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-2400 | Number of Edges in Forest | Let $F = \struct {V, E}$ be a forest with $n$ nodes and $m$ components.
Then $F$ contains $n - m$ edges. | By definition, a forest is a disconnected graph whose components are all trees.
Let the number of nodes in each component of $F$ be $n_1, n_2, \ldots, n_m$ where of course $\ds \sum_{i \mathop = 1}^m n_i = n$.
From Finite Connected Simple Graph is Tree iff Size is One Less than Order, the number of edges in tree $i$ is... | Let $F = \struct {V, E}$ be a [[Definition:Forest|forest]] with $n$ [[Definition:Node of Tree|nodes]] and $m$ [[Definition:Component of Graph|components]].
Then $F$ contains $n - m$ [[Definition:Edge of Graph|edges]]. | By definition, a [[Definition:Forest|forest]] is a [[Definition:Connected Graph|disconnected graph]] whose [[Definition:Component of Graph|components]] are all [[Definition:Tree (Graph Theory)|trees]].
Let the number of [[Definition:Node of Tree|nodes]] in each [[Definition:Component of Graph|component]] of $F$ be $n_... | Number of Edges in Forest | https://proofwiki.org/wiki/Number_of_Edges_in_Forest | https://proofwiki.org/wiki/Number_of_Edges_in_Forest | [
"Forests"
] | [
"Definition:Forest",
"Definition:Tree (Graph Theory)/Node",
"Definition:Component of Graph",
"Definition:Graph (Graph Theory)/Edge"
] | [
"Definition:Forest",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Component of Graph",
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Component of Graph",
"Finite Connected Simple Graph is Tree iff Size is One Less than Order",
"Definition:Graph (Graph... |
proofwiki-2401 | Cayley's Formula | The number of distinct labeled trees with $n$ nodes is $n^{n - 2}$. | {{MissingLinks}}
Follows directly from Bijection between Prüfer Sequences and Labeled Trees.
This shows that there is a bijection between the set of labeled trees with $n$ nodes and the set of all Prüfer sequences of the form:
:$\tuple {\mathbf a_1, \mathbf a_2, \ldots, \mathbf a_{n - 2} }$
where each of the $\mathbf a... | The number of [[Definition:Distinct Elements|distinct]] [[Definition:Labeled Graph|labeled]] [[Definition:Tree (Graph Theory)|trees]] with $n$ [[Definition:Node of Tree|nodes]] is $n^{n - 2}$. | {{MissingLinks}}
Follows directly from [[Bijection between Prüfer Sequences and Labeled Trees]].
This shows that there is a [[Definition:Bijection|bijection]] between the set of [[Definition:Labeled Graph|labeled]] [[Definition:Tree (Graph Theory)|trees]] with $n$ [[Definition:Node of Tree|nodes]] and the [[Definitio... | Cayley's Formula | https://proofwiki.org/wiki/Cayley's_Formula | https://proofwiki.org/wiki/Cayley's_Formula | [
"Cayley's Formula",
"Graph Theory",
"Combinatorics",
"Cayley's Theorems"
] | [
"Definition:Distinct/Plural",
"Definition:Labeled Graph",
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node"
] | [
"Bijection between Prüfer Sequences and Labeled Trees",
"Definition:Bijection",
"Definition:Labeled Graph",
"Definition:Tree (Graph Theory)",
"Definition:Tree (Graph Theory)/Node",
"Definition:Set",
"Definition:Prüfer Sequence",
"Definition:Integer",
"Equivalence of Mappings between Finite Sets of S... |
proofwiki-2402 | Bijection between Prüfer Sequences and Labeled Trees | There is a one-to-one correspondence between Prüfer sequences and labeled trees.
That is, every labeled tree has a unique Prüfer sequence that defines it, and every Prüfer sequence defines just one labeled tree. | Let $T$ be the set of all labeled trees of order $n$.
Let $P$ be the set of all Prüfer sequence of length $n-2$.
Let $\phi: T \to P$ be the mapping that maps each tree to its Prüfer sequence.
* From Prüfer Sequence from Labeled Tree, $\phi$ is clearly well-defined, as every element of $T$ maps uniquely to an element of... | There is a [[Definition:Bijection|one-to-one correspondence]] between [[Definition:Prüfer Sequence|Prüfer sequences]] and [[Definition:Labeled Graph|labeled]] [[Definition:Tree (Graph Theory)|trees]].
That is, every [[Definition:Labeled Graph|labeled]] [[Definition:Tree (Graph Theory)|tree]] has a unique [[Definition:... | Let $T$ be the set of all [[Definition:Labeled Graph|labeled]] [[Definition:Tree (Graph Theory)|trees]] of [[Definition:Order of Graph|order $n$]].
Let $P$ be the set of all [[Definition:Prüfer Sequence|Prüfer sequence]] of length $n-2$.
Let $\phi: T \to P$ be the [[Definition:Mapping|mapping]] that maps each tree t... | Bijection between Prüfer Sequences and Labeled Trees | https://proofwiki.org/wiki/Bijection_between_Prüfer_Sequences_and_Labeled_Trees | https://proofwiki.org/wiki/Bijection_between_Prüfer_Sequences_and_Labeled_Trees | [
"Tree Theory",
"Combinatorics"
] | [
"Definition:Bijection",
"Definition:Prüfer Sequence",
"Definition:Labeled Graph",
"Definition:Tree (Graph Theory)",
"Definition:Labeled Graph",
"Definition:Tree (Graph Theory)",
"Definition:Prüfer Sequence",
"Definition:Prüfer Sequence",
"Definition:Labeled Graph",
"Definition:Tree (Graph Theory)"... | [
"Definition:Labeled Graph",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Prüfer Sequence",
"Definition:Mapping",
"Prüfer Sequence from Labeled Tree",
"Definition:Well-Defined/Mapping",
"Labeled Tree from Prüfer Sequence",
"Definition:Well-Defined/Mapping",
... |
proofwiki-2403 | Tree has Center or Bicenter | Every tree has either:
:$(1): \quad$ Exactly one center
or:
:$(2): \quad$ Exactly one bicenter,
but never both.
That is, every tree is either central or bicentral. | A tree whose order is $1$ or $2$ is already trivially central or bicentral.
Let $T$ be a tree of order at least $3$.
First we establish that the construction of a center or bicenter actually works.
From Finite Tree has Leaf Nodes, there are always at least two nodes of degree $1$ to be removed.
By the Handshake Lemma, ... | Every [[Definition:Tree (Graph Theory)|tree]] has either:
:$(1): \quad$ Exactly one [[Definition:Center of Tree|center]]
or:
:$(2): \quad$ Exactly one [[Definition:Bicenter of Tree|bicenter]],
but never both.
That is, every tree is either [[Definition:Central Tree|central]] or [[Definition:Bicentral Tree|bicentral]]... | A [[Definition:Tree (Graph Theory)|tree]] whose [[Definition:Order of Graph|order]] is $1$ or $2$ is already trivially [[Definition:Central Tree|central]] or [[Definition:Bicentral Tree|bicentral]].
Let $T$ be a [[Definition:Tree (Graph Theory)|tree]] of [[Definition:Order of Graph|order]] at least $3$.
First we es... | Tree has Center or Bicenter | https://proofwiki.org/wiki/Tree_has_Center_or_Bicenter | https://proofwiki.org/wiki/Tree_has_Center_or_Bicenter | [
"Tree Theory"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Center of Tree",
"Definition:Center of Tree",
"Definition:Central Tree",
"Definition:Bicentral Tree"
] | [
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Central Tree",
"Definition:Bicentral Tree",
"Definition:Tree (Graph Theory)",
"Definition:Graph (Graph Theory)/Order",
"Definition:Center of Tree",
"Definition:Center of Tree",
"Finite Tree has Leaf Nodes",
"Def... |
proofwiki-2404 | Sophie Germain's Identity | :$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$ | {{begin-eqn}}
{{eqn | o =
| r = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}
| c =
}}
{{eqn | r = x^4 + x^2 \cdot 2 y^2 - x^2 \cdot 2 x y + x^2 \cdot 2 y^2 + 4 y^2 - 2 y^2 \cdot 2 x y + x^2 \cdot 2 x y + 2 y^2 \cdot 2 x y - 2 x y \cdot 2 x y
| c =
}}
{{eqn | r = x^4 + 4 y^4
| c =... | :$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$ | {{begin-eqn}}
{{eqn | o =
| r = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}
| c =
}}
{{eqn | r = x^4 + x^2 \cdot 2 y^2 - x^2 \cdot 2 x y + x^2 \cdot 2 y^2 + 4 y^2 - 2 y^2 \cdot 2 x y + x^2 \cdot 2 x y + 2 y^2 \cdot 2 x y - 2 x y \cdot 2 x y
| c =
}}
{{eqn | r = x^4 + 4 y^4
| c =... | Sophie Germain's Identity/Proof 1 | https://proofwiki.org/wiki/Sophie_Germain's_Identity | https://proofwiki.org/wiki/Sophie_Germain's_Identity/Proof_1 | [
"Fourth Powers",
"Sophie Germain's Identity"
] | [] | [] |
proofwiki-2405 | Sophie Germain's Identity | :$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$ | {{begin-eqn}}
{{eqn | o =
| r = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}
| c =
}}
{{eqn | r = \paren {x^2 + 2 y^2}^2 - \paren {2 x y}^2
| c = Difference of Two Squares
}}
{{eqn | r = x^4 + 2 \cdot x^2 \cdot 2 y^2 + 4 y^2 - 2 x y \cdot 2 x y
| c =
}}
{{eqn | r = x^4 + 4 y^4
... | :$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$ | {{begin-eqn}}
{{eqn | o =
| r = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}
| c =
}}
{{eqn | r = \paren {x^2 + 2 y^2}^2 - \paren {2 x y}^2
| c = [[Difference of Two Squares]]
}}
{{eqn | r = x^4 + 2 \cdot x^2 \cdot 2 y^2 + 4 y^2 - 2 x y \cdot 2 x y
| c =
}}
{{eqn | r = x^4 + 4 y^... | Sophie Germain's Identity/Proof 2 | https://proofwiki.org/wiki/Sophie_Germain's_Identity | https://proofwiki.org/wiki/Sophie_Germain's_Identity/Proof_2 | [
"Fourth Powers",
"Sophie Germain's Identity"
] | [] | [
"Difference of Two Squares"
] |
proofwiki-2406 | Prim's Algorithm produces Minimum Spanning Tree | Prim's Algorithm always produces a minimum spanning tree. | Suppose that Prim's Algorithm produces a tree $T$.
Let there exist another spanning tree $S$ with a smaller total weight.
Let $e$ be an edge of smallest weight which lies in $T$ but not $S$.
If we add $e$ to $S$, we obtain a cycle, from Equivalent Definitions for Finite Tree.
This cycle contains an edge $e'$ which is i... | [[Prim's Algorithm]] always produces a [[Definition:Minimum Spanning Tree|minimum spanning tree]]. | Suppose that [[Prim's Algorithm]] produces a [[Definition:Tree (Graph Theory)|tree]] $T$.
Let there exist another [[Definition:Spanning Tree|spanning tree]] $S$ with a smaller total [[Definition:Weight (Network Theory)|weight]].
Let $e$ be an [[Definition:Edge of Graph|edge]] of smallest [[Definition:Weight (Network ... | Prim's Algorithm produces Minimum Spanning Tree | https://proofwiki.org/wiki/Prim's_Algorithm_produces_Minimum_Spanning_Tree | https://proofwiki.org/wiki/Prim's_Algorithm_produces_Minimum_Spanning_Tree | [
"Prim's Algorithm",
"Tree Theory"
] | [
"Prim's Algorithm",
"Definition:Minimum Spanning Tree"
] | [
"Prim's Algorithm",
"Definition:Tree (Graph Theory)",
"Definition:Spanning Tree",
"Definition:Network (Graph Theory)/Weight",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Network (Graph Theory)/Weight",
"Definition:Cycle (Graph Theory)",
"Characteristics of Finite Tree",
"Definition:Cycle (Gr... |
proofwiki-2407 | Number of Hamilton Cycles in Complete Graph | For all $n \ge 3$, the number of distinct Hamilton cycles in the complete graph $K_n$ is $\dfrac {\paren {n - 1}!} 2$. | In a complete graph, every vertex is adjacent to every other vertex.
Therefore, if we were to take all the vertices in a complete graph in any order, there will be a path through those vertices in that order.
Joining either end of that path gives us a Hamilton cycle.
From Cardinality of Set of Bijections, there are $n!... | For all $n \ge 3$, the number of [[Definition:Distinct Elements|distinct]] [[Definition:Hamilton Cycle|Hamilton cycles]] in the [[Definition:Complete Graph|complete graph]] $K_n$ is $\dfrac {\paren {n - 1}!} 2$. | In a [[Definition:Complete Graph|complete graph]], every [[Definition:Vertex of Graph|vertex]] is [[Definition:Adjacent Vertices (Undirected Graph)|adjacent]] to every other [[Definition:Vertex of Graph|vertex]].
Therefore, if we were to take all the [[Definition:Vertex of Graph|vertices]] in a [[Definition:Complete G... | Number of Hamilton Cycles in Complete Graph | https://proofwiki.org/wiki/Number_of_Hamilton_Cycles_in_Complete_Graph | https://proofwiki.org/wiki/Number_of_Hamilton_Cycles_in_Complete_Graph | [
"Hamiltonian Graphs",
"Combinatorics"
] | [
"Definition:Distinct/Plural",
"Definition:Hamilton Cycle",
"Definition:Complete Graph"
] | [
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Complete Graph",
"Definition:Path (Graph Theory)",
"Definition:Graph (G... |
proofwiki-2408 | Coreflexive Relation is Subset of Diagonal Relation | A coreflexive relation is a subset of the diagonal relation. | Let $\RR \subseteq S \times S$ be a coreflexive relation.
Let $\tuple {x, y} \in \RR$.
By definition of coreflexive, it follows that $x = y$, and hence $\tuple {x, y} = \tuple {x, x}$.
So by definition of the diagonal relation:
:$\tuple {x, y} \in \Delta_S$
Hence the result.
{{qed}}
Category:Coreflexive Relations
Categ... | A [[Definition:Coreflexive Relation|coreflexive relation]] is a [[Definition:Subset|subset]] of the [[Definition:Diagonal Relation|diagonal relation]]. | Let $\RR \subseteq S \times S$ be a [[Definition:Coreflexive Relation|coreflexive relation]].
Let $\tuple {x, y} \in \RR$.
By definition of [[Definition:Coreflexive Relation|coreflexive]], it follows that $x = y$, and hence $\tuple {x, y} = \tuple {x, x}$.
So by definition of the [[Definition:Diagonal Relation|diago... | Coreflexive Relation is Subset of Diagonal Relation | https://proofwiki.org/wiki/Coreflexive_Relation_is_Subset_of_Diagonal_Relation | https://proofwiki.org/wiki/Coreflexive_Relation_is_Subset_of_Diagonal_Relation | [
"Coreflexive Relations",
"Diagonal Relation"
] | [
"Definition:Coreflexive Relation",
"Definition:Subset",
"Definition:Diagonal Relation"
] | [
"Definition:Coreflexive Relation",
"Definition:Coreflexive Relation",
"Definition:Diagonal Relation",
"Category:Coreflexive Relations",
"Category:Diagonal Relation"
] |
proofwiki-2409 | Reflexive Euclidean Relation is Equivalence | A relation is an equivalence {{iff}} it is either left-Euclidean or right-Euclidean, and also reflexive. | === Sufficient Condition ===
Let $\RR$ be an equivalence relation.
Then by definition it is reflexive.
It is also transitive and symmetric.
So, let $x, y, z \in \RR$ such that $x \mathrel \RR y$ and $x \mathrel \RR z$.
From symmetry, we have $y \mathrel \RR x$, and by transitivity it follows that $y \mathrel \RR z$.
He... | A [[Definition:Relation|relation]] is an [[Definition:Equivalence Relation|equivalence]] {{iff}} it is either [[Definition:Left-Euclidean Relation|left-Euclidean]] or [[Definition:Right-Euclidean Relation|right-Euclidean]], and also [[Definition:Reflexive Relation|reflexive]]. | === Sufficient Condition ===
Let $\RR$ be an [[Definition:Equivalence Relation|equivalence relation]].
Then by definition it is [[Definition:Reflexive Relation|reflexive]].
It is also [[Definition:Transitive Relation|transitive]] and [[Definition:Symmetric Relation|symmetric]].
So, let $x, y, z \in \RR$ such that... | Reflexive Euclidean Relation is Equivalence | https://proofwiki.org/wiki/Reflexive_Euclidean_Relation_is_Equivalence | https://proofwiki.org/wiki/Reflexive_Euclidean_Relation_is_Equivalence | [
"Reflexive Relations",
"Euclidean Relations",
"Equivalence Relations"
] | [
"Definition:Relation",
"Definition:Equivalence Relation",
"Definition:Euclidean Relation/Left-Euclidean",
"Definition:Euclidean Relation/Right-Euclidean",
"Definition:Reflexive Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Euclidean Relation/Right-Euclidean",
"Definition:Symmetric Relation",
"Definition:Tran... |
proofwiki-2410 | Reflexive Relation is Serial | Every reflexive relation is also a serial relation. | Let $\RR \subseteq S \times S$ be a relation in $S$.
We have that $\RR$ is serial {{iff}}:
:$\forall x \in S: \exists y \in S: \tuple {x, y} \in \RR$
That is, {{iff}} every element relates to at least one element.
We have that $\RR$ is reflexive {{iff}}:
:$\forall x \in S: \tuple {x, x} \in \RR$
Hence if $\RR$ is refle... | Every [[Definition:Reflexive Relation|reflexive relation]] is also a [[Definition:Serial Relation|serial relation]]. | Let $\RR \subseteq S \times S$ be a [[Definition:Relation|relation in $S$]].
We have that $\RR$ is [[Definition:Serial Relation|serial]] {{iff}}:
:$\forall x \in S: \exists y \in S: \tuple {x, y} \in \RR$
That is, {{iff}} every element relates to at least one element.
We have that $\RR$ is [[Definition:Reflexive Re... | Reflexive Relation is Serial | https://proofwiki.org/wiki/Reflexive_Relation_is_Serial | https://proofwiki.org/wiki/Reflexive_Relation_is_Serial | [
"Reflexive Relations",
"Serial Relations"
] | [
"Definition:Reflexive Relation",
"Definition:Serial Relation"
] | [
"Definition:Relation",
"Definition:Serial Relation",
"Definition:Reflexive Relation",
"Definition:Reflexive Relation",
"Definition:Serial Relation",
"Category:Reflexive Relations",
"Category:Serial Relations"
] |
proofwiki-2411 | Complement of Complete Bipartite Graph | Let $K_{p, q}$ be a complete bipartite graph.
The complement of $K_{p, q}$ consists of a disconnected graph with two components:
:The complete graph $K_p$
:The complete graph $K_q$. | By definition, the complete bipartite graph $K_{p, q}$ consists of two sets of vertices: $A$ of cardinality $p$, and $B$ of cardinality $q$, such that:
:Every vertex in $A$ is adjacent to every vertex in $B$
:No vertex in $A$ is adjacent to any other vertex in $A$
:No vertex in $B$ is adjacent to any other vertex in $B... | Let $K_{p, q}$ be a [[Definition:Complete Bipartite Graph|complete bipartite graph]].
