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-10700 | There Exists No Universal Set | There exists no set which is an absolutely universal set.
That is:
:$\map \neg {\exists \, \UU: \forall T: T \in \UU}$
where $T$ is any arbitrary object at all.
That is, a set that contains ''everything'' cannot exist. | {{AimForCont}} such a $\UU$ exists.
Using the Axiom of Specification, we can create the set of all ordinals:
:$\set {x \in \UU: x \text{ is an ordinal} }$
But from Burali-Forti Paradox, this set cannot exist, which is a contradiction.
{{qed}} | There exists no [[Definition:Set|set]] which is an absolutely [[Definition:Universal Set|universal set]].
That is:
:$\map \neg {\exists \, \UU: \forall T: T \in \UU}$
where $T$ is any arbitrary [[Definition:Object|object]] at all.
That is, a [[Definition:Set|set]] that contains ''everything'' cannot exist. | {{AimForCont}} such a $\UU$ exists.
Using the [[Axiom:Axiom of Specification (Sets)|Axiom of Specification]], we can create the [[Definition:Set|set]] of all [[Definition:Ordinal|ordinals]]:
:$\set {x \in \UU: x \text{ is an ordinal} }$
But from [[Burali-Forti Paradox]], this [[Definition:Set|set]] cannot exist, whic... | There Exists No Universal Set/Proof 4 | https://proofwiki.org/wiki/There_Exists_No_Universal_Set | https://proofwiki.org/wiki/There_Exists_No_Universal_Set/Proof_4 | [
"There Exists No Universal Set",
"Universal Set",
"Naive Set Theory"
] | [
"Definition:Set",
"Definition:Universal Set",
"Definition:Object",
"Definition:Set"
] | [
"Axiom:Axiom of Specification/Set Theory",
"Definition:Set",
"Definition:Ordinal",
"Burali-Forti Paradox",
"Definition:Set",
"Definition:Contradiction"
] |
proofwiki-10701 | Union of Disjoint Singletons is Doubleton | Let $\set a$ and $\set b$ be singletons such that $a \ne b$.
Then:
:$\set a \cup \set b = \set {a, b}$ | Let $x \in \set a \cup \set b$.
Then by the Axiom of Unions:
:$x = a \lor x = b$
It follows from the Axiom of Pairing that:
:$x \in \set {a, b}$
Thus by definition of subset:
:$\set a \cup \set b \subseteq \set {a, b}$
{{qed|lemma}}
Let $x \in \set {a, b}$.
Then by the Axiom of Pairing:
:$x = a \lor x = b$
So by the Ax... | Let $\set a$ and $\set b$ be [[Definition:Singleton|singletons]] such that $a \ne b$.
Then:
:$\set a \cup \set b = \set {a, b}$ | Let $x \in \set a \cup \set b$.
Then by the [[Axiom:Axiom of Unions (Set Theory)|Axiom of Unions]]:
:$x = a \lor x = b$
It follows from the [[Axiom:Axiom of Pairing (Set Theory)|Axiom of Pairing]] that:
:$x \in \set {a, b}$
Thus by definition of [[Definition:Subset|subset]]:
:$\set a \cup \set b \subseteq \set {a, ... | Union of Disjoint Singletons is Doubleton/Proof 1 | https://proofwiki.org/wiki/Union_of_Disjoint_Singletons_is_Doubleton | https://proofwiki.org/wiki/Union_of_Disjoint_Singletons_is_Doubleton/Proof_1 | [
"Set Union",
"Singletons",
"Doubletons",
"Union of Disjoint Singletons is Doubleton"
] | [
"Definition:Singleton"
] | [
"Axiom:Axiom of Unions/Set Theory",
"Axiom:Axiom of Pairing/Set Theory",
"Definition:Subset",
"Axiom:Axiom of Pairing/Set Theory",
"Axiom:Axiom of Unions/Set Theory",
"Definition:Subset",
"Definition:Set Equality/Definition 2"
] |
proofwiki-10702 | Union of Disjoint Singletons is Doubleton | Let $\set a$ and $\set b$ be singletons such that $a \ne b$.
Then:
:$\set a \cup \set b = \set {a, b}$ | Straightforward from Union of Unordered Tuples.
{{qed}} | Let $\set a$ and $\set b$ be [[Definition:Singleton|singletons]] such that $a \ne b$.
Then:
:$\set a \cup \set b = \set {a, b}$ | Straightforward from [[Union of Unordered Tuples]].
{{qed}} | Union of Disjoint Singletons is Doubleton/Proof 2 | https://proofwiki.org/wiki/Union_of_Disjoint_Singletons_is_Doubleton | https://proofwiki.org/wiki/Union_of_Disjoint_Singletons_is_Doubleton/Proof_2 | [
"Set Union",
"Singletons",
"Doubletons",
"Union of Disjoint Singletons is Doubleton"
] | [
"Definition:Singleton"
] | [
"Union of Unordered Tuples"
] |
proofwiki-10703 | Heine-Borel Theorem/Real Line/Closed and Bounded Set | Let $F$ be a closed and bounded real set.
Let $C$ be a set of open real sets.
Let $C$ be a cover of $F$.
Then there is a finite subset of $C$ that covers $F$. | We are given that $C$ is a set of open real sets that covers $F$.
In other words, $C$ is an open cover of $F$.
We need to show that there is a finite subset of $C$ that covers $F$.
In other words, we need to show that $C$ has a finite subcover.
Let $F_o$ be the complement of $F$ in $\R$.
By definition of closed real se... | Let $F$ be a [[Definition:Closed Set (Real Analysis)|closed]] and [[Definition:Bounded Subset of Real Numbers|bounded]] [[Definition:Real Number|real set]].
Let $C$ be a [[Definition:Set|set]] of [[Definition:Open Set (Real Analysis)|open real sets]].
Let $C$ be a [[Definition:Cover of Set|cover]] of $F$.
Then ther... | We are given that $C$ is a [[Definition:Set|set]] of [[Definition:Open Set (Real Analysis)|open real sets]] that [[Definition:Cover of Set|covers]] $F$.
In other words, $C$ is an [[Definition:Open Cover|open cover]] of $F$.
We need to show that there is a [[Definition:Finite Subset|finite subset]] of $C$ that [[Defin... | Heine-Borel Theorem/Real Line/Closed and Bounded Set | https://proofwiki.org/wiki/Heine-Borel_Theorem/Real_Line/Closed_and_Bounded_Set | https://proofwiki.org/wiki/Heine-Borel_Theorem/Real_Line/Closed_and_Bounded_Set | [
"Real Analysis"
] | [
"Definition:Closed Set/Real Analysis",
"Definition:Bounded Set/Real Numbers",
"Definition:Real Number",
"Definition:Set",
"Definition:Open Set/Real Analysis",
"Definition:Cover of Set",
"Definition:Finite Subset",
"Definition:Cover of Set"
] | [
"Definition:Set",
"Definition:Open Set/Real Analysis",
"Definition:Cover of Set",
"Definition:Open Cover",
"Definition:Finite Subset",
"Definition:Cover of Set",
"Definition:Subcover/Finite",
"Definition:Relative Complement",
"Definition:Closed Set/Real Analysis",
"Definition:Open Set/Real Analysi... |
proofwiki-10704 | Cardinality of Set of All Mappings from Empty Set | Let $T$ be a set.
Let $T^\O$ denote the set of all mappings from $\O$ to $S$.
Then:
:$\card {T^\O} = 1$
where $\card {T^\O}$ denotes the cardinality of $\O^S$. | The only element of $T^\O$ is the null relation:
:$\O \times T$
From Null Relation is Mapping iff Domain is Empty Set, $\O \times T$ is a mapping from $\O$ to $T$.
The result follows from Empty Mapping is Unique.
That is:
:$\card {T^\O} = 1$
{{qed}} | Let $T$ be a [[Definition:Set|set]].
Let $T^\O$ denote the [[Definition:Set of All Mappings|set of all mappings]] from $\O$ to $S$.
