
{{eqn | l = \paren {k + 1} \paren {2 n - k}
| r = 2 n k + 2 n - k^2 - k
}}
{{eqn | o = >
| r = 2 k^2 +... | Let $n \in \N$.
Then $\paren {2 n - 1}! \, \paren {2 n}! > \paren {3 n - 1}!$ for all $n > 1$. | Let $n, k \in \N_{> 0}$.
Suppose $n > 1$ and $n > k$.
We show that $\paren {k + 1} \paren {2 n - k} > 2 n + k$.
For $k = 1$:
:$2 \paren {2 n - 1} = 4 n - 2 \ge 2 n + 2 > 2 n + 1$
For $k > 1$:
{{begin-eqn}}
{{eqn | l = \paren {k + 1} \paren {2 n - k}
| r = 2 n k + 2 n - k^2 - k
}}
{{eqn | o = >
| r = ... | Factorial as Product of Consecutive Factorials/Lemma 1 | https://proofwiki.org/wiki/Factorial_as_Product_of_Consecutive_Factorials/Lemma_1 | https://proofwiki.org/wiki/Factorial_as_Product_of_Consecutive_Factorials/Lemma_1 | [
"Factorials",
"Factorial as Product of Consecutive Factorials"
] | [] | [
"Category:Factorials",
"Category:Factorial as Product of Consecutive Factorials"
] |
proofwiki-17146 | Entire Function with Bounded Real Part is Constant | Let $f : \C \to \C$ be an entire function.
Let the real part of $f$ be bounded.
That is, there exists a positive real number $M$ such that:
:$\cmod {\map \Re {\map f z} } < M$
for all $z \in \C$, where $\map \Re {\map f z}$ denotes the real part of $\map f z$.
Then $f$ is constant. | Let $g : \C \to \C$ be a complex function with:
:$\ds \map g z = e^{\map f z}$
By Derivative of Complex Composite Function, $g$ is entire with derivative:
:$\ds \map {g'} z = \map {f'} z e^{\map f z}$
We have:
{{begin-eqn}}
{{eqn | l = \cmod {\map g z}
| r = e^{\map \Re {\map f z} }
| c = Modulus of Positive Real Num... | Let $f : \C \to \C$ be an [[Definition:Entire Function|entire function]].
Let the [[Definition:Real Part|real part]] of $f$ be [[Definition:Bounded Mapping|bounded]].
That is, there exists a [[Definition:Positive Real Number|positive real number]] $M$ such that:
:$\cmod {\map \Re {\map f z} } < M$
for all $z \in ... | Let $g : \C \to \C$ be a [[Definition:Complex Function|complex function]] with:
:$\ds \map g z = e^{\map f z}$
By [[Derivative of Complex Composite Function]], $g$ is [[Definition:Entire Function|entire]] with [[Definition:Derivative|derivative]]:
:$\ds \map {g'} z = \map {f'} z e^{\map f z}$
We have:
{{begin-eqn}... | Entire Function with Bounded Real Part is Constant | https://proofwiki.org/wiki/Entire_Function_with_Bounded_Real_Part_is_Constant | https://proofwiki.org/wiki/Entire_Function_with_Bounded_Real_Part_is_Constant | [
"Complex Analysis"
] | [
"Definition:Entire Function",
"Definition:Complex Number/Real Part",
"Definition:Bounded Mapping",
"Definition:Positive/Real Number",
"Definition:Complex Number/Real Part",
"Definition:Constant Mapping"
] | [
"Definition:Complex Function",
"Derivative of Complex Composite Function",
"Definition:Entire Function",
"Definition:Derivative",
"Modulus of Positive Real Number to Complex Power is Positive Real Number to Power of Real Part",
"Exponential is Strictly Increasing",
"Exponential is Strictly Increasing",
... |
proofwiki-17147 | Mass of Mole of Isotope of Element | Let $S$ be a substance made up entirely of a particular isotope $Q$ of a particular element.
Let one atom of $Q$ contain $n$ neutrons and $p$ protons.
Then one mole of $S$ has a mass of approximately $n + p$ grams. | Let $m_Q$ grams be the mass of one atom of $Q$.
Let $m_C$ grams be the mass of one atom of carbon-12.
Let $M_Q$ grams be the mass of one mole of $Q$.
Let $M_C$ grams be the mass of one mole of carbon-12.
The mass of a proton and the mass of a neutron are approximately equal.
Together, the protons and neutrons make up t... | Let $S$ be a [[Definition:Substance|substance]] made up entirely of a particular [[Definition:Isotope|isotope]] $Q$ of a particular [[Definition:Chemical Element|element]].
Let one [[Definition:Atom (Physics)|atom]] of $Q$ contain $n$ [[Definition:Neutron|neutrons]] and $p$ [[Definition:Proton|protons]].
Then one [[D... | Let $m_Q$ [[Definition:Gram|grams]] be the [[Definition:Mass|mass]] of one [[Definition:Atom (Physics)|atom]] of $Q$.
Let $m_C$ [[Definition:Gram|grams]] be the [[Definition:Mass|mass]] of one [[Definition:Atom (Physics)|atom]] of [[Definition:Carbon/12|carbon-12]].
Let $M_Q$ [[Definition:Gram|grams]] be the [[Defin... | Mass of Mole of Isotope of Element | https://proofwiki.org/wiki/Mass_of_Mole_of_Isotope_of_Element | https://proofwiki.org/wiki/Mass_of_Mole_of_Isotope_of_Element | [
"Substance"
] | [
"Definition:Substance",
"Definition:Chemical Element/Isotope",
"Definition:Chemical Element",
"Definition:Atom (Physics)",
"Definition:Neutron",
"Definition:Proton",
"Definition:Mole",
"Definition:Mass",
"Definition:Metric System/Mass/Gram"
] | [
"Definition:Metric System/Mass/Gram",
"Definition:Mass",
"Definition:Atom (Physics)",
"Definition:Metric System/Mass/Gram",
"Definition:Mass",
"Definition:Atom (Physics)",
"Definition:Carbon/12",
"Definition:Metric System/Mass/Gram",
"Definition:Mass",
"Definition:Mole",
"Definition:Metric Syste... |
proofwiki-17148 | Mass of Mole of Substance | Let $S$ be a substance with molecular weight $W_S$.
Then one mole of $S$ has a mass of $W_S$ grams. | Let $m_S$ grams be the mean mass of one molecule of $S$.
Let $m_C$ grams be the mass of one atom of carbon-12.
Let $M_S$ grams be the mass of one mole of $S$.
Let $M_C$ grams be the mass of one mole of carbon-12.
Then:
{{begin-eqn}}
{{eqn | l = W_S
| r = m_S \times \dfrac {12} {m_C}
| c = {{Defof|Molecular ... | Let $S$ be a [[Definition:Substance|substance]] with [[Definition:Molecular Weight|molecular weight]] $W_S$.
Then one [[Definition:Mole|mole]] of $S$ has a [[Definition:Mass|mass]] of $W_S$ [[Definition:Gram|grams]]. | Let $m_S$ [[Definition:Gram|grams]] be the [[Definition:Arithmetic Mean|mean]] [[Definition:Mass|mass]] of one [[Definition:Molecule|molecule]] of $S$.
Let $m_C$ [[Definition:Gram|grams]] be the [[Definition:Mass|mass]] of one [[Definition:Atom (Physics)|atom]] of [[Definition:Carbon/12|carbon-12]].
Let $M_S$ [[Defi... | Mass of Mole of Substance | https://proofwiki.org/wiki/Mass_of_Mole_of_Substance | https://proofwiki.org/wiki/Mass_of_Mole_of_Substance | [
"Substance"
] | [
"Definition:Substance",
"Definition:Molecular Weight",
"Definition:Mole",
"Definition:Mass",
"Definition:Metric System/Mass/Gram"
] | [
"Definition:Metric System/Mass/Gram",
"Definition:Arithmetic Mean",
"Definition:Mass",
"Definition:Molecule",
"Definition:Metric System/Mass/Gram",
"Definition:Mass",
"Definition:Atom (Physics)",
"Definition:Carbon/12",
"Definition:Metric System/Mass/Gram",
"Definition:Mass",
"Definition:Mole",
... |
proofwiki-17149 | Linear Second Order ODE/y'' = y' | The second order ODE:
:$(1): \quad y'' = y'$
has the general solution:
:$y = A_1 e^x + A_2$ | The proof proceeds by using Solution of Second Order Differential Equation with Missing Dependent Variable.
Substitute $p$ for $y'$ in $(1)$:
{{begin-eqn}}
{{eqn | l = \dfrac {\d p} {\d x}
| r = p
| c = where $p = \dfrac {\d y} {\d x}$
}}
{{eqn | ll= \leadsto
| l = \int \rd x
| r = \int \frac {\... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' = y'$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = A_1 e^x + A_2$ | The proof proceeds by using [[Solution of Second Order Differential Equation with Missing Dependent Variable]].
Substitute $p$ for $y'$ in $(1)$:
{{begin-eqn}}
{{eqn | l = \dfrac {\d p} {\d x}
| r = p
| c = where $p = \dfrac {\d y} {\d x}$
}}
{{eqn | ll= \leadsto
| l = \int \rd x
| r = \int \f... | Linear Second Order ODE/y'' = y'/Proof 1 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_=_y' | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_=_y'/Proof_1 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' = y'"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution of Second Order Differential Equation with Missing Dependent Variable",
"Solution to Separable Differential Equation",
"Primitive of Reciprocal",
"Solution to Separable Differential Equation",
"Primitive of Exponential Function"
] |
proofwiki-17150 | Linear Second Order ODE/y'' = y' | The second order ODE:
:$(1): \quad y'' = y'$
has the general solution:
:$y = A_1 e^x + A_2$ | Using Solution of Second Order Differential Equation with Missing Independent Variable, $(1)$ can be expressed as:
{{begin-eqn}}
{{eqn | l = p \frac {\d p} {\d y}
| r = p
| c = where $p = \dfrac {\d y} {\d x}$
}}
{{eqn | ll= \leadsto
| l = \int \rd y
| r = \int \frac {p \rd p} p
| c = Solu... | The [[Definition:Second Order ODE|second order ODE]]:
:$(1): \quad y'' = y'$
has the [[Definition:General Solution to Differential Equation|general solution]]:
:$y = A_1 e^x + A_2$ | Using [[Solution of Second Order Differential Equation with Missing Independent Variable]], $(1)$ can be expressed as:
{{begin-eqn}}
{{eqn | l = p \frac {\d p} {\d y}
| r = p
| c = where $p = \dfrac {\d y} {\d x}$
}}
{{eqn | ll= \leadsto
| l = \int \rd y
| r = \int \frac {p \rd p} p
| c =... | Linear Second Order ODE/y'' = y'/Proof 2 | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_=_y' | https://proofwiki.org/wiki/Linear_Second_Order_ODE/y''_=_y'/Proof_2 | [
"Examples of Constant Coefficient LSOODEs",
"Linear Second Order ODE/y'' = y'"
] | [
"Definition:Second Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Solution of Second Order Differential Equation with Missing Independent Variable",
"Solution to Separable Differential Equation",
"Primitive of Constant",
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation/Solution by Integrating F... |
proofwiki-17151 | First Order ODE/x y' = 2 y | The first order ODE:
:$x y' = 2 y$
has the general solution:
:$y = C x^2$
where $C$ is an arbitrary constant. | {{begin-eqn}}
{{eqn | l = x y'
| r = 2 y
| c =
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d y} y
| r = 2 \int \dfrac {\d x} x
| c = Solution to Separable Differential Equation
}}
{{eqn | ll= \leadsto
| l = \ln y
| r = 2 \ln x + \ln C
| c = Primitive of Reciprocal
}}
{{e... | The [[Definition:First Order ODE|first order ODE]]:
:$x y' = 2 y$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = C x^2$
where $C$ is an [[Definition:Arbitrary Constant|arbitrary constant]]. | {{begin-eqn}}
{{eqn | l = x y'
| r = 2 y
| c =
}}
{{eqn | ll= \leadsto
| l = \int \dfrac {\d y} y
| r = 2 \int \dfrac {\d x} x
| c = [[Solution to Separable Differential Equation]]
}}
{{eqn | ll= \leadsto
| l = \ln y
| r = 2 \ln x + \ln C
| c = [[Primitive of Reciprocal]... | First Order ODE/x y' = 2 y | https://proofwiki.org/wiki/First_Order_ODE/x_y'_=_2_y | https://proofwiki.org/wiki/First_Order_ODE/x_y'_=_2_y | [
"First Order ODE/x dy = k y dx"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Solution to Separable Differential Equation",
"Primitive of Reciprocal"
] |
proofwiki-17152 | Singleton in Normed Vector Space is Closed | Let $X$ be a vector space over $\R$ or $\C$.