The [[Definition:Complement of Simple Graph|complement]] of $K_{p, q}$ consists of a [[Definition:Connected Graph|disconnected graph]] with two [[Definition:Component of Graph|components]]:
:The [[Definition:Complete Graph|complete ... | By definition, the [[Definition:Complete Bipartite Graph|complete bipartite graph]] $K_{p, q}$ consists of two sets of [[Definition:Vertex of Graph|vertices]]: $A$ of [[Definition:Cardinality|cardinality]] $p$, and $B$ of [[Definition:Cardinality|cardinality]] $q$, such that:
:Every [[Definition:Vertex of Graph|vertex... | Complement of Complete Bipartite Graph | https://proofwiki.org/wiki/Complement_of_Complete_Bipartite_Graph | https://proofwiki.org/wiki/Complement_of_Complete_Bipartite_Graph | [
"Bipartite Graphs"
] | [
"Definition:Complete Bipartite Graph",
"Definition:Complement of Graph/Simple Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Component of Graph",
"Definition:Complete Graph",
"Definition:Complete Graph"
] | [
"Definition:Complete Bipartite Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Cardinality",
"Definition:Cardinality",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Gr... |
proofwiki-2412 | Complement of Strict Total Ordering | Let $\struct {S, \prec}$ be a relational structure such that $\prec$ is a strict total ordering.
Then the complement of $\prec$ is a weak total ordering. | We need to show that $\nprec$ is a weak total ordering.
First we check in turn each of the criteria for an ordering: | Let $\struct {S, \prec}$ be a [[Definition:Relational Structure|relational structure]] such that $\prec$ is a [[Definition:Strict Total Ordering|strict total ordering]].
Then the [[Definition:Complement of Relation|complement]] of $\prec$ is a [[Definition:Weak Total Ordering|weak total ordering]]. | We need to show that $\nprec$ is a [[Definition:Weak Total Ordering|weak total ordering]].
First we check in turn each of the criteria for an [[Definition:Ordering|ordering]]: | Complement of Strict Total Ordering | https://proofwiki.org/wiki/Complement_of_Strict_Total_Ordering | https://proofwiki.org/wiki/Complement_of_Strict_Total_Ordering | [
"Total Orderings"
] | [
"Definition:Relational Structure",
"Definition:Strict Total Ordering",
"Definition:Complement of Relation",
"Definition:Total Ordering"
] | [
"Definition:Total Ordering",
"Definition:Ordering"
] |
proofwiki-2413 | Strict Weak Ordering Induces Partition | Let $\struct {S, \prec}$ be a relational structure such that $\prec$ is a strict weak ordering on $S$.
Then $S$ can be partitioned into equivalence classes whose equivalence relation is "is non-comparable".
That is, each of the partitions $A$ of $S$ is a relational structure $\struct {\mathbb S, <}$ such that:
:$\mathb... | From the definition of strict weak ordering, we define the symbol $\Bumpeq$ as:
:$a \Bumpeq b := \neg a \prec b \land \neg b \prec a$
that is, $a \Bumpeq b$ means "$a$ and $b$ are non-comparable".
Checking in turn each of the criteria for equivalence: | Let $\struct {S, \prec}$ be a [[Definition:Relational Structure|relational structure]] such that $\prec$ is a [[Definition:Strict Weak Ordering|strict weak ordering]] on $S$.
Then $S$ can be [[Definition:Partition (Set Theory)|partitioned]] into [[Definition:Equivalence Class|equivalence classes]] whose [[Definition:... | From the definition of [[Definition:Strict Weak Ordering|strict weak ordering]], we define the symbol $\Bumpeq$ as:
:$a \Bumpeq b := \neg a \prec b \land \neg b \prec a$
that is, $a \Bumpeq b$ means "$a$ and $b$ are [[Definition:Non-Comparable Elements|non-comparable]]".
Checking in turn each of the criteria for [[Def... | Strict Weak Ordering Induces Partition | https://proofwiki.org/wiki/Strict_Weak_Ordering_Induces_Partition | https://proofwiki.org/wiki/Strict_Weak_Ordering_Induces_Partition | [
"Order Theory",
"Equivalence Relations"
] | [
"Definition:Relational Structure",
"Definition:Strict Weak Ordering",
"Definition:Set Partition",
"Definition:Equivalence Class",
"Definition:Equivalence Relation",
"Definition:Non-Comparable Elements",
"Definition:Set Partition",
"Definition:Relational Structure",
"Definition:Set Partition",
"Def... | [
"Definition:Strict Weak Ordering",
"Definition:Non-Comparable Elements",
"Definition:Equivalence Relation",
"Definition:Strict Weak Ordering"
] |
proofwiki-2414 | Mahler's Inequality | The geometric mean of the termwise sum of two finite sequences of positive real numbers is never less than the sum of their two separate geometric means:
:$\ds \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n} \ge \prod_{k \mathop = 1}^n x_k^{1/n} + \prod_{k \mathop = 1}^n y_k^{1/n}$
where $x_k, y_k > 0$ for all $k$. | {{begin-eqn}}
{{eqn | l = \prod_{k \mathop = 1}^n \paren {\frac {x_k} {x_k + y_k} }^{1/n}
| o = \le
| r = \frac 1 n \sum_{k \mathop = 1}^n \frac {x_k} {x_k + y_k}
| c = Cauchy's Mean Theorem
}}
{{eqn | l = \prod_{k \mathop = 1}^n \paren {\frac {y_k} {x_k + y_k} }^{1/n}
| o = \le
| r = \fra... | The [[Definition:Geometric Mean|geometric mean]] of the termwise sum of two [[Definition:Sequence|finite sequences]] of [[Definition:Positive Real Number|positive real numbers]] is never less than the sum of their two separate [[Definition:Geometric Mean|geometric means]]:
:$\ds \prod_{k \mathop = 1}^n \paren {x_k + y... | {{begin-eqn}}
{{eqn | l = \prod_{k \mathop = 1}^n \paren {\frac {x_k} {x_k + y_k} }^{1/n}
| o = \le
| r = \frac 1 n \sum_{k \mathop = 1}^n \frac {x_k} {x_k + y_k}
| c = [[Cauchy's Mean Theorem]]
}}
{{eqn | l = \prod_{k \mathop = 1}^n \paren {\frac {y_k} {x_k + y_k} }^{1/n}
| o = \le
| r = ... | Mahler's Inequality | https://proofwiki.org/wiki/Mahler's_Inequality | https://proofwiki.org/wiki/Mahler's_Inequality | [
"Algebra",
"Measures of Central Tendency"
] | [
"Definition:Geometric Mean",
"Definition:Sequence",
"Definition:Positive/Real Number",
"Definition:Geometric Mean"
] | [
"Cauchy's Mean Theorem",
"Cauchy's Mean Theorem",
"Category:Algebra",
"Category:Measures of Central Tendency"
] |
proofwiki-2415 | Equivalence of Definitions of Transitive Closure of Relation/Intersection is Smallest | Let $\RR$ be a relation on a set $S$.
Then the intersection of all transitive relations on $S$ that contain $\RR$ is the smallest transitive relation on $S$ that contains $\RR$. | Note that the trivial relation $\TT = S \times S$ on $S$ contains $\RR$, by the definition of a relation on $S$.
Further, $\TT$ is transitive by Trivial Relation is Equivalence.
Thus there is at least one transitive relation on $S$ that contains $\RR$.
Now define $\RR^+$ as the intersection of all transitive relations ... | Let $\RR$ be a [[Definition:Relation|relation]] on a [[Definition:Set|set]] $S$.
Then the [[Definition:Set Intersection|intersection]] of all [[Definition:Transitive Relation|transitive relations]] on $S$ that [[Definition:Subset|contain]] $\RR$ is the [[Definition:Smallest Set by Set Inclusion|smallest]] transitive ... | Note that the [[Definition:Trivial Relation|trivial relation]] $\TT = S \times S$ on $S$ contains $\RR$, by the definition of a [[Definition:Endorelation|relation]] on $S$.
Further, $\TT$ is [[Definition:Transitive Relation|transitive]] by [[Trivial Relation is Equivalence]].
Thus there is at least one [[Definition:... | Equivalence of Definitions of Transitive Closure of Relation/Intersection is Smallest/Proof | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Transitive_Closure_of_Relation/Intersection_is_Smallest | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Transitive_Closure_of_Relation/Intersection_is_Smallest/Proof | [
"Equivalence of Definitions of Transitive Closure of Relation"
] | [
"Definition:Relation",
"Definition:Set",
"Definition:Set Intersection",
"Definition:Transitive Relation",
"Definition:Subset",
"Definition:Smallest Set by Set Inclusion"
] | [
"Definition:Trivial Relation",
"Definition:Endorelation",
"Definition:Transitive Relation",
"Trivial Relation is Equivalence",
"Definition:Transitive Relation",
"Definition:Set Intersection",
"Definition:Transitive Relation",
"Intersection of Transitive Relations is Transitive",
"Definition:Transiti... |
proofwiki-2416 | Intersection of Transitive Relations is Transitive | The intersection of two transitive relations is also a transitive relation. | Let $\RR_1$ and $\RR_2$ be transitive relations on an arbitrary set $S$.
Let $\tuple {s_1, s_2} \in \RR_1 \cap \RR_2$ and $\tuple {s_2, s_3} \in \RR_1 \cap \RR_2$.
Then by definition of intersection:
:$\tuple {s_1, s_2} \in \RR_1$ and $\tuple {s_1, s_2} \in \RR_2$
:$\tuple {s_2, s_3} \in \RR_1$ and $\tuple {s_2, s_3} \... | The [[Definition:Set Intersection|intersection]] of two [[Definition:Transitive Relation|transitive relations]] is also a [[Definition:Transitive Relation|transitive relation]]. | Let $\RR_1$ and $\RR_2$ be [[Definition:Transitive Relation|transitive relations]] on an arbitrary [[Definition:Set|set]] $S$.
Let $\tuple {s_1, s_2} \in \RR_1 \cap \RR_2$ and $\tuple {s_2, s_3} \in \RR_1 \cap \RR_2$.
Then by definition of [[Definition:Set Intersection|intersection]]:
:$\tuple {s_1, s_2} \in \RR_1$ ... | Intersection of Transitive Relations is Transitive | https://proofwiki.org/wiki/Intersection_of_Transitive_Relations_is_Transitive | https://proofwiki.org/wiki/Intersection_of_Transitive_Relations_is_Transitive | [
"Intersection of Transitive Relations is Transitive",
"Transitive Relations",
"Set Intersection"
] | [
"Definition:Set Intersection",
"Definition:Transitive Relation",
"Definition:Transitive Relation"
] | [
"Definition:Transitive Relation",
"Definition:Set",
"Definition:Set Intersection",
"Definition:Transitive Relation",
"Definition:Set Intersection",
"Definition:Transitive Relation",
"Category:Intersection of Transitive Relations is Transitive",
"Category:Transitive Relations",
"Category:Set Intersec... |
proofwiki-2417 | Intersection of Reflexive Relations is Reflexive | The intersection of two reflexive relations is also a reflexive relation. | Let $\RR_1$ and $\RR_2$ be reflexive relations on a set $S$.
From Relation Contains Diagonal Relation iff Reflexive, we have that:
: $\Delta_S \subseteq \RR_1$
: $\Delta_S \subseteq \RR_2$
Hence from Intersection is Largest Subset:
: $\Delta_S \subseteq \RR_1 \cap \RR_2$
Hence the result, from Relation Contains Diagona... | The [[Definition:Set Intersection|intersection]] of two [[Definition:Reflexive Relation|reflexive relations]] is also a [[Definition:Reflexive Relation|reflexive relation]]. | Let $\RR_1$ and $\RR_2$ be [[Definition:Reflexive Relation|reflexive relations]] on a set $S$.
From [[Relation Contains Diagonal Relation iff Reflexive]], we have that:
: $\Delta_S \subseteq \RR_1$
: $\Delta_S \subseteq \RR_2$
Hence from [[Intersection is Largest Subset]]:
: $\Delta_S \subseteq \RR_1 \cap \RR_2$
Hen... | Intersection of Reflexive Relations is Reflexive | https://proofwiki.org/wiki/Intersection_of_Reflexive_Relations_is_Reflexive | https://proofwiki.org/wiki/Intersection_of_Reflexive_Relations_is_Reflexive | [
"Reflexive Relations",
"Set Intersection"
] | [
"Definition:Set Intersection",
"Definition:Reflexive Relation",
"Definition:Reflexive Relation"
] | [
"Definition:Reflexive Relation",
"Equivalence of Definitions of Reflexive Relation",
"Intersection is Largest Subset",
"Equivalence of Definitions of Reflexive Relation",
"Category:Reflexive Relations",
"Category:Set Intersection"
] |
proofwiki-2418 | Intersection of Symmetric Relations is Symmetric | The intersection of two symmetric relations is also a symmetric relation. | Let $\RR_1$ and $\RR_2$ be symmetric relations on a set $S$.
Let $\RR_3 = \RR_1 \cap \RR_2$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = \RR_3
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {x, y}
| o = \in
| r = \RR_1
| c = {{Defof|Set Intersection}}
}}
{{eqn | l... | The [[Definition:Set Intersection|intersection]] of two [[Definition:Symmetric Relation|symmetric relations]] is also a [[Definition:Symmetric Relation|symmetric relation]]. | Let $\RR_1$ and $\RR_2$ be [[Definition:Symmetric Relation|symmetric relations]] on a set $S$.
Let $\RR_3 = \RR_1 \cap \RR_2$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = \RR_3
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {x, y}
| o = \in
| r = \RR_1
| c = {{D... | Intersection of Symmetric Relations is Symmetric | https://proofwiki.org/wiki/Intersection_of_Symmetric_Relations_is_Symmetric | https://proofwiki.org/wiki/Intersection_of_Symmetric_Relations_is_Symmetric | [
"Symmetric Relations",
"Set Intersection"
] | [
"Definition:Set Intersection",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation"
] | [
"Definition:Symmetric Relation",
"Category:Symmetric Relations",
"Category:Set Intersection"
] |
proofwiki-2419 | Union of Reflexive Relations is Reflexive | The union of two reflexive relations is also a reflexive relation. | Let $\RR_1$ and $\RR_2$ be reflexive relations on a set $S$.
From Relation Contains Diagonal Relation iff Reflexive, we have that:
: $\Delta_S \subseteq \RR_1$
: $\Delta_S \subseteq \RR_2$
Hence from Subset Relation is Transitive:
: $\Delta_S \subseteq \RR_1 \cup \RR_2$
Hence the result, by definition of reflexive rela... | The [[Definition:Set Union|union]] of two [[Definition:Reflexive Relation|reflexive relations]] is also a [[Definition:Reflexive Relation|reflexive relation]]. | Let $\RR_1$ and $\RR_2$ be [[Definition:Reflexive Relation|reflexive relations]] on a set $S$.
From [[Relation Contains Diagonal Relation iff Reflexive]], we have that:
: $\Delta_S \subseteq \RR_1$
: $\Delta_S \subseteq \RR_2$
Hence from [[Subset Relation is Transitive]]:
: $\Delta_S \subseteq \RR_1 \cup \RR_2$
Henc... | Union of Reflexive Relations is Reflexive | https://proofwiki.org/wiki/Union_of_Reflexive_Relations_is_Reflexive | https://proofwiki.org/wiki/Union_of_Reflexive_Relations_is_Reflexive | [
"Reflexive Relations",
"Set Union"
] | [
"Definition:Set Union",
"Definition:Reflexive Relation",
"Definition:Reflexive Relation"
] | [
"Definition:Reflexive Relation",
"Equivalence of Definitions of Reflexive Relation",
"Subset Relation is Transitive",
"Definition:Reflexive Relation/Definition 2",
"Category:Reflexive Relations",
"Category:Set Union"
] |
proofwiki-2420 | Union of Symmetric Relations is Symmetric | The union of two symmetric relations is also a symmetric relation. | Let $\RR_1$ and $\RR_2$ be symmetric relations on a set $S$.
Let $\RR_3 = \RR_1 \cup \RR_2$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = \RR_3
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {x, y}
| o = \in
| r = \RR_1
| c = {{Defof|Set Union}}
}}
{{eqn | lo= \lor... | The [[Definition:Set Union|union]] of two [[Definition:Symmetric Relation|symmetric relations]] is also a [[Definition:Symmetric Relation|symmetric relation]]. | Let $\RR_1$ and $\RR_2$ be [[Definition:Symmetric Relation|symmetric relations]] on a set $S$.
Let $\RR_3 = \RR_1 \cup \RR_2$.
Then:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = \RR_3
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {x, y}
| o = \in
| r = \RR_1
| c = {{D... | Union of Symmetric Relations is Symmetric | https://proofwiki.org/wiki/Union_of_Symmetric_Relations_is_Symmetric | https://proofwiki.org/wiki/Union_of_Symmetric_Relations_is_Symmetric | [
"Symmetric Relations",
"Set Union"
] | [
"Definition:Set Union",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation"
] | [
"Definition:Symmetric Relation",
"Category:Symmetric Relations",
"Category:Set Union"
] |
proofwiki-2421 | Union of Transitive Relations Not Always Transitive | The union of transitive relations is not necessarily itself transitive. | Proof by counterexample:
Let $S = \set {a, b, c, d}$.
Let $\RR_1$ be the transitive relation $\set {\tuple {a, b}, \tuple {b, c}, \tuple {a, c} }$.
Let $\RR_2$ be the transitive relation $\set {\tuple {b, c}, \tuple {c, d}, \tuple {b, d} }$.
Then we have that $\tuple {a, b} \in \RR_1 \cup \RR_2$ and $\tuple {b, d} \in ... | The [[Definition:Set Union|union]] of [[Definition:Transitive Relation|transitive relations]] is not necessarily itself [[Definition:Transitive Relation|transitive]]. | [[Proof by Counterexample|Proof by counterexample]]:
Let $S = \set {a, b, c, d}$.
Let $\RR_1$ be the [[Definition:Transitive Relation|transitive relation]] $\set {\tuple {a, b}, \tuple {b, c}, \tuple {a, c} }$.
Let $\RR_2$ be the [[Definition:Transitive Relation|transitive relation]] $\set {\tuple {b, c}, \tuple {c,... | Union of Transitive Relations Not Always Transitive | https://proofwiki.org/wiki/Union_of_Transitive_Relations_Not_Always_Transitive | https://proofwiki.org/wiki/Union_of_Transitive_Relations_Not_Always_Transitive | [
"Transitive Relations",
"Set Union"
] | [
"Definition:Set Union",
"Definition:Transitive Relation",
"Definition:Transitive Relation"
] | [
"Proof by Counterexample",
"Definition:Transitive Relation",
"Definition:Transitive Relation",
"Definition:Transitive Relation",
"Category:Transitive Relations",
"Category:Set Union"
] |
proofwiki-2422 | Set System Closed under Intersection is Commutative Semigroup | Let $\SS$ be a system of sets.
Let $\SS$ be such that:
:$\forall A, B \in \SS: A \cap B \in \SS$
Then $\struct {\SS, \cap}$ is a commutative semigroup. | === Closure ===
We have {{hypothesis}} that $\struct {\SS, \cap}$ is closed. | Let $\SS$ be a [[Definition:System of Sets|system of sets]].
Let $\SS$ be such that:
:$\forall A, B \in \SS: A \cap B \in \SS$
Then $\struct {\SS, \cap}$ is a [[Definition:Commutative Semigroup|commutative semigroup]]. | === Closure ===
We have {{hypothesis}} that $\struct {\SS, \cap}$ is [[Definition:Closed Algebraic Structure|closed]]. | Set System Closed under Intersection is Commutative Semigroup | https://proofwiki.org/wiki/Set_System_Closed_under_Intersection_is_Commutative_Semigroup | https://proofwiki.org/wiki/Set_System_Closed_under_Intersection_is_Commutative_Semigroup | [
"Set Systems",
"Set Intersection",
"Commutative Semigroups"
] | [
"Definition:Set of Sets",
"Definition:Commutative Semigroup"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-2423 | Set System Closed under Union is Commutative Semigroup | Let $\SS$ be a system of sets.
Let $\SS$ be such that:
:$\forall A, B \in \SS: A \cup B \in \SS$
Then $\struct {\SS, \cup}$ is a commutative semigroup. | === Closure ===
We have {{hypothesis}} that $\struct {\SS, \cup}$ is closed. | Let $\SS$ be a [[Definition:System of Sets|system of sets]].