Then:
:$\card {T^\O} = 1$
where $\card {T^\O}$ denotes the [[Definition:Cardinality|cardinality]] of $\O^S$. | The only element of $T^\O$ is the [[Definition:Null Relation|null relation]]:
:$\O \times T$
From [[Null Relation is Mapping iff Domain is Empty Set]], $\O \times T$ is a [[Definition:Mapping|mapping]] from $\O$ to $T$.
The result follows from [[Empty Mapping is Unique]].
That is:
:$\card {T^\O} = 1$
{{qed}} | Cardinality of Set of All Mappings from Empty Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_All_Mappings_from_Empty_Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_All_Mappings_from_Empty_Set | [
"Combinatorics",
"Mapping Theory"
] | [
"Definition:Set",
"Definition:Set of All Mappings",
"Definition:Cardinality"
] | [
"Definition:Null Relation",
"Null Relation is Mapping iff Domain is Empty Set",
"Definition:Mapping",
"Empty Mapping is Unique"
] |
proofwiki-10705 | Cardinality of Set of All Mappings to Empty Set | Let $S$ be a set.
Let $\O^S$ be the set of all mappings from $S$ to $\O$.
Then:
:$\card {\O^S} = \begin{cases}
1 & : S = \O \\
0 & : S \ne \O
\end{cases}$
where $\card {\O^S}$ denotes the cardinality of $\O^S$. | From Null Relation is Mapping iff Domain is Empty Set, the null relation:
:$\RR = \O \subseteq S \times T$
is not a mapping unless $S = \O$.
So if $S \ne \O$:
:$\card {\O^S} = 0$
If $S = \O$:
:$\card {\O^S} = 1$
Hence the result.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\O^S$ be the [[Definition:Set of All Mappings|set of all mappings]] from $S$ to $\O$.
Then:
:$\card {\O^S} = \begin{cases}
1 & : S = \O \\
0 & : S \ne \O
\end{cases}$
where $\card {\O^S}$ denotes the [[Definition:Cardinality|cardinality]] of $\O^S$. | From [[Null Relation is Mapping iff Domain is Empty Set]], the [[Definition:Null Relation|null relation]]:
:$\RR = \O \subseteq S \times T$
is not a [[Definition:Mapping|mapping]] unless $S = \O$.
So if $S \ne \O$:
:$\card {\O^S} = 0$
If $S = \O$:
:$\card {\O^S} = 1$
Hence the result.
{{qed}} | Cardinality of Set of All Mappings to Empty Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_All_Mappings_to_Empty_Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_All_Mappings_to_Empty_Set | [
"Combinatorics",
"Mapping Theory"
] | [
"Definition:Set",
"Definition:Set of All Mappings",
"Definition:Cardinality"
] | [
"Null Relation is Mapping iff Domain is Empty Set",
"Definition:Null Relation",
"Definition:Mapping"
] |
proofwiki-10706 | Intersection Distributes over Union/Family of Sets/Corollary | Let $I$ and $J$ be indexing sets.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$.
Then:
:$\ds \bigcup_{\tuple {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta} = \paren {\bigcup_{\alpha \mathop \in I} A_\alph... | {{begin-eqn}}
{{eqn | l = \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}
| r = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B
| c = Intersection Distributes over Union of Family
}}
{{eqn | ll= \leadsto
| l = \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap \paren {\bigcup_{\beta ... | Let $I$ and $J$ be [[Definition:Indexing Set|indexing sets]].
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be [[Definition:Indexed Family of Subsets|indexed families of subsets]] of a [[Definition:Set|set]] $S$.
Then:
:$\ds \bigcup_{\tuple {\alpha, \beta} \mathop \in ... | {{begin-eqn}}
{{eqn | l = \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}
| r = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B
| c = [[Intersection Distributes over Union of Family]]
}}
{{eqn | ll= \leadsto
| l = \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap \paren {\bigcup_{\b... | Intersection Distributes over Union/Family of Sets/Corollary | https://proofwiki.org/wiki/Intersection_Distributes_over_Union/Family_of_Sets/Corollary | https://proofwiki.org/wiki/Intersection_Distributes_over_Union/Family_of_Sets/Corollary | [
"Intersection Distributes over Union"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Set",
"Definition:Set Union/Family of Sets"
] | [
"Intersection Distributes over Union/Family of Sets",
"Intersection Distributes over Union/Family of Sets"
] |
proofwiki-10707 | Union Distributes over Intersection/Family of Sets/Corollary | Let $I$ and $J$ be indexing sets.
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be indexed families of subsets of a set $S$.
Then:
:$\ds \bigcap_{\tuple{\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cup B_\beta} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha... | {{begin-eqn}}
{{eqn | l = \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup B}
| r = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B
| c = Union Distributes over Intersection of Family
}}
{{eqn | ll= \leadsto
| l = \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup \paren {\bigcap_{\beta ... | Let $I$ and $J$ be [[Definition:Indexing Set|indexing sets]].
Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\beta}_{\beta \mathop \in J}$ be [[Definition:Indexed Family of Subsets|indexed families of subsets]] of a [[Definition:Set|set]] $S$.
Then:
:$\ds \bigcap_{\tuple{\alpha, \beta} \mathop \in I... | {{begin-eqn}}
{{eqn | l = \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup B}
| r = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B
| c = [[Union Distributes over Intersection of Family]]
}}
{{eqn | ll= \leadsto
| l = \bigcap_{\alpha \mathop \in I} \paren {A_\alpha \cup \paren {\bigcap_{\b... | Union Distributes over Intersection/Family of Sets/Corollary | https://proofwiki.org/wiki/Union_Distributes_over_Intersection/Family_of_Sets/Corollary | https://proofwiki.org/wiki/Union_Distributes_over_Intersection/Family_of_Sets/Corollary | [
"Union Distributes over Intersection"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Subsets",
"Definition:Set",
"Definition:Set Intersection/Family of Sets"
] | [
"Union Distributes over Intersection/Family of Sets",
"Union Distributes over Intersection/Family of Sets"
] |
proofwiki-10708 | Cartesian Product of Unions/General Result | Let $I$ and $J$ be indexing sets.
Let $\family {A_i}_{i \mathop \in I}$ and $\family {B_j}_{j \mathop \in J}$ be families of sets indexed by $I$ and $J$ respectively.
Then:
:$\ds \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j} = \bigcup_{\tuple {i, j} \mathop \in I \times J} \paren ... | {{begin-eqn}}
{{eqn | o =
| r = \tuple {x, y} \in \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {\exists i \in I: x \in A_i}
}}
{{eqn | o = \land
| r = \paren {\exists j \in J: y \in B_j}
| c = {{Def... | Let $I$ and $J$ be [[Definition:Indexing Set|indexing sets]].
Let $\family {A_i}_{i \mathop \in I}$ and $\family {B_j}_{j \mathop \in J}$ be [[Definition:Indexed Family of Sets|families of sets]] [[Definition:Indexing Set|indexed]] by $I$ and $J$ respectively.
Then:
:$\ds \paren {\bigcup_{i \mathop \in I} A_i} \times... | {{begin-eqn}}
{{eqn | o =
| r = \tuple {x, y} \in \paren {\bigcup_{i \mathop \in I} A_i} \times \paren {\bigcup_{j \mathop \in J} B_j}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {\exists i \in I: x \in A_i}
}}
{{eqn | o = \land
| r = \paren {\exists j \in J: y \in B_j}
| c = {{Def... | Cartesian Product of Unions/General Result | https://proofwiki.org/wiki/Cartesian_Product_of_Unions/General_Result | https://proofwiki.org/wiki/Cartesian_Product_of_Unions/General_Result | [
"Cartesian Product of Unions"
] | [
"Definition:Indexing Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Indexing Set",
"Definition:Set Union/Family of Sets",
"Definition:Cartesian Product"
] | [] |
proofwiki-10709 | Preimages All Exist iff Surjection/Corollary | :$\forall B \subseteq T, B \ne \O: f^{-1} \sqbrk B \ne \O$
{{iff}}:
:$f$ is a surjection
where $f^{-1} \sqbrk B$ denotes the preimage of $B \subseteq T$. | === Necessary Condition ===
Let $f$ be a surjection.