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $\set x \subseteq X$ be a singleton.
Then $\set x$ is closed. | Note that either $\set x = X$ or $\set x \subset X$.
Suppose, $X = \set x$.
Then:
:$X \setminus \set x = \O$
We have that Empty Set is Open in Normed Vector Space.
$\set x$ is a complement of $\O$ in $X$.
By definition, $x$ is closed.
Suppose, $X \ne \set x$, i.e. $\set x \subset X$.
Define $U := X \setminus \set x$.
L... | Let $X$ be a [[Definition:Vector Space|vector space]] over $\R$ or $\C$.
Let $\struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\set x \subseteq X$ be a [[Definition:Singleton|singleton]].
Then $\set x$ is [[Definition:Closed Set in Normed Vector Space|closed]]. | Note that either $\set x = X$ or $\set x \subset X$.
Suppose, $X = \set x$.
Then:
:$X \setminus \set x = \O$
We have that [[Empty Set is Open in Normed Vector Space]].
$\set x$ is a [[Definition:Relative Complement|complement]] of $\O$ in $X$.
By definition, $x$ is [[Definition:Closed Set in Normed Vector Space|c... | Singleton in Normed Vector Space is Closed | https://proofwiki.org/wiki/Singleton_in_Normed_Vector_Space_is_Closed | https://proofwiki.org/wiki/Singleton_in_Normed_Vector_Space_is_Closed | [
"Normed Vector Spaces"
] | [
"Definition:Vector Space",
"Definition:Normed Vector Space",
"Definition:Singleton",
"Definition:Closed Set/Normed Vector Space"
] | [
"Empty Set is Open in Normed Vector Space",
"Definition:Relative Complement",
"Definition:Closed Set/Normed Vector Space",
"Normed Vector Space is Open in Itself",
"Definition:Relative Complement",
"Definition:Open Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space"
] |
proofwiki-17153 | Singleton Set is Nowhere Dense in Rational Space | Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$.
Then every singleton subset of $\Q$ is nowhere dense in $\struct {\Q, \tau_d}$. | Let $x \in \Q$.
By definition of nowhere dense, we need to show that:
:$\paren {\set x^-}^\circ = \O$
where $S^-$ denotes the closure of a set $S$ and $S^\circ$ denotes its interior.
By Real Number is Closed in Real Number Line, $\set x$ is closed in $\struct {\R, \tau_d}$.
$\struct {\Q, \tau_d}$ is a subspace of $\str... | Let $\struct {\Q, \tau_d}$ be the [[Definition:Rational Number Space|rational number space]] under the [[Definition:Euclidean Topology on Real Number Line|Euclidean topology]] $\tau_d$.
Then every [[Definition:Singleton|singleton]] [[Definition:Subset|subset]] of $\Q$ is [[Definition:Nowhere Dense|nowhere dense]] in $... | Let $x \in \Q$.
By definition of [[Definition:Nowhere Dense|nowhere dense]], we need to show that:
:$\paren {\set x^-}^\circ = \O$
where $S^-$ denotes the [[Definition:Closure (Topology)|closure]] of a [[Definition:Set|set]] $S$ and $S^\circ$ denotes its [[Definition:Interior (Topology)|interior]].
By [[Real Number ... | Singleton Set is Nowhere Dense in Rational Space | https://proofwiki.org/wiki/Singleton_Set_is_Nowhere_Dense_in_Rational_Space | https://proofwiki.org/wiki/Singleton_Set_is_Nowhere_Dense_in_Rational_Space | [
"Rational Number Space",
"Singletons",
"Examples of Nowhere Dense"
] | [
"Definition:Rational Number Space",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Singleton",
"Definition:Subset",
"Definition:Nowhere Dense"
] | [
"Definition:Nowhere Dense",
"Definition:Closure (Topology)",
"Definition:Set",
"Definition:Interior (Topology)",
"Real Number is Closed in Real Number Line",
"Definition:Closed Set/Topology",
"Definition:Topological Subspace",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
... |
proofwiki-17154 | Real Function of Two Variables represents Surface in Cartesian 3-Space | Let $S$ and $T$ be subsets of the set of real numbers $\R$.
Let $f: S \times T \to \R$ be a real function of two variables.
Then the locus of $f$ describes a surface embedded in the Cartesian space $\R^3$. | {{ProofWanted|The book says "clearly represents", but there is a concern that $f$ may need to be defined as continuous. And while it's intuitively obvious, it may not be so easy to prove rigorously.}} | Let $S$ and $T$ be [[Definition:Subset|subsets]] of the [[Definition:Real Number|set of real numbers]] $\R$.
Let $f: S \times T \to \R$ be a [[Definition:Real Function of Two Variables|real function of two variables]].
Then the [[Definition:Locus|locus]] of $f$ describes a [[Definition:Surface|surface]] embedded in ... | {{ProofWanted|The book says "clearly represents", but there is a concern that $f$ may need to be defined as continuous. And while it's intuitively obvious, it may not be so easy to prove rigorously.}} | Real Function of Two Variables represents Surface in Cartesian 3-Space | https://proofwiki.org/wiki/Real_Function_of_Two_Variables_represents_Surface_in_Cartesian_3-Space | https://proofwiki.org/wiki/Real_Function_of_Two_Variables_represents_Surface_in_Cartesian_3-Space | [
"Real Functions"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Function/Two Variables",
"Definition:Locus",
"Definition:Surface",
"Definition:Cartesian Product/Cartesian Space"
] | [] |
proofwiki-17155 | Cauchy's Lemma (Number Theory) | Let $a$ and $b$ be odd positive integers.
Suppose $a$ and $b$ satisfy:
{{begin-eqn}}
{{eqn | n = 1
| l = b^2
| o = <
| r = 4 a
}}
{{eqn | n = 2
| l = 3 a
| o = <
| r = b^2 + 2 b + 4
}}
{{end-eqn}}
Then there exist non-negative integers $s, t, u, v$ such that:
{{begin-eqn}}
{{eqn | ... | Because $a$ is odd, we can write:
:$a = 2 k + 1$
for some positive integer $k$.
Then:
{{begin-eqn}}
{{eqn | l = 4 a - b^2
| r = 4 \paren {2 k + 1} - b^2
}}
{{eqn | o = \equiv
| r = 8 k + 4 - 1
| rr= \pmod 8
| c = Odd Square Modulo 8
}}
{{eqn | o = \equiv
| r = 3
| rr= \pmod 8
}}
{{en... | Let $a$ and $b$ be [[Definition:Odd Integer|odd]] [[Definition:Positive Integer|positive integers]].
Suppose $a$ and $b$ satisfy:
{{begin-eqn}}
{{eqn | n = 1
| l = b^2
| o = <
| r = 4 a
}}
{{eqn | n = 2
| l = 3 a
| o = <
| r = b^2 + 2 b + 4
}}
{{end-eqn}}
Then there exist [[Defin... | Because $a$ is [[Definition:Odd Integer|odd]], we can write:
:$a = 2 k + 1$
for some [[Definition:Positive Integer|positive integer]] $k$.
Then:
{{begin-eqn}}
{{eqn | l = 4 a - b^2
| r = 4 \paren {2 k + 1} - b^2
}}
{{eqn | o = \equiv
| r = 8 k + 4 - 1
| rr= \pmod 8
| c = [[Odd Square Modulo 8]... | Cauchy's Lemma (Number Theory) | https://proofwiki.org/wiki/Cauchy's_Lemma_(Number_Theory) | https://proofwiki.org/wiki/Cauchy's_Lemma_(Number_Theory) | [
"Integer as Sum of Polygonal Numbers"
] | [
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Positive/Integer"
] | [
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Odd Square Modulo 8",
"Definition:Positive/Integer",
"Integer as Sum of Three Odd Squares",
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer"... |
proofwiki-17156 | Finite Subset of Normed Vector Space is Closed | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $F \subseteq X$ be finite.
Then $F$ is closed in $M$. | Suppose $F$ is empty.
By Empty Set is Closed in Normed Vector Space, $F$ is closed.
Suppose, for some $n \in \N$, that:
:$\ds F = \bigcup_{i \mathop = 1}^n \set {x_i}$
We have that Singleton in Normed Vector Space is Closed.
Hence $F$ is a finite union of closed sets.
By Finite Union of Closed Sets is Closed in Normed ... | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $F \subseteq X$ be [[Definition:Finite Set|finite]].
Then $F$ is [[Definition:Closed Set in Normed Vector Space|closed]] in $M$. | Suppose $F$ is [[Definition:Empty Set|empty]].
By [[Empty Set is Closed in Normed Vector Space]], $F$ is [[Definition:Closed Set in Normed Vector Space|closed]].
Suppose, for some $n \in \N$, that:
:$\ds F = \bigcup_{i \mathop = 1}^n \set {x_i}$
We have that [[Singleton in Normed Vector Space is Closed]].
Hence $... | Finite Subset of Normed Vector Space is Closed | https://proofwiki.org/wiki/Finite_Subset_of_Normed_Vector_Space_is_Closed | https://proofwiki.org/wiki/Finite_Subset_of_Normed_Vector_Space_is_Closed | [
"Closed Sets",
"Normed Vector Spaces",
"Closed Sets (Normed Vector Spaces)",
"Closed Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Finite Set",
"Definition:Closed Set/Normed Vector Space"
] | [
"Definition:Empty Set",
"Empty Set is Closed/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Singleton in Normed Vector Space is Closed",
"Definition:Set Union/Finite Union",
"Definition:Closed Set/Normed Vector Space",
"Finite Union of Closed Sets is Closed/Normed Vector Space",
"... |
proofwiki-17157 | Finite Union of Closed Sets is Closed/Normed Vector Space | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Then the union of finitely many closed sets of $M$ is itself closed. | Let $\ds \bigcup_{i \mathop = 1}^n F_i$ be the union of a finite number of closed sets of $M$.
By definition of closed set, each of the $X \setminus F_i$ is by definition open in $M$.
Then from De Morgan's laws:
:$\ds X \setminus \bigcup_{i \mathop = 1}^n F_i = \bigcap_{i \mathop = 1}^n \paren {X \setminus F_i}$
We hav... | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Set Union|union]] of [[Definition:Finite Set|finitely many]] [[Definition:Closed Set in Normed Vector Space|closed sets]] of $M$ is itself [[Definition:Closed Set in Normed Vector Space|close... | Let $\ds \bigcup_{i \mathop = 1}^n F_i$ be the [[Definition:Set Union|union]] of a [[Definition:Finite Set|finite]] number of [[Definition:Closed Set in Normed Vector Space|closed sets]] of $M$.