Let $\SS$ be such that:
:$\forall A, B \in \SS: A \cup B \in \SS$
Then $\struct {\SS, \cup}$ is a [[Definition:Commutative Semigroup|commutative semigroup]]. | === Closure ===
We have {{hypothesis}} that $\struct {\SS, \cup}$ is [[Definition:Closed Algebraic Structure|closed]]. | Set System Closed under Union is Commutative Semigroup | https://proofwiki.org/wiki/Set_System_Closed_under_Union_is_Commutative_Semigroup | https://proofwiki.org/wiki/Set_System_Closed_under_Union_is_Commutative_Semigroup | [
"Set Systems",
"Set Union",
"Commutative Semigroups"
] | [
"Definition:Set of Sets",
"Definition:Commutative Semigroup"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-2424 | Set System Closed under Symmetric Difference is Abelian Group | Let $\SS$ be a system of sets.
Let $\SS$ be such that:
:$\forall A, B \in \SS: A \symdif B \in \SS$
where $A \symdif B$ denotes the symmetric difference between $A$ and $B$.
Then $\struct {\SS, \symdif}$ is an abelian group. | === {{Group-axiom|0|nolink}} ===
By presupposition on $\SS$, $\struct {\SS, \symdif}$ is closed.
{{qed|lemma}} | Let $\SS$ be a [[Definition:System of Sets|system of sets]].
Let $\SS$ be such that:
:$\forall A, B \in \SS: A \symdif B \in \SS$
where $A \symdif B$ denotes the [[Definition:Symmetric Difference|symmetric difference]] between $A$ and $B$.
Then $\struct {\SS, \symdif}$ is an [[Definition:Abelian Group|abelian group]... | === {{Group-axiom|0|nolink}} ===
By presupposition on $\SS$, $\struct {\SS, \symdif}$ is [[Definition:Closed Algebraic Structure|closed]].
{{qed|lemma}} | Set System Closed under Symmetric Difference is Abelian Group | https://proofwiki.org/wiki/Set_System_Closed_under_Symmetric_Difference_is_Abelian_Group | https://proofwiki.org/wiki/Set_System_Closed_under_Symmetric_Difference_is_Abelian_Group | [
"Abelian Groups",
"Symmetric Difference",
"Set Systems"
] | [
"Definition:Set of Sets",
"Definition:Symmetric Difference",
"Definition:Abelian Group"
] | [
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-2425 | Ring of Sets is Commutative Ring | A ring of sets $\struct {\RR, \symdif, \cap}$ is a commutative ring whose zero is $\O$. | By definition, the operations $\cap$ and $\symdif$ are closed in $\RR$.
Hence we can apply the following results:
* Set System Closed under Symmetric Difference is Abelian Group: $\struct {\RR, \symdif}$ is an abelian group.
* Set System Closed under Intersection is Commutative Semigroup: $\struct {\RR, \cap}$ is a com... | A [[Definition:Ring of Sets|ring of sets]] $\struct {\RR, \symdif, \cap}$ is a [[Definition:Commutative Ring|commutative ring]] whose [[Definition:Ring Zero|zero]] is $\O$. | By [[Definition:Ring of Sets|definition]], the operations $\cap$ and $\symdif$ are [[Definition:Closed Operation|closed]] in $\RR$.
Hence we can apply the following results:
* [[Set System Closed under Symmetric Difference is Abelian Group]]: $\struct {\RR, \symdif}$ is an [[Definition:Abelian Group|abelian group]].
... | Ring of Sets is Commutative Ring | https://proofwiki.org/wiki/Ring_of_Sets_is_Commutative_Ring | https://proofwiki.org/wiki/Ring_of_Sets_is_Commutative_Ring | [
"Commutative Rings",
"Rings of Sets"
] | [
"Definition:Ring of Sets",
"Definition:Commutative Ring",
"Definition:Ring Zero"
] | [
"Definition:Ring of Sets",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Set System Closed under Symmetric Difference is Abelian Group",
"Definition:Abelian Group",
"Set System Closed under Intersection is Commutative Semigroup",
"Definition:Commutative Semigroup",
"Intersection Distribu... |
proofwiki-2426 | Union as Symmetric Difference with Intersection | Let $A$ and $B$ be sets.
Then:
:$A \cup B = \paren {A \symdif B} \symdif \paren {A \cap B}$
where:
:$A \cup B$ denotes set union
:$A \cap B$ denotes set intersection
:$A \symdif B$ denotes set symmetric difference | From the definition of symmetric difference:
:$\paren {A \symdif B} \symdif \paren {A \cap B} = \paren {\paren {A \symdif B} \cup \paren {A \cap B} } \setminus \paren {\paren {A \symdif B} \cap \paren {A \cap B} }$
Also from the definition of symmetric difference:
:$\paren {A \symdif B} \cap \paren {A \cap B} = \paren ... | Let $A$ and $B$ be [[Definition:Set|sets]].
Then:
:$A \cup B = \paren {A \symdif B} \symdif \paren {A \cap B}$
where:
:$A \cup B$ denotes [[Definition:Set Union|set union]]
:$A \cap B$ denotes [[Definition:Set Intersection|set intersection]]
:$A \symdif B$ denotes [[Definition:Symmetric Difference|set symmetric differ... | From the definition of [[Definition:Symmetric Difference/Definition 2|symmetric difference]]:
:$\paren {A \symdif B} \symdif \paren {A \cap B} = \paren {\paren {A \symdif B} \cup \paren {A \cap B} } \setminus \paren {\paren {A \symdif B} \cap \paren {A \cap B} }$
Also from the definition of [[Definition:Symmetric Diff... | Union as Symmetric Difference with Intersection | https://proofwiki.org/wiki/Union_as_Symmetric_Difference_with_Intersection | https://proofwiki.org/wiki/Union_as_Symmetric_Difference_with_Intersection | [
"Set Union",
"Set Intersection",
"Symmetric Difference"
] | [
"Definition:Set",
"Definition:Set Union",
"Definition:Set Intersection",
"Definition:Symmetric Difference"
] | [
"Definition:Symmetric Difference/Definition 2",
"Definition:Symmetric Difference/Definition 2",
"Set Difference Intersection with Second Set is Empty Set",
"Set Difference with Empty Set is Self",
"Set Union is Idempotent",
"Union is Commutative",
"Union is Associative",
"Set Difference Union Intersec... |
proofwiki-2427 | Set Difference as Symmetric Difference with Intersection | :$S \setminus T = S \symdif \paren {S \cap T}$
where:
:$S \setminus T$ denotes set difference
:$S \symdif T$ denotes set symmetric difference
:$S \cap T$ denotes set intersection. | {{begin-eqn}}
{{eqn | l = S \symdif \paren {S \cap T}
| r = \paren {S \setminus \paren {S \cap T} } \cup \paren {\paren {S \cap T} \setminus S}
| c = {{Defof|Symmetric Difference}}
}}
{{eqn | r = \paren {S \setminus \paren {S \cap T} } \cup \O
| c = Set Difference of Intersection with Set is Empty Set... | :$S \setminus T = S \symdif \paren {S \cap T}$
where:
:$S \setminus T$ denotes [[Definition:Set Difference|set difference]]
:$S \symdif T$ denotes [[Definition:Symmetric Difference|set symmetric difference]]
:$S \cap T$ denotes [[Definition:Set Intersection|set intersection]]. | {{begin-eqn}}
{{eqn | l = S \symdif \paren {S \cap T}
| r = \paren {S \setminus \paren {S \cap T} } \cup \paren {\paren {S \cap T} \setminus S}
| c = {{Defof|Symmetric Difference}}
}}
{{eqn | r = \paren {S \setminus \paren {S \cap T} } \cup \O
| c = [[Set Difference of Intersection with Set is Empty S... | Set Difference as Symmetric Difference with Intersection | https://proofwiki.org/wiki/Set_Difference_as_Symmetric_Difference_with_Intersection | https://proofwiki.org/wiki/Set_Difference_as_Symmetric_Difference_with_Intersection | [
"Set Intersection",
"Set Difference",
"Symmetric Difference"
] | [
"Definition:Set Difference",
"Definition:Symmetric Difference",
"Definition:Set Intersection"
] | [
"Set Difference of Intersection with Set is Empty Set",
"Set Difference with Intersection is Difference",
"Union with Empty Set",
"Category:Set Intersection",
"Category:Set Difference",
"Category:Symmetric Difference"
] |
proofwiki-2428 | Unit of System of Sets is Unique | The unit of a system of sets, if it exists, is unique.
If $U$ is the unit of a system of sets $\SS$, then $\forall A \in \SS: A \subseteq U$. | Let $\SS$ be a system of sets.
Suppose $U$ and $U'$ are both units of $\SS$.
Then, by definition:
:$\forall A \in \SS: A \cap U = A$
:$\forall A \in \SS: A \cap U' = A$
This applies to both $U$ and $U'$, of course.
So $U \cap U' = U$ and $U' \cap U = U'$.
From Intersection with Subset is Subset it follows that $U \subs... | The [[Definition:Unit of System of Sets|unit of a system of sets]], if it exists, is unique.
If $U$ is the [[Definition:Unit of System of Sets|unit of a system of sets]] $\SS$, then $\forall A \in \SS: A \subseteq U$. | Let $\SS$ be a [[Definition:System of Sets|system of sets]].
Suppose $U$ and $U'$ are both [[Definition:Unit of System of Sets|units]] of $\SS$.
Then, by definition:
:$\forall A \in \SS: A \cap U = A$
:$\forall A \in \SS: A \cap U' = A$
This applies to both $U$ and $U'$, of course.
So $U \cap U' = U$ and $U' \cap U... | Unit of System of Sets is Unique | https://proofwiki.org/wiki/Unit_of_System_of_Sets_is_Unique | https://proofwiki.org/wiki/Unit_of_System_of_Sets_is_Unique | [
"Set Systems"
] | [
"Definition:Unit of System of Sets",
"Definition:Unit of System of Sets"
] | [
"Definition:Set of Sets",
"Definition:Unit of System of Sets",
"Intersection with Subset is Subset",
"Definition:Set Equality/Definition 2",
"Intersection with Subset is Subset",
"Category:Set Systems"
] |
proofwiki-2429 | Power Set is Algebra of Sets | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then $\powerset S$ is an algebra of sets where $S$ is the unit. | From Power Set is Closed under Intersection and Power Set is Closed under Symmetric Difference, we have that:
:$(1): \quad \forall A, B \in \powerset S: A \cap B \in \powerset S$
:$(2): \quad \forall A, B \in \powerset S: A * B \in \powerset S$
From the definition of power set:
:$\forall A \in \powerset S: A \subseteq ... | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then $\powerset S$ is an [[Definition:Algebra of Sets|algebra of sets]] where $S$ is the [[Definition:Unit of System of Sets|unit]]. | From [[Power Set is Closed under Intersection]] and [[Power Set is Closed under Symmetric Difference]], we have that:
:$(1): \quad \forall A, B \in \powerset S: A \cap B \in \powerset S$
:$(2): \quad \forall A, B \in \powerset S: A * B \in \powerset S$
From the definition of [[Definition:Power Set|power set]]:
:$\for... | Power Set is Algebra of Sets | https://proofwiki.org/wiki/Power_Set_is_Algebra_of_Sets | https://proofwiki.org/wiki/Power_Set_is_Algebra_of_Sets | [
"Power Set",
"Algebras of Sets"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Algebra of Sets",
"Definition:Unit of System of Sets"
] | [
"Power Set is Closed under Intersection",
"Power Set is Closed under Symmetric Difference",
"Definition:Power Set",
"Definition:Unit of System of Sets",
"Definition:Ring of Sets",
"Definition:Unit of System of Sets",
"Definition:Algebra of Sets/Definition 2",
"Category:Power Set",
"Category:Algebras... |
proofwiki-2430 | Power Set is Closed under Intersection | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$ | Let $A, B \in \powerset S$.
Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.
From Intersection is Subset we have that $A \cap B \subseteq A$.
It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.
Thus $A \cap B \in \powerset S$ and closure is proved.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$ | Let $A, B \in \powerset S$.
Then by the definition of [[Definition:Power Set|power set]], $A \subseteq S$ and $B \subseteq S$.
From [[Intersection is Subset]] we have that $A \cap B \subseteq A$.
It follows from [[Subset Relation is Transitive]] that $A \cap B \subseteq S$.
Thus $A \cap B \in \powerset S$ and [[Def... | Power Set is Closed under Intersection | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Intersection | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Intersection | [
"Power Set",
"Set Intersection"
] | [
"Definition:Set",
"Definition:Power Set"
] | [
"Definition:Power Set",
"Intersection is Subset",
"Subset Relation is Transitive",
"Definition:Closure (Abstract Algebra)"
] |
proofwiki-2431 | Power Set is Closed under Symmetric Difference | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A, B \in \powerset S: A \symdif B \in \powerset S$
where $A \symdif B$ is the symmetric difference between $A$ and $B$. | Let $A, B \in \powerset S$.
Then by definition of power set:
:$A, B \subseteq S$
Then:
{{begin-eqn}}
{{eqn | l = A \symdif B
| o = \subseteq
| r = A \cup B
| c = Symmetric Difference is Subset of Union
}}
{{eqn | o = \subseteq
| r = S
| c = Union is Smallest Superset
}}
{{eqn | ll= \leadst... | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then:
:$\forall A, B \in \powerset S: A \symdif B \in \powerset S$
where $A \symdif B$ is the [[Definition:Symmetric Difference|symmetric difference]] between $A$ and $B$. | Let $A, B \in \powerset S$.
Then by definition of [[Definition:Power Set|power set]]:
:$A, B \subseteq S$
Then:
{{begin-eqn}}
{{eqn | l = A \symdif B
| o = \subseteq
| r = A \cup B
| c = [[Symmetric Difference is Subset of Union]]
}}
{{eqn | o = \subseteq
| r = S
| c = [[Union is Smalle... | Power Set is Closed under Symmetric Difference | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Symmetric_Difference | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Symmetric_Difference | [
"Power Set",
"Symmetric Difference"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Symmetric Difference"
] | [
"Definition:Power Set",
"Symmetric Difference is Subset of Union",
"Union is Smallest Superset"
] |
proofwiki-2432 | Intersection of Rings of Sets | Let $\RR_k$ be a ring of sets, where $k$ is an element of an arbitrary set of indices.
Then their intersection $\ds \RR = \bigcap_k \RR_k$ is itself a ring of sets. | Consider the set $\ds \SS = \bigcup_k \RR_k$.
Let $S = \set {X \in Y: Y \in \SS}$.
This contains all the elements of all the sets contained in all the $\RR_k$.
Now consider the power set $\powerset S$ of $S$.
By Power Set is Algebra of Sets and the definition of algebra of sets, we have that $\powerset S$ is a ring of ... | Let $\RR_k$ be a [[Definition:Ring of Sets|ring of sets]], where $k$ is an element of an arbitrary [[Definition:Indexing Set|set of indices]].
Then their [[Definition:Set Intersection|intersection]] $\ds \RR = \bigcap_k \RR_k$ is itself a [[Definition:Ring of Sets|ring of sets]]. | Consider the [[Definition:Set|set]] $\ds \SS = \bigcup_k \RR_k$.
Let $S = \set {X \in Y: Y \in \SS}$.
This contains all the [[Definition:Element|elements]] of all the [[Definition:Set|sets]] contained in all the $\RR_k$.
Now consider the [[Definition:Power Set|power set]] $\powerset S$ of $S$.
By [[Power Set is Alg... | Intersection of Rings of Sets | https://proofwiki.org/wiki/Intersection_of_Rings_of_Sets | https://proofwiki.org/wiki/Intersection_of_Rings_of_Sets | [
"Rings of Sets"
] | [
"Definition:Ring of Sets",
"Definition:Indexing Set",
"Definition:Set Intersection",
"Definition:Ring of Sets"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Set",
"Definition:Power Set",
"Power Set is Algebra of Sets",
"Definition:Algebra of Sets",
"Definition:Ring of Sets",
"Definition:Ring (Abstract Algebra)",
"Definition:Abstract Algebra",
"Definition:Subring",
"Intersection of Subrings is Subri... |
proofwiki-2433 | Minimal Ring Generated by System of Sets | Let $\SS$ be a non-empty system of sets.
Then there is a unique ring of sets $\map \RR \SS$ which:
:$(1): \quad$ contains $\SS$
:$(2): \quad$ is contained by every ring of sets which also contains $\SS$.
This ring of sets $\map \RR \SS$ is called the '''minimal ring generated by $\SS$'''. | === Uniqueness ===
Suppose there were two such rings of sets $\map \RR \SS$ and $\map \RR \SS'$.
Then by definition their intersection $\map \RR \SS \cap \map \RR \SS'$ would also contain $\SS$.
By Intersection of Rings of Sets, $\map \RR \SS \cap \map \RR \SS'$ is also a ring of sets.
From Intersection is Subset, $\ma... | Let $\SS$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:System of Sets|system of sets]].
Then there is a unique [[Definition:Ring of Sets|ring of sets]] $\map \RR \SS$ which:
:$(1): \quad$ [[Definition:Subset|contains]] $\SS$
:$(2): \quad$ is [[Definition:Subset|contained by]] every [[Definition:Ring of Se... | === Uniqueness ===
Suppose there were two such [[Definition:Ring of Sets|rings of sets]] $\map \RR \SS$ and $\map \RR \SS'$.
Then by definition their [[Definition:Set Intersection|intersection]] $\map \RR \SS \cap \map \RR \SS'$ would also contain $\SS$.
By [[Intersection of Rings of Sets]], $\map \RR \SS \cap \map ... | Minimal Ring Generated by System of Sets | https://proofwiki.org/wiki/Minimal_Ring_Generated_by_System_of_Sets | https://proofwiki.org/wiki/Minimal_Ring_Generated_by_System_of_Sets | [
"Set Systems"
] | [
"Definition:Non-Empty Set",
"Definition:Set of Sets",
"Definition:Ring of Sets",
"Definition:Subset",
"Definition:Subset",
"Definition:Ring of Sets",
"Definition:Ring of Sets"
] | [
"Definition:Ring of Sets",
"Definition:Set Intersection",
"Intersection of Rings of Sets",
"Definition:Ring of Sets",
"Intersection is Subset",
"Definition:Set",
"Definition:Ring of Sets",
"Definition:Set",
"Definition:Ring of Sets",
"Definition:Set Intersection",
"Definition:Ring of Sets",
"D... |
proofwiki-2434 | Menelaus's Theorem | Let $ABC$ be a triangle.
Let points $D, E, F$ lie on lines $BC, AC, AB$ respectively (produced if necessary).
Then $D, E$ and $F$ are collinear {{iff}}:
: $\dfrac {AF} {FB} \cdot \dfrac {BD} {DC} \cdot \dfrac {CE} {EA} = -1$
In the above, the line segments $AF, BD, EA$ are determined to have negative length if they lie... | === Necessary Condition ===
First we check that the equation works out negative.
From Pasch's Axiom, the line $DEF$ must intersect either two sides of $\triangle ABC$:
:500px
or none of them:
:500px
That means there is an odd number of negative contributions to the product.
Hence the equation works out to be negative.
... | Let $ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let points $D, E, F$ lie on [[Definition:Straight Line|lines]] $BC, AC, AB$ respectively ([[Definition:Production|produced]] if necessary).
Then $D, E$ and $F$ are [[Definition:Collinear Points|collinear]] {{iff}}:
: $\dfrac {AF} {FB} \cdot \dfrac {BD} {DC}... | === Necessary Condition ===
First we check that the equation works out [[Definition:Negative|negative]].