Let $B \subseteq T$ such that $B \ne \O$.
Then:
:$\exists t \in T: t \in B$
From Preimages All Exist iff Surjection:
:$\map {f^{-1} } t \ne \O$
As $t \in B$ it follows from Preimage of Subset is Subset of Preimage that:
:$f^{-1} \sqbrk B \ne \O$
$B$ is arbitrary, so:... | :$\forall B \subseteq T, B \ne \O: f^{-1} \sqbrk B \ne \O$
{{iff}}:
:$f$ is a [[Definition:Surjection|surjection]]
where $f^{-1} \sqbrk B$ denotes the [[Definition:Preimage of Subset under Mapping|preimage]] of $B \subseteq T$. | === Necessary Condition ===
Let $f$ be a [[Definition:Surjection|surjection]].
Let $B \subseteq T$ such that $B \ne \O$.
Then:
:$\exists t \in T: t \in B$
From [[Preimages All Exist iff Surjection]]:
:$\map {f^{-1} } t \ne \O$
As $t \in B$ it follows from [[Preimage of Subset is Subset of Preimage]] that:
:$f^{-1}... | Preimages All Exist iff Surjection/Corollary | https://proofwiki.org/wiki/Preimages_All_Exist_iff_Surjection/Corollary | https://proofwiki.org/wiki/Preimages_All_Exist_iff_Surjection/Corollary | [
"Surjections",
"Inverse Mappings"
] | [
"Definition:Surjection",
"Definition:Preimage/Mapping/Subset"
] | [
"Definition:Surjection",
"Preimages All Exist iff Surjection",
"Preimage of Subset is Subset of Preimage",
"Definition:Surjection",
"Definition:Surjection"
] |
proofwiki-10710 | Image of Preimage under Relation is Subset | Let $\RR \subseteq S \times T$ be a relation.
Then:
:$B \subseteq T \implies \paren {\RR \circ \RR^{-1} } \sqbrk B \subseteq B$
where:
:$\RR \sqbrk B$ denotes the image of $B$ under $\RR$
:$\RR^{-1} \sqbrk B$ denotes the preimage of $B$ under $\RR$
:$\RR \circ \RR^{-1}$ denotes composition of $\RR$ and $\RR^{-1}$. | Let $B \subseteq T$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \paren {\RR \circ \RR^{-1} } \sqbrk B
}}
{{eqn | ll= \leadsto
| l = y
| o = \in
| r = \RR \sqbrk {\RR^{-1} \sqbrk B}
| c = {{Defof|Composition of Relations}}
}}
{{eqn | ll= \leadsto
| q = \exists x \in \RR^{... | Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]].
Then:
:$B \subseteq T \implies \paren {\RR \circ \RR^{-1} } \sqbrk B \subseteq B$
where:
:$\RR \sqbrk B$ denotes the [[Definition:Image of Subset under Relation|image of $B$ under $\RR$]]
:$\RR^{-1} \sqbrk B$ denotes the [[Definition:Preimage of Su... | Let $B \subseteq T$.
Then:
{{begin-eqn}}
{{eqn | l = y
| o = \in
| r = \paren {\RR \circ \RR^{-1} } \sqbrk B
}}
{{eqn | ll= \leadsto
| l = y
| o = \in
| r = \RR \sqbrk {\RR^{-1} \sqbrk B}
| c = {{Defof|Composition of Relations}}
}}
{{eqn | ll= \leadsto
| q = \exists x \in \RR... | Image of Preimage under Relation is Subset | https://proofwiki.org/wiki/Image_of_Preimage_under_Relation_is_Subset | https://proofwiki.org/wiki/Image_of_Preimage_under_Relation_is_Subset | [
"Preimages under Relations",
"Composite Relations"
] | [
"Definition:Relation",
"Definition:Image (Set Theory)/Relation/Subset",
"Definition:Preimage/Relation/Subset",
"Definition:Composition of Relations"
] | [
"Definition:Subset"
] |
proofwiki-10711 | Inverse of Direct Image Mapping does not necessarily equal Inverse Image Mapping | Let $S$ and $T$ be sets.
Let $\RR \subseteq S \times T$ be a relation.
Let $\RR^\to$ be the direct image mapping of $\RR$.
Let $\RR^\gets$ be the inverse image mapping of $\RR$.
Then it is not necessarily the case that:
:$\paren {\RR^\to}^{-1} = \RR^\gets$
where $\paren {\RR^\to}^{-1}$ denote the inverse of $\RR^\to$.
... | Proof by Counterexample:
Let $S = T = \set {0, 1}$.
Let $\RR = \set {\tuple {0, 0}, \tuple {0, 1} }$.
We have that:
:$\RR^{-1} = \set {\tuple {0, 0}, \tuple {1, 0} }$
:$\powerset S = \powerset T = \set {\O, \set 0, \set 1, \set {0, 1} }$
Thus, by inspection:
{{begin-eqn}}
{{eqn | l = \map {\RR^\to} \O
| r = \O
... | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]].
Let $\RR^\to$ be the [[Definition:Direct Image Mapping of Relation|direct image mapping]] of $\RR$.
Let $\RR^\gets$ be the [[Definition:Inverse Image Mapping of Relation|inverse image mapping]] of $\RR$.... | [[Proof by Counterexample]]:
Let $S = T = \set {0, 1}$.
Let $\RR = \set {\tuple {0, 0}, \tuple {0, 1} }$.
We have that:
:$\RR^{-1} = \set {\tuple {0, 0}, \tuple {1, 0} }$
:$\powerset S = \powerset T = \set {\O, \set 0, \set 1, \set {0, 1} }$
Thus, by inspection:
{{begin-eqn}}
{{eqn | l = \map {\RR^\to} \O
... | Inverse of Direct Image Mapping does not necessarily equal Inverse Image Mapping | https://proofwiki.org/wiki/Inverse_of_Direct_Image_Mapping_does_not_necessarily_equal_Inverse_Image_Mapping | https://proofwiki.org/wiki/Inverse_of_Direct_Image_Mapping_does_not_necessarily_equal_Inverse_Image_Mapping | [
"Direct Image Mappings",
"Inverse Image Mappings"
] | [
"Definition:Set",
"Definition:Relation",
"Definition:Direct Image Mapping/Relation",
"Definition:Inverse Image Mapping/Relation",
"Definition:Inverse of Mapping",
"Definition:Inverse Relation",
"Definition:Direct Image Mapping/Relation",
"Definition:Inverse Image Mapping/Relation"
] | [
"Proof by Counterexample",
"Definition:Inverse of Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Mapping",
"Category:Direct Image Mappings",
"Category:Inverse Image Mappings"
] |
proofwiki-10712 | Equivalence of Definitions of Maximal Element | Let $\struct {S, \RR}$ be a relational structure.
Let $T \subseteq S$ be a subset of $S$.
{{TFAE|def = Maximal Element}} | === Definition 1 implies Definition 2 ===
Let $x$ be an maximal element by definition 1.
That is:
:$(1): \quad \forall y \in T: x \mathrel \RR y \implies x = y$
{{AimForCont}} that:
:$\exists y \in T: x \mathrel {\RR^\ne} y$
Then by definition:
:$x \mathrel \RR y \land x \ne y$
which contradicts $(1)$.
Thus by Proof by... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
{{TFAE|def = Maximal Element}} | === Definition 1 implies Definition 2 ===
Let $x$ be an [[Definition:Maximal Element/Definition 1|maximal element by definition 1]].