By definition of [[Definition:Closed Set in Normed Vector Space|closed set]], each of the $X \setminus F_i$ is by definition... | Finite Union of Closed Sets is Closed/Normed Vector Space | https://proofwiki.org/wiki/Finite_Union_of_Closed_Sets_is_Closed/Normed_Vector_Space | https://proofwiki.org/wiki/Finite_Union_of_Closed_Sets_is_Closed/Normed_Vector_Space | [
"Closed Sets (Normed Vector Spaces)",
"Closed Sets",
"Finite Union of Closed Sets is Closed",
"Finite Union of Closed Sets is Closed",
"Closed Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Set Union",
"Definition:Finite Set",
"Definition:Closed Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space"
] | [
"Definition:Set Union",
"Definition:Finite Set",
"Definition:Closed Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"De Morgan's Laws (Set Theory)",
"Definition:Set Intersection",
"Definition:Finite Set",
"Definition:Open Set/Normed V... |
proofwiki-17158 | Primitive of x fourth by Cosine of a x | :$\ds \int x^4 \cos a x \rd x = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12 \sin a x} {a^3} x^2 - \frac {24 \cos a x} {a^4} x + \frac {24 \sin a x} {a^5} + C$ | {{begin-eqn}}
{{eqn | l = \int x^4 \cos a x \rd x
| r = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^{4 - 1} - \frac {4 \paren {4 - 1} } {a^2} \int x^{4 - 2} \cos a x \rd x + C
| c = Primitive of $x^n \cos a x$
}}
{{eqn | r = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12} {a^2} \i... | :$\ds \int x^4 \cos a x \rd x = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^3 - \frac {12 \sin a x} {a^3} x^2 - \frac {24 \cos a x} {a^4} x + \frac {24 \sin a x} {a^5} + C$ | {{begin-eqn}}
{{eqn | l = \int x^4 \cos a x \rd x
| r = \frac {\sin a x} a x^4 + \frac {4 \cos a x} {a^2} x^{4 - 1} - \frac {4 \paren {4 - 1} } {a^2} \int x^{4 - 2} \cos a x \rd x + C
| c = [[Primitive of Power of x by Cosine of a x|Primitive of $x^n \cos a x$]]
}}
{{eqn | r = \frac {\sin a x} a x^4 + \frac... | Primitive of x fourth by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_fourth_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_fourth_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Power of x by Cosine of a x",
"Primitive of x squared by Cosine of a x"
] |
proofwiki-17159 | Primitive of x sixth by Cosine of a x | :$\ds \int x^6 \cos a x \rd x = \frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^5 - \frac {30 \sin a x} {a^3} x^4 - \frac {120 \cos a x} {a^4} x^3 + \frac {360 \sin a x} {a^5} x^2 + \frac {720 \cos a x} {a^6} x - \frac {720 \sin a x} {a^7} + C$ | {{begin-eqn}}
{{eqn | l = \int x^6 \cos a x \rd x
| r = \frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^{6 - 1} - \frac {6 \paren {6 - 1} } {a^2} \int x^{6 - 2} \cos a x \rd x + C
| c = Primitive of $x^n \cos a x$
}}
{{eqn | r = \frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^5 - \frac {30} {a^2} \i... | :$\ds \int x^6 \cos a x \rd x = \frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^5 - \frac {30 \sin a x} {a^3} x^4 - \frac {120 \cos a x} {a^4} x^3 + \frac {360 \sin a x} {a^5} x^2 + \frac {720 \cos a x} {a^6} x - \frac {720 \sin a x} {a^7} + C$ | {{begin-eqn}}
{{eqn | l = \int x^6 \cos a x \rd x
| r = \frac {\sin a x} a x^6 + \frac {6 \cos a x} {a^2} x^{6 - 1} - \frac {6 \paren {6 - 1} } {a^2} \int x^{6 - 2} \cos a x \rd x + C
| c = [[Primitive of Power of x by Cosine of a x|Primitive of $x^n \cos a x$]]
}}
{{eqn | r = \frac {\sin a x} a x^6 + \frac... | Primitive of x sixth by Cosine of a x | https://proofwiki.org/wiki/Primitive_of_x_sixth_by_Cosine_of_a_x | https://proofwiki.org/wiki/Primitive_of_x_sixth_by_Cosine_of_a_x | [
"Primitives involving Cosine Function"
] | [] | [
"Primitive of Power of x by Cosine of a x",
"Primitive of x fourth by Cosine of a x"
] |
proofwiki-17160 | Fourier Series/Sixth Power of x over Minus Pi to Pi | :$\ds x^6 = \frac {\pi^6} 7 + \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 + 1440} {n^6} \cos n \pi \cos n x$ | Since $x^6 = \paren {-x}^6$, $x^6$ is an even function.
By Fourier Series for Even Function over Symmetric Range, the Fourier series of $\map f x$ can be expressed as:
:$x^6 \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
{{begin-eqn}}
{{eqn | l = a_n
| r = \df... | :$\ds x^6 = \frac {\pi^6} 7 + \sum_{n \mathop = 1}^\infty \frac {12 n^4 \pi^4 - 240 n^2 \pi^2 + 1440} {n^6} \cos n \pi \cos n x$ | Since $x^6 = \paren {-x}^6$, $x^6$ is an [[Definition:Even Function|even function]].
By [[Fourier Series for Even Function over Symmetric Range]], the [[Definition:Fourier Series|Fourier series]] of $\map f x$ can be expressed as:
:$x^6 \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for a... | Fourier Series/Sixth Power of x over Minus Pi to Pi | https://proofwiki.org/wiki/Fourier_Series/Sixth_Power_of_x_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Sixth_Power_of_x_over_Minus_Pi_to_Pi | [
"Examples of Fourier Series"
] | [] | [
"Definition:Even Function",
"Fourier Series for Even Function over Symmetric Range",
"Definition:Fourier Series",
"Primitive of x sixth by Cosine of a x",
"Primitive of Power"
] |
proofwiki-17161 | Derivative of x to the a x | :$\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$ | Let $y := x^{a x}$.
As $x > 0$, we can take the natural logarithm of both sides:
{{begin-eqn}}
{{eqn | l = \ln y
| r = \ln x^{a x}
}}
{{eqn | r = a x \ln x
| c = Laws of Logarithms
}}
{{eqn | l = \map {\frac \d {\d x} } {\ln y}
| r = \map {\frac \d {\d x} } {a x \ln x}
| c = differentiating both... | :$\dfrac \d {\d x} x^{a x} = a x^{a x} \paren {\ln x + 1}$ | Let $y := x^{a x}$.
As $x > 0$, we can take the [[Definition:Natural Logarithm|natural logarithm]] of both sides:
{{begin-eqn}}
{{eqn | l = \ln y
| r = \ln x^{a x}
}}
{{eqn | r = a x \ln x
| c = [[Laws of Logarithms]]
}}
{{eqn | l = \map {\frac \d {\d x} } {\ln y}
| r = \map {\frac \d {\d x} } {a x ... | Derivative of x to the a x | https://proofwiki.org/wiki/Derivative_of_x_to_the_a_x | https://proofwiki.org/wiki/Derivative_of_x_to_the_a_x | [
"Derivative of x to the x"
] | [] | [
"Definition:Natural Logarithm",
"Laws of Logarithms",
"Derivative of Composite Function",
"Derivative of Natural Logarithm Function",
"Product Rule for Derivatives",
"Derivative of Identity Function",
"Derivative of Natural Logarithm Function",
"Category:Derivative of x to the x"
] |
proofwiki-17162 | Integer as Sum of Polygonal Numbers/Lemma 1 | Let $n, m \in \N_{>0}$ such that $m \ge 3$.
Let $n < 116 m$.
Then $n$ can be expressed as a sum of at most $m + 2$ polygonal numbers of order $m + 2$. | From Closed Form for Polygonal Numbers:
:$\map P {m + 2, k} = \dfrac m 2 \paren {k^2 - k} + k = m T_{k - 1} + k$
Where $T_{k - 1}$ are triangular numbers.
The first few $\paren {m + 2}$-gonal numbers less than $116 m$ are:
:$0, 1, m + 2, 3 m + 3, 6 m + 4, 10 m + 5, 15 m + 6, 21 m + 7, 28 m + 8, 36 m + 9, 45 m + 10, 55 ... | Let $n, m \in \N_{>0}$ such that $m \ge 3$.
Let $n < 116 m$.
Then $n$ can be expressed as a [[Definition:Integer Addition|sum]] of at most $m + 2$ [[Definition:Polygonal Number|polygonal numbers of order $m + 2$]]. | From [[Closed Form for Polygonal Numbers]]:
:$\map P {m + 2, k} = \dfrac m 2 \paren {k^2 - k} + k = m T_{k - 1} + k$
Where $T_{k - 1}$ are [[Definition:Triangular Number|triangular numbers]].
The first few [[Definition:Polygonal Number|$\paren {m + 2}$-gonal numbers]] less than $116 m$ are:
:$0, 1, m + 2, 3 m + 3,... | Integer as Sum of Polygonal Numbers/Lemma 1 | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers/Lemma_1 | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers/Lemma_1 | [
"Integer as Sum of Polygonal Numbers"
] | [
"Definition:Addition/Integers",
"Definition:Polygonal Number"
] | [
"Closed Form for Polygonal Numbers",
"Definition:Triangular Number",
"Definition:Polygonal Number",
"Category:Integer as Sum of Polygonal Numbers"
] |
proofwiki-17163 | Integer as Sum of Polygonal Numbers/Lemma 3 | Let $n, m, r \in \R_{>0}$.
Suppose $\dfrac n m > 1$.
Let $b \in \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$.
Define:
:$a = 2 \paren {\dfrac {n - b - r} m} + b = \paren {1 - \dfrac 2 m} b + 2 \paren {\dfrac {n - r} m}$
Then $a, b$ satisfy:
:$b^2 < 4 a$
:$... | === $b^2 < 4 a$ ===
:$b^2 - 4 a = b^2 - 4 \paren {1 - \dfrac 2 m} b - 8 \paren {\dfrac {n - r} m}$
By the Quadratic Formula, $b^2 - 4 a < 0$ when $b$ is between:
{{begin-eqn}}
{{eqn | o =
| r = \frac 1 2 \paren {4 \paren {1 - \frac 2 m} \pm \sqrt {16 \paren {1 - \frac 2 m}^2 + 32 \paren {\frac {n - r} m} } }
}}
{... | Let $n, m, r \in \R_{>0}$.
Suppose $\dfrac n m > 1$.
Let $b \in \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$.
Define:
:$a = 2 \paren {\dfrac {n - b - r} m} + b = \paren {1 - \dfrac 2 m} b + 2 \paren {\dfrac {n - r} m}$
Then $a, b$ satisfy:
:$b^2 < 4... | === $b^2 < 4 a$ ===
:$b^2 - 4 a = b^2 - 4 \paren {1 - \dfrac 2 m} b - 8 \paren {\dfrac {n - r} m}$
By the [[Quadratic Formula]], $b^2 - 4 a < 0$ when $b$ is between:
{{begin-eqn}}
{{eqn | o =
| r = \frac 1 2 \paren {4 \paren {1 - \frac 2 m} \pm \sqrt {16 \paren {1 - \frac 2 m}^2 + 32 \paren {\frac {n - r} m} }... | Integer as Sum of Polygonal Numbers/Lemma 3 | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers/Lemma_3 | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers/Lemma_3 | [
"Integer as Sum of Polygonal Numbers"
] | [] | [
"Solution to Quadratic Equation",
"Solution to Quadratic Equation"
] |
proofwiki-17164 | Integer as Sum of Polygonal Numbers/Lemma 2 | Let $n, m \in \R_{>0}$ such that $\dfrac n m \ge 1$.
Define $I$ to be the open real interval:
:$I = \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$
Then:
:For $\dfrac n m \ge 116$, the length of $I$ is greater than $4$. | We need to show that $\paren {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } - \paren {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3}} > 4$ when $\dfrac n m \ge 116$.
Let $x = \dfrac n m - 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\frac 2 3 + \sqrt {8 \paren {\frac n m} - 8} } - \paren {\frac 1 2 + \sqrt {6 \paren... | Let $n, m \in \R_{>0}$ such that $\dfrac n m \ge 1$.
Define $I$ to be the [[Definition:Open Real Interval|open real interval]]:
:$I = \openint {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3} }$
Then:
:For $\dfrac n m \ge 116$, the [[Definition:Length of Real Interva... | We need to show that $\paren {\dfrac 2 3 + \sqrt {8 \paren {\dfrac n m} - 8} } - \paren {\dfrac 1 2 + \sqrt {6 \paren {\dfrac n m} - 3}} > 4$ when $\dfrac n m \ge 116$.
Let $x = \dfrac n m - 1$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {\frac 2 3 + \sqrt {8 \paren {\frac n m} - 8} } - \paren {\frac 1 2 + \sqrt {6 \pa... | Integer as Sum of Polygonal Numbers/Lemma 2 | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers/Lemma_2 | https://proofwiki.org/wiki/Integer_as_Sum_of_Polygonal_Numbers/Lemma_2 | [
"Integer as Sum of Polygonal Numbers"
] | [
"Definition:Real Interval/Open",
"Definition:Real Interval/Length"
] | [] |
proofwiki-17165 | Integer as Sum of Three Odd Squares | Let $r$ be a positive integer.