From [[Axiom:Pasch's Axiom|Pasch's Axiom]], the line $DEF$ must [[Definition:Intersection (Geometry)|intersect]] either two [[Definition:Side of Polygon|sides]] of $\triangle ABC$:
:[[File:MenelausTheorem1.png|500... | Menelaus's Theorem | https://proofwiki.org/wiki/Menelaus's_Theorem | https://proofwiki.org/wiki/Menelaus's_Theorem | [
"Menelaus's Theorem",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Line/Straight Line",
"Definition:Production",
"Definition:Collinear/Points",
"Definition:Line/Segment",
"Definition:Negative",
"Definition:Linear Measure/Length",
"Definition:Line/Segment"
] | [
"Definition:Negative",
"Axiom:Pasch's Axiom",
"Definition:Intersection (Geometry)",
"Definition:Polygon/Side",
"File:MenelausTheorem1.png",
"File:MenelausTheorem2.png",
"Definition:Odd Integer",
"Definition:Negative",
"Definition:Multiplication",
"Definition:Negative",
"File:MenelausTheorem3.png... |
proofwiki-2435 | Lines Joining Equal and Parallel Straight Lines are Parallel | The straight lines joining equal and parallel straight lines at their endpoints, in the same direction, are themselves equal and parallel.
{{:Euclid:Proposition/I/33}} | :300px
Let $AB, CD$ be equal and parallel.
Let $AC, BD$ join their endpoints in the same direction.
Draw $BC$.
From Parallelism implies Equal Alternate Angles:
:$\angle ABC = \angle BCD$
We have that $AB, BC$ are equal to $DC, CB$ and $\angle ABC = \angle BCD$.
It follows from Triangle Side-Angle-Side Congruence that $... | The [[Definition:Straight Line|straight lines]] joining equal and [[Definition:Parallel Lines|parallel]] [[Definition:Straight Line|straight lines]] at their [[Definition:Endpoint of Line|endpoints]], in the same direction, are themselves equal and [[Definition:Parallel Lines|parallel]].
{{:Euclid:Proposition/I/33}} | :[[File:Euclid-I-33.png|300px]]
Let $AB, CD$ be equal and [[Definition:Parallel Lines|parallel]].
Let $AC, BD$ join their [[Definition:Endpoint of Line|endpoints]] in the same direction.
Draw $BC$.
From [[Parallelism implies Equal Alternate Angles]]:
:$\angle ABC = \angle BCD$
We have that $AB, BC$ are equal to ... | Lines Joining Equal and Parallel Straight Lines are Parallel | https://proofwiki.org/wiki/Lines_Joining_Equal_and_Parallel_Straight_Lines_are_Parallel | https://proofwiki.org/wiki/Lines_Joining_Equal_and_Parallel_Straight_Lines_are_Parallel | [
"Parallel Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Line/Endpoint",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Euclid-I-33.png",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Endpoint",
"Parallelism implies Equal Alternate Angles",
"Triangle Side-Angle-Side Congruence",
"Definition:Line/Straight Line",
"Definition:Transversal (Geometry)/Alternate Angles",
"Equal Alternate Angles implies Parall... |
proofwiki-2436 | Opposite Sides and Angles of Parallelogram are Equal | The opposite sides and angles of a parallelogram are equal to one another, and either of its diameters bisects its area.
{{:Euclid:Proposition/I/34}} | :300px
Let $ACDB$ be a parallelogram, and let $BC$ be a diameter.
By definition of parallelogram, $AB \parallel CD$, and $BC$ intersects both.
So by Parallelism implies Equal Alternate Angles:
:$\angle ABC = \angle BCD$
Similarly, by definition of parallelogram, $AC \parallel BD$, and $BC$ intersects both.
So by Parall... | The [[Definition:Opposite Sides|opposite sides]] and [[Definition:Opposite Angles|angles]] of a [[Definition:Parallelogram|parallelogram]] are equal to one another, and either of its [[Definition:Diameter of Parallelogram|diameters]] [[Definition:Bisection|bisects]] its [[Definition:Area|area]].
{{:Euclid:Proposition... | :[[File:Euclid-I-34.png|300px]]
Let $ACDB$ be a [[Definition:Parallelogram|parallelogram]], and let $BC$ be a [[Definition:Diameter of Parallelogram|diameter]].
By definition of [[Definition:Parallelogram|parallelogram]], $AB \parallel CD$, and $BC$ intersects both.
So by [[Parallelism implies Equal Alternate Angles... | Opposite Sides and Angles of Parallelogram are Equal | https://proofwiki.org/wiki/Opposite_Sides_and_Angles_of_Parallelogram_are_Equal | https://proofwiki.org/wiki/Opposite_Sides_and_Angles_of_Parallelogram_are_Equal | [
"Parallelograms"
] | [
"Definition:Polygon/Opposite",
"Definition:Polygon/Opposite",
"Definition:Quadrilateral/Parallelogram",
"Definition:Diameter of Parallelogram",
"Definition:Bisection",
"Definition:Area"
] | [
"File:Euclid-I-34.png",
"Definition:Quadrilateral/Parallelogram",
"Definition:Diameter of Parallelogram",
"Definition:Quadrilateral/Parallelogram",
"Parallelism implies Equal Alternate Angles",
"Definition:Quadrilateral/Parallelogram",
"Parallelism implies Equal Alternate Angles",
"Triangle Angle-Side... |
proofwiki-2437 | Equal Sized Triangles on Same Base have Same Height | Triangles of equal area which are on the same base, and on the same side of it, are also in the same parallels.
{{:Euclid:Proposition/I/39}} | :300px
Let $ABC$ and $DBC$ be equal-area triangles which are on the same base $BC$ and on the same side as it.
Let $AD$ be joined.
Suppose $AD$ were not parallel to $BC$.
Then, by Construction of Parallel Line we draw $AE$ parallel to $BC$.
So by Triangles with Same Base and Same Height have Equal Area:
: $\triangle AB... | [[Definition:Triangle (Geometry)|Triangles]] of equal area which are on the same [[Definition:Base of Triangle|base]], and on the same side of it, are also in the same [[Definition:Parallel Lines|parallels]].
{{:Euclid:Proposition/I/39}} | :[[File:Euclid-I-39.png|300px]]
Let $ABC$ and $DBC$ be equal-area [[Definition:Triangle (Geometry)|triangles]] which are on the same [[Definition:Base of Triangle|base]] $BC$ and on the same side as it.
Let $AD$ be joined.
Suppose $AD$ were not [[Definition:Parallel Lines|parallel]] to $BC$.
Then, by [[Constructio... | Equal Sized Triangles on Same Base have Same Height | https://proofwiki.org/wiki/Equal_Sized_Triangles_on_Same_Base_have_Same_Height | https://proofwiki.org/wiki/Equal_Sized_Triangles_on_Same_Base_have_Same_Height | [
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Base",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Euclid-I-39.png",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Base",
"Definition:Parallel (Geometry)/Lines",
"Construction of Parallel Line",
"Triangles with Same Base and Same Height have Equal Area",
"Definition:Parallel (Geometry)/Lines",
"Definition:Parallel (Geometry)/... |
proofwiki-2438 | Equal Sized Triangles on Equal Base have Same Height | Triangles of equal area which are on equal bases, and on the same side of it, are also in the same parallels.
{{:Euclid:Proposition/I/40}} | :300px
Let $ABC$ and $CDE$ be equal-area triangles which are on equal bases $BC$ and $CD$, and on the same side.
Let $AE$ be joined.
Suppose $AE$ were not parallel to $BC$.
Then, by Construction of Parallel Line we draw $AF$ parallel to $BD$.
So by Triangles with Equal Base and Same Height have Equal Area, $\triangle A... | [[Definition:Triangle (Geometry)|Triangles]] of equal area which are on equal [[Definition:Base of Triangle|bases]], and on the same side of it, are also in the same [[Definition:Parallel Lines|parallels]].
{{:Euclid:Proposition/I/40}} | :[[File:Euclid-I-40.png|300px]]
Let $ABC$ and $CDE$ be equal-area [[Definition:Triangle (Geometry)|triangles]] which are on equal [[Definition:Base of Triangle|bases]] $BC$ and $CD$, and on the same side.
Let $AE$ be joined.
Suppose $AE$ were not [[Definition:Parallel Lines|parallel]] to $BC$.
Then, by [[Construct... | Equal Sized Triangles on Equal Base have Same Height | https://proofwiki.org/wiki/Equal_Sized_Triangles_on_Equal_Base_have_Same_Height | https://proofwiki.org/wiki/Equal_Sized_Triangles_on_Equal_Base_have_Same_Height | [
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Base",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Euclid-I-40.png",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Base",
"Definition:Parallel (Geometry)/Lines",
"Construction of Parallel Line",
"Triangles with Equal Base and Same Height have Equal Area",
"Definition:Parallel (Geometry)/Lines",
"Definition:Converse Statement"... |
proofwiki-2439 | Parallelogram on Same Base as Triangle has Twice its Area | A parallelogram on the same base as a triangle, and in the same parallels, has twice the area of the triangle.
{{:Euclid:Proposition/I/41}} | :300px
Let $ABCD$ be a parallelogram on the same base $BC$ as a triangle $EBC$, between the same parallels $BC$ and $AE$.
Join $AC$.
Then $\triangle ABC = \triangle EBC$ from Triangles with Same Base and Same Height have Equal Area.
But from Opposite Sides and Angles of Parallelogram are Equal, $AC$ bisects $ABCD$.
So ... | A [[Definition:Parallelogram|parallelogram]] on the same [[Definition:Base of Parallelogram|base]] as a [[Definition:Base of Triangle|triangle]], and in the same [[Definition:Parallel Lines|parallels]], has twice the area of the [[Definition:Triangle (Geometry)|triangle]].
{{:Euclid:Proposition/I/41}} | :[[File:Euclid-I-41.png|300px]]
Let $ABCD$ be a [[Definition:Parallelogram|parallelogram]] on the same base $BC$ as a [[Definition:Triangle (Geometry)|triangle]] $EBC$, between the same [[Definition:Parallel Lines|parallels]] $BC$ and $AE$.
Join $AC$.
Then $\triangle ABC = \triangle EBC$ from [[Triangles with Same ... | Parallelogram on Same Base as Triangle has Twice its Area | https://proofwiki.org/wiki/Parallelogram_on_Same_Base_as_Triangle_has_Twice_its_Area | https://proofwiki.org/wiki/Parallelogram_on_Same_Base_as_Triangle_has_Twice_its_Area | [
"Area of Parallelogram",
"Areas of Triangles"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Parallelogram/Base",
"Definition:Triangle (Geometry)/Base",
"Definition:Parallel (Geometry)/Lines",
"Definition:Triangle (Geometry)"
] | [
"File:Euclid-I-41.png",
"Definition:Quadrilateral/Parallelogram",
"Definition:Triangle (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Triangles with Same Base and Same Height have Equal Area",
"Opposite Sides and Angles of Parallelogram are Equal",
"Definition:Bisection",
"Definition:Quadrilate... |
proofwiki-2440 | Construction of Parallelogram equal to Triangle in Given Angle | A parallelogram can be constructed in a given angle the same size as any given triangle.
{{:Euclid:Proposition/I/42}} | :400px
Let $ABC$ be the given triangle, and $D$ the given angle.
Bisect $BC$ at $E$, and join $AE$.
Construct $AG$ parallel to $EC$.
Construct $\angle CEF$ equal to $\angle D$.
Construct $CG$ parallel to $EF$.
Then $FEGC$ is a parallelogram.
Since $BE = EC$, from Triangles with Equal Base and Same Height have Equal Are... | A [[Definition:Parallelogram|parallelogram]] can be constructed in a given [[Definition:Rectilineal Angle|angle]] the same size as any given [[Definition:Triangle (Geometry)|triangle]].
{{:Euclid:Proposition/I/42}} | :[[File:Euclid-I-42.png|400px]]
Let $ABC$ be the given [[Definition:Triangle (Geometry)|triangle]], and $D$ the given [[Definition:Rectilineal Angle|angle]].
[[Bisection of Straight Line|Bisect]] $BC$ at $E$, and join $AE$.
[[Construction of Parallel Line|Construct]] $AG$ [[Definition:Parallel Lines|parallel]] to $E... | Construction of Parallelogram equal to Triangle in Given Angle | https://proofwiki.org/wiki/Construction_of_Parallelogram_equal_to_Triangle_in_Given_Angle | https://proofwiki.org/wiki/Construction_of_Parallelogram_equal_to_Triangle_in_Given_Angle | [
"Triangles",
"Parallelograms"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Angle/Rectilineal",
"Definition:Triangle (Geometry)"
] | [
"File:Euclid-I-42.png",
"Definition:Triangle (Geometry)",
"Definition:Angle/Rectilineal",
"Bisection of Straight Line",
"Construction of Parallel Line",
"Definition:Parallel (Geometry)/Lines",
"Construction of Equal Angle",
"Construction of Parallel Line",
"Definition:Parallel (Geometry)/Lines",
"... |
proofwiki-2441 | Construction of Parallelogram in Given Angle equal to Given Polygon | A parallelogram can be constructed in a given angle the same size as any given polygon.
{{:Euclid:Proposition/I/45}} | :600px
Let $ABCD$ be the given polygon, and let $E$ be the given angle.
Join $DB$, and construct the parallelogram $FGHK$ equal in size to $\triangle ABD$, in $\angle HKF = \angle E$.
Then construct the parallelogram $GLMH$ equal in area to $\triangle BCD$ on the line segment $GH$, in $\angle GHM = \angle E$.
We now ne... | A [[Definition:Parallelogram|parallelogram]] can be constructed in a given [[Definition:Rectilinear Angle|angle]] the same size as any given [[Definition:Polygon|polygon]].
{{:Euclid:Proposition/I/45}} | :[[File:Euclid-I-45.png|600px]]
Let $ABCD$ be the given [[Definition:Polygon|polygon]], and let $E$ be the given [[Definition:Rectilinear Angle|angle]].
Join $DB$, and [[Construction of Parallelogram equal to Triangle in Given Angle|construct the parallelogram]] $FGHK$ equal in size to $\triangle ABD$, in $\angle HKF... | Construction of Parallelogram in Given Angle equal to Given Polygon | https://proofwiki.org/wiki/Construction_of_Parallelogram_in_Given_Angle_equal_to_Given_Polygon | https://proofwiki.org/wiki/Construction_of_Parallelogram_in_Given_Angle_equal_to_Given_Polygon | [
"Parallelograms",
"Polygons"
] | [
"Definition:Quadrilateral/Parallelogram",
"Definition:Angle/Rectilineal",
"Definition:Polygon"
] | [
"File:Euclid-I-45.png",
"Definition:Polygon",
"Definition:Angle/Rectilineal",
"Construction of Parallelogram equal to Triangle in Given Angle",
"Construction of Parallelogram on Given Line equal to Triangle in Given Angle",
"Definition:Line/Segment",
"Definition:Quadrilateral/Parallelogram",
"Parallel... |
proofwiki-2442 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows directly and trivially from Square of Sum:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^2
| r = x^2 + 2 x y + y^2
| c = Square of Sum
}}
{{eqn | ll= \leadsto
| l = \paren {x + y}^2 - x^2
| r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows directly and trivially from [[Square of Sum]]:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^2
| r = x^2 + 2 x y + y^2
| c = [[Square of Sum]]
}}
{{eqn | ll= \leadsto
| l = \paren {x + y}^2 - x^2
| r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | Square of Sum less Square/Algebraic Proof 1 | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum_less_Square/Algebraic_Proof_1 | [
"Square of Sum",
"Square Function"
] | [] | [
"Square of Sum",
"Square of Sum"
] |
proofwiki-2443 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{begin-eqn}}
{{eqn | l = \paren {x + y}^2 - x^2
| r = \paren {x + y + x} \paren {x + y - x}
| c = Difference of Two Squares
}}
{{eqn | r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{begin-eqn}}
{{eqn | l = \paren {x + y}^2 - x^2
| r = \paren {x + y + x} \paren {x + y - x}
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | Square of Sum less Square/Algebraic Proof 2 | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum_less_Square/Algebraic_Proof_2 | [
"Square of Sum",
"Square Function"
] | [] | [
"Difference of Two Squares"
] |
proofwiki-2444 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{:Euclid:Proposition/II/6}}
Let $AB$ be bisected at $C$, and let $BD$ be added to it in a straight line.
Then the rectangle contained by $AD$ and $BD$ together with the square on $BC$ equals the square on $CD$.
:400px
The proof is as follows.
Construct the square $CEFD$ on $CD$, and join $DE$.
Construct $BG$ parallel ... | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{:Euclid:Proposition/II/6}}
Let $AB$ be [[Bisection of Straight Line|bisected at $C$]], and let $BD$ be added to it in a straight line.
Then the [[Definition:Containment of Rectangle|rectangle contained]] by $AD$ and $BD$ together with the [[Definition:Square (Geometry)|square]] on $BC$ equals the [[Definition:Squar... | Square of Sum less Square/Geometric Proof | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum_less_Square/Geometric_Proof | [
"Square of Sum",
"Square Function"
] | [] | [
"Bisection of Straight Line",
"Definition:Quadrilateral/Rectangle/Containment",
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"File:Euclid-II-6.png",
"Construction of Square on Given Straight Line",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Constructi... |
proofwiki-2445 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows from the distribution of multiplication over addition:
{{begin-eqn}}
{{eqn | l = \paren {a + 2 b}^2
| r = \paren {a + 2 b} \cdot \paren {a + 2 b}
| c =
}}
{{eqn | r = a \cdot \paren {a + 2 b} + 2 b \cdot \paren {a + 2 b}
| c = Real Multiplication Distributes over Addition
}}
{{eqn | r = a \cd... | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows from [[Real Multiplication Distributes over Addition|the distribution of multiplication over addition]]:
{{begin-eqn}}
{{eqn | l = \paren {a + 2 b}^2
| r = \paren {a + 2 b} \cdot \paren {a + 2 b}
| c =
}}
{{eqn | r = a \cdot \paren {a + 2 b} + 2 b \cdot \paren {a + 2 b}
| c = [[Real Multipli... | Square of Sum with Double/Algebraic Proof 1 | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum_with_Double/Algebraic_Proof_1 | [
"Square of Sum",
"Square Function"
] | [] | [
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition"
] |
proofwiki-2446 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | A direct application of the Binomial Theorem:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2, x = a, y = 2 b$.
{{qed}} | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | A direct application of the [[Binomial Theorem]]:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2, x = a, y = 2 b$.
{{qed}} | Square of Sum with Double/Algebraic Proof 2 | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum_with_Double/Algebraic_Proof_2 | [
"Square of Sum",
"Square Function"
] | [] | [
"Binomial Theorem"
] |
proofwiki-2447 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{:Euclid:Proposition/II/8}}
That is: $4 \paren {a + b} b + a^2 = \paren {a + 2 b}^2$.
:400px
Let the straight line $AB$ be cut at random at $C$.
Then four times the rectangle contained by $AB$ and $BC$ together with the square on $AC$ equals the square on $AB$ and $BC$ as a single straight line.
The proof is as follo... | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{:Euclid:Proposition/II/8}}
That is: $4 \paren {a + b} b + a^2 = \paren {a + 2 b}^2$.
:[[File:Euclid-II-8.png|400px]]
Let the [[Definition:Straight Line|straight line]] $AB$ be cut at random at $C$.
Then four times the [[Definition:Containment of Rectangle|rectangle contained]] by $AB$ and $BC$ together with the ... | Square of Sum with Double/Geometric Proof | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum_with_Double/Geometric_Proof | [
"Square of Sum",
"Square Function"
] | [] | [
"File:Euclid-II-8.png",
"Definition:Line/Straight Line",
"Definition:Quadrilateral/Rectangle/Containment",
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"Definition:Line/Straight Line",
"Definition:Production",
"Construction of Square on Given Straight Line",
"Square of Sum",... |
proofwiki-2448 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows from the distribution of multiplication over addition:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^2
| r = \paren {x + y} \cdot \paren {x + y}
| c = {{Defof|Square Function}}
}}
{{eqn | r = x \cdot \paren {x + y} + y \cdot \paren {x + y}
| c = Real Multiplication Distributes over Addition
}}
{{eq... | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows from [[Real Multiplication Distributes over Addition|the distribution of multiplication over addition]]:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^2
| r = \paren {x + y} \cdot \paren {x + y}
| c = {{Defof|Square Function}}
}}
{{eqn | r = x \cdot \paren {x + y} + y \cdot \paren {x + y}
| c = [[... | Square of Sum/Algebraic Proof 1 | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum/Algebraic_Proof_1 | [
"Square of Sum",
"Square Function"
] | [] | [
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition",
"Real Multiplication is Commutative"
] |
proofwiki-2449 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows directly from the Binomial Theorem:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2$.