That is:
:$(1): \quad \forall y \in T: x \mathrel \RR y \implies x = y$
{{AimForCont}} that:
:$\exists y \in T: x \mathrel {\RR^\ne} y$
Then by definition:
:$x \mathrel \RR y \land x ... | Equivalence of Definitions of Maximal Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Maximal_Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Maximal_Element | [
"Maximal Elements"
] | [
"Definition:Relational Structure",
"Definition:Subset"
] | [
"Definition:Maximal Element/Definition 1",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Maximal Element/Definition 2",
"Definition:Maximal Element/Definition 2",
"Definition:Contradiction",
"Definition:Maximal Element/Definition 1"
] |
proofwiki-10713 | Equivalence of Definitions of Minimal Element | Let $\struct {S, \RR}$ be a relational structure.
Let $T \subseteq S$ be a subset of $S$.
{{TFAE|def = Minimal Element}} | === Definition 1 implies Definition 2 ===
Let $x$ be an minimal element by definition 1.
That is:
:$(1): \quad \forall y \in T: y \mathrel \RR x \implies x = y$
{{AimForCont}}:
:$\exists y \in T: y \mathrel {\RR^\ne} x$
Then by definition:
:$y \mathrel \RR x \land x \ne y$
which contradicts $(1)$.
Thus by Proof by Cont... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
{{TFAE|def = Minimal Element}} | === Definition 1 implies Definition 2 ===
Let $x$ be an [[Definition:Minimal Element/Definition 1|minimal element by definition 1]].
That is:
:$(1): \quad \forall y \in T: y \mathrel \RR x \implies x = y$
{{AimForCont}}:
:$\exists y \in T: y \mathrel {\RR^\ne} x$
Then by definition:
:$y \mathrel \RR x \land x \ne y... | Equivalence of Definitions of Minimal Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Minimal_Element | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Minimal_Element | [
"Order Theory"
] | [
"Definition:Relational Structure",
"Definition:Subset"
] | [
"Definition:Minimal Element/Definition 1",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Minimal Element/Definition 2",
"Definition:Minimal Element/Definition 2",
"Definition:Contradiction",
"Definition:Minimal Element/Definition 1"
] |
proofwiki-10714 | Singleton of Power Set less Empty Set is Minimal Subset | Let $S$ be a set which is non-empty.
Let $\CC = \powerset S \setminus \O$, that is, the power set of $S$ without the empty set.
Let $x \in S$.
Then $\set x$ is a minimal element of the ordered structure $\struct {\CC, \subseteq}$. | Let $y \in \CC$ such that $y \subseteq \set x$.
We have that $\O \notin \CC$.
Therefore:
:$\exists z \in S: z \in y$
But as $y \subseteq \set x$ it follows that:
:$z \in \set x$
and so by definition of singleton:
:$z = x$
and so:
:$y = \set x$
and so:
:$y = x$
Thus, by definition, $\set x$ is a minimal element of $\str... | Let $S$ be a [[Definition:Set|set]] which is [[Definition:Non-Empty Set|non-empty]].
Let $\CC = \powerset S \setminus \O$, that is, the [[Definition:Power Set|power set]] of $S$ without the [[Definition:Empty Set|empty set]].
Let $x \in S$.
Then $\set x$ is a [[Definition:Minimal Element|minimal element]] of the [[... | Let $y \in \CC$ such that $y \subseteq \set x$.
We have that $\O \notin \CC$.
Therefore:
:$\exists z \in S: z \in y$
But as $y \subseteq \set x$ it follows that:
:$z \in \set x$
and so by definition of [[Definition:Singleton|singleton]]:
:$z = x$
and so:
:$y = \set x$
and so:
:$y = x$
Thus, by definition, $\set x$... | Singleton of Power Set less Empty Set is Minimal Subset | https://proofwiki.org/wiki/Singleton_of_Power_Set_less_Empty_Set_is_Minimal_Subset | https://proofwiki.org/wiki/Singleton_of_Power_Set_less_Empty_Set_is_Minimal_Subset | [
"Power Set",
"Singletons"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Power Set",
"Definition:Empty Set",
"Definition:Minimal/Element",
"Definition:Ordered Structure"
] | [
"Definition:Singleton",
"Definition:Minimal/Element"
] |
proofwiki-10715 | Power Set less Empty Set has no Smallest Element iff not Singleton | Let $S$ be a set which is non-empty.
Let $\CC = \powerset S \setminus \O$, that is, the power set of $S$ without the empty set.
Then the ordered structure $\struct {\CC, \subseteq}$ has no smallest element {{iff}} $S$ is not a singleton. | === Necessary Condition ===
Let $S$ not be a singleton.
Then $\exists x, y \in S: x \ne y$.
Let $Z \in \CC$ be the smallest element of $\CC$.
Then:
:$\forall T \in \CC: Z \subseteq T$
But by Singleton of Power Set less Empty Set is Minimal Subset, both $\set x$ and $\set y$ are minimal elements of $\struct {\CC, \subse... | Let $S$ be a [[Definition:Set|set]] which is [[Definition:Non-Empty Set|non-empty]].
Let $\CC = \powerset S \setminus \O$, that is, the [[Definition:Power Set|power set]] of $S$ without the [[Definition:Empty Set|empty set]].
Then the [[Definition:Ordered Structure|ordered structure]] $\struct {\CC, \subseteq}$ has ... | === Necessary Condition ===
Let $S$ not be a [[Definition:Singleton|singleton]].
Then $\exists x, y \in S: x \ne y$.
Let $Z \in \CC$ be the [[Definition:Smallest Element|smallest element]] of $\CC$.
Then:
:$\forall T \in \CC: Z \subseteq T$
But by [[Singleton of Power Set less Empty Set is Minimal Subset]], both ... | Power Set less Empty Set has no Smallest Element iff not Singleton | https://proofwiki.org/wiki/Power_Set_less_Empty_Set_has_no_Smallest_Element_iff_not_Singleton | https://proofwiki.org/wiki/Power_Set_less_Empty_Set_has_no_Smallest_Element_iff_not_Singleton | [
"Power Set"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Power Set",
"Definition:Empty Set",
"Definition:Ordered Structure",
"Definition:Smallest Element",
"Definition:Singleton"
] | [
"Definition:Singleton",
"Definition:Smallest Element",
"Singleton of Power Set less Empty Set is Minimal Subset",
"Definition:Minimal/Element",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Singleton",
"Definition:Smallest Element",
"Definition:Singleton"
] |
proofwiki-10716 | Natural Numbers under Multiplication form Semigroup | Let $\N$ be the set of natural numbers.
Let $\times$ denote the operation of multiplication on $\N$.
The structure $\struct {\N, \times}$ forms a semigroup. | === {{Semigroup-axiom|0|nolink}} ===
We have that Natural Number Multiplication is Closed.
That is, $\struct {\N, \times}$ is closed.
{{qed|lemma}} | Let $\N$ be the set of [[Definition:Natural Numbers|natural numbers]].
Let $\times$ denote the operation of [[Definition:Natural Number Multiplication|multiplication]] on $\N$.
The [[Definition:Algebraic Structure with One Operation|structure]] $\struct {\N, \times}$ forms a [[Definition:Semigroup|semigroup]]. | === {{Semigroup-axiom|0|nolink}} ===
We have that [[Natural Number Multiplication is Closed]].
That is, $\struct {\N, \times}$ is [[Definition:Closed Algebraic Structure|closed]].
{{qed|lemma}} | Natural Numbers under Multiplication form Semigroup | https://proofwiki.org/wiki/Natural_Numbers_under_Multiplication_form_Semigroup | https://proofwiki.org/wiki/Natural_Numbers_under_Multiplication_form_Semigroup | [
"Natural Number Multiplication",
"Examples of Semigroups"
] | [
"Definition:Natural Numbers",
"Definition:Multiplication/Natural Numbers",
"Definition:Algebraic Structure/One Operation",
"Definition:Semigroup"
] | [
"Natural Number Multiplication is Closed",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] |
proofwiki-10717 | Non-Zero Natural Numbers under Addition form Semigroup | Let $\N_{>0}$ be the set of natural numbers without zero, that is:
:$\N_{>0} = \N \setminus \set 0$
Let $+$ denote natural number addition.
The structure $\struct {\N_{>0}, +}$ forms a semigroup. | This is a specific instance of Natural Numbers Bounded Below under Addition form Commutative Semigroup.