Then:
:$r \equiv 3 \pmod 8$
{{iff}}:
:$r$ is the sum of $3$ odd squares. | === Sufficient Condition ===
From Integer as Sum of Three Squares, every positive integer not of the form $4^n \paren {8 m + 7}$ can be expressed as the sum of three squares.
Hence every positive integer $r$ such that $r \equiv 3 \pmod 8$ can likewise be expressed as the sum of three squares.
From Square Modulo 8, the ... | Let $r$ be a [[Definition:Positive Integer|positive integer]].
Then:
:$r \equiv 3 \pmod 8$
{{iff}}:
:$r$ is the sum of $3$ [[Definition:Odd Square|odd squares]]. | === Sufficient Condition ===
From [[Integer as Sum of Three Squares]], every [[Definition:Positive Integer|positive integer]] not of the form $4^n \paren {8 m + 7}$ can be expressed as the sum of three [[Definition:Square Number|squares]].
Hence every [[Definition:Positive Integer|positive integer]] $r$ such that $r ... | Integer as Sum of Three Odd Squares | https://proofwiki.org/wiki/Integer_as_Sum_of_Three_Odd_Squares | https://proofwiki.org/wiki/Integer_as_Sum_of_Three_Odd_Squares | [
"Odd Squares",
"Number Theory"
] | [
"Definition:Positive/Integer",
"Definition:Odd Square"
] | [
"Integer as Sum of Three Squares",
"Definition:Positive/Integer",
"Definition:Square Number",
"Definition:Positive/Integer",
"Definition:Square Number",
"Square Modulo 8",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Square Number",
"Definition:Congruence (Number Theory)",
... |
proofwiki-17166 | Binet Form/Second Form | The recursive sequence:
:$V_n = m V_{n - 1} + V_{n - 2}$
where:
{{begin-eqn}}
{{eqn | l = V_0
| r = 2
| c =
}}
{{eqn | l = V_1
| r = m
| c =
}}
{{end-eqn}}
has the closed-form solution:
:$V_n = \alpha^n + \beta^n$
where $\Delta, \alpha, \beta$ are as for the first form. | Proof by induction:
Let $\map P n$ be the proposition:
:$V_n = \alpha^n + \beta^n$ | The [[Definition:Recursive Sequence|recursive sequence]]:
:$V_n = m V_{n - 1} + V_{n - 2}$
where:
{{begin-eqn}}
{{eqn | l = V_0
| r = 2
| c =
}}
{{eqn | l = V_1
| r = m
| c =
}}
{{end-eqn}}
has the [[Definition:Closed-Form Solution|closed-form solution]]:
:$V_n = \alpha^n + \beta^n$
where $\... | Proof by [[Principle of Mathematical Induction|induction]]:
Let $\map P n$ be the proposition:
:$V_n = \alpha^n + \beta^n$ | Binet Form/Second Form | https://proofwiki.org/wiki/Binet_Form/Second_Form | https://proofwiki.org/wiki/Binet_Form/Second_Form | [
"Binet Form"
] | [
"Definition:Recursive Sequence",
"Definition:Closed Form Solution",
"Binet Form"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-17167 | Join Semilattice Ordered Subset Not Always Subsemilattice | Let $\struct {S, \circ}$ be a semilattice.
Let $\preceq$ be the ordering on $S$ defined by:
:$a \preceq b \iff \paren {a \circ b} = b$
Let $T$ be a subset of $S$.
Let the ordered subset $\struct {T, \preceq \restriction_T}$ be a join semilattice.
Let $\vee$ be the binary operation on $T$ defined by:
:for all $a, b \in ... | Let $S = \set {p, q, r, s}$.
Let $\circ: S \times S \to S$ be defined by:
:$p \circ q = q \circ p = r$
:$p \circ r = r \circ p = r$
:$q \circ r = r \circ q = r$
:$p \circ s = s \circ p = s$
:$q \circ s = s \circ q = s$
:$r \circ s = s \circ r = s$
:$p \circ p = p$
:$q \circ q = q$
:$r \circ r = r$
:$s \circ s = s$
Then... | Let $\struct {S, \circ}$ be a [[Definition:Semilattice|semilattice]].
Let $\preceq$ be the [[Definition:Ordering|ordering]] on $S$ defined by:
:$a \preceq b \iff \paren {a \circ b} = b$
Let $T$ be a [[Definition:Subset|subset]] of $S$.
Let the [[Definition:Ordered Subset|ordered subset]] $\struct {T, \preceq \restr... | Let $S = \set {p, q, r, s}$.
Let $\circ: S \times S \to S$ be defined by:
:$p \circ q = q \circ p = r$
:$p \circ r = r \circ p = r$
:$q \circ r = r \circ q = r$
:$p \circ s = s \circ p = s$
:$q \circ s = s \circ q = s$
:$r \circ s = s \circ r = s$
:$p \circ p = p$
:$q \circ q = q$
:$r \circ r = r$
:$s \circ s = s$
Th... | Join Semilattice Ordered Subset Not Always Subsemilattice | https://proofwiki.org/wiki/Join_Semilattice_Ordered_Subset_Not_Always_Subsemilattice | https://proofwiki.org/wiki/Join_Semilattice_Ordered_Subset_Not_Always_Subsemilattice | [
"Subsemilattices"
] | [
"Definition:Semilattice",
"Definition:Ordering",
"Definition:Subset",
"Definition:Ordered Subset",
"Definition:Join Semilattice",
"Definition:Operation/Binary Operation",
"Definition:Join (Order Theory)",
"Definition:Subsemilattice"
] | [
"Definition:Semilattice",
"Definition:Ordering",
"Definition:Join (Order Theory)",
"Definition:Join Semilattice",
"Definition:Restriction/Operation",
"Definition:Subsemilattice"
] |
proofwiki-17168 | Closure of Subset of Closed Set of Topological Space is Subset | Let $T$ = $\struct {S, \tau}$ be a topological space.
Let $F$ be a closed set of $T$.
Let $H \subseteq F$ be a subset of $F$.
Let $H^-$ denote the closure of $H$.
Then $H^- \subseteq F$. | {{begin-eqn}}
{{eqn | l = H
| o = \subseteq
| r = F
}}
{{eqn | ll= \leadsto
| l = H^-
| o = \subseteq
| r = F^-
| c = Topological Closure of Subset is Subset of Topological Closure
}}
{{eqn | r = F
| c = Set is Closed iff Equals Topological Closure
}}
{{end-eqn}}
{{qed}} | Let $T$ = $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $F$ be a [[Definition:Closed Set (Topology)|closed set]] of $T$.
Let $H \subseteq F$ be a [[Definition:Subset|subset]] of $F$.
Let $H^-$ denote the [[Definition:Closure (Topology)|closure]] of $H$.
Then $H^- \subseteq F$. | {{begin-eqn}}
{{eqn | l = H
| o = \subseteq
| r = F
}}
{{eqn | ll= \leadsto
| l = H^-
| o = \subseteq
| r = F^-
| c = [[Topological Closure of Subset is Subset of Topological Closure]]
}}
{{eqn | r = F
| c = [[Set is Closed iff Equals Topological Closure]]
}}
{{end-eqn}}
{{qed}... | Closure of Subset of Closed Set of Topological Space is Subset/Proof 1 | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Topological_Space_is_Subset | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Topological_Space_is_Subset/Proof_1 | [
"Set Closures",
"Closure of Subset of Closed Set of Topological Space is Subset"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology",
"Definition:Subset",
"Definition:Closure (Topology)"
] | [
"Topological Closure of Subset is Subset of Topological Closure",
"Set is Closed iff Equals Topological Closure"
] |
proofwiki-17169 | Closure of Subset of Closed Set of Topological Space is Subset | Let $T$ = $\struct {S, \tau}$ be a topological space.
Let $F$ be a closed set of $T$.
Let $H \subseteq F$ be a subset of $F$.
Let $H^-$ denote the closure of $H$.
Then $H^- \subseteq F$. | Let $x \notin F$.
Then $x$ is in the open set $S \setminus F$.
From Set Difference with Subset is Superset of Set Difference:
:$S \setminus F \subseteq S \setminus H$
From Subsets of Disjoint Sets are Disjoint:
:$S \setminus F \cap H = \O$
From Set is Open iff Neighborhood of all its Points:
:$S \setminus F$ is an neig... | Let $T$ = $\struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $F$ be a [[Definition:Closed Set (Topology)|closed set]] of $T$.
Let $H \subseteq F$ be a [[Definition:Subset|subset]] of $F$.
Let $H^-$ denote the [[Definition:Closure (Topology)|closure]] of $H$.
Then $H^- \subseteq F$. | Let $x \notin F$.
Then $x$ is in the [[Definition:Open Set (Topology)|open set]] $S \setminus F$.
From [[Set Difference with Subset is Superset of Set Difference]]:
:$S \setminus F \subseteq S \setminus H$
From [[Subsets of Disjoint Sets are Disjoint]]:
:$S \setminus F \cap H = \O$
From [[Set is Open iff Neighborho... | Closure of Subset of Closed Set of Topological Space is Subset/Proof 2 | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Topological_Space_is_Subset | https://proofwiki.org/wiki/Closure_of_Subset_of_Closed_Set_of_Topological_Space_is_Subset/Proof_2 | [
"Set Closures",
"Closure of Subset of Closed Set of Topological Space is Subset"
] | [
"Definition:Topological Space",
"Definition:Closed Set/Topology",
"Definition:Subset",
"Definition:Closure (Topology)"
] | [
"Definition:Open Set/Topology",
"Set Difference with Subset is Superset of Set Difference",
"Subsets of Disjoint Sets are Disjoint",
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point",
"Definition:Closure (Topology)/Definition 6",
"Set Difference with Subset is ... |
proofwiki-17170 | Number of Partial Derivatives of Order n | Let $u = \map f {x_1, x_2, \ldots, x_m}$ be a function of the $m$ independent variables $x_1, x_2, \ldots, x_m$.
There are $m^n$ partial derivatives of $u$ of order $n$. | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:There are $m^n$ partial derivatives of $u$ of order $n$.
$\map P 0$ is the degenerate case where $f$ is not partially differentiated at all:
:$\map f {x_1, x_2, \ldots, x_m}$
and it is apparent that there is only $1 = m^0$ ... | Let $u = \map f {x_1, x_2, \ldots, x_m}$ be a [[Definition:Real-Valued Function|function]] of the $m$ [[Definition:Real Independent Variable|independent variables]] $x_1, x_2, \ldots, x_m$.
There are $m^n$ [[Definition:Partial Derivative|partial derivatives]] of $u$ of [[Definition:Order of Partial Derivative|order $... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:There are $m^n$ [[Definition:Partial Derivative|partial derivatives]] of $u$ of [[Definition:Order of Partial Derivative|order $n$]].
$\map P 0$ is the [... | Number of Partial Derivatives of Order n | https://proofwiki.org/wiki/Number_of_Partial_Derivatives_of_Order_n | https://proofwiki.org/wiki/Number_of_Partial_Derivatives_of_Order_n | [
"Partial Differentiation",
"Number of Partial Derivatives of Order n"
] | [
"Definition:Real-Valued Function",
"Definition:Independent Variable/Real Function",
"Definition:Partial Derivative",
"Definition:Partial Derivative/Order"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Partial Derivative",
"Definition:Partial Derivative/Order",
"Definition:Degenerate Case",
"Definition:Partial Derivative",
"Definition:Partial Derivative",
"Definition:Partial Derivative/Order",
"Definition:Partial Derivati... |
proofwiki-17171 | Normed Vector Space is Open in Itself | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Then the set $X$ is an open set of $M$. | By definition, an open set $S \subseteq A$ is one where every point inside it is an element of an open ball contained entirely within that set.
Let $x \in X$.
An open ball of $x$ in $M$ is by definition a subset of $X$.
Hence the result.
{{qed}} | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Set|set]] $X$ is an [[Definition:Open Set in Normed Vector Space|open set]] of $M$. | By definition, an [[Definition:Open Set in Normed Vector Space|open set]] $S \subseteq A$ is one where every point inside it is an element of an [[Definition:Open Ball in Normed Vector Space|open ball]] contained entirely within that set.
Let $x \in X$.