{{qed}} | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | Follows directly from the [[Binomial Theorem]]:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2$.
{{qed}} | Square of Sum/Algebraic Proof 2 | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum/Algebraic_Proof_2 | [
"Square of Sum",
"Square Function"
] | [] | [
"Binomial Theorem"
] |
proofwiki-2450 | Square of Sum | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{:Euclid:Proposition/II/4}}
:250px
Let the straight line $AB$ be cut at random at $C$.
Construct the square $ADEB$ on $AB$ and join $DB$.
Construct $CF$ parallel to $AD$ through $C$.
Construct $HK$ parallel to $AB$ through $G$.
From Parallelism implies Equal Alternate Angles, $\angle CGB = \angle ADB$.
We have that $B... | :$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$ | {{:Euclid:Proposition/II/4}}
:[[File:Euclid-II-4.png|250px]]
Let the [[Definition:Straight Line|straight line]] $AB$ be cut at random at $C$.
[[Construction of Square on Given Straight Line|Construct the square $ADEB$]] on $AB$ and join $DB$.
[[Construction of Parallel Line|Construct $CF$ parallel]] to $AD$ through... | Square of Sum/Geometric Proof | https://proofwiki.org/wiki/Square_of_Sum | https://proofwiki.org/wiki/Square_of_Sum/Geometric_Proof | [
"Square of Sum",
"Square Function"
] | [] | [
"File:Euclid-II-4.png",
"Definition:Line/Straight Line",
"Construction of Square on Given Straight Line",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Parallelism implies Equal Alternate Angles",
"Isosceles Triangle has Two Equal Angles",
"Triangle with Two Equal Angles is Isosce... |
proofwiki-2451 | Square of Sum less Square | :$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$ | Follows directly and trivially from Square of Sum:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^2
| r = x^2 + 2 x y + y^2
| c = Square of Sum
}}
{{eqn | ll= \leadsto
| l = \paren {x + y}^2 - x^2
| r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | :$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$ | Follows directly and trivially from [[Square of Sum]]:
{{begin-eqn}}
{{eqn | l = \paren {x + y}^2
| r = x^2 + 2 x y + y^2
| c = [[Square of Sum]]
}}
{{eqn | ll= \leadsto
| l = \paren {x + y}^2 - x^2
| r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | Square of Sum less Square/Algebraic Proof 1 | https://proofwiki.org/wiki/Square_of_Sum_less_Square | https://proofwiki.org/wiki/Square_of_Sum_less_Square/Algebraic_Proof_1 | [
"Algebra",
"Square of Sum less Square"
] | [] | [
"Square of Sum",
"Square of Sum"
] |
proofwiki-2452 | Square of Sum less Square | :$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$ | {{begin-eqn}}
{{eqn | l = \paren {x + y}^2 - x^2
| r = \paren {x + y + x} \paren {x + y - x}
| c = Difference of Two Squares
}}
{{eqn | r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | :$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$ | {{begin-eqn}}
{{eqn | l = \paren {x + y}^2 - x^2
| r = \paren {x + y + x} \paren {x + y - x}
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \paren {2 x + y} y
| c =
}}
{{end-eqn}}
{{qed}} | Square of Sum less Square/Algebraic Proof 2 | https://proofwiki.org/wiki/Square_of_Sum_less_Square | https://proofwiki.org/wiki/Square_of_Sum_less_Square/Algebraic_Proof_2 | [
"Algebra",
"Square of Sum less Square"
] | [] | [
"Difference of Two Squares"
] |
proofwiki-2453 | Square of Sum less Square | :$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$ | {{:Euclid:Proposition/II/6}}
Let $AB$ be bisected at $C$, and let $BD$ be added to it in a straight line.
Then the rectangle contained by $AD$ and $BD$ together with the square on $BC$ equals the square on $CD$.
:400px
The proof is as follows.
Construct the square $CEFD$ on $CD$, and join $DE$.
Construct $BG$ parallel ... | :$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$ | {{:Euclid:Proposition/II/6}}
Let $AB$ be [[Bisection of Straight Line|bisected at $C$]], and let $BD$ be added to it in a straight line.
Then the [[Definition:Containment of Rectangle|rectangle contained]] by $AD$ and $BD$ together with the [[Definition:Square (Geometry)|square]] on $BC$ equals the [[Definition:Squar... | Square of Sum less Square/Geometric Proof | https://proofwiki.org/wiki/Square_of_Sum_less_Square | https://proofwiki.org/wiki/Square_of_Sum_less_Square/Geometric_Proof | [
"Algebra",
"Square of Sum less Square"
] | [] | [
"Bisection of Straight Line",
"Definition:Quadrilateral/Rectangle/Containment",
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"File:Euclid-II-6.png",
"Construction of Square on Given Straight Line",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Constructi... |
proofwiki-2454 | Square of Difference | :$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$ | {{begin-eqn}}
{{eqn | l = \paren {x - y}^2 =
| r = \paren {x - y} \cdot \paren {x - y}
| c =
}}
{{eqn | r = x \cdot \paren {x - y} - y \cdot \paren {x - y}
| c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x - x \cdot y - y \cdot x + y \cdot y
| c = Real Multiplication... | :$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$ | {{begin-eqn}}
{{eqn | l = \paren {x - y}^2 =
| r = \paren {x - y} \cdot \paren {x - y}
| c =
}}
{{eqn | r = x \cdot \paren {x - y} - y \cdot \paren {x - y}
| c = [[Real Multiplication Distributes over Addition]]
}}
{{eqn | r = x \cdot x - x \cdot y - y \cdot x + y \cdot y
| c = [[Real Multipli... | Square of Difference/Algebraic Proof 1 | https://proofwiki.org/wiki/Square_of_Difference | https://proofwiki.org/wiki/Square_of_Difference/Algebraic_Proof_1 | [
"Square Function",
"Square of Difference"
] | [] | [
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition"
] |
proofwiki-2455 | Square of Difference | :$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$ | Follows directly from the Binomial Theorem:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2$ and $y = -y$.
{{qed}} | :$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$ | Follows directly from the [[Binomial Theorem]]:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2$ and $y = -y$.
{{qed}} | Square of Difference/Algebraic Proof 2 | https://proofwiki.org/wiki/Square_of_Difference | https://proofwiki.org/wiki/Square_of_Difference/Algebraic_Proof_2 | [
"Square Function",
"Square of Difference"
] | [] | [
"Binomial Theorem"
] |
proofwiki-2456 | Square of Difference | :$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$ | {{:Euclid:Proposition/II/7}}
:250px
That is:
:$x^2 + y^2 = \paren {x - y}^2 + 2 x y$
Let the straight line $AB$ be cut at random at $C$.
Construct the square $ADEB$ on $AB$ and join $DB$.
Construct $CN$ parallel to $AD$ through $C$ and let it cross $DB$ at $G$.
Construct $HF$ parallel to $AB$ through $G$.
From Complem... | :$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$ | {{:Euclid:Proposition/II/7}}
:[[File:Euclid-II-7.png|250px]]
That is:
:$x^2 + y^2 = \paren {x - y}^2 + 2 x y$
Let the [[Definition:Straight Line|straight line]] $AB$ be cut at random at $C$.
[[Construction of Square on Given Straight Line|Construct the square $ADEB$]] on $AB$ and join $DB$.
[[Construction of Pa... | Square of Difference/Geometric Proof | https://proofwiki.org/wiki/Square_of_Difference | https://proofwiki.org/wiki/Square_of_Difference/Geometric_Proof | [
"Square Function",
"Square of Difference"
] | [] | [
"File:Euclid-II-7.png",
"Definition:Line/Straight Line",
"Construction of Square on Given Straight Line",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Complements of Parallelograms are Equal",
"Definition:Gnomon",
"Definition:Gnomon",
"Definition:Quadrilateral/Rectangle/Contain... |
proofwiki-2457 | Square of Sum with Double | :$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$ | Follows from the distribution of multiplication over addition:
{{begin-eqn}}
{{eqn | l = \paren {a + 2 b}^2
| r = \paren {a + 2 b} \cdot \paren {a + 2 b}
| c =
}}
{{eqn | r = a \cdot \paren {a + 2 b} + 2 b \cdot \paren {a + 2 b}
| c = Real Multiplication Distributes over Addition
}}
{{eqn | r = a \cd... | :$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$ | Follows from [[Real Multiplication Distributes over Addition|the distribution of multiplication over addition]]:
{{begin-eqn}}
{{eqn | l = \paren {a + 2 b}^2
| r = \paren {a + 2 b} \cdot \paren {a + 2 b}
| c =
}}
{{eqn | r = a \cdot \paren {a + 2 b} + 2 b \cdot \paren {a + 2 b}
| c = [[Real Multipli... | Square of Sum with Double/Algebraic Proof 1 | https://proofwiki.org/wiki/Square_of_Sum_with_Double | https://proofwiki.org/wiki/Square_of_Sum_with_Double/Algebraic_Proof_1 | [
"Algebra",
"Square of Sum with Double"
] | [] | [
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition",
"Real Multiplication Distributes over Addition"
] |
proofwiki-2458 | Square of Sum with Double | :$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$ | A direct application of the Binomial Theorem:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2, x = a, y = 2 b$.
{{qed}} | :$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$ | A direct application of the [[Binomial Theorem]]:
:$\ds \forall n \in \Z_{\ge 0}: \paren {x + y}^n = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k$
putting $n = 2, x = a, y = 2 b$.
{{qed}} | Square of Sum with Double/Algebraic Proof 2 | https://proofwiki.org/wiki/Square_of_Sum_with_Double | https://proofwiki.org/wiki/Square_of_Sum_with_Double/Algebraic_Proof_2 | [
"Algebra",
"Square of Sum with Double"
] | [] | [
"Binomial Theorem"
] |
proofwiki-2459 | Square of Sum with Double | :$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$ | {{:Euclid:Proposition/II/8}}
That is: $4 \paren {a + b} b + a^2 = \paren {a + 2 b}^2$.
:400px
Let the straight line $AB$ be cut at random at $C$.
Then four times the rectangle contained by $AB$ and $BC$ together with the square on $AC$ equals the square on $AB$ and $BC$ as a single straight line.
The proof is as follo... | :$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$ | {{:Euclid:Proposition/II/8}}
That is: $4 \paren {a + b} b + a^2 = \paren {a + 2 b}^2$.
:[[File:Euclid-II-8.png|400px]]
Let the [[Definition:Straight Line|straight line]] $AB$ be cut at random at $C$.
Then four times the [[Definition:Containment of Rectangle|rectangle contained]] by $AB$ and $BC$ together with the ... | Square of Sum with Double/Geometric Proof | https://proofwiki.org/wiki/Square_of_Sum_with_Double | https://proofwiki.org/wiki/Square_of_Sum_with_Double/Geometric_Proof | [
"Algebra",
"Square of Sum with Double"
] | [] | [
"File:Euclid-II-8.png",
"Definition:Line/Straight Line",
"Definition:Quadrilateral/Rectangle/Containment",
"Definition:Quadrilateral/Square",
"Definition:Quadrilateral/Square",
"Definition:Line/Straight Line",
"Definition:Production",
"Construction of Square on Given Straight Line",
"Square of Sum",... |
proofwiki-2460 | Sum of Squares of Sum and Difference | :$\forall a, b \in \R: \paren {a + b}^2 + \paren {a - b}^2 = 2 \paren {a^2 + b^2}$ | {{begin-eqn}}
{{eqn | o =
| r = \left({a + b}\right)^2 + \left({a - b}\right)^2
| c =
}}
{{eqn | r = a^2 + 2 a b + b^2 + a^2 - 2 a b + b^2
| c = Square of Sum and Square of Difference
}}
{{eqn | r = a^2 + b^2 + a^2 + b^2
}}
{{eqn | r = 2 \left({a^2 + b^2}\right)
| c =
}}
{{end-eqn}}
{{qed}} | :$\forall a, b \in \R: \paren {a + b}^2 + \paren {a - b}^2 = 2 \paren {a^2 + b^2}$ | {{begin-eqn}}
{{eqn | o =
| r = \left({a + b}\right)^2 + \left({a - b}\right)^2
| c =
}}
{{eqn | r = a^2 + 2 a b + b^2 + a^2 - 2 a b + b^2
| c = [[Square of Sum]] and [[Square of Difference]]
}}
{{eqn | r = a^2 + b^2 + a^2 + b^2
}}
{{eqn | r = 2 \left({a^2 + b^2}\right)
| c =
}}
{{end-eqn}}
{... | Sum of Squares of Sum and Difference/Algebraic Proof | https://proofwiki.org/wiki/Sum_of_Squares_of_Sum_and_Difference | https://proofwiki.org/wiki/Sum_of_Squares_of_Sum_and_Difference/Algebraic_Proof | [
"Algebra",
"Squares",
"Sum of Squares of Sum and Difference"
] | [] | [
"Square of Sum",
"Square of Difference"
] |
proofwiki-2461 | Sum of Squares of Sum and Difference | :$\forall a, b \in \R: \paren {a + b}^2 + \paren {a - b}^2 = 2 \paren {a^2 + b^2}$ | {{:Euclid:Proposition/II/9}}
:400px
That is, from the above diagram:
: $\left({AC + CD}\right)^2 + \left({AC - CD}\right)^2 = 2 \left({AC^2 + CD^2}\right)$
Let $AB$ be bisected at $C$, and let $D$ be another random point on $AB$.
Then the squares on $AD$ and $DB$ are double the squares on $AC$ and $CD$.
The proof is as... | :$\forall a, b \in \R: \paren {a + b}^2 + \paren {a - b}^2 = 2 \paren {a^2 + b^2}$ | {{:Euclid:Proposition/II/9}}
:[[File:Euclid-II-9.png|400px]]
That is, from the above diagram:
: $\left({AC + CD}\right)^2 + \left({AC - CD}\right)^2 = 2 \left({AC^2 + CD^2}\right)$
Let $AB$ be [[Bisection of Straight Line|bisected at $C$]], and let $D$ be another random point on $AB$.
Then the [[Definition:Square ... | Sum of Squares of Sum and Difference/Geometric Proof 1 | https://proofwiki.org/wiki/Sum_of_Squares_of_Sum_and_Difference | https://proofwiki.org/wiki/Sum_of_Squares_of_Sum_and_Difference/Geometric_Proof_1 | [
"Algebra",
"Squares",
"Sum of Squares of Sum and Difference"
] | [] | [
"File:Euclid-II-9.png",
"Bisection of Straight Line",
"Definition:Quadrilateral/Square",
"Construction of Perpendicular Line",
"Construction of Equal Straight Lines from Unequal",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Isosceles Triangle has Two Equal Angles",
"Definition... |
proofwiki-2462 | Sum of Squares of Sum and Difference | :$\forall a, b \in \R: \paren {a + b}^2 + \paren {a - b}^2 = 2 \paren {a^2 + b^2}$ | {{:Euclid:Proposition/II/10}}
:400px
That is, from the above diagram:
:$\paren {AC + CD}^2 + \paren {CD - AC}^2 = 2 \paren {AC^2 + CD^2}$
Let $AB$ be bisected at $C$, and let $AB$ be produced to some point $D$.
Then the squares on $AD$ and $BD$ are double the squares on $AC$ and $CD$.
The proof is as follows.
Construct... | :$\forall a, b \in \R: \paren {a + b}^2 + \paren {a - b}^2 = 2 \paren {a^2 + b^2}$ | {{:Euclid:Proposition/II/10}}
:[[File:Euclid-II-10.png|400px]]
That is, from the above diagram:
:$\paren {AC + CD}^2 + \paren {CD - AC}^2 = 2 \paren {AC^2 + CD^2}$
Let $AB$ be [[Bisection of Straight Line|bisected at $C$]], and let $AB$ be [[Definition:Production|produced]] to some point $D$.
Then the [[Definition... | Sum of Squares of Sum and Difference/Geometric Proof 2 | https://proofwiki.org/wiki/Sum_of_Squares_of_Sum_and_Difference | https://proofwiki.org/wiki/Sum_of_Squares_of_Sum_and_Difference/Geometric_Proof_2 | [
"Algebra",
"Squares",
"Sum of Squares of Sum and Difference"
] | [] | [
"File:Euclid-II-10.png",
"Bisection of Straight Line",
"Definition:Production",
"Definition:Quadrilateral/Square",
"Construction of Perpendicular Line",
"Construction of Equal Straight Lines from Unequal",
"Construction of Parallel Line",
"Construction of Parallel Line",
"Parallelism implies Supplem... |
proofwiki-2463 | Construction of Square equal to Given Polygon | A square can be constructed the same size as any given polygon.
{{:Euclid:Proposition/II/14}} | :500px
Let $A$ be the given polygon.
Construct the rectangle $BCDE$ equal to the given polygon.
If it so happens that $BE = ED$, then $BCDE$ is a square, and the construction is complete.
Suppose $BE \ne ED$. Then WLOG suppose $BE > ED$.
Produce $BE$ from $E$ and construct on it $EF = ED$.
Bisect $BF$ at $G$.
Construct... | A [[Definition:Square (Geometry)|square]] can be constructed the same size as any given [[Definition:Polygon|polygon]].
{{:Euclid:Proposition/II/14}} | :[[File:Euclid-II-14.png|500px]]
Let $A$ be the given [[Definition:Polygon|polygon]].
[[Construction of Parallelogram in Given Angle equal to Given Polygon|Construct the rectangle $BCDE$]] equal to the given [[Definition:Polygon|polygon]].
If it so happens that $BE = ED$, then $BCDE$ is a [[Definition:Square (Geome... | Construction of Square equal to Given Polygon | https://proofwiki.org/wiki/Construction_of_Square_equal_to_Given_Polygon | https://proofwiki.org/wiki/Construction_of_Square_equal_to_Given_Polygon | [
"Squares",
"Polygons"
] | [
"Definition:Quadrilateral/Square",
"Definition:Polygon"
] | [
"File:Euclid-II-14.png",
"Definition:Polygon",
"Construction of Parallelogram in Given Angle equal to Given Polygon",
"Definition:Polygon",
"Definition:Quadrilateral/Square",
"Definition:WLOG",
"Definition:Production",
"Construction of Equal Straight Lines from Unequal",
"Bisection of Straight Line"... |
proofwiki-2464 | Finding Center of Circle | For any given circle, it is possible to find its center.
{{:Euclid:Proposition/III/1}} | :300px
Draw any chord $AB$ on the circle in question.
Bisect $AB$ at $D$.
Construct $CE$ perpendicular to $AB$ at $D$, where $C$ and $E$ are where this perpendicular meets the circle.
Bisect $CE$ at $F$.
Then $F$ is the center of the circle.
The proof is as follows.
Suppose $F$ were not the center of the circle, but th... | For any given [[Definition:Circle|circle]], it is possible to find its [[Definition:Center of Circle|center]].
{{:Euclid:Proposition/III/1}} | :[[File:Euclid-III-1.png|300px]]
Draw any [[Definition:Chord of Circle|chord]] $AB$ on the circle in question.
[[Bisection of Straight Line|Bisect $AB$]] at $D$.
[[Construction of Perpendicular Line|Construct $CE$ perpendicular]] to $AB$ at $D$, where $C$ and $E$ are where this [[Definition:Perpendicular|perpendicul... | Finding Center of Circle/Proof 1 | https://proofwiki.org/wiki/Finding_Center_of_Circle | https://proofwiki.org/wiki/Finding_Center_of_Circle/Proof_1 | [
"Circles",
"Finding Center of Circle"
] | [
"Definition:Circle",
"Definition:Circle/Center"
] | [
"File:Euclid-III-1.png",
"Definition:Circle/Chord",
"Bisection of Straight Line",
"Construction of Perpendicular Line",
"Definition:Right Angle/Perpendicular",
"Definition:Circle",
"Bisection of Straight Line",
"Definition:Circle/Center",
"Definition:Circle/Center",
"Definition:Circle/Center",
"... |
proofwiki-2465 | Finding Center of Circle | For any given circle, it is possible to find its center.