{{qed}} | Let $\N_{>0}$ be the set of [[Definition:Natural Numbers|natural numbers]] without [[Definition:Zero (Number)|zero]], that is:
:$\N_{>0} = \N \setminus \set 0$
Let $+$ denote [[Definition:Natural Number Addition|natural number addition]].
The [[Definition:Algebraic Structure with One Operation|structure]] $\struct {\... | This is a specific instance of [[Natural Numbers Bounded Below under Addition form Commutative Semigroup]].
{{qed}} | Non-Zero Natural Numbers under Addition form Semigroup | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Addition_form_Semigroup | https://proofwiki.org/wiki/Non-Zero_Natural_Numbers_under_Addition_form_Semigroup | [
"Natural Number Addition",
"Examples of Semigroups"
] | [
"Definition:Natural Numbers",
"Definition:Zero (Number)",
"Definition:Addition/Natural Numbers",
"Definition:Algebraic Structure/One Operation",
"Definition:Semigroup"
] | [
"Natural Numbers Bounded Below under Addition form Commutative Semigroup"
] |
proofwiki-10718 | Bernoulli's Hanging Chain Problem | Consider a uniform chain $C$ whose physical properties are as follows:
:$C$ is of length $L$
:The mass per unit length of $C$ is $\lambda$
:$C$ is of zero stiffness.
Let $C$ be suspended in a vertical line from a fixed point and otherwise free to move.
Let $C$ be slightly disturbed in a vertical plane from its position... | {{tidy|minor stylistic stuff, no big deal}}
The following plot shows the chain at a single instant of its motion.
thumb600pxalt=The picture shows an x-y plot of the hanging [[Definition:Chain (Physics)chain. The chain is attached at the origin and hangs along the $x$-axis. The chain lies along a curving path to show it... | Consider a [[Definition:Uniform|uniform]] [[Definition:Chain (Physics)|chain]] $C$ whose [[Definition:Physical Property|physical properties]] are as follows:
:$C$ is of [[Definition:Length (Linear Measure)|length]] $L$
:The [[Definition:Mass|mass]] per unit [[Definition:Length (Linear Measure)|length]] of $C$ is $\lamb... | {{tidy|minor stylistic stuff, no big deal}}
The following plot shows the [[Definition:Chain (Physics)|chain]] at a [[Definition:Instantaneous|single instant]] of its [[Definition:Motion|motion]].
[[File:Bernoulli chain.png|thumb|600px|alt=The picture shows an x-y plot of the hanging [[Definition:Chain (Physics)|chain... | Bernoulli's Hanging Chain Problem | https://proofwiki.org/wiki/Bernoulli's_Hanging_Chain_Problem | https://proofwiki.org/wiki/Bernoulli's_Hanging_Chain_Problem | [
"Second Order ODEs"
] | [
"Definition:Uniform",
"Definition:Chain (Physics)",
"Definition:Physical Property",
"Definition:Linear Measure/Length",
"Definition:Mass",
"Definition:Linear Measure/Length",
"Definition:Zero (Number)",
"Definition:Stiffness (Physics)",
"Definition:Vertical Line",
"Definition:Point",
"Definition... | [
"Definition:Chain (Physics)",
"Definition:Instantaneous",
"Definition:Motion",
"File:Bernoulli chain.png",
"Definition:Chain (Physics)",
"Definition:Chain (Physics)",
"Definition:Coordinate System/Origin",
"Definition:Axis/X-Axis",
"Definition:Chain (Physics)",
"Definition:Curved Path",
"Definit... |
proofwiki-10719 | Real Line Continuity by Inverse of Mapping | Let $f$ be a real function.
Let the domain of $f$ be open.
Let $f^{-1}$ be the inverse of $f$.
Then $f$ is continuous {{iff}}:
:for every open real set $O$ that overlaps with the image of $f$, the preimage $f^{-1} \sqbrk O$ is open. | === Necessary Condition ===
Let $\Dom f$ be the domain of $f$.
Let $\Img f$ be the image of $f$.
Let $f^{-1} \sqbrk O$ be the preimage of $O$ under $f$.
Thus by definition:
: $\Img f$ is the set of points $q$ in the codomain of $f$ satisfying $q = \map f p$ for a point $p$ in $\Dom f$.
:$f^{-1} \sqbrk O$ is the set of ... | Let $f$ be a [[Definition:Real Function|real function]].
Let the [[Definition:Domain of Mapping|domain]] of $f$ be [[Definition:Open Set (Real Analysis)|open]].
Let $f^{-1}$ be the [[Definition:Inverse of Mapping|inverse]] of $f$.
Then $f$ is [[Definition:Continuous Real Function|continuous]] {{iff}}:
:for every [[... | === Necessary Condition ===
Let $\Dom f$ be the [[Definition:Domain of Mapping|domain]] of $f$.
Let $\Img f$ be the [[Definition:Image of Mapping|image]] of $f$.
Let $f^{-1} \sqbrk O$ be the [[Definition:Preimage of Subset under Mapping|preimage]] of $O$ under $f$.
Thus by definition:
: $\Img f$ is the [[Definitio... | Real Line Continuity by Inverse of Mapping | https://proofwiki.org/wiki/Real_Line_Continuity_by_Inverse_of_Mapping | https://proofwiki.org/wiki/Real_Line_Continuity_by_Inverse_of_Mapping | [
"Real Analysis",
"Continuity"
] | [
"Definition:Real Function",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Open Set/Real Analysis",
"Definition:Inverse of Mapping",
"Definition:Continuous Real Function",
"Definition:Open Set/Real Analysis",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Preimage/Mapping/Subset",... | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Preimage/Mapping/Subset",
"Definition:Set",
"Definition:Element",
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Set",
"Definition:Element",
"Definition:Continuous Real Function",
"Defin... |
proofwiki-10720 | Trisecting the Angle/Neusis Construction | Let $\alpha$ be an angle which is to be trisected.
This can be achieved by means of a neusis construction. | We have that $\angle BCD + \angle ACB$ make a straight angle.
As $CD = AB$ by construction, $CD = BC$ by definition of radius of circle.
Thus $\triangle BCD$ is isosceles.
By Isosceles Triangle has Two Equal Angles:
:$\angle CBD = \angle CDB$
From Sum of Angles of Triangle equals Two Right Angles:
:$\angle BCD + 2 \ang... | Let $\alpha$ be an [[Definition:Angle|angle]] which is to be [[Definition:Trisection|trisected]].
This can be achieved by means of a [[Definition:Neusis Construction|neusis construction]]. | We have that $\angle BCD + \angle ACB$ make a [[Definition:Straight Angle|straight angle]].
As $CD = AB$ by construction, $CD = BC$ by definition of [[Definition:Radius of Circle|radius of circle]].
Thus $\triangle BCD$ is [[Definition:Isosceles Triangle|isosceles]].
By [[Isosceles Triangle has Two Equal Angles]]:
:... | Trisecting the Angle/Neusis Construction | https://proofwiki.org/wiki/Trisecting_the_Angle/Neusis_Construction | https://proofwiki.org/wiki/Trisecting_the_Angle/Neusis_Construction | [
"Trisecting the Angle",
"Neusis Constructions"
] | [
"Definition:Angle",
"Definition:Trisection",
"Definition:Neusis Construction"
] | [
"Definition:Straight Angle",
"Definition:Circle/Radius",
"Definition:Triangle (Geometry)/Isosceles",
"Isosceles Triangle has Two Equal Angles",
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Right Angle",
"Isosceles Triangle has Two Equal Angles",
"Sum of Angles of Triangle equals Tw... |
proofwiki-10721 | Trisecting the Angle by Compass and Straightedge Construction is Impossible | There is no compass and straightedge construction for the trisection of the general angle. | Let $OA$ and $OB$ intersect at $O$.
It will be shown that there is no general method using a compass and straightedge construction to construct $OC$ such that $\angle AOB = 3 \times \angle AOC$.