An [[Definition:Open Ball in Normed Vector Space|open ball]] of ... | Normed Vector Space is Open in Itself/Proof 1 | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself/Proof_1 | [
"Open Sets (Normed Vector Spaces)",
"Normed Vector Space is Open in Itself",
"Open Sets (Normed Vector Spaces)",
"Normed Vector Space is Open in Itself"
] | [
"Definition:Normed Vector Space",
"Definition:Set",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Subset"
] |
proofwiki-17172 | Normed Vector Space is Open in Itself | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Then the set $X$ is an open set of $M$. | By definition, a subset $S \subseteq X$ is open if:
:$\forall x \in X : \exists \epsilon \in \R_{>0} : \map {B_\epsilon} x \subseteq S$
Let $S = X$.
{{AimForCont}} $X$ is not open.
By De Morgan's laws:
:$\exists x \in X : \forall \epsilon \in \R_{>0} : \map {B_\epsilon} x \cap \paren {X \setminus S} \ne \O$
Note that:
... | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Set|set]] $X$ is an [[Definition:Open Set in Normed Vector Space|open set]] of $M$. | By [[Definition:Definition|definition]], a [[Definition:Subset|subset]] $S \subseteq X$ is [[Definition:Open Set in Normed Vector Space|open]] if:
:$\forall x \in X : \exists \epsilon \in \R_{>0} : \map {B_\epsilon} x \subseteq S$
Let $S = X$.
{{AimForCont}} $X$ is [[Definition:Logical Not|not]] [[Definition:Open Se... | Normed Vector Space is Open in Itself/Proof 2 | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself/Proof_2 | [
"Open Sets (Normed Vector Spaces)",
"Normed Vector Space is Open in Itself",
"Open Sets (Normed Vector Spaces)",
"Normed Vector Space is Open in Itself"
] | [
"Definition:Normed Vector Space",
"Definition:Set",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Definition",
"Definition:Subset",
"Definition:Open Set/Normed Vector Space",
"Definition:Logical Not",
"Definition:Open Set/Normed Vector Space",
"De Morgan's Laws (Predicate Logic)",
"Intersection with Empty Set",
"Empty Set is Unique",
"Definition:Contradiction"
] |
proofwiki-17173 | Normed Vector Space is Open in Itself/Proof 1 | Let $M = \struct{X, \norm {\, \cdot \,}}$ be a normed vector space.
Then the set $X$ is an open set of $M$. | By definition, an open set $S \subseteq A$ is one where every point inside it is an element of an open ball contained entirely within that set.
Let $x \in X$.
An open ball of $x$ in $M$ is by definition a subset of $X$.
Hence the result.
{{qed}} | Let $M = \struct{X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Set|set]] $X$ is an [[Definition:Open Set in Normed Vector Space|open set]] of $M$. | By definition, an [[Definition:Open Set in Normed Vector Space|open set]] $S \subseteq A$ is one where every point inside it is an element of an [[Definition:Open Ball in Normed Vector Space|open ball]] contained entirely within that set.
Let $x \in X$.
An [[Definition:Open Ball in Normed Vector Space|open ball]] of ... | Normed Vector Space is Open in Itself/Proof 1 | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself/Proof_1 | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself/Proof_1 | [
"Normed Vector Spaces",
"Open Sets",
"Normed Vector Space is Open in Itself",
"Normed Vector Space is Open in Itself"
] | [
"Definition:Normed Vector Space",
"Definition:Set",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Subset"
] |
proofwiki-17174 | Normed Vector Space is Open in Itself/Proof 2 | Let $M = \struct{X, \norm {\, \cdot \,}}$ be a normed vector space.
Then the set $X$ is an open set of $M$. | By definition, a subset $S \subseteq X$ is open if:
:$\forall x \in X : \exists \epsilon \in \R_{>0} : \map {B_\epsilon} x \subseteq S$
Let $S = X$.
{{AimForCont}} $X$ is not open.
By De Morgan's laws:
:$\exists x \in X : \forall \epsilon \in \R_{>0} : \map {B_\epsilon} x \cap \paren {X \setminus S} \ne \O$
Note that:
... | Let $M = \struct{X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Set|set]] $X$ is an [[Definition:Open Set in Normed Vector Space|open set]] of $M$. | By [[Definition:Definition|definition]], a [[Definition:Subset|subset]] $S \subseteq X$ is [[Definition:Open Set in Normed Vector Space|open]] if:
:$\forall x \in X : \exists \epsilon \in \R_{>0} : \map {B_\epsilon} x \subseteq S$
Let $S = X$.
{{AimForCont}} $X$ is [[Definition:Logical Not|not]] [[Definition:Open Se... | Normed Vector Space is Open in Itself/Proof 2 | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself/Proof_2 | https://proofwiki.org/wiki/Normed_Vector_Space_is_Open_in_Itself/Proof_2 | [
"Normed Vector Space is Open in Itself"
] | [
"Definition:Normed Vector Space",
"Definition:Set",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Definition",
"Definition:Subset",
"Definition:Open Set/Normed Vector Space",
"Definition:Logical Not",
"Definition:Open Set/Normed Vector Space",
"De Morgan's Laws (Predicate Logic)",
"Intersection with Empty Set",
"Empty Set is Unique",
"Definition:Contradiction"
] |
proofwiki-17175 | Empty Set is Closed/Normed Vector Space | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Then the empty set $\O$ is closed in $M$. | From Normed Vector Space is Open in Itself, $X$ is open in $M$.
But:
:$\O = \relcomp X X$
where $\complement_X$ denotes the set complement relative to $X$.
The result follows by definition of closed set.
{{qed}} | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Empty Set|empty set]] $\O$ is [[Definition:Closed Set in Normed Vector Space|closed]] in $M$. | From [[Normed Vector Space is Open in Itself]], $X$ is [[Definition:Open Set in Normed Vector Space|open]] in $M$.
But:
:$\O = \relcomp X X$
where $\complement_X$ denotes the [[Definition:Relative Complement|set complement relative to $X$]].
The result follows by definition of [[Definition:Closed Set in Normed Vector... | Empty Set is Closed/Normed Vector Space | https://proofwiki.org/wiki/Empty_Set_is_Closed/Normed_Vector_Space | https://proofwiki.org/wiki/Empty_Set_is_Closed/Normed_Vector_Space | [
"Empty Set",
"Empty Set is Closed",
"Empty Set is Closed",
"Closed Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Empty Set",
"Definition:Closed Set/Normed Vector Space"
] | [
"Normed Vector Space is Open in Itself",
"Definition:Open Set/Normed Vector Space",
"Definition:Relative Complement",
"Definition:Closed Set/Normed Vector Space"
] |
proofwiki-17176 | Empty Set is Open in Normed Vector Space | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Then the empty set $\O$ is an open set of $M$. | By definition, an open set $S \subseteq X$ is one where every point inside it is an element of an open ball contained entirely within that set.
That is, there are no points in $S$ which have an open ball some of whose elements are not in $S$.
As there are no elements in $\O$, the result follows vacuously.
{{qed}} | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Empty Set|empty set]] $\O$ is an [[Definition:Open Set in Normed Vector Space|open set]] of $M$. | By definition, an [[Definition:Open Set in Normed Vector Space|open set]] $S \subseteq X$ is one where every [[Definition:Point|point]] inside it is an [[Definition:Element|element]] of an [[Definition:Open Ball|open ball]] contained entirely within that [[Definition:Set|set]].
That is, there are no [[Definition:Point... | Empty Set is Open in Normed Vector Space | https://proofwiki.org/wiki/Empty_Set_is_Open_in_Normed_Vector_Space | https://proofwiki.org/wiki/Empty_Set_is_Open_in_Normed_Vector_Space | [
"Normed Vector Spaces",
"Open Sets",
"Open Sets (Normed Vector Spaces)",
"Empty Set",
"Open Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Empty Set",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Open Set/Normed Vector Space",
"Definition:Point",
"Definition:Element",
"Definition:Open Ball",
"Definition:Set",
"Definition:Point",
"Definition:Open Ball/Normed Vector Space",
"Definition:Element",
"Definition:Element",
"Definition:Vacuous Truth"
] |
proofwiki-17177 | Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.
Then $\sequence {f_n}$ is uniformly Cauchy on $S$ {{iff}} $\sequence {f_n}$ converges uniformly on $S$. | === Sufficient Condition ===
Let $\sequence {f_n}$ be uniformly Cauchy on $S$.
{{:Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition}} | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $S \to \R$.
Then $\sequence {f_n}$ is [[Definition:Uniform Cauchy Criterion|uniformly Cauchy]] on $S$ {{iff}} $\sequence {f_n}$ [[Definition:Uniformly Convergent Real Sequence|converges u... | === [[Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition|Sufficient Condition]] ===
Let $\sequence {f_n}$ be [[Definition:Uniform Cauchy Criterion|uniformly Cauchy]] on $S$.
{{:Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition}} | Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent | https://proofwiki.org/wiki/Sequence_of_Functions_is_Uniformly_Cauchy_iff_Uniformly_Convergent | https://proofwiki.org/wiki/Sequence_of_Functions_is_Uniformly_Cauchy_iff_Uniformly_Convergent | [
"Uniform Convergence",
"Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Uniform Cauchy Criterion",
"Definition:Uniform Convergence/Real Sequence"
] | [
"Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition",
"Definition:Uniform Cauchy Criterion"
] |
proofwiki-17178 | Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.
Let $\sequence {f_n}$ be uniformly Cauchy on $S$.
Then $\sequence {f_n}$ is uniformly convergent on $S$. | Let $\epsilon \in \R_{> 0}$ be arbitrary.
As $\sequence {f_n}$ is uniformly Cauchy on $S$, there exists $N \in \N$ such that:
:$\size {\map {f_n} x - \map {f_m} x} < \dfrac \epsilon 2$
for all $n, m > N$ and $x \in S$.
Let $x \in S$ be fixed.
Then:
:$\size {\map {f_n} x - \map {f_m} x} < \epsilon$
for all $n, m > N$.... | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $S \to \R$.
Let $\sequence {f_n}$ be [[Definition:Uniform Cauchy Criterion|uniformly Cauchy]] on $S$.
Then $\sequence {f_n}$ is [[Definition:Uniform Convergence|uniformly convergent]] on... | Let $\epsilon \in \R_{> 0}$ be arbitrary.
As $\sequence {f_n}$ is [[Definition:Uniform Cauchy Criterion|uniformly Cauchy]] on $S$, there exists $N \in \N$ such that:
:$\size {\map {f_n} x - \map {f_m} x} < \dfrac \epsilon 2$
for all $n, m > N$ and $x \in S$.
Let $x \in S$ be fixed.
Then:
:$\size {\map {f_n} x ... | Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Sufficient Condition | https://proofwiki.org/wiki/Sequence_of_Functions_is_Uniformly_Cauchy_iff_Uniformly_Convergent/Sufficient_Condition | https://proofwiki.org/wiki/Sequence_of_Functions_is_Uniformly_Cauchy_iff_Uniformly_Convergent/Sufficient_Condition | [
"Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Uniform Cauchy Criterion",
"Definition:Uniform Convergence"
] | [
"Definition:Uniform Cauchy Criterion",
"Definition:Sequence",
"Definition:Cauchy Criterion",
"Cauchy's Convergence Criterion/Real Numbers",
"Definition:Convergent Sequence",
"Definition:Uniform Convergence",
"Definition:Uniform Convergence"
] |
proofwiki-17179 | Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Necessary Condition | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.
Let $\sequence {f_n}$ be uniformly convergent on $S$.
Then $\sequence {f_n}$ is uniformly Cauchy on $S$. | Let $\epsilon \in \R_{> 0}$ be arbitrary.
Since $f_n \to f$ uniformly, there exists some $N \in \N$ such that:
:$\size {\map {f_n} x - \map f x} < \dfrac \epsilon 2$
for all $x \in S$ and $n > N$.
Then if $x \in S$ and $n, m > N$, we have:
{{begin-eqn}}
{{eqn | l = \size {\map {f_n} x - \map {f_m} x}
| r = \size {\m... | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $S \to \R$.
Let $\sequence {f_n}$ be [[Definition:Uniform Convergence|uniformly convergent]] on $S$.
Then $\sequence {f_n}$ is [[Definition:Uniform Cauchy Criterion|uniformly Cauchy]] on ... | Let $\epsilon \in \R_{> 0}$ be arbitrary.