{{:Euclid:Proposition/III/1}} | From Perpendicular Bisector of Chord Passes Through Center, $CE$ passes through the center of the circle.
The center must be the point $F$ such that $FE = FC$.
That is, $F$ is the bisector of $CE$. | For any given [[Definition:Circle|circle]], it is possible to find its [[Definition:Center of Circle|center]].
{{:Euclid:Proposition/III/1}} | From [[Perpendicular Bisector of Chord Passes Through Center]], $CE$ passes through the [[Definition:Center of Circle|center of the circle]].
The [[Definition:Center of Circle|center]] must be the point $F$ such that $FE = FC$.
That is, $F$ is the [[Definition:Bisection|bisector]] of $CE$. | Finding Center of Circle/Proof 2 | https://proofwiki.org/wiki/Finding_Center_of_Circle | https://proofwiki.org/wiki/Finding_Center_of_Circle/Proof_2 | [
"Circles",
"Finding Center of Circle"
] | [
"Definition:Circle",
"Definition:Circle/Center"
] | [
"Perpendicular Bisector of Chord Passes Through Center",
"Definition:Circle/Center",
"Definition:Circle/Center",
"Definition:Bisection"
] |
proofwiki-2466 | Conditions for Diameter to be Perpendicular Bisector | If in a circle a diameter bisects a chord (which is itself not a diameter), then it cuts it at right angles, and if it cuts it at right angles then it bisects it.
{{:Euclid:Proposition/III/3}} | :300px
Let $ABC$ be a circle, in which $AB$ is a chord which is not a diameter (i.e. it does not pass through the center). | If in a [[Definition:Circle|circle]] a [[Definition:Diameter of Circle|diameter]] [[Definition:Bisection|bisects]] a [[Definition:Chord of Circle|chord]] (which is itself not a [[Definition:Diameter of Circle|diameter]]), then it cuts it at [[Definition:Right Angle|right angles]], and if it cuts it at [[Definition:Righ... | :[[File:Euclid-III-3.png|300px]]
Let $ABC$ be a [[Definition:Circle|circle]], in which $AB$ is a [[Definition:Chord of Circle|chord]] which is not a [[Definition:Diameter of Circle|diameter]] (i.e. it does not pass through the [[Definition:Center of Circle|center]]). | Conditions for Diameter to be Perpendicular Bisector | https://proofwiki.org/wiki/Conditions_for_Diameter_to_be_Perpendicular_Bisector | https://proofwiki.org/wiki/Conditions_for_Diameter_to_be_Perpendicular_Bisector | [
"Circles"
] | [
"Definition:Circle",
"Definition:Circle/Diameter",
"Definition:Bisection",
"Definition:Circle/Chord",
"Definition:Circle/Diameter",
"Definition:Right Angle",
"Definition:Right Angle",
"Definition:Bisection"
] | [
"File:Euclid-III-3.png",
"Definition:Circle",
"Definition:Circle/Chord",
"Definition:Circle/Diameter",
"Definition:Circle/Center",
"Definition:Circle/Diameter",
"Definition:Circle/Diameter",
"Definition:Circle/Diameter"
] |
proofwiki-2467 | Chords do not Bisect Each Other | If in a circle two chords (which are not diameters) cut one another, then they do not bisect one another.
{{:Euclid:Proposition/III/4}} | :250px
Let $ABCD$ be a circle, in which $AC$ and $BD$ are chords which are not diameters (i.e. they do not pass through the center).
Let $AC$ and $BD$ intersect at $E$.
{{AimForCont}} they were able to bisect one another, such that $AE = EC$ and $BE = ED$.
Find the center $F$ of the circle, and join $FE$.
From Conditio... | If in a [[Definition:Circle|circle]] two [[Definition:Chord of Circle|chords]] (which are not [[Definition:Diameter of Circle|diameters]]) cut one another, then they do not [[Definition:Bisection|bisect]] one another.
{{:Euclid:Proposition/III/4}} | :[[File:Euclid-III-4.png|250px]]
Let $ABCD$ be a circle, in which $AC$ and $BD$ are [[Definition:Chord of Circle|chords]] which are not [[Definition:Diameter of Circle|diameters]] (i.e. they do not pass through the [[Definition:Center of Circle|center]]).
Let $AC$ and $BD$ [[Definition:Intersection (Geometry)|interse... | Chords do not Bisect Each Other | https://proofwiki.org/wiki/Chords_do_not_Bisect_Each_Other | https://proofwiki.org/wiki/Chords_do_not_Bisect_Each_Other | [
"Circles"
] | [
"Definition:Circle",
"Definition:Circle/Chord",
"Definition:Circle/Diameter",
"Definition:Bisection"
] | [
"File:Euclid-III-4.png",
"Definition:Circle/Chord",
"Definition:Circle/Diameter",
"Definition:Circle/Center",
"Definition:Intersection (Geometry)",
"Definition:Bisection",
"Finding Center of Circle",
"Conditions for Diameter to be Perpendicular Bisector",
"Definition:Bisection",
"Definition:Right ... |
proofwiki-2468 | Concentric Circles do not Intersect | If two circles are concentric, they share no points on their circumferences.
Alternatively, this can be worded:
:If two concentric circles share one point on their circumferences, then they share them all (that is, they are the same circle). | This follows directly from:
:Intersecting Circles have Different Centers
:Touching Circles have Different Centers.
{{qed}}
Category:Concentric Circles
jrlz100khvixyaze2kcnsi6uaptwglk | If two [[Definition:Circle|circles]] are [[Definition:Concentric Circles|concentric]], they share no [[Definition:Point|points]] on their [[Definition:Circumference of Circle|circumferences]].
Alternatively, this can be worded:
:If two [[Definition:Concentric Circles|concentric circles]] share one point on their [[De... | This follows directly from:
:[[Intersecting Circles have Different Centers]]
:[[Touching Circles have Different Centers]].
{{qed}}
[[Category:Concentric Circles]]
jrlz100khvixyaze2kcnsi6uaptwglk | Concentric Circles do not Intersect | https://proofwiki.org/wiki/Concentric_Circles_do_not_Intersect | https://proofwiki.org/wiki/Concentric_Circles_do_not_Intersect | [
"Concentric Circles"
] | [
"Definition:Circle",
"Definition:Concentric/Circles",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Concentric/Circles",
"Definition:Circle/Circumference"
] | [
"Intersecting Circles have Different Centers",
"Touching Circles have Different Centers",
"Category:Concentric Circles"
] |
proofwiki-2469 | Existence of Real Logarithm | Let $b, y \in \R$ such that $b > 1$ and $y > 0$.
Then there exists a unique real $x \in \R$ such that $b^x = y$.
This $x$ is called the '''logarithm of $y$ to the base $b$'''.
Also see the definition of a (general) logarithm. | We start by establishing a lemma: | Let $b, y \in \R$ such that $b > 1$ and $y > 0$.
Then there exists a unique real $x \in \R$ such that $b^x = y$.
This $x$ is called the '''logarithm of $y$ to the base $b$'''.
Also see [[Definition:General Logarithm|the definition of a (general) logarithm]]. | We start by establishing a lemma: | Existence of Real Logarithm | https://proofwiki.org/wiki/Existence_of_Real_Logarithm | https://proofwiki.org/wiki/Existence_of_Real_Logarithm | [
"Logarithms"
] | [
"Definition:General Logarithm"
] | [] |
proofwiki-2470 | Baire Category Theorem/Complete Metric Space | Let $M = \struct {A, d}$ be a complete metric space.
Then $M = \struct {A, d}$ is also a Baire space. | Let $U_n$ be a countable set of open sets of $M$ all of which are everywhere dense.
The strategy of this proof is to show that the intersection $\bigcap U_n$ is everywhere dense.
Let $W \subseteq A$ be a non-empty open set of $M$.
From Open Set Characterization of Denseness, since $U_1$ is everywhere dense:
:$W \cap U_... | Let $M = \struct {A, d}$ be a [[Definition:Complete Metric Space|complete metric space]].
Then $M = \struct {A, d}$ is also a [[Definition:Baire Space (Topology)|Baire space]]. | Let $U_n$ be a [[Definition:Countable Set|countable set]] of [[Definition:Open Set (Topology)|open sets]] of $M$ all of which are [[Definition:Everywhere Dense|everywhere dense]].
The strategy of this proof is to show that the [[Definition:Set Intersection|intersection]] $\bigcap U_n$ is [[Definition:Everywhere Dense|... | Baire Category Theorem/Complete Metric Space | https://proofwiki.org/wiki/Baire_Category_Theorem/Complete_Metric_Space | https://proofwiki.org/wiki/Baire_Category_Theorem/Complete_Metric_Space | [
"Baire Category Theorem",
"Complete Metric Spaces",
"Baire Spaces",
"Examples of Use of Axiom of Dependent Choice"
] | [
"Definition:Complete Metric Space",
"Definition:Baire Space (Topology)"
] | [
"Definition:Countable Set",
"Definition:Open Set/Topology",
"Definition:Everywhere Dense",
"Definition:Set Intersection",
"Definition:Everywhere Dense",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Open Set Characterization of Denseness",
"Definition:Everywhere Dense",
"Definition:... |
proofwiki-2471 | Baire Characterisation Theorem | A real-valued function $f$ defined on a Banach space $X$ is a Baire-$1$ function {{iff}}:
: for every non-empty closed subset $K$ of $X$, the restriction of $f$ to $K$ has a point of continuity relative to the topology of $K$.
{{explain|What is Baire-$1$? The linked definition does not have the numeration.}} | {{ProofWanted}}
{{Namedfor|René-Louis Baire|cat = Baire}}
Category:Topology
4olj66tipmyrbw8qpve0xhj63f91bde | A [[Definition:Real-Valued Function|real-valued function]] $f$ defined on a [[Definition:Banach Space|Banach space]] $X$ is a [[Definition:Baire Function|Baire-$1$ function]] {{iff}}:
: for every [[Definition:Empty Set|non-empty]] [[Definition:Closure (Topology)|closed subset]] $K$ of $X$, the [[Definition:Restriction ... | {{ProofWanted}}
{{Namedfor|René-Louis Baire|cat = Baire}}
[[Category:Topology]]
4olj66tipmyrbw8qpve0xhj63f91bde | Baire Characterisation Theorem | https://proofwiki.org/wiki/Baire_Characterisation_Theorem | https://proofwiki.org/wiki/Baire_Characterisation_Theorem | [
"Topology"
] | [
"Definition:Real-Valued Function",
"Definition:Banach Space",
"Definition:Baire Function",
"Definition:Empty Set",
"Definition:Closure (Topology)",
"Definition:Restriction/Mapping",
"Definition:Continuous Mapping (Topology)/Point",
"Definition:Topology"
] | [
"Category:Topology"
] |
proofwiki-2472 | Ramsey's Theorem | In any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs.
For $2$ colors, Ramsey's theorem states that for any pair of positive integers $\tuple {r, s}$, there exists a least positive integer $\map R {r, s}$ such that for any complete graph on $\map R {r, s}$ v... | First we prove Ramsey's Theorem for the 2-color case, by induction on $r + s$.
It is clear from the definition that
:$\forall n \in \N: \map R {n, 1} = \map R {1, n} = 1$
because the complete graph on one node has no edges.
This is the basis for the induction.
We prove that $\map R {r, s}$ exists by finding an explicit... | In any coloring of the edges of a sufficiently large [[Definition:Complete Graph|complete graph]], one will find [[Definition:Monochromatic Graph|monochromatic]] [[Definition:Complete Graph|complete]] [[Definition:Subgraph|subgraphs]].
For $2$ colors, Ramsey's theorem states that for any pair of [[Definition:Positive... | First we prove [[Ramsey's Theorem]] for the 2-color case, by [[Principle of Mathematical Induction|induction]] on $r + s$.
It is clear from the definition that
:$\forall n \in \N: \map R {n, 1} = \map R {1, n} = 1$
because the complete graph on one node has no edges.
This is the [[Definition:Basis for the Induction|b... | Ramsey's Theorem | https://proofwiki.org/wiki/Ramsey's_Theorem | https://proofwiki.org/wiki/Ramsey's_Theorem | [
"Ramsey Theory"
] | [
"Definition:Complete Graph",
"Definition:Monochromatic Graph",
"Definition:Complete Graph",
"Definition:Subgraph",
"Definition:Positive/Integer",
"Definition:Complete Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Edge",
"... | [
"Ramsey's Theorem",
"Principle of Mathematical Induction",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis"
] |
proofwiki-2473 | Schur's Theorem (Ramsey Theory) | Let $r$ be a positive integer.
Then there exists a positive integer $S$ such that:
:for every partition of the integers $\set {1, \ldots, S}$ into $r$ parts, one of the parts contains integers $x$, $y$ and $z$ such that:
::$x + y = z$ | Let:
:$n = \map R {3, \ldots, 3}$
where $\map R {3, \ldots, 3}$ denotes the Ramsey number on $r$ colors.
Take $S$ to be $n$.
{{refactor|Extract the below process of "coloring" a partition into its own page|level = medium}}
partition the integers $\set {1, \ldots, n}$ into $r$ parts, which we denote by '''colors'''.
Tha... | Let $r$ be a [[Definition:Positive Integer|positive integer]].
Then there exists a [[Definition:Positive Integer|positive integer]] $S$ such that:
:for every [[Definition:Set Partition|partition]] of the [[Definition:Integer|integers]] $\set {1, \ldots, S}$ into $r$ parts, one of the parts contains integers $x$, $y$ a... | Let:
:$n = \map R {3, \ldots, 3}$
where $\map R {3, \ldots, 3}$ denotes the [[Definition:Ramsey Number|Ramsey number]] on $r$ colors.
Take $S$ to be $n$.
{{refactor|Extract the below process of "coloring" a partition into its own page|level = medium}}
[[Definition:Set Partition|partition]] the [[Definition:Integer|i... | Schur's Theorem (Ramsey Theory) | https://proofwiki.org/wiki/Schur's_Theorem_(Ramsey_Theory) | https://proofwiki.org/wiki/Schur's_Theorem_(Ramsey_Theory) | [
"Combinatorics",
"Ramsey Theory"
] | [
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Set Partition",
"Definition:Integer"
] | [
"Definition:Ramsey Number",
"Definition:Set Partition",
"Definition:Integer",
"Definition:Ramsey Theory",
"Definition:Complete Graph",
"Definition:Coloring",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Coloring",
"Ramsey's Theorem",
"Definition:Mon... |
proofwiki-2474 | Brahmagupta's Formula | The area of a cyclic quadrilateral with sides of lengths $a, b, c, d$ is:
:$\sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} }$
where $s$ is the semiperimeter:
:$s = \dfrac {a + b + c + d} 2$ | Let $ABCD$ be a cyclic quadrilateral with sides $a, b, c, d$.
:300px
Area of $ABCD$ = Area of $\triangle ABC$ + Area of $\triangle ADC$
From Area of Triangle in Terms of Two Sides and Angle:
{{begin-eqn}}
{{eqn | l = \triangle ABC
| r = \frac 1 2 a b \sin \angle ABC
| c =
}}
{{eqn | l = \triangle ADC
... | The [[Definition:Area|area]] of a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]] with [[Definition:Side of Polygon|sides]] of [[Definition:Length (Linear Measure)|lengths]] $a, b, c, d$ is:
:$\sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} }$
where $s$ is the [[Definition:Semiperimeter|se... | Let $ABCD$ be a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]] with [[Definition:Side of Polygon|sides]] $a, b, c, d$.
:[[File:BrahmaguptasFormula.png|300px]]
Area of $ABCD$ = Area of $\triangle ABC$ + Area of $\triangle ADC$
From [[Area of Triangle in Terms of Two Sides and Angle]]:
{{begin-eqn}}
{{eqn |... | Brahmagupta's Formula | https://proofwiki.org/wiki/Brahmagupta's_Formula | https://proofwiki.org/wiki/Brahmagupta's_Formula | [
"Brahmagupta's Formula",
"Areas of Quadrilaterals"
] | [
"Definition:Area",
"Definition:Cyclic Quadrilateral",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Semiperimeter"
] | [
"Definition:Cyclic Quadrilateral",
"Definition:Polygon/Side",
"File:BrahmaguptasFormula.png",
"Area of Triangle in Terms of Two Sides and Angle",
"Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles",
"Definition:Right Angle",
"Definition:Supplementary Angles",
"Sine and Cosine of Suppleme... |
proofwiki-2475 | Euler-Binet Formula | The Fibonacci numbers have a closed-form solution:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n} {\phi - \paren {1 - \phi}}$
where $\phi$ is the golden mean.
Puttin... | Proof by induction:
For all $n \in \N$, let $\map P n$ be the proposition:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
=== Basis for the Induction ===
$\map P 0$ is true, as this just says:
:$\dfrac {\phi^0 - \hat \phi^0} {\sqrt 5} = \dfrac {1 - 1} {\sqrt 5} = 0 = F_0$
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn |... | The [[Definition:Fibonacci Number|Fibonacci numbers]] have a [[Definition:Closed-Form Solution|closed-form solution]]:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n}... | Proof by [[Second Principle of Mathematical Induction|induction]]:
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
=== Basis for the Induction ===
$\map P 0$ is true, as this just says:
:$\dfrac {\phi^0 - \hat \phi^0} {\sqrt 5} = \df... | Euler-Binet Formula/Proof 1 | https://proofwiki.org/wiki/Euler-Binet_Formula | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_1 | [
"Euler-Binet Formula",
"Fibonacci Numbers",
"Golden Mean",
"Closed Forms"
] | [
"Definition:Fibonacci Number",
"Definition:Closed Form Solution",
"Definition:Golden Mean"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Definition:Golden Mean",
"Solution to Quadratic Equation",
"Definition:Quadratic Equation",
"Euler-Binet Formula/Proof 1",
... |
proofwiki-2476 | Euler-Binet Formula | The Fibonacci numbers have a closed-form solution:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n} {\phi - \paren {1 - \phi}}$
where $\phi$ is the golden mean.
Puttin... | Let $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$.
First by {{Lemma|Cassini's Identity|proof = yes}}:
:$(1): \quad \forall n \in \Z_{>1}: A^n = \begin {bmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{bmatrix}$
Recall from Eigenvalues of Real Square Matrix are Roots of Characteristic Equation, we can find the eige... | The [[Definition:Fibonacci Number|Fibonacci numbers]] have a [[Definition:Closed-Form Solution|closed-form solution]]:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n}... | Let $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$.
First by {{Lemma|Cassini's Identity|proof = yes}}:
:$(1): \quad \forall n \in \Z_{>1}: A^n = \begin {bmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{bmatrix}$
Recall from [[Eigenvalues of Real Square Matrix are Roots of Characteristic Equation]], we can find t... | Euler-Binet Formula/Proof 2 | https://proofwiki.org/wiki/Euler-Binet_Formula | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_2 | [
"Euler-Binet Formula",
"Fibonacci Numbers",
"Golden Mean",
"Closed Forms"
] | [
"Definition:Fibonacci Number",
"Definition:Closed Form Solution",
"Definition:Golden Mean"
] | [
"Eigenvalues of Real Square Matrix are Roots of Characteristic Equation",
"Definition:Eigenvalue/Real Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Equation",
"Determinant of Matrix Product",
"Square of Golden Mean equals One plus Golden Mean",
"Defini... |
proofwiki-2477 | Euler-Binet Formula | The Fibonacci numbers have a closed-form solution:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n} {\phi - \paren {1 - \phi}}$
where $\phi$ is the golden mean.