It is sufficient to demonstrate that this is impossible for one specific angle.
Hence we choose $\angle AOB = 60 \degrees$.
L... | There is no [[Definition:Compass and Straightedge Construction|compass and straightedge construction]] for the [[Definition:Trisection|trisection]] of the general [[Definition:Angle|angle]]. | Let $OA$ and $OB$ [[Definition:Intersection (Geometry)|intersect]] at $O$.
It will be shown that there is no general method using a [[Definition:Compass and Straightedge Construction|compass and straightedge construction]] to construct $OC$ such that $\angle AOB = 3 \times \angle AOC$.
It is [[Definition:Sufficient C... | Trisecting the Angle by Compass and Straightedge Construction is Impossible | https://proofwiki.org/wiki/Trisecting_the_Angle_by_Compass_and_Straightedge_Construction_is_Impossible | https://proofwiki.org/wiki/Trisecting_the_Angle_by_Compass_and_Straightedge_Construction_is_Impossible | [
"Trisecting the Angle",
"Compass and Straightedge Constructions"
] | [
"Definition:Compass and Straightedge Construction",
"Definition:Trisection",
"Definition:Angle"
] | [
"Definition:Intersection (Geometry)",
"Definition:Compass and Straightedge Construction",
"Definition:Conditional/Sufficient Condition",
"Definition:Angle",
"Definition:Point",
"Definition:Unit Circle",
"Definition:Circle/Center",
"Definition:Axis/X-Axis",
"Definition:Trisection",
"Definition:Poin... |
proofwiki-10722 | Doubling the Cube by Compass and Straightedge Construction is Impossible | There is no compass and straightedge construction to allow a cube to be constructed whose volume is double that of a given cube. | Suppose it is possible.
Then from a cube of edge length $L$ we can construct a new cube with edge length $\sqrt [3] 2 L$.
$\sqrt [3] 2$ is algebraic of degree $3$.
This contradicts Constructible Length with Compass and Straightedge.
{{qed}} | There is no [[Definition:Compass and Straightedge Construction|compass and straightedge construction]] to allow a [[Definition:Cube (Geometry)|cube]] to be constructed whose [[Definition:Volume|volume]] is double that of a given [[Definition:Cube (Geometry)|cube]]. | Suppose it is possible.
Then from a [[Definition:Cube (Geometry)|cube]] of [[Definition:Edge of Polyhedron|edge]] [[Definition:Length of Line|length]] $L$ we can construct a new [[Definition:Cube (Geometry)|cube]] with [[Definition:Edge of Polyhedron|edge]] [[Definition:Length of Line|length]] $\sqrt [3] 2 L$.
$\sqrt... | Doubling the Cube by Compass and Straightedge Construction is Impossible | https://proofwiki.org/wiki/Doubling_the_Cube_by_Compass_and_Straightedge_Construction_is_Impossible | https://proofwiki.org/wiki/Doubling_the_Cube_by_Compass_and_Straightedge_Construction_is_Impossible | [
"Doubling the Cube",
"Compass and Straightedge Constructions",
"Field Extensions"
] | [
"Definition:Compass and Straightedge Construction",
"Definition:Cube/Geometry",
"Definition:Volume",
"Definition:Cube/Geometry"
] | [
"Definition:Cube/Geometry",
"Definition:Polyhedron/Edge",
"Definition:Linear Measure/Length",
"Definition:Cube/Geometry",
"Definition:Polyhedron/Edge",
"Definition:Linear Measure/Length",
"Definition:Algebraic Number",
"Definition:Algebraic Number/Degree",
"Constructible Length with Compass and Stra... |
proofwiki-10723 | Squaring the Circle by Compass and Straightedge Construction is Impossible | There is no compass and straightedge construction to allow a square to be constructed whose area is equal to that of a given circle. | Squaring the Circle consists of constructing a line segment of length $\sqrt \pi$ of another.
From Constructible Length with Compass and Straightedge, any such line segment has a length which is an algebraic number of degree $2$.
But $\pi$ is transcendental.
Hence $\pi$ and therefore $\sqrt \pi$ is not such an algebrai... | There is no [[Definition:Compass and Straightedge Construction|compass and straightedge construction]] to allow a [[Definition:Square (Geometry)|square]] to be constructed whose [[Definition:Area|area]] is equal to that of a given [[Definition:Circle|circle]]. | [[Squaring the Circle]] consists of constructing a [[Definition:Line Segment|line segment]] of [[Definition:Length (Linear Measure)|length]] $\sqrt \pi$ of another.
From [[Constructible Length with Compass and Straightedge]], any such [[Definition:Line Segment|line segment]] has a [[Definition:Length (Linear Measure)|... | Squaring the Circle by Compass and Straightedge Construction is Impossible | https://proofwiki.org/wiki/Squaring_the_Circle_by_Compass_and_Straightedge_Construction_is_Impossible | https://proofwiki.org/wiki/Squaring_the_Circle_by_Compass_and_Straightedge_Construction_is_Impossible | [
"Compass and Straightedge Constructions",
"Squaring the Circle"
] | [
"Definition:Compass and Straightedge Construction",
"Definition:Quadrilateral/Square",
"Definition:Area",
"Definition:Circle"
] | [
"Squaring the Circle",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Constructible Length with Compass and Straightedge",
"Definition:Line/Segment",
"Definition:Linear Measure/Length",
"Definition:Algebraic Number",
"Definition:Algebraic Number/Degree",
"Pi is Transcendental",
"D... |
proofwiki-10724 | Time Taken for Body to Fall at Earth's Surface | Let an object $m$ be released above ground from a point near the Earth's surface and allowed to fall freely.
Let $m$ fall a distance $s$ in time $t$.
Then:
:$s = \dfrac 1 2 g t^2$
or:
:$t = \sqrt {\dfrac {2 s} g}$
where $g$ is the Acceleration Due to Gravity at the height through which $m$ falls.
It is supposed that th... | From Equations of Motion with Constant Acceleration: Distance after Time:
:$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$
Here the body falls from rest, so:
:$\mathbf u = \mathbf 0$
Thus:
:$\mathbf s = \dfrac {\mathbf g t^2} 2$
and so taking magnitudes:
:$s = \dfrac {g t^2} 2$
It follows by multiplying by $\dfrac... | Let an [[Definition:Object|object]] $m$ be released above ground from a point near the [[Definition:Earth|Earth's]] surface and allowed to fall freely.
Let $m$ fall a [[Definition:Displacement|distance]] $s$ in [[Definition:Time|time]] $t$.
Then:
:$s = \dfrac 1 2 g t^2$
or:
:$t = \sqrt {\dfrac {2 s} g}$
where $g$ is ... | From [[Equations of Motion with Constant Acceleration/Distance after Time|Equations of Motion with Constant Acceleration: Distance after Time]]:
:$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$
Here the body falls from [[Definition:Stationary|rest]], so:
:$\mathbf u = \mathbf 0$
Thus:
:$\mathbf s = \dfrac {\mat... | Time Taken for Body to Fall at Earth's Surface | https://proofwiki.org/wiki/Time_Taken_for_Body_to_Fall_at_Earth's_Surface | https://proofwiki.org/wiki/Time_Taken_for_Body_to_Fall_at_Earth's_Surface | [
"Mechanics",
"Gravity"
] | [
"Definition:Object",
"Definition:Earth",
"Definition:Displacement",
"Definition:Time",
"Acceleration Due to Gravity"
] | [
"Equations of Motion with Constant Acceleration/Distance after Time",
"Definition:Stationary"
] |
proofwiki-10725 | Length of Chord of Circle | Let $C$ be a circle of radius $r$.
Let $AB$ be a chord which joins the endpoints of the arc $ADB$.
Then:
:$AB = 2 r \sin \dfrac \theta 2$
where $\theta$ is the angle subtended by $AB$ at the center of $C$. | :300px
Let $O$ be the center of $C$.
Let $AB$ be bisected by $OD$.
Consider the pair of triangles $\triangle AOE$ and $\triangle BOE$.