Since $f_n \to f$ [[Definition:Uniform Convergence|uniformly]], there exists some $N \in \N$ such that:
:$\size {\map {f_n} x - \map f x} < \dfrac \epsilon 2$
for all $x \in S$ and $n > N$.
Then if $x \in S$ and $n, m > N$, we have:
{{begin-eqn}}
{{eqn | l = \size {\map ... | Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent/Necessary Condition | https://proofwiki.org/wiki/Sequence_of_Functions_is_Uniformly_Cauchy_iff_Uniformly_Convergent/Necessary_Condition | https://proofwiki.org/wiki/Sequence_of_Functions_is_Uniformly_Cauchy_iff_Uniformly_Convergent/Necessary_Condition | [
"Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Uniform Convergence",
"Definition:Uniform Cauchy Criterion"
] | [
"Definition:Uniform Convergence",
"Triangle Inequality/Real Numbers",
"Definition:Uniform Cauchy Criterion"
] |
proofwiki-17180 | Open Ball of Point Inside Open Ball/Normed Vector Space | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct {X, \norm {\, \cdot \,} }$.
Let $y \in \map {B_\epsilon} x$.
Then:
:$\exists \delta \in \R: \map {B_\delta} y \subseteq \map {B_\epsilon} x$
That is, for every point in an open $\ep... | Let $\delta = \epsilon - \norm {x - y}$.
From the definition of open ball, this is strictly positive, since $y \in \map {B_\epsilon} x$.
Let $z \in \map {B_\delta} y$ be arbitrary.
We have:
{{begin-eqn}}
{{eqn | l = z
| o = \in
| r = \map {B_\delta} y
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
... | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\map {B_\epsilon} x$ be an [[Definition:Open Ball in Normed Vector Space|open $\epsilon$-ball]] in $M = \struct {X, \norm {\, \cdot \,} }$.
Let $y \in \map {B_\epsilon} x$.
Then:
:$\exists \delta \in \R: \m... | Let $\delta = \epsilon - \norm {x - y}$.
From the definition of [[Definition:Open Ball in Normed Vector Space|open ball]], this is [[Definition:Strictly Positive|strictly positive]], since $y \in \map {B_\epsilon} x$.
Let $z \in \map {B_\delta} y$ be arbitrary.
We have:
{{begin-eqn}}
{{eqn | l = z
| o = \in
... | Open Ball of Point Inside Open Ball/Normed Vector Space | https://proofwiki.org/wiki/Open_Ball_of_Point_Inside_Open_Ball/Normed_Vector_Space | https://proofwiki.org/wiki/Open_Ball_of_Point_Inside_Open_Ball/Normed_Vector_Space | [
"Open Balls"
] | [
"Definition:Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space"
] | [
"Definition:Open Ball/Normed Vector Space",
"Definition:Strictly Positive",
"Triangle Inequality"
] |
proofwiki-17181 | Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function | Let $S \subseteq \R$.
Let $x \in S$.
Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$ converging uniformly to $f: S \to \R$.
Let $f_n$ be continuous at $x$ for all $n \in \N$.
Then $f$ is continuous at $x$. | Let $\epsilon \in \R_{> 0}$.
Since $f_n \to f$ uniformly, there exists some $N \in \N$ such that:
:$\size {\map {f_n} x - \map f x} < \dfrac \epsilon 3$
for all $x \in S$ and $n \ge N$.
Since $f_N$ is continuous at $x$, there exists some $\delta > 0$ such that:
:for all $y$ with $\size {x - y} < \delta$, we have $\s... | Let $S \subseteq \R$.
Let $x \in S$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $S \to \R$ [[Definition:Uniformly Convergent Real Sequence|converging uniformly]] to $f: S \to \R$.
Let $f_n$ be [[Definition:Continuous Real Function|continuous]] at $x$ fo... | Let $\epsilon \in \R_{> 0}$.
Since $f_n \to f$ [[Definition:Uniformly Convergent Real Sequence|uniformly]], there exists some $N \in \N$ such that:
:$\size {\map {f_n} x - \map f x} < \dfrac \epsilon 3$
for all $x \in S$ and $n \ge N$.
Since $f_N$ is [[Definition:Continuous Real Function|continuous]] at $x$, ther... | Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_of_Continuous_Functions_Converges_to_Continuous_Function | https://proofwiki.org/wiki/Uniformly_Convergent_Sequence_of_Continuous_Functions_Converges_to_Continuous_Function | [
"Uniform Convergence",
"Continuous Functions"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Uniform Convergence/Real Sequence",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function"
] | [
"Definition:Uniform Convergence/Real Sequence",
"Definition:Continuous Real Function",
"Triangle Inequality/Real Numbers",
"Definition:Continuous Real Function"
] |
proofwiki-17182 | Open Ball is Open Set/Normed Vector Space | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Let $x \in X$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball of $x$ in $M$.
Then $\map {B_\epsilon} x$ is an open set of $M$. | Let $y \in \map {B_\epsilon} x$.
From Open Ball of Point Inside Open Ball in Normed Vector Space, there exists $\delta \in \R_{>0}$ such that $\map {B_\delta} y \subseteq \map {B_\epsilon} x$
The result follows from the definition of open set.
{{qed}} | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $x \in X$.
Let $\epsilon \in \R_{>0}$.
Let $\map {B_\epsilon} x$ be an [[Definition:Open Ball in Normed Vector Space|open $\epsilon$-ball]] of $x$ in $M$.
Then $\map {B_\epsilon} x$ is an [[Definition:Open S... | Let $y \in \map {B_\epsilon} x$.
From [[Open Ball of Point Inside Open Ball in Normed Vector Space]], there exists $\delta \in \R_{>0}$ such that $\map {B_\delta} y \subseteq \map {B_\epsilon} x$
The result follows from the definition of [[Definition:Open Set in Normed Vector Space|open set]].
{{qed}} | Open Ball is Open Set/Normed Vector Space | https://proofwiki.org/wiki/Open_Ball_is_Open_Set/Normed_Vector_Space | https://proofwiki.org/wiki/Open_Ball_is_Open_Set/Normed_Vector_Space | [
"Open Balls",
"Open Ball is Open Set",
"Open Ball is Open Set",
"Open Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Open Ball/Normed Vector Space",
"Definition:Open Set/Normed Vector Space"
] | [
"Open Ball of Point Inside Open Ball/Normed Vector Space",
"Definition:Open Set/Normed Vector Space"
] |
proofwiki-17183 | Finite Intersection of Open Sets of Normed Vector Space is Open | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $U_1, U_2, \ldots, U_n$ be open in $M$.
Then $\ds \bigcap_{i \mathop = 1}^n U_i$ is open in $M$.
That is, a finite intersection of open subsets is open. | Let $\ds x \in \bigcap_{i \mathop = 1}^n U_i$.
For each $i \in \closedint 1 n$, we have $x \in U_i$.
Thus:
:$\exists \epsilon_i > 0: \map {B_{\epsilon_i}} x \subseteq U_i$
where $\map {B_{\epsilon_i}} x$ is the open $\epsilon_i$-ball of $x$.
Let $\ds \epsilon = \min_{i \mathop = 1}^n \set {\epsilon_i}$.
So:
:$\epsilon ... | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $U_1, U_2, \ldots, U_n$ be [[Definition:Open Set in Normed Vector Space|open]] in $M$.
Then $\ds \bigcap_{i \mathop = 1}^n U_i$ is [[Definition:Open Set in Normed Vector Space|open]] in $M$.
That is, a [[De... | Let $\ds x \in \bigcap_{i \mathop = 1}^n U_i$.
For each $i \in \closedint 1 n$, we have $x \in U_i$.
Thus:
:$\exists \epsilon_i > 0: \map {B_{\epsilon_i}} x \subseteq U_i$
where $\map {B_{\epsilon_i}} x$ is the [[Definition:Open Ball in Normed Vector Space|open $\epsilon_i$-ball]] of $x$.
Let $\ds \epsilon = \min_{... | Finite Intersection of Open Sets of Normed Vector Space is Open | https://proofwiki.org/wiki/Finite_Intersection_of_Open_Sets_of_Normed_Vector_Space_is_Open | https://proofwiki.org/wiki/Finite_Intersection_of_Open_Sets_of_Normed_Vector_Space_is_Open | [
"Open Sets",
"Open Sets (Normed Vector Spaces)",
"Set Intersection",
"Open Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Set Intersection/Finite Intersection",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Open Ball/Normed Vector Space"
] |
proofwiki-17184 | Union of Open Sets of Normed Vector Space is Open | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
The union of a set of open sets of $M$ is open in $M$. | Let $I$ be any indexing set.
Let $U_i$ be open in $M$ for all $i \in I$.
Let $\ds x \in \bigcup_{i \mathop \in I} U_i$.
Then $x \in U_k$ for some $k \in I$.
Since $U_k$ is open in $M$:
:$\ds \exists \epsilon > 0: \map {B_\epsilon} x \subseteq U_k \subseteq \bigcup_{i \mathop \in I} U_i$
where $\map {B_\epsilon} x$ is t... | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
The [[Definition:Set Union|union]] of a [[Definition:Set|set]] of [[Definition:Open Set in Normed Vector Space|open sets]] of $M$ is [[Definition:Open Set in Normed Vector Space|open in $M$]]. | Let $I$ be any [[Definition:Indexing Set|indexing set]].
Let $U_i$ be [[Definition:Open Set in Normed Vector Space|open in $M$]] for all $i \in I$.
Let $\ds x \in \bigcup_{i \mathop \in I} U_i$.
Then $x \in U_k$ for some $k \in I$.
Since $U_k$ is [[Definition:Open Set in Normed Vector Space|open in $M$]]:
:$\ds \... | Union of Open Sets of Normed Vector Space is Open | https://proofwiki.org/wiki/Union_of_Open_Sets_of_Normed_Vector_Space_is_Open | https://proofwiki.org/wiki/Union_of_Open_Sets_of_Normed_Vector_Space_is_Open | [
"Open Sets (Normed Vector Spaces)",
"Set Union"
] | [
"Definition:Normed Vector Space",
"Definition:Set Union",
"Definition:Set",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space"
] | [
"Definition:Indexing Set",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Open Ball/Normed Vector Space"
] |
proofwiki-17185 | Definite Integral of Limit of Uniformly Convergent Sequence of Integrable Functions | Let $a, b \in \R$ with $a < b$.
Let $\sequence {f_n}$ be a sequence of Riemann integrable real functions $\closedint a b \to \R$ converging uniformly to $f : \closedint a b \to \R$.
Then $f$ is integrable, and:
:$\ds \int_a^b \map f x \rd x = \lim_{n \mathop \to \infty} \int_a^b \map {f_n} x \rd x$ | By Limit of Uniformly Convergent Sequence of Integrable Functions is Integrable, $f$ is integrable.
We have:
{{begin-eqn}}
{{eqn | l = \size {\int_a^b \map f x \rd x - \int_a^b \map {f_n} x \rd x}
| r = \size {\int_a^b \paren {\map f x - \map {f_n} x} \rd x}
}}
{{eqn | o = \le
| r = \int_a^b \size {\map f x... | Let $a, b \in \R$ with $a < b$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Riemann Integrable Function|Riemann integrable]] [[Definition:Real Function|real functions]] $\closedint a b \to \R$ [[Definition:Uniform Convergence|converging uniformly]] to $f : \closedint a b \to \R$.
The... | By [[Limit of Uniformly Convergent Sequence of Integrable Functions is Integrable]], $f$ is integrable.
We have:
{{begin-eqn}}
{{eqn | l = \size {\int_a^b \map f x \rd x - \int_a^b \map {f_n} x \rd x}
| r = \size {\int_a^b \paren {\map f x - \map {f_n} x} \rd x}
}}
{{eqn | o = \le
| r = \int_a^b \size {\m... | Definite Integral of Limit of Uniformly Convergent Sequence of Integrable Functions | https://proofwiki.org/wiki/Definite_Integral_of_Limit_of_Uniformly_Convergent_Sequence_of_Integrable_Functions | https://proofwiki.org/wiki/Definite_Integral_of_Limit_of_Uniformly_Convergent_Sequence_of_Integrable_Functions | [
"Integral Calculus",
"Uniform Convergence"
] | [
"Definition:Sequence",
"Definition:Definite Integral/Riemann",
"Definition:Real Function",
"Definition:Uniform Convergence",
"Definition:Definite Integral/Riemann"
] | [
"Limit of Uniformly Convergent Sequence of Integrable Functions is Integrable",
"Triangle Inequality for Integrals",
"Darboux's Theorem",
"Definition:Uniform Convergence",
"Definition:Uniform Convergence",
"Category:Integral Calculus",
"Category:Uniform Convergence"
] |
proofwiki-17186 | Events One of Which equals Union | Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.