Puttin... | This follows as a direct application of the first Binet form:
:$U_n = m U_{n - 1} + U_{n - 2}$
where:
{{begin-eqn}}
{{eqn | l = U_0
| r = 0
}}
{{eqn | l = U_1
| r = 1
}}
{{end-eqn}}
has the closed-form solution:
:$U_n = \dfrac {\alpha^n - \beta^n} {\Delta}$
where:
{{begin-eqn}}
{{eqn | l = \Delta
| r ... | The [[Definition:Fibonacci Number|Fibonacci numbers]] have a [[Definition:Closed-Form Solution|closed-form solution]]:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n}... | This follows as a direct application of the [[Binet Form#First Form|first Binet form]]:
:$U_n = m U_{n - 1} + U_{n - 2}$
where:
{{begin-eqn}}
{{eqn | l = U_0
| r = 0
}}
{{eqn | l = U_1
| r = 1
}}
{{end-eqn}}
has the [[Definition:Closed-Form Solution|closed-form solution]]:
:$U_n = \dfrac {\alpha^n - \be... | Euler-Binet Formula/Proof 3 | https://proofwiki.org/wiki/Euler-Binet_Formula | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_3 | [
"Euler-Binet Formula",
"Fibonacci Numbers",
"Golden Mean",
"Closed Forms"
] | [
"Definition:Fibonacci Number",
"Definition:Closed Form Solution",
"Definition:Golden Mean"
] | [
"Binet Form",
"Definition:Closed Form Solution"
] |
proofwiki-2478 | Euler-Binet Formula | The Fibonacci numbers have a closed-form solution:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n} {\phi - \paren {1 - \phi}}$
where $\phi$ is the golden mean.
Puttin... | From Generating Function for Fibonacci Numbers, a generating function for the Fibonacci numbers is:
:$\map G z = \dfrac z {1 - z - z^2}$
Hence:
{{begin-eqn}}
{{eqn | l = \map G z
| r = \dfrac z {1 - z - z^2}
| c =
}}
{{eqn | r = \dfrac 1 {\sqrt 5} \paren {\dfrac 1 {1 - \phi z} - \dfrac 1 {1 - \hat \phi z} ... | The [[Definition:Fibonacci Number|Fibonacci numbers]] have a [[Definition:Closed-Form Solution|closed-form solution]]:
:$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5} = \dfrac {\phi^n - \paren {1 - \phi}^n}... | From [[Generating Function for Fibonacci Numbers]], a [[Definition:Generating Function|generating function]] for the [[Definition:Fibonacci Number|Fibonacci numbers]] is:
:$\map G z = \dfrac z {1 - z - z^2}$
Hence:
{{begin-eqn}}
{{eqn | l = \map G z
| r = \dfrac z {1 - z - z^2}
| c =
}}
{{eqn | r = \df... | Euler-Binet Formula/Proof 4 | https://proofwiki.org/wiki/Euler-Binet_Formula | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_4 | [
"Euler-Binet Formula",
"Fibonacci Numbers",
"Golden Mean",
"Closed Forms"
] | [
"Definition:Fibonacci Number",
"Definition:Closed Form Solution",
"Definition:Golden Mean"
] | [
"Generating Function for Fibonacci Numbers",
"Definition:Generating Function",
"Definition:Fibonacci Number",
"Definition:Partial Fractions Expansion",
"Sum of Infinite Geometric Sequence",
"Definition:Coefficient",
"Definition:Fibonacci Number"
] |
proofwiki-2479 | Hölder's Inequality for Sums | Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
=== Formulation $1$ ===
{{:Hölder's Inequality for Sums/Formulation 1}}
=== Formulation $2$ ===
{{:Hölder's... | Let $\sequence {x_k}_{k \mathop \in \N}$ and $\sequence {y_k}_{k \mathop \in \N}$ be infinite sequences in $\GF$ such that:
:$\forall m > n: x_m = y_m = 0$
Then we have:
{{begin-eqn}}
{{eqn | l = \sum \limits_{k \mathop = 1}^n \size {x_k y_k}
| r = \sum \limits_{k \mathop \in \N} \size {x_k y_k}
| c = {{hyp... | Let $p, q \in \R_{>0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]] such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the [[Definition:Set|set]] of either the [[Definition:Real Number|real numbers]] or the [[Definition:Complex Number|comp... | Let $\sequence {x_k}_{k \mathop \in \N}$ and $\sequence {y_k}_{k \mathop \in \N}$ be [[Definition:Infinite Sequence|infinite sequences]] in $\GF$ such that:
:$\forall m > n: x_m = y_m = 0$
Then we have:
{{begin-eqn}}
{{eqn | l = \sum \limits_{k \mathop = 1}^n \size {x_k y_k}
| r = \sum \limits_{k \mathop \in \N... | Hölder's Inequality for Sums/Finite/Proof | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums | https://proofwiki.org/wiki/Hölder's_Inequality_for_Sums/Finite/Proof | [
"Hölder's Inequality for Sums",
"Hölder's Inequality",
"Analysis",
"Inequalities"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Set",
"Definition:Real Number",
"Definition:Complex Number",
"Hölder's Inequality for Sums/Formulation 1",
"Hölder's Inequality for Sums/Formulation 2"
] | [
"Definition:Sequence/Infinite Sequence",
"Hölder's Inequality for Sums"
] |
proofwiki-2480 | Binet Form | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | Proof by induction:
For all $n \in \N$, let $\map P n$ be the proposition:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
=== Basis for the Induction ===
$\map P 0$ is true, as this just says:
:$\dfrac {\phi^0 - \hat \phi^0} {\sqrt 5} = \dfrac {1 - 1} {\sqrt 5} = 0 = F_0$
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn |... | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | Proof by [[Second Principle of Mathematical Induction|induction]]:
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
=== Basis for the Induction ===
$\map P 0$ is true, as this just says:
:$\dfrac {\phi^0 - \hat \phi^0} {\sqrt 5} = \df... | Euler-Binet Formula/Proof 1 | https://proofwiki.org/wiki/Binet_Form | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_1 | [
"Analysis",
"Binet Form"
] | [] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Definition:Golden Mean",
"Solution to Quadratic Equation",
"Definition:Quadratic Equation",
"Euler-Binet Formula/Proof 1",
... |
proofwiki-2481 | Binet Form | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | Let $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$.
First by {{Lemma|Cassini's Identity|proof = yes}}:
:$(1): \quad \forall n \in \Z_{>1}: A^n = \begin {bmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{bmatrix}$
Recall from Eigenvalues of Real Square Matrix are Roots of Characteristic Equation, we can find the eige... | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | Let $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$.
First by {{Lemma|Cassini's Identity|proof = yes}}:
:$(1): \quad \forall n \in \Z_{>1}: A^n = \begin {bmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{bmatrix}$
Recall from [[Eigenvalues of Real Square Matrix are Roots of Characteristic Equation]], we can find t... | Euler-Binet Formula/Proof 2 | https://proofwiki.org/wiki/Binet_Form | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_2 | [
"Analysis",
"Binet Form"
] | [] | [
"Eigenvalues of Real Square Matrix are Roots of Characteristic Equation",
"Definition:Eigenvalue/Real Square Matrix",
"Definition:Determinant/Matrix",
"Definition:Matrix/Square Matrix",
"Definition:Equation",
"Determinant of Matrix Product",
"Square of Golden Mean equals One plus Golden Mean",
"Defini... |
proofwiki-2482 | Binet Form | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | This follows as a direct application of the first Binet form:
:$U_n = m U_{n - 1} + U_{n - 2}$
where:
{{begin-eqn}}
{{eqn | l = U_0
| r = 0
}}
{{eqn | l = U_1
| r = 1
}}
{{end-eqn}}
has the closed-form solution:
:$U_n = \dfrac {\alpha^n - \beta^n} {\Delta}$
where:
{{begin-eqn}}
{{eqn | l = \Delta
| r ... | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | This follows as a direct application of the [[Binet Form#First Form|first Binet form]]:
:$U_n = m U_{n - 1} + U_{n - 2}$
where:
{{begin-eqn}}
{{eqn | l = U_0
| r = 0
}}
{{eqn | l = U_1
| r = 1
}}
{{end-eqn}}
has the [[Definition:Closed-Form Solution|closed-form solution]]:
:$U_n = \dfrac {\alpha^n - \be... | Euler-Binet Formula/Proof 3 | https://proofwiki.org/wiki/Binet_Form | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_3 | [
"Analysis",
"Binet Form"
] | [] | [
"Binet Form",
"Definition:Closed Form Solution"
] |
proofwiki-2483 | Binet Form | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | From Generating Function for Fibonacci Numbers, a generating function for the Fibonacci numbers is:
:$\map G z = \dfrac z {1 - z - z^2}$
Hence:
{{begin-eqn}}
{{eqn | l = \map G z
| r = \dfrac z {1 - z - z^2}
| c =
}}
{{eqn | r = \dfrac 1 {\sqrt 5} \paren {\dfrac 1 {1 - \phi z} - \dfrac 1 {1 - \hat \phi z} ... | Let $m \in \R$.
Define:
{{begin-eqn}}
{{eqn | l = \Delta
| r = \sqrt {m^2 + 4}
}}
{{eqn | l = \alpha
| r = \frac {m + \Delta} 2
}}
{{eqn | l = \beta
| r = \frac {m - \Delta} 2
}}
{{end-eqn}} | From [[Generating Function for Fibonacci Numbers]], a [[Definition:Generating Function|generating function]] for the [[Definition:Fibonacci Number|Fibonacci numbers]] is:
:$\map G z = \dfrac z {1 - z - z^2}$
Hence:
{{begin-eqn}}
{{eqn | l = \map G z
| r = \dfrac z {1 - z - z^2}
| c =
}}
{{eqn | r = \df... | Euler-Binet Formula/Proof 4 | https://proofwiki.org/wiki/Binet_Form | https://proofwiki.org/wiki/Euler-Binet_Formula/Proof_4 | [
"Analysis",
"Binet Form"
] | [] | [
"Generating Function for Fibonacci Numbers",
"Definition:Generating Function",
"Definition:Fibonacci Number",
"Definition:Partial Fractions Expansion",
"Sum of Infinite Geometric Sequence",
"Definition:Coefficient",
"Definition:Fibonacci Number"
] |
proofwiki-2484 | Binet-Cauchy Identity | :$\ds \paren {\sum_{i \mathop = 1}^n a_i c_i} \paren {\sum_{j \mathop = 1}^n b_j d_j} = \paren {\sum_{i \mathop = 1}^n a_i d_i} \paren {\sum_{j \mathop = 1}^n b_j c_j} + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}$
where all of the $a, b, c, d$ are elements of ... | Expanding the last term:
{{begin-eqn}}
{{eqn | o =
| r = \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}
| c =
}}
{{eqn | r = \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i c_i b_j d_j + a_j c_j b_i d_i}
| c =
}}
{{eqn | o =
... | :$\ds \paren {\sum_{i \mathop = 1}^n a_i c_i} \paren {\sum_{j \mathop = 1}^n b_j d_j} = \paren {\sum_{i \mathop = 1}^n a_i d_i} \paren {\sum_{j \mathop = 1}^n b_j c_j} + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}$
where all of the $a, b, c, d$ are [[Definitio... | Expanding the last term:
{{begin-eqn}}
{{eqn | o =
| r = \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}
| c =
}}
{{eqn | r = \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i c_i b_j d_j + a_j c_j b_i d_i}
| c =
}}
{{eqn | o =
... | Binet-Cauchy Identity/Proof 1 | https://proofwiki.org/wiki/Binet-Cauchy_Identity | https://proofwiki.org/wiki/Binet-Cauchy_Identity/Proof_1 | [
"Binet-Cauchy Identity",
"Summations"
] | [
"Definition:Element",
"Definition:Commutative Ring"
] | [] |
proofwiki-2485 | Binet-Cauchy Identity | :$\ds \paren {\sum_{i \mathop = 1}^n a_i c_i} \paren {\sum_{j \mathop = 1}^n b_j d_j} = \paren {\sum_{i \mathop = 1}^n a_i d_i} \paren {\sum_{j \mathop = 1}^n b_j c_j} + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}$
where all of the $a, b, c, d$ are elements of ... | This is a special case of the Cauchy-Binet Formula:
:$\ds \map \det {\mathbf A \mathbf B} = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \map \det {\mathbf A_{j_1 j_2 \ldots j_m} } \map \det {\mathbf B_{j_1 j_2 \ldots j_m} }$
where:
:$\mathbf A$ is an $m \times n$ matrix
:$\mathbf B$ is a... | :$\ds \paren {\sum_{i \mathop = 1}^n a_i c_i} \paren {\sum_{j \mathop = 1}^n b_j d_j} = \paren {\sum_{i \mathop = 1}^n a_i d_i} \paren {\sum_{j \mathop = 1}^n b_j c_j} + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}$
where all of the $a, b, c, d$ are [[Definitio... | This is a special case of the [[Cauchy-Binet Formula]]:
:$\ds \map \det {\mathbf A \mathbf B} = \sum_{1 \mathop \le j_1 \mathop < j_2 \mathop < \cdots \mathop < j_m \le n} \map \det {\mathbf A_{j_1 j_2 \ldots j_m} } \map \det {\mathbf B_{j_1 j_2 \ldots j_m} }$
where:
:$\mathbf A$ is an [[Definition:Matrix|$m \times n$... | Binet-Cauchy Identity/Proof 2 | https://proofwiki.org/wiki/Binet-Cauchy_Identity | https://proofwiki.org/wiki/Binet-Cauchy_Identity/Proof_2 | [
"Binet-Cauchy Identity",
"Summations"
] | [
"Definition:Element",
"Definition:Commutative Ring"
] | [
"Cauchy-Binet Formula",
"Definition:Matrix",
"Definition:Matrix",
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Matrix",
"Definition:Matrix/Row"
] |
proofwiki-2486 | Nicomachus's Theorem | {{begin-eqn}}
{{eqn | l = 1^3
| r = 1
}}
{{eqn | l = 2^3
| r = 3 + 5
}}
{{eqn | l = 3^3
| r = 7 + 9 + 11
}}
{{eqn | l = 4^3
| r = 13 + 15 + 17 + 19
}}
{{eqn | l = \vdots
| o =
}}
{{end-eqn}}
In general:
:$\forall n \in \N_{>0}: n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \dotsb + ... | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \dotsb + \paren {n^2 + n - 1}$
=== Basis for the Induction ===
$\map P 1$ is true, as this just says $1^3 = 1$.
This is our basis for the induction.
=== Induction Hypothesis ===
Now we n... | {{begin-eqn}}
{{eqn | l = 1^3
| r = 1
}}
{{eqn | l = 2^3
| r = 3 + 5
}}
{{eqn | l = 3^3
| r = 7 + 9 + 11
}}
{{eqn | l = 4^3
| r = 13 + 15 + 17 + 19
}}
{{eqn | l = \vdots
| o =
}}
{{end-eqn}}
In general:
:$\forall n \in \N_{>0}: n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \dotsb ... | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \dotsb + \paren {n^2 + n - 1}$
=== Basis for the Induction ===
$\map P 1$ is true, as this just says $1^3 = 1$.
Th... | Nicomachus's Theorem/Proof 1 | https://proofwiki.org/wiki/Nicomachus's_Theorem | https://proofwiki.org/wiki/Nicomachus's_Theorem/Proof_1 | [
"Algebra",
"Sums of Sequences",
"Nicomachus's Theorem"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Nicomachus's Theorem/Proof 1",
"Principle of Mathematical Induction"
] |
proofwiki-2487 | Nicomachus's Theorem | {{begin-eqn}}
{{eqn | l = 1^3
| r = 1
}}
{{eqn | l = 2^3
| r = 3 + 5
}}
{{eqn | l = 3^3
| r = 7 + 9 + 11
}}
{{eqn | l = 4^3
| r = 13 + 15 + 17 + 19
}}
{{eqn | l = \vdots
| o =
}}
{{end-eqn}}
In general:
:$\forall n \in \N_{>0}: n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \dotsb + ... | From the definition:
:$\paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 + n - 1}$
can be written:
:$\paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 - n + 2 n - 1}$
Writing this in sum notation:
{{begin-eqn}}
{{eqn | o =
| r = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldo... | {{begin-eqn}}
{{eqn | l = 1^3
| r = 1
}}
{{eqn | l = 2^3
| r = 3 + 5
}}
{{eqn | l = 3^3
| r = 7 + 9 + 11
}}
{{eqn | l = 4^3
| r = 13 + 15 + 17 + 19
}}
{{eqn | l = \vdots
| o =
}}
{{end-eqn}}
In general:
:$\forall n \in \N_{>0}: n^3 = \paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \dotsb ... | From the definition:
:$\paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 + n - 1}$
can be written:
:$\paren {n^2 - n + 1} + \paren {n^2 - n + 3} + \ldots + \paren {n^2 - n + 2 n - 1}$
Writing this in [[Definition:Summation|sum notation]]:
{{begin-eqn}}
{{eqn | o =
| r = \paren {n^2 - n + 1} +... | Nicomachus's Theorem/Proof 2 | https://proofwiki.org/wiki/Nicomachus's_Theorem | https://proofwiki.org/wiki/Nicomachus's_Theorem/Proof_2 | [
"Algebra",
"Sums of Sequences",
"Nicomachus's Theorem"
] | [] | [
"Definition:Summation",
"Odd Number Theorem"
] |
proofwiki-2488 | Principle of Conservation of Angular Momentum | Newton's Laws of Motion imply the conservation of angular momentum in systems of masses in which no external force is acting. | We start by stating Newton's Third Law of Motion in all its detail.
We consider a collection of massive bodies denoted by the subscripts $1$ to $N$.
These bodies interact with each other and exert forces on each other and these forces occur in equal and opposite pairs.
The force $F_{i j}$ exerted by body $i$ on body $j... | [[Newton's Laws of Motion]] imply the conservation of [[Definition:Angular Momentum|angular momentum]] in systems of [[Definition:Mass|masses]] in which no [[Definition:External Force|external force]] is acting. | We start by stating [[Newton's Third Law of Motion]] in all its detail.
We consider a collection of massive [[Definition:Body|bodies]] denoted by the subscripts $1$ to $N$.
These [[Definition:Body|bodies]] interact with each other and exert [[Definition:Force|forces]] on each other and these [[Definition:Force|forces... | Principle of Conservation of Angular Momentum | https://proofwiki.org/wiki/Principle_of_Conservation_of_Angular_Momentum | https://proofwiki.org/wiki/Principle_of_Conservation_of_Angular_Momentum | [
"Principle of Conservation of Angular Momentum",
"Angular Momentum",
"Classical Mechanics"
] | [
"Newton's Laws of Motion",
"Definition:Angular Momentum",
"Definition:Mass",
"Definition:External Force"
] | [
"Newton's Laws of Motion/Third Law",
"Definition:Body",
"Definition:Body",
"Definition:Force",
"Definition:Force",
"Definition:Force",
"Definition:Body",
"Definition:Body",
"Definition:Force",
"Definition:Body",
"Definition:Body",
"Newton's Laws of Motion/Third Law",
"Definition:Force",
"D... |
proofwiki-2489 | Lexicographic Order on Products of Well-Ordered Sets | Let $S$ be a set which is well-ordered by $\preccurlyeq$.
Let $\preccurlyeq_l$ be the lexicographic order on the set $T$ of all ordered tuples of $S$.
Then:
:$(1): \quad$ For a given $n \in \N: n > 0$, $\preccurlyeq_l$ is a well-ordering on the set $T_n$ of all ordered $n$-tuples of $S$;
:$(2): \quad \preccurlyeq_l$ is... | It is straightforward to show that $\preccurlyeq_l$ is a total ordering on both $T_n$ and $T$.
{{handwaving|there ought to be a result here that should be linked to}}
It remains to investigate the nature of $\preccurlyeq_l$ as to whether or not it is a well-ordering. | Let $S$ be a [[Definition:Set|set]] which is [[Definition:Well-Ordered Set|well-ordered]] by $\preccurlyeq$.
Let $\preccurlyeq_l$ be the [[Definition:Lexicographic Order|lexicographic order]] on the set $T$ of all [[Definition:Ordered Tuple|ordered tuples]] of $S$.