We see that:
:$AE = EB$ since $AB$ is bisected by $OD$
:$AO = BO$ since they are radii
:$OE = OE$ since they are common sides.
By Triangle Side-Side-Side Congruence:
:$\triangle AOE = \... | Let $C$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$.
Let $AB$ be a [[Definition:Chord of Circle|chord]] which joins the [[Definition:Endpoint of Line|endpoints]] of the [[Definition:Arc of Circle|arc]] $ADB$.
Then:
:$AB = 2 r \sin \dfrac \theta 2$
where $\theta$ is the [[Definition... | :[[File:LengthOfChord.png|300px]]
Let $O$ be the [[Definition:Center of Circle|center]] of $C$.
Let $AB$ be [[Definition:Bisection|bisected]] by $OD$.
Consider the pair of [[Definition:Triangle (Geometry)|triangles]] $\triangle AOE$ and $\triangle BOE$.
We see that:
:$AE = EB$ since $AB$ is [[Definition:Bisection... | Length of Chord of Circle/Proof 1 | https://proofwiki.org/wiki/Length_of_Chord_of_Circle | https://proofwiki.org/wiki/Length_of_Chord_of_Circle/Proof_1 | [
"Circles",
"Length of Chord of Circle"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Chord",
"Definition:Line/Endpoint",
"Definition:Circle/Arc",
"Definition:Angle",
"Definition:Subtend",
"Definition:Circle/Center"
] | [
"File:LengthOfChord.png",
"Definition:Circle/Center",
"Definition:Bisection",
"Definition:Triangle (Geometry)",
"Definition:Bisection",
"Definition:Radius",
"Triangle Side-Side-Side Congruence"
] |
proofwiki-10726 | Length of Chord of Circle | Let $C$ be a circle of radius $r$.
Let $AB$ be a chord which joins the endpoints of the arc $ADB$.
Then:
:$AB = 2 r \sin \dfrac \theta 2$
where $\theta$ is the angle subtended by $AB$ at the center of $C$. | We have $AO = BO$ since they are radii.
Therefore $\triangle AOB$ is isosceles.
By Isosceles Triangle has Two Equal Angles:
:$\angle OAB = \angle OBA$
By Sum of Angles of Triangle equals Two Right Angles:
:$\angle OAB + \angle OBA + \theta = 180 \degrees$
Therefore $\angle OAB = \dfrac {180 \degrees - \theta} 2 = 90 \d... | Let $C$ be a [[Definition:Circle|circle]] of [[Definition:Radius of Circle|radius]] $r$.
Let $AB$ be a [[Definition:Chord of Circle|chord]] which joins the [[Definition:Endpoint of Line|endpoints]] of the [[Definition:Arc of Circle|arc]] $ADB$.
Then:
:$AB = 2 r \sin \dfrac \theta 2$
where $\theta$ is the [[Definition... | We have $AO = BO$ since they are [[Definition:Radius|radii]].
Therefore $\triangle AOB$ is [[Definition:Isosceles Triangle|isosceles]].
By [[Isosceles Triangle has Two Equal Angles]]:
:$\angle OAB = \angle OBA$
By [[Sum of Angles of Triangle equals Two Right Angles]]:
:$\angle OAB + \angle OBA + \theta = 180 \degr... | Length of Chord of Circle/Proof 2 | https://proofwiki.org/wiki/Length_of_Chord_of_Circle | https://proofwiki.org/wiki/Length_of_Chord_of_Circle/Proof_2 | [
"Circles",
"Length of Chord of Circle"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Chord",
"Definition:Line/Endpoint",
"Definition:Circle/Arc",
"Definition:Angle",
"Definition:Subtend",
"Definition:Circle/Center"
] | [
"Definition:Radius",
"Definition:Triangle (Geometry)/Isosceles",
"Isosceles Triangle has Two Equal Angles",
"Sum of Angles of Triangle equals Two Right Angles",
"Law of Sines",
"Double Angle Formulas/Sine",
"Sine of Supplementary Angle"
] |
proofwiki-10727 | Ptolemy's Theorem | Let $ABCD$ be a cyclic quadrilateral.
Then:
:$AB \times CD + AD \times BC = AC \times BD$ | 250px
Let $\Box ABCD$ be a cyclic quadrilateral, with diagonals $AC$ and $BD$.
By Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles:
:$\angle ABC$ is supplementary to $\angle ADC$
As well:
:$\angle BAD$ is supplementary to $\angle BCD$
Construct two triangles $\triangle A'B'C'$ and $\triangle C'D'E'$ co... | Let $ABCD$ be a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]].
Then:
:$AB \times CD + AD \times BC = AC \times BD$ | [[File:Ptproof7.png|250px]]
Let $\Box ABCD$ be a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]], with [[Definition:Diameter of Quadrilateral|diagonals]] $AC$ and $BD$.
By [[Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles]]:
:$\angle ABC$ is [[Definition:Supplementary Angles|supplementary]]... | Ptolemy's Theorem/Proof 2 | https://proofwiki.org/wiki/Ptolemy's_Theorem | https://proofwiki.org/wiki/Ptolemy's_Theorem/Proof_2 | [
"Ptolemy's Theorem",
"Cyclic Quadrilaterals"
] | [
"Definition:Cyclic Quadrilateral"
] | [
"File:Ptproof7.png",
"Definition:Cyclic Quadrilateral",
"Definition:Diameter of Quadrilateral",
"Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles",
"Definition:Supplementary Angles",
"Definition:Supplementary Angles",
"Definition:Triangle (Geometry)",
"Definition:Congruence (Geometry)",... |
proofwiki-10728 | Ptolemy's Theorem | Let $ABCD$ be a cyclic quadrilateral.
Then:
:$AB \times CD + AD \times BC = AC \times BD$ | 400px
Let an arbitrary circle $K$ be drawn in the plane.
Let $A$, $B$, $C$, and $D$ be arbitrary points on $K$.
By definition, $\Box ABCD$ is a cyclic quadrilateral.
We are to show that $AB \cdot CD + BC \cdot AD = AC \cdot BD$.
Let $T$ be an inversive transformation such that:
:the inversion center of $T$ is $D$
:the ... | Let $ABCD$ be a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]].
Then:
:$AB \times CD + AD \times BC = AC \times BD$ | [[File:Ptproof16.png|400px]]
Let an arbitrary [[Definition:Circle|circle]] $K$ be drawn in the [[Definition:Cartesian Plane|plane]].
Let $A$, $B$, $C$, and $D$ be arbitrary [[Definition:Point|points]] on $K$.
By definition, $\Box ABCD$ is a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]].
We are to show t... | Ptolemy's Theorem/Proof 3 | https://proofwiki.org/wiki/Ptolemy's_Theorem | https://proofwiki.org/wiki/Ptolemy's_Theorem/Proof_3 | [
"Ptolemy's Theorem",
"Cyclic Quadrilaterals"
] | [
"Definition:Cyclic Quadrilateral"
] | [
"File:Ptproof16.png",
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Cyclic Quadrilateral",
"Definition:Inversive Transformation",
"Definition:Inversive Transformation/Inversion Center",
"Definition:Inversive Transformation/Inversion Circle",
"Definition:Interior ... |
proofwiki-10729 | Spherical Law of Sines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$ | {{begin-eqn}}
{{eqn | l = \sin b \sin c \cos A
| r = \cos a - \cos b \cos c
| c = Spherical Law of Cosines
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \cos^2 A
| r = \cos^2 a - 2 \cos a \cos b \cos c + \cos^2 b \cos^2 c
| c =
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \pare... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | {{begin-eqn}}
{{eqn | l = \sin b \sin c \cos A
| r = \cos a - \cos b \cos c
| c = [[Spherical Law of Cosines]]
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \cos^2 A
| r = \cos^2 a - 2 \cos a \cos b \cos c + \cos^2 b \cos^2 c
| c =
}}
{{eqn | ll= \leadsto
| l = \sin^2 b \sin^2 c \... | Spherical Law of Sines/Proof 1 | https://proofwiki.org/wiki/Spherical_Law_of_Sines | https://proofwiki.org/wiki/Spherical_Law_of_Sines/Proof_1 | [
"Spherical Trigonometry",
"Named Theorems",
"Spherical Law of Sines"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"Spherical Law of Cosines",
"Sum of Squares of Sine and Cosine",
"Sum of Squares of Sine and Cosine",
"Definition:Spherical Triangle",
"Definition:Spherical Triangle/Side",
"Definition:Angular Measure/Radian",
"Definition:Spherical Angle",
"Shape of Sine Function",
"Definition:Square Root/Negative",... |
proofwiki-10730 | Spherical Law of Sines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\dfrac {\sin a} {\sin A} = \dfrac {\sin b} {\sin B} = \dfrac {\sin c} {\sin C}$ | :500px
Let $A$, $B$ and $C$ be the vertices of a spherical triangle on the surface of a sphere $S$.