Let $A$ and $B$ be such that:
:$A \cup B = A$
Then whenever $B$ occurs, it is always the case that $A$ occurs as well. | From Union with Superset is Superset:
:$A \cup B = A \iff B \subseteq A$
Let $B$ occur.
Let $\omega$ be the outcome of $\EE$.
Let $\omega \in B$.
That is, by definition of occurrence of event, $B$ occurs.
Then by definition of subset:
:$\omega \in A$
Thus by definition of occurrence of event, $A$ occurs.
Hence the resu... | Let the [[Definition:Probability Space|probability space]] of an [[Definition:Experiment|experiment]] $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be [[Definition:Event|events]] of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.
Let $A$ and $B$ be such that:
:$A \cup B = A$
Then when... | From [[Union with Superset is Superset]]:
:$A \cup B = A \iff B \subseteq A$
Let $B$ [[Definition:Occurrence of Event|occur]].
Let $\omega$ be the [[Definition:Outcome|outcome]] of $\EE$.
Let $\omega \in B$.
That is, by definition of [[Definition:Occurrence of Event|occurrence of event]], $B$ [[Definition:Occurren... | Events One of Which equals Union | https://proofwiki.org/wiki/Events_One_of_Which_equals_Union | https://proofwiki.org/wiki/Events_One_of_Which_equals_Union | [
"Events One of Which equals Union",
"Unions of Events"
] | [
"Definition:Probability Space",
"Definition:Experiment",
"Definition:Event",
"Definition:Event/Occurrence",
"Definition:Event/Occurrence"
] | [
"Union with Superset is Superset",
"Definition:Event/Occurrence",
"Definition:Elementary Event",
"Definition:Event/Occurrence",
"Definition:Event/Occurrence",
"Definition:Subset",
"Definition:Event/Occurrence",
"Definition:Event/Occurrence"
] |
proofwiki-17187 | De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Corollary | :$T_1 \cup T_2 = \overline {\overline T_1 \cap \overline T_2}$ | {{begin-eqn}}
{{eqn | l = T_1 \cup T_2
| r = \overline {\overline {T_1 \cup T_2} }
| c = Complement of Complement
}}
{{eqn | r = \overline {\overline T_1 \cap \overline T_2}
| c = De Morgan's Laws: Complement of Union
}}
{{end-eqn}}
{{qed}} | :$T_1 \cup T_2 = \overline {\overline T_1 \cap \overline T_2}$ | {{begin-eqn}}
{{eqn | l = T_1 \cup T_2
| r = \overline {\overline {T_1 \cup T_2} }
| c = [[Complement of Complement]]
}}
{{eqn | r = \overline {\overline T_1 \cap \overline T_2}
| c = [[De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|De Morgan's Laws: Complement of Union]]
}}
{{end-eq... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Corollary | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Union/Corollary | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Union/Corollary | [
"De Morgan's Laws"
] | [] | [
"Complement of Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Union"
] |
proofwiki-17188 | Intersection of Closed Sets is Closed/Normed Vector Space | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Then the intersection of an arbitrary number of closed sets of $M$ (either finitely or infinitely many) is itself closed. | Let $I$ be an indexing set (either finite or infinite).
Let $\ds \bigcap_{i \mathop \in I} V_i$ be the intersection of a indexed family of closed sets of $M$ indexed by $I$.
By definition of closed set, each of $X \setminus V_i$ are by definition open in $M$.
From De Morgan's laws: Difference with Intersection:
:$\ds X... | Let $M = \struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]].
Then the [[Definition:Set Intersection|intersection]] of an arbitrary number of [[Definition:Closed Set in Normed Vector Space|closed sets]] of $M$ (either [[Definition:Finite Set|finitely]] or [[Definition:Infini... | Let $I$ be an [[Definition:Indexing Set|indexing set]] (either [[Definition:Finite|finite]] or [[Definition:Infinite|infinite]]).
Let $\ds \bigcap_{i \mathop \in I} V_i$ be the [[Definition:Set Intersection|intersection]] of a [[Definition:Indexed Family of Subsets|indexed family]] of [[Definition:Closed Set in Normed... | Intersection of Closed Sets is Closed/Normed Vector Space | https://proofwiki.org/wiki/Intersection_of_Closed_Sets_is_Closed/Normed_Vector_Space | https://proofwiki.org/wiki/Intersection_of_Closed_Sets_is_Closed/Normed_Vector_Space | [
"Closed Sets (Normed Vector Spaces)",
"Closed Sets",
"Intersection of Closed Sets is Closed",
"Intersection of Closed Sets is Closed",
"Closed Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Set Intersection",
"Definition:Closed Set/Normed Vector Space",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Closed Set/Normed Vector Space"
] | [
"Definition:Indexing Set",
"Definition:Finite",
"Definition:Infinite",
"Definition:Set Intersection",
"Definition:Indexing Set/Family of Subsets",
"Definition:Closed Set/Normed Vector Space",
"Definition:Closed Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"De Morgan's Laws (S... |
proofwiki-17189 | Normed Vector Space is Closed in Itself | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.
Then $X$ is closed in $M$. | From Empty Set is Open in Normed Vector Space, $\O$ is open in $M$.
But:
:$X = \relcomp X \O$
where $\complement_X$ denotes the set complement relative to $X$.
The result follows by definition of closed set.
{{qed}} | Let $M = \struct {X, \norm {\, \cdot \,}}$ be a [[Definition:Normed Vector Space|normed vector space]].
Then $X$ is [[Definition:Closed Set in Normed Vector Space|closed]] in $M$. | From [[Empty Set is Open in Normed Vector Space]], $\O$ is [[Definition:Open Set in Normed Vector Space|open]] in $M$.
But:
:$X = \relcomp X \O$
where $\complement_X$ denotes the [[Definition:Relative Complement|set complement relative to $X$]].
The result follows by definition of [[Definition:Closed Set in Normed Ve... | Normed Vector Space is Closed in Itself | https://proofwiki.org/wiki/Normed_Vector_Space_is_Closed_in_Itself | https://proofwiki.org/wiki/Normed_Vector_Space_is_Closed_in_Itself | [
"Normed Vector Spaces",
"Closed Sets",
"Closed Sets (Normed Vector Spaces)",
"Closed Sets (Normed Vector Spaces)"
] | [
"Definition:Normed Vector Space",
"Definition:Closed Set/Normed Vector Space"
] | [
"Empty Set is Open in Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Relative Complement",
"Definition:Closed Set/Normed Vector Space"
] |
proofwiki-17190 | Infinite Series of Functions is Uniformly Convergent iff Sequence of Partial Sums is Uniformly Cauchy | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a sequence of real functions $S \to \R$.
Then the infinite series:
:$\ds \sum_{n \mathop = 1}^\infty f_n$
converges uniformly on $S$ {{iff}} for all $\varepsilon \in \R_{> 0}$ there exists $N \in \N$ such that:
:$\ds \size {\sum_{k \mathop = m + 1}^n \map {f_k} x} < \vare... | Let $\sequence {s_n}$ be a sequence of real functions $S \to \R$ with:
:$\ds \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_k} x$
for each $n \in \N$ and $x \in S$.
By definition of uniform convergence of an infinite series:
:$\ds \sum_{n \mathop = 1}^\infty f_n$ is uniformly convergent {{iff}} $\sequence {s_n}$ is uni... | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $S \to \R$.
Then the infinite series:
:$\ds \sum_{n \mathop = 1}^\infty f_n$
[[Definition:Uniformly Convergent Infinite Series|converges uniformly]] on $S$ {{iff}} for all $\varepsilon ... | Let $\sequence {s_n}$ be a sequence of real functions $S \to \R$ with:
:$\ds \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_k} x$
for each $n \in \N$ and $x \in S$.
By definition of [[Definition:Uniformly Convergent Infinite Series|uniform convergence of an infinite series]]:
:$\ds \sum_{n \mathop = 1}^\infty f_n$ ... | Infinite Series of Functions is Uniformly Convergent iff Sequence of Partial Sums is Uniformly Cauchy | https://proofwiki.org/wiki/Infinite_Series_of_Functions_is_Uniformly_Convergent_iff_Sequence_of_Partial_Sums_is_Uniformly_Cauchy | https://proofwiki.org/wiki/Infinite_Series_of_Functions_is_Uniformly_Convergent_iff_Sequence_of_Partial_Sums_is_Uniformly_Cauchy | [
"Uniform Convergence"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Uniform Convergence/Infinite Series"
] | [
"Definition:Uniform Convergence/Infinite Series",
"Definition:Uniform Convergence/Infinite Series",
"Definition:Uniform Convergence/Real Sequence",
"Sequence of Functions is Uniformly Cauchy iff Uniformly Convergent",
"Definition:Uniform Convergence/Infinite Series",
"Definition:Uniform Cauchy Criterion"
... |
proofwiki-17191 | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Corollary | :$T_1 \cap T_2 = \overline {\overline T_1 \cup \overline T_2}$ | {{begin-eqn}}
{{eqn | l = T_1 \cap T_2
| r = \overline {\overline {T_1 \cap T_2} }
| c = Complement of Complement
}}
{{eqn | r = \overline {\overline T_1 \cup \overline T_2}
| c = De Morgan's Laws: Complement of Intersection
}}
{{end-eqn}}
{{qed}} | :$T_1 \cap T_2 = \overline {\overline T_1 \cup \overline T_2}$ | {{begin-eqn}}
{{eqn | l = T_1 \cap T_2
| r = \overline {\overline {T_1 \cap T_2} }
| c = [[Complement of Complement]]
}}
{{eqn | r = \overline {\overline T_1 \cup \overline T_2}
| c = [[De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection|De Morgan's Laws: Complement of Intersection... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Corollary | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Corollary | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Corollary | [
"De Morgan's Laws"
] | [] | [
"Complement of Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection"
] |
proofwiki-17192 | LCM of 3 Integers in terms of GCDs of Pairs of those Integers/Lemma | Let $a, b, c \in \Z_{>0}$ be strictly positive integers.
Then:
:$\gcd \set {\gcd \set {a, b}, \gcd \set {a, c} } = \gcd \set {a, b, c}$ | Let $\gcd \set {a, b, c} = d_1$.
From definition:
:$d_1 \divides a$, $d_1 \divides b$ and $d_1 \divides c$.
By Common Divisor Divides GCD:
:$d_1 \divides \gcd \set {a, b}$ and $d_1 \divides \gcd \set {a, c}$.
By Common Divisor Divides GCD again:
:$d_1 \divides \gcd \set {\gcd \set {a, b}, \gcd \set {a, c} }$.
On the ot... | Let $a, b, c \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Then:
:$\gcd \set {\gcd \set {a, b}, \gcd \set {a, c} } = \gcd \set {a, b, c}$ | Let $\gcd \set {a, b, c} = d_1$.
From [[Definition:Greatest Common Divisor of Integers|definition]]:
:$d_1 \divides a$, $d_1 \divides b$ and $d_1 \divides c$.
By [[Common Divisor Divides GCD]]:
:$d_1 \divides \gcd \set {a, b}$ and $d_1 \divides \gcd \set {a, c}$.
By [[Common Divisor Divides GCD]] again:
:$d_1 \divid... | LCM of 3 Integers in terms of GCDs of Pairs of those Integers/Lemma | https://proofwiki.org/wiki/LCM_of_3_Integers_in_terms_of_GCDs_of_Pairs_of_those_Integers/Lemma | https://proofwiki.org/wiki/LCM_of_3_Integers_in_terms_of_GCDs_of_Pairs_of_those_Integers/Lemma | [
"Greatest Common Divisor",
"LCM of 3 Integers in terms of GCDs of Pairs of those Integers"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Definition:Greatest Common Divisor/Integers",
"Common Divisor Divides GCD",
"Common Divisor Divides GCD",
"Definition:Greatest Common Divisor/Integers",
"Definition:Greatest Common Divisor/Integers",
"Absolute Value of Integer is not less than Divisors",
"Category:Greatest Common Divisor",
"Category:... |
proofwiki-17193 | Events One of Which equals Intersection | Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be events of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.