Then:
:$(1): \quad$ For a given $n \in \N: n > 0$, ... | It is straightforward to show that $\preccurlyeq_l$ is a [[Definition:Total Ordering|total ordering]] on both $T_n$ and $T$.
{{handwaving|there ought to be a result here that should be linked to}}
It remains to investigate the nature of $\preccurlyeq_l$ as to whether or not it is a [[Definition:Well-Ordering|well-ord... | Lexicographic Order on Products of Well-Ordered Sets | https://proofwiki.org/wiki/Lexicographic_Order_on_Products_of_Well-Ordered_Sets | https://proofwiki.org/wiki/Lexicographic_Order_on_Products_of_Well-Ordered_Sets | [
"Lexicographic Order",
"Well-Orderings"
] | [
"Definition:Set",
"Definition:Well-Ordered Set",
"Definition:Lexicographic Order",
"Definition:Ordered Tuple",
"Definition:Well-Ordering",
"Definition:Ordered Tuple",
"Definition:Well-Ordering",
"Definition:Ordered Tuple"
] | [
"Definition:Total Ordering",
"Definition:Well-Ordering"
] |
proofwiki-2490 | Basis Expansion of Irrational Number | A basis expansion of an irrational number never terminates and does not recur. | We use a Proof by Contraposition.
Thus, we show that if a basis expansion of a (real) number terminates or recurs, then that number is rational.
{{AimForCont}} $x \in \R$ were to terminate in some number base $b$.
Then (using the notation of that definition):
:$\exists k \in \N: f_k = 0$
and so we can express $x$ preci... | A [[Definition:Basis Expansion|basis expansion]] of an [[Definition:Irrational Number|irrational number]] never [[Definition:Termination of Basis Expansion|terminates]] and does not [[Definition:Recurrence of Basis Expansion|recur]]. | We use a [[Proof by Contraposition]].
Thus, we show that if a [[Definition:Basis Expansion|basis expansion]] of a [[Definition:Real Number|(real) number]] [[Definition:Termination of Basis Expansion|terminates]] or [[Definition:Recurrence of Basis Expansion|recurs]], then that number is [[Definition:Rational Number|ra... | Basis Expansion of Irrational Number | https://proofwiki.org/wiki/Basis_Expansion_of_Irrational_Number | https://proofwiki.org/wiki/Basis_Expansion_of_Irrational_Number | [
"Irrational Numbers",
"Basis Expansions",
"Arithmetic"
] | [
"Definition:Basis Expansion",
"Definition:Irrational Number",
"Definition:Basis Expansion/Termination",
"Definition:Basis Expansion/Recurrence"
] | [
"Proof by Contraposition",
"Definition:Basis Expansion",
"Definition:Real Number",
"Definition:Basis Expansion/Termination",
"Definition:Basis Expansion/Recurrence",
"Definition:Rational Number",
"Definition:Basis Expansion/Termination",
"Definition:Number Base",
"Definition:Fraction/Numerator",
"... |
proofwiki-2491 | Determinant of Combinatorial Matrix | Let $C_n$ be the combinatorial matrix of order $n$ given by:
:<nowiki>$C_n = \begin{bmatrix}
x + y & y & \cdots & y \\
y & x + y & \cdots & y \\
\vdots & \vdots & \ddots & \vdots \\
y & y & \cdots & x + y
\end{bmatrix}$</nowiki>
Then the determinant of $C_n$ is given by:
:$\map \det {C_n} = x^{n - 1} \paren {x + n y}$ | Take the determinant $\map \det {C_n}$:
:<nowiki>$\map \det {C_n} = \begin{vmatrix}
x + y & y & y & \cdots & y \\
y & x + y & y & \cdots & y \\
y & y & x + y & \cdots & y \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
y & y & y & \cdots & x + y
\end{vmatrix}$</nowiki>
Subtract column $1$ from columns $2$ to $n$.
From... | Let $C_n$ be the [[Definition:Combinatorial Matrix|combinatorial matrix]] of [[Definition:Order of Square Matrix|order $n$]] given by:
:<nowiki>$C_n = \begin{bmatrix}
x + y & y & \cdots & y \\
y & x + y & \cdots & y \\
\vdots & \vdots & \ddots & \vdots \\
y & y & \cdots & x + y
\end{bmatrix}$</nowiki>
Then the [[Def... | Take the [[Definition:Determinant of Matrix|determinant]] $\map \det {C_n}$:
:<nowiki>$\map \det {C_n} = \begin{vmatrix}
x + y & y & y & \cdots & y \\
y & x + y & y & \cdots & y \\
y & y & x + y & \cdots & y \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
y & y & y & \cdots & x + y
\end{vmatrix}$</nowiki>
Subtract ... | Determinant of Combinatorial Matrix | https://proofwiki.org/wiki/Determinant_of_Combinatorial_Matrix | https://proofwiki.org/wiki/Determinant_of_Combinatorial_Matrix | [
"Combinatorial Matrix",
"Determinants"
] | [
"Definition:Combinatorial Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Determinant/Matrix"
] | [
"Definition:Determinant/Matrix",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Multiple of Row Added to Row of Determinant",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Multiple of Row Added to Row of Determinant",
"Definition:Determinant/Matrix",
"Definition:Triangular Matrix/Lowe... |
proofwiki-2492 | Power Set is Closed under Union | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A, B \in \powerset S: A \cup B \in \powerset S$ | Let $A, B \in \powerset S$.
Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.
We also have $A \cup B \subseteq S \iff A \subseteq S \land B \subseteq S$ from Union is Smallest Superset.
Thus $A \cup B \in \powerset S$, and closure is proved.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then:
:$\forall A, B \in \powerset S: A \cup B \in \powerset S$ | Let $A, B \in \powerset S$.
Then by the definition of [[Definition:Power Set|power set]], $A \subseteq S$ and $B \subseteq S$.
We also have $A \cup B \subseteq S \iff A \subseteq S \land B \subseteq S$ from [[Union is Smallest Superset]].
Thus $A \cup B \in \powerset S$, and [[Definition:Closure (Abstract Algebra)|c... | Power Set is Closed under Union | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Union | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Union | [
"Power Set",
"Set Union",
"Closed Algebraic Structures"
] | [
"Definition:Set",
"Definition:Power Set"
] | [
"Definition:Power Set",
"Union is Smallest Superset",
"Definition:Closure (Abstract Algebra)"
] |
proofwiki-2493 | Matrix Product with Adjugate Matrix | Let $R$ be a commutative ring with unity.
Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$.
Let $\adj {\mathbf A}$ be its adjugate matrix.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A \cdot \adj {\mathbf A}
| r = \map \det {\mathbf A} \cdot \mathbf I_n
}}
{{eqn | l = \adj {\mathbf A} \cdot \mathbf ... | Let $\mathbf A = \paren {a_{i j} }$.
Let $A_{i j}$ denote the cofactor of $a_{i j} \in \mathbf A$. | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\mathbf A \in R^{n \times n}$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order]] $n$.
Let $\adj {\mathbf A}$ be its [[Definition:Adjugate Matrix|adjugate matrix]].
Then:
{{begin-eq... | Let $\mathbf A = \paren {a_{i j} }$.
Let $A_{i j}$ denote the [[Definition:Cofactor of Element|cofactor]] of $a_{i j} \in \mathbf A$. | Matrix Product with Adjugate Matrix | https://proofwiki.org/wiki/Matrix_Product_with_Adjugate_Matrix | https://proofwiki.org/wiki/Matrix_Product_with_Adjugate_Matrix | [
"Inverse Matrices",
"Conventional Matrix Multiplication",
"Adjugate Matrices",
"Determinants"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Adjugate Matrix",
"Definition:Determinant/Matrix",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order"
] | [
"Definition:Cofactor/Element"
] |
proofwiki-2494 | Square of Ones Matrix | Let $\mathbf J = \sqbrk 1_n$ be a square ones matrix of order $n$.
Then $\mathbf J^2 = n \mathbf J$.
That is:
:<nowiki>$\begin{bmatrix}
1 & 1 & \cdots & 1 \\
1 & 1 & \cdots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \cdots & 1
\end{bmatrix}^2 = \begin{bmatrix}
n & n & \cdots & n \\
n & n & \cdots & n \\
\vdots... | Follows directly from the definition of matrix multiplication:
:$\ds \forall i \in \closedint 1 m, j \in \closedint 1 p: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$
In this case, $m = n$ and $a_{i k} = b_{k j} = 1$.
Hence:
:$\ds c_{i j} = \sum_{k \mathop = 1}^n 1 \times 1 = n$
{{qed}}
Category:Matrix Algebr... | Let $\mathbf J = \sqbrk 1_n$ be a [[Definition:Square Ones Matrix|square ones matrix]] of [[Definition:Order of Square Matrix|order $n$]].
Then $\mathbf J^2 = n \mathbf J$.
That is:
:<nowiki>$\begin{bmatrix}
1 & 1 & \cdots & 1 \\
1 & 1 & \cdots & 1 \\
\vdots & \vdots & \ddots & \vdots \\
1 & 1 & \cdots & 1
\end{bmat... | Follows directly from the definition of [[Definition:Matrix Product (Conventional)|matrix multiplication]]:
:$\ds \forall i \in \closedint 1 m, j \in \closedint 1 p: c_{i j} = \sum_{k \mathop = 1}^n a_{i k} \circ b_{k j}$
In this case, $m = n$ and $a_{i k} = b_{k j} = 1$.
Hence:
:$\ds c_{i j} = \sum_{k \mathop = 1}^... | Square of Ones Matrix | https://proofwiki.org/wiki/Square_of_Ones_Matrix | https://proofwiki.org/wiki/Square_of_Ones_Matrix | [
"Matrix Algebra"
] | [
"Definition:Ones Matrix/Square",
"Definition:Matrix/Square Matrix/Order"
] | [
"Definition:Matrix Product (Conventional)",
"Category:Matrix Algebra"
] |
proofwiki-2495 | Inverse of Combinatorial Matrix | Let $C_n$ be the combinatorial matrix of order $n$ given by:
:$C_n = \begin{bmatrix}
x + y & y & \cdots & y \\
y & x + y & \cdots & y \\
\vdots & \vdots & \ddots & \vdots \\
y & y & \cdots & x + y
\end{bmatrix}$
Then its inverse $C_n^{-1} = \sqbrk b_n$ can be specified as:
:$b_{i j} = \dfrac {-y + \delta_{i j} \paren {... | From the definition of the combinatorial matrix:
:$C_n = x \mathbf I_n + y \mathbf J_n$
where:
:$\mathbf I_n$ is the unit matrix of order $n$
:$\mathbf J_n$ is the square ones matrix of order $n$.
From Square of Ones Matrix we have $\mathbf J_n^2 = n \mathbf J_n$.
Hence:
{{begin-eqn}}
{{eqn | l = \paren {x \mathbf I_n ... | Let $C_n$ be the [[Definition:Combinatorial Matrix|combinatorial matrix]] of [[Definition:Order of Square Matrix|order $n$]] given by:
:$C_n = \begin{bmatrix}
x + y & y & \cdots & y \\
y & x + y & \cdots & y \\
\vdots & \vdots & \ddots & \vdots \\
y & y & \cdots & x + y
\end{bmatrix}$
Then its [[Definition:Inverse M... | From the definition of the [[Definition:Combinatorial Matrix|combinatorial matrix]]:
:$C_n = x \mathbf I_n + y \mathbf J_n$
where:
:$\mathbf I_n$ is the [[Definition:Unit Matrix|unit matrix]] of [[Definition:Order of Square Matrix|order $n$]]
:$\mathbf J_n$ is the [[Definition:Square Ones Matrix|square ones matrix]] of... | Inverse of Combinatorial Matrix | https://proofwiki.org/wiki/Inverse_of_Combinatorial_Matrix | https://proofwiki.org/wiki/Inverse_of_Combinatorial_Matrix | [
"Combinatorial Matrix"
] | [
"Definition:Combinatorial Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Inverse Matrix",
"Definition:Kronecker Delta"
] | [
"Definition:Combinatorial Matrix",
"Definition:Unit Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Ones Matrix/Square",
"Definition:Matrix/Square Matrix/Order",
"Square of Ones Matrix"
] |
proofwiki-2496 | Inverse of Vandermonde Matrix | Let $V_n$ be the Vandermonde matrix of order $n$ given by:
:<nowiki>$V_n = \begin {bmatrix}
x_1 & x_2 & \cdots & x_n \\
x_1^2 & x_2^2 & \cdots & x_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
x_1^n & x_2^n & \cdots & x_n^n
\end{bmatrix}$</nowiki>
Then its inverse $V_n^{-1} = \sqbrk b_n$ can be specified as:
:<nowi... | First consider the classical form of the Vandermonde matrix:
:<nowiki>$W_n = \begin{bmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n - 1} \\
1 & x_2 & x_2^2 & \cdots & x_2^{n - 1} \\
\vdots & \vdots & \ddots & \vdots \\
1 & x_n & x_n^2 & \cdots & x_n^{n - 1} \\
\end{bmatrix}$</nowiki>
By Value of Vandermonde Determinan... | Let $V_n$ be the [[Definition:Vandermonde Matrix/Formulation 2|Vandermonde matrix]] of [[Definition:Order of Square Matrix|order $n$]] given by:
:<nowiki>$V_n = \begin {bmatrix}
x_1 & x_2 & \cdots & x_n \\
x_1^2 & x_2^2 & \cdots & x_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
x_1^n & x_2^n & \cdots & x_n^n
\end{... | First consider the classical form of the [[Definition:Vandermonde Matrix/Formulation 1|Vandermonde matrix]]:
:<nowiki>$W_n = \begin{bmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n - 1} \\
1 & x_2 & x_2^2 & \cdots & x_2^{n - 1} \\
\vdots & \vdots & \ddots & \vdots \\
1 & x_n & x_n^2 & \cdots & x_n^{n - 1} \\
\end{bmat... | Inverse of Vandermonde Matrix/Proof 1 | https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix | https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix/Proof_1 | [
"Vandermonde Matrices",
"Inverse of Vandermonde Matrix"
] | [
"Definition:Vandermonde Matrix/Formulation 2",
"Definition:Matrix/Square Matrix/Order",
"Definition:Inverse Matrix"
] | [
"Definition:Vandermonde Matrix/Formulation 1",
"Value of Vandermonde Determinant/Formulation 1",
"Definition:Determinant/Matrix",
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Definition:Inverse Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Inverse Matrix",
"D... |
proofwiki-2497 | Inverse of Vandermonde Matrix | Let $V_n$ be the Vandermonde matrix of order $n$ given by:
:<nowiki>$V_n = \begin {bmatrix}
x_1 & x_2 & \cdots & x_n \\
x_1^2 & x_2^2 & \cdots & x_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
x_1^n & x_2^n & \cdots & x_n^n
\end{bmatrix}$</nowiki>
Then its inverse $V_n^{-1} = \sqbrk b_n$ can be specified as:
:<nowi... | === Definition 1 ===
:<nowiki>$V_n = \begin{bmatrix}
x_1 & \cdots & x_n \\
x_1^2 & \cdots & x_n^2 \\
\vdots & \ddots & \vdots \\
x_1^{n} & \cdots & x_n^{n} \\
\end{bmatrix} \quad$</nowiki> {{Defof|Vandermonde Matrix|subdef = Formulation 2}}
:<nowiki>$V = \begin{bmatrix}
1 & \cdots & 1 \... | Let $V_n$ be the [[Definition:Vandermonde Matrix/Formulation 2|Vandermonde matrix]] of [[Definition:Order of Square Matrix|order $n$]] given by:
:<nowiki>$V_n = \begin {bmatrix}
x_1 & x_2 & \cdots & x_n \\
x_1^2 & x_2^2 & \cdots & x_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
x_1^n & x_2^n & \cdots & x_n^n
\end{... | === Definition 1 ===
:<nowiki>$V_n = \begin{bmatrix}
x_1 & \cdots & x_n \\
x_1^2 & \cdots & x_n^2 \\
\vdots & \ddots & \vdots \\
x_1^{n} & \cdots & x_n^{n} \\
\end{bmatrix} \quad$</nowiki> {{Defof|Vandermonde Matrix|subdef = Formulation 2}}
:<nowiki>$V = \begin{bmatrix}
1 & \cdots & 1... | Inverse of Vandermonde Matrix/Proof 2 | https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix | https://proofwiki.org/wiki/Inverse_of_Vandermonde_Matrix/Proof_2 | [
"Vandermonde Matrices",
"Inverse of Vandermonde Matrix"
] | [
"Definition:Vandermonde Matrix/Formulation 2",
"Definition:Matrix/Square Matrix/Order",
"Definition:Inverse Matrix"
] | [
"Definition:Diagonal Matrix",
"Definition:Symmetric Function/Elementary",
"Inverse of Matrix Product",
"Viète's Formulas",
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Left or Right Inverse of Matrix is Inverse"
] |
proofwiki-2498 | Floor Function is Idempotent | :$\floor {\floor x} = \floor x$ | Let $y = \floor x$.
By Floor Function is Integer, $y$ is an integer.
Then from Real Number is Integer iff equals Floor:
:$\floor y = y$
So:
:$\floor {\floor x} = \floor x$
{{qed}} | :$\floor {\floor x} = \floor x$ | Let $y = \floor x$.
By [[Floor Function is Integer]], $y$ is an [[Definition:Integer|integer]].
Then from [[Real Number is Integer iff equals Floor]]:
:$\floor y = y$
So:
:$\floor {\floor x} = \floor x$
{{qed}} | Floor Function is Idempotent | https://proofwiki.org/wiki/Floor_Function_is_Idempotent | https://proofwiki.org/wiki/Floor_Function_is_Idempotent | [
"Floor Function"
] | [] | [
"Floor Function is Integer",
"Definition:Integer",
"Real Number is Integer iff equals Floor"
] |
proofwiki-2499 | Kronecker's Lemma | Let $\sequence {x_n}$ be an infinite sequence of real numbers such that:
:$\ds \sum_{n \mathop = 1}^\infty x_n = s$
exists and is finite.
Then for $0 < b_1 \le b_2 \le b_3 \le \ldots$ and $b_n \to \infty$:
:$\ds \lim_{n \mathop \to \infty} \frac 1 {b_n} \sum_{k \mathop = 1}^n b_k x_k = 0$ | Let $S_k$ denote the partial sums of the $x$s.
Using Summation by Parts:
:$\ds \frac 1 {b_n} \sum_{k \mathop = 1}^n b_k x_k = S_n - \frac 1 {b_n} \sum_{k \mathop = 1}^{n - 1} \paren {b_{k + 1} - b_k} S_k$
Now, pick any $\epsilon \in \R_{>0}$.
Choose $N$ such that $S_k$ is $\epsilon$-close to $s$ for $k > N$.
This can b... | Let $\sequence {x_n}$ be an [[Definition:Infinite Sequence|infinite sequence]] of [[Definition:Real Number|real numbers]] such that:
:$\ds \sum_{n \mathop = 1}^\infty x_n = s$
exists and is finite.
Then for $0 < b_1 \le b_2 \le b_3 \le \ldots$ and $b_n \to \infty$:
:$\ds \lim_{n \mathop \to \infty} \frac 1 {b_n} \s... | Let $S_k$ denote the [[Definition:Partial Sum|partial sums]] of the $x$s.
Using [[Summation by Parts]]:
:$\ds \frac 1 {b_n} \sum_{k \mathop = 1}^n b_k x_k = S_n - \frac 1 {b_n} \sum_{k \mathop = 1}^{n - 1} \paren {b_{k + 1} - b_k} S_k$
Now, pick any $\epsilon \in \R_{>0}$.
Choose $N$ such that $S_k$ is [[Definition... | Kronecker's Lemma | https://proofwiki.org/wiki/Kronecker's_Lemma | https://proofwiki.org/wiki/Kronecker's_Lemma | [
"Real Analysis"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Real Number"
] | [
"Definition:Series/Sequence of Partial Sums",
"Abel's Lemma",
"Definition:Epsilon-Close"
] |
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