Let $A$ and $B$ be such that:
:$A \cap B = A$
Then whenever $A$ occurs, it is always the case that $B$ occurs as well. | From Intersection with Subset is Subset:
:$A \cap B = A \iff A \subseteq B$
Let $A$ occur.
Let $\omega$ be the outcome of $\EE$.
Let $\omega \in A$.
That is, by definition of occurrence of event, $A$ occurs.
Then by definition of subset:
:$\omega \in B$
Thus by definition of occurrence of event, $B$ occurs.
Hence the r... | Let the [[Definition:Probability Space|probability space]] of an [[Definition:Experiment|experiment]] $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be [[Definition:Event|events]] of $\EE$, so that $A \subseteq \Omega$ and $B \subseteq \Omega$.
Let $A$ and $B$ be such that:
:$A \cap B = A$
Then when... | From [[Intersection with Subset is Subset]]:
:$A \cap B = A \iff A \subseteq B$
Let $A$ [[Definition:Occurrence of Event|occur]].
Let $\omega$ be the [[Definition:Outcome|outcome]] of $\EE$.
Let $\omega \in A$.
That is, by definition of [[Definition:Occurrence of Event|occurrence of event]], $A$ [[Definition:Occur... | Events One of Which equals Intersection | https://proofwiki.org/wiki/Events_One_of_Which_equals_Intersection | https://proofwiki.org/wiki/Events_One_of_Which_equals_Intersection | [
"Events One of Which equals Intersection",
"Intersections of Events"
] | [
"Definition:Probability Space",
"Definition:Experiment",
"Definition:Event",
"Definition:Event/Occurrence",
"Definition:Event/Occurrence"
] | [
"Intersection with Subset is Subset",
"Definition:Event/Occurrence",
"Definition:Elementary Event",
"Definition:Event/Occurrence",
"Definition:Event/Occurrence",
"Definition:Subset",
"Definition:Event/Occurrence",
"Definition:Event/Occurrence"
] |
proofwiki-17194 | Union of Event with Complement is Certainty | Let the probability space of an experiment $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A \in \Sigma$ be an events of $\EE$, so that $A \subseteq \Omega$.
Then the union of $A$ with its complementary event can be evaluated as:
:$A \cup \overline A = \Omega$
where $\overline A$ denotes the complementary event to $A$.
... | By definition:
:$A \subseteq \Omega$
and:
:$\overline A = \relcomp \Omega A$
From Union with Relative Complement:
:$A \cup \overline A = \Omega$
We then have from Kolmogorov axiom $(2)$ that:
:$\map \Pr \Omega = 1$
The result follows by definition of certainty.
{{qed}}
{{LEM|Union with Relative Complement}} | Let the [[Definition:Probability Space|probability space]] of an [[Definition:Experiment|experiment]] $\EE$ be $\struct {\Omega, \Sigma, \Pr}$.
Let $A \in \Sigma$ be an [[Definition:Event|events]] of $\EE$, so that $A \subseteq \Omega$.
Then the [[Definition:Union of Events|union]] of $A$ with its [[Definition:Compl... | By definition:
:$A \subseteq \Omega$
and:
:$\overline A = \relcomp \Omega A$
From [[Union with Relative Complement]]:
:$A \cup \overline A = \Omega$
We then have from [[Axiom:Kolmogorov Axioms|Kolmogorov axiom $(2)$]] that:
:$\map \Pr \Omega = 1$
The result follows by definition of [[Definition:Certain Event|certai... | Union of Event with Complement is Certainty | https://proofwiki.org/wiki/Union_of_Event_with_Complement_is_Certainty | https://proofwiki.org/wiki/Union_of_Event_with_Complement_is_Certainty | [
"Complementary Events",
"Unions of Events",
"Certain Events"
] | [
"Definition:Probability Space",
"Definition:Experiment",
"Definition:Event",
"Definition:Event/Occurrence/Union",
"Definition:Complementary Event",
"Definition:Complementary Event",
"Definition:Event/Occurrence/Certainty"
] | [
"Union with Relative Complement",
"Axiom:Kolmogorov Axioms",
"Definition:Event/Occurrence/Certainty"
] |
proofwiki-17195 | Equivalence of Definitions of Adherent Point of Set/Definition by Open Neighborhood iff Definition by Neighborhood | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
{{TFAE|def = Adherent Point of Set|view = adherent point of $H$}} | === Necessary Condition ===
Let every open neighborhood $U$ of $x$ satisfy:
:$H \cap U \ne \O$
Let $N$ be an arbitrary neighborhood of $x$.
By definition of a neighborhood:
:$\exists V \in \tau: x \in V \subseteq N$
From Set is Open iff Neighborhood of all its Points, $V$ is an open neighborhood of $x$.
Thus:
:$H \cap ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$.
{{TFAE|def = Adherent Point of Set|view = adherent point of $H$}} | === Necessary Condition ===
Let every [[Definition:Open Neighborhood of Point|open neighborhood]] $U$ of $x$ satisfy:
:$H \cap U \ne \O$
Let $N$ be an arbitrary [[Definition:Neighborhood of Point|neighborhood]] of $x$.
By definition of a [[Definition:Neighborhood of Point|neighborhood]]:
:$\exists V \in \tau: x \in ... | Equivalence of Definitions of Adherent Point of Set/Definition by Open Neighborhood iff Definition by Neighborhood | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Adherent_Point_of_Set/Definition_by_Open_Neighborhood_iff_Definition_by_Neighborhood | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Adherent_Point_of_Set/Definition_by_Open_Neighborhood_iff_Definition_by_Neighborhood | [
"Equivalence of Definitions of Adherent Point of Set"
] | [
"Definition:Topological Space"
] | [
"Definition:Open Neighborhood/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Set is Open iff Neighborhood of all its Points",
"Definition:Open Neighborhood/Point",
"Definition:Contrapositive Statement",
"Subsets of Disjoint Sets are Disjoint",
"Definit... |
proofwiki-17196 | Compact Sets in Fortissimo Space | A subset of a Fortissimo space is compact {{iff}} it is finite. | Let $T = \struct {S, \tau}$ be a Fortissimo space. | A [[Definition:Subset|subset]] of a [[Definition:Fortissimo Space|Fortissimo space]] is [[Definition:Compact Topological Subspace|compact]] {{iff}} it is [[Definition:Finite Set|finite]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Fortissimo Space|Fortissimo space]]. | Compact Sets in Fortissimo Space | https://proofwiki.org/wiki/Compact_Sets_in_Fortissimo_Space | https://proofwiki.org/wiki/Compact_Sets_in_Fortissimo_Space | [
"Fortissimo Spaces",
"Examples of Compact Topological Spaces"
] | [
"Definition:Subset",
"Definition:Fortissimo Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Finite Set"
] | [
"Definition:Fortissimo Space",
"Definition:Fortissimo Space"
] |
proofwiki-17197 | Equivalence of Definitions of Boundary | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
{{TFAE|def = Boundary (Topology)|view = boundary of $H$}} | === Definition $1$ is equivalent to Definition $3$ ===
This is demonstrated in Boundary is Intersection of Closure with Closure of Complement.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$.
{{TFAE|def = Boundary (Topology)|view = boundary of $H$}} | === Definition $1$ is equivalent to Definition $3$ ===
This is demonstrated in [[Boundary is Intersection of Closure with Closure of Complement]].
{{qed|lemma}} | Equivalence of Definitions of Boundary | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boundary | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Boundary | [
"Set Boundaries"
] | [
"Definition:Topological Space"
] | [
"Boundary is Intersection of Closure with Closure of Complement"
] |
proofwiki-17198 | Uniformly Convergent Series of Continuous Functions Converges to Continuous Function | Let $S \subseteq \R$.
Let $x \in S$.
Let $\sequence {f_n}$ be a sequence of real functions.
Let $f_n$ be continuous at $x$ for all $n \in \N$.
Let the infinite series:
:$\ds \sum_{n \mathop = 1}^\infty f_n$
be uniformly convergent to a real function $f : S \to \R$.
Then $f$ is continuous at $x$. | Let $\sequence {s_n}$ be sequence of real functions $S \to \R$ such that:
:$\ds \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_n} x$
for each $n \in \N$ and $x \in S$.
By Sum Rule for Continuous Real Functions:
:$s_n$ is continuous at $x$ for all $n \in \N$.
Since additionally $s_n \to f$ uniformly, we have by Uniforml... | Let $S \subseteq \R$.
Let $x \in S$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]].
Let $f_n$ be [[Definition:Continuous Real Function|continuous]] at $x$ for all $n \in \N$.
Let the [[Definition:Infinite Series|infinite series]]:
:$\ds \sum_{n \mathop ... | Let $\sequence {s_n}$ be [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]] $S \to \R$ such that:
:$\ds \map {s_n} x = \sum_{k \mathop = 1}^n \map {f_n} x$
for each $n \in \N$ and $x \in S$.
By [[Sum Rule for Continuous Real Functions]]:
:$s_n$ is [[Definition:Continuous Real Function|... | Uniformly Convergent Series of Continuous Functions Converges to Continuous Function | https://proofwiki.org/wiki/Uniformly_Convergent_Series_of_Continuous_Functions_Converges_to_Continuous_Function | https://proofwiki.org/wiki/Uniformly_Convergent_Series_of_Continuous_Functions_Converges_to_Continuous_Function | [
"Uniform Convergence",
"Continuous Real Functions"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Series",
"Definition:Uniform Convergence/Infinite Series",
"Definition:Real Function",
"Definition:Continuous Real Function"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Combination Theorem for Continuous Functions/Real/Sum Rule",
"Definition:Continuous Real Function",
"Definition:Uniform Convergence/Infinite Series",
"Uniformly Convergent Sequence of Continuous Functions Converges to Continuous Function",
"Definition:... |
proofwiki-17199 | Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a sequence of real functions.
Let $f_n$ be continuous for all $n \in \N$.
Let the infinite series:
:$\ds \sum_{n \mathop = 1}^\infty f_n$
be uniformly convergent to a real function $f : S \to \R$.
Then $f$ is continuous. | Let $x \in S$.
Then $f_n$ is continuous at $x$ for all $n \in \N$.
Since:
:$\ds \sum_{n \mathop = 1}^\infty f_n$
converges uniformly to $f$, we have by Uniformly Convergent Series of Continuous Functions Converges to Continuous Function:
:$f$ is continuous at $x$.
As $x \in S$ was arbitrary, we have that:
:$f$ is cont... | Let $S \subseteq \R$.
Let $\sequence {f_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Function|real functions]].
Let $f_n$ be [[Definition:Continuous Real Function|continuous]] for all $n \in \N$.
Let the [[Definition:Infinite Series|infinite series]]:
:$\ds \sum_{n \mathop = 1}^\infty f_n$
be [[D... | Let $x \in S$.
Then $f_n$ is [[Definition:Continuous Real Function|continuous]] at $x$ for all $n \in \N$.
Since:
:$\ds \sum_{n \mathop = 1}^\infty f_n$
[[Definition:Uniformly Convergent Infinite Series|converges uniformly]] to $f$, we have by [[Uniformly Convergent Series of Continuous Functions Converges to Conti... | Uniformly Convergent Series of Continuous Functions Converges to Continuous Function/Corollary | https://proofwiki.org/wiki/Uniformly_Convergent_Series_of_Continuous_Functions_Converges_to_Continuous_Function/Corollary | https://proofwiki.org/wiki/Uniformly_Convergent_Series_of_Continuous_Functions_Converges_to_Continuous_Function/Corollary | [
"Uniform Convergence",
"Continuous Functions"
] | [
"Definition:Sequence",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Series",
"Definition:Uniform Convergence/Infinite Series",
"Definition:Real Function",
"Definition:Continuous Real Function"
] | [
"Definition:Continuous Real Function",
"Definition:Uniform Convergence/Infinite Series",
"Uniformly Convergent Series of Continuous Functions Converges to Continuous Function",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Category:Uniform Convergence",
"Category:Continu... |
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