By definition of a spherical triangle, $AB$, $BC$ and $AC$ are arcs of great circles on $S$.
By definition of a great circle, the center of each of these great circles is $O$.
Let $O$ be joined to each of $A$, $B$ and $C... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Spherical-Cosine-Formula-2.png|500px]]
Let $A$, $B$ and $C$ be the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] $S$.
By definition of a [[Definition:Spherical Triangle|spherical triangle]], $AB$,... | Spherical Law of Sines/Proof 2 | https://proofwiki.org/wiki/Spherical_Law_of_Sines | https://proofwiki.org/wiki/Spherical_Law_of_Sines/Proof_2 | [
"Spherical Trigonometry",
"Named Theorems",
"Spherical Law of Sines"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:Spherical-Cosine-Formula-2.png",
"Definition:Polygon/Vertex",
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Spherical Triangle",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Great Circle",
"Definition:Circle/Center",
"Definition:Great Circle",
... |
proofwiki-10731 | Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\cos a = \cos b \cos c + \sin b \sin c \cos A$ | {{begin-eqn}}
{{eqn | l = \sin c \sin a \cos B
| r = \cos b - \cos c \cos a
| c = Spherical Law of Cosines
}}
{{eqn | r = \cos b - \cos c \paren {\cos b \cos c + \sin b \sin c \cos A}
| c = Spherical Law of Cosines
}}
{{eqn | r = \cos b \paren {1 - \cos^2 c} - \sin b \sin c \cos c \cos A
| c = r... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | {{begin-eqn}}
{{eqn | l = \sin c \sin a \cos B
| r = \cos b - \cos c \cos a
| c = [[Spherical Law of Cosines]]
}}
{{eqn | r = \cos b - \cos c \paren {\cos b \cos c + \sin b \sin c \cos A}
| c = [[Spherical Law of Cosines]]
}}
{{eqn | r = \cos b \paren {1 - \cos^2 c} - \sin b \sin c \cos c \cos A
... | Analogue Formula for Spherical Law of Cosines/Proof 1 | https://proofwiki.org/wiki/Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Proof_1 | [
"Spherical Law of Cosines",
"Spherical Trigonometry",
"Named Theorems"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"Spherical Law of Cosines",
"Spherical Law of Cosines",
"Sum of Squares of Sine and Cosine",
"Spherical Law of Cosines",
"Spherical Law of Cosines",
"Sum of Squares of Sine and Cosine"
] |
proofwiki-10732 | Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\cos a = \cos b \cos c + \sin b \sin c \cos A$ | :500px
Suppose $c$ is less than $\dfrac \pi 2$.
Let $BA$ be produced to $D$ so that $BD = \dfrac \pi 2$.
Then:
:$AD = \dfrac \pi 2 - c$
and:
:$\angle CAD = pi - A$
Let $C$ and $D$ be joined by an arc of a great circle, denoted $x$.
From the triangle $\sphericalangle DAC$, using the Spherical Law of Cosines:
{{begin-eqn... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Spherical-Cosine-Formula-Analog.png|500px]]
Suppose $c$ is less than $\dfrac \pi 2$.
Let $BA$ be [[Definition:Production|produced]] to $D$ so that $BD = \dfrac \pi 2$.
Then:
:$AD = \dfrac \pi 2 - c$
and:
:$\angle CAD = pi - A$
Let $C$ and $D$ be joined by an [[Definition:Arc of Circle|arc]] of a [[Definitio... | Analogue Formula for Spherical Law of Cosines/Proof 2 | https://proofwiki.org/wiki/Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Proof_2 | [
"Spherical Law of Cosines",
"Spherical Trigonometry",
"Named Theorems"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:Spherical-Cosine-Formula-Analog.png",
"Definition:Production",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Spherical Triangle",
"Spherical Law of Cosines",
"Definition:Spherical Triangle",
"Spherical Law of Cosines",
"Definition:Point"
] |
proofwiki-10733 | Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\cos a = \cos b \cos c + \sin b \sin c \cos A$ | :500px
Let $A$, $B$ and $C$ be the vertices of a spherical triangle on the surface of a sphere $S$.
By definition of a spherical triangle, $AB$, $BC$ and $AC$ are arcs of great circles on $S$.
By definition of a great circle, the center of each of these great circles is $O$.
Let $O$ be joined to each of $A$, $B$ and $C... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Spherical-Cosine-Formula-2.png|500px]]
Let $A$, $B$ and $C$ be the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] $S$.
By definition of a [[Definition:Spherical Triangle|spherical triangle]], $AB$,... | Analogue Formula for Spherical Law of Cosines/Proof 3 | https://proofwiki.org/wiki/Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Analogue_Formula_for_Spherical_Law_of_Cosines/Proof_3 | [
"Spherical Law of Cosines",
"Spherical Trigonometry",
"Named Theorems"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:Spherical-Cosine-Formula-2.png",
"Definition:Polygon/Vertex",
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Spherical Triangle",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Great Circle",
"Definition:Circle/Center",
"Definition:Great Circle",
... |
proofwiki-10734 | Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\cos a = \cos b \cos c + \sin b \sin c \cos A$ | :500px
Let $A$, $B$ and $C$ be the vertices of a spherical triangle on the surface of a sphere $S$.
By definition of a spherical triangle, $AB$, $BC$ and $AC$ are arcs of great circles on $S$.
By definition of a great circle, the center of each of these great circles is $O$.
Let $AD$ be the tangent to the great circle ... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Spherical-Cosine-Formula.png|500px]]
Let $A$, $B$ and $C$ be the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] $S$.
By definition of a [[Definition:Spherical Triangle|spherical triangle]], $AB$, $... | Spherical Law of Cosines/Proof 1 | https://proofwiki.org/wiki/Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Spherical_Law_of_Cosines/Proof_1 | [
"Spherical Law of Cosines",
"Spherical Trigonometry",
"Named Theorems"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:Spherical-Cosine-Formula.png",
"Definition:Polygon/Vertex",
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Spherical Triangle",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Great Circle",
"Definition:Circle/Center",
"Definition:Great Circle",
... |
proofwiki-10735 | Spherical Law of Cosines | Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.
Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.
Then:
:$\cos a = \cos b \cos c + \sin b \sin c \cos A$ | :500px
Let $A$, $B$ and $C$ be the vertices of a spherical triangle on the surface of a sphere $S$.
By definition of a spherical triangle, $AB$, $BC$ and $AC$ are arcs of great circles on $S$.
By definition of a great circle, the center of each of these great circles is $O$.
Let $O$ be joined to each of $A$, $B$ and $C... | Let $\triangle ABC$ be a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] whose [[Definition:Center of Sphere|center]] is $O$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a, b, c$ of $\triangle ABC$ be measured by the [[Definition:Subtend|ang... | :[[File:Spherical-Cosine-Formula-2.png|500px]]
Let $A$, $B$ and $C$ be the [[Definition:Vertex of Polygon|vertices]] of a [[Definition:Spherical Triangle|spherical triangle]] on the surface of a [[Definition:Sphere (Geometry)|sphere]] $S$.
By definition of a [[Definition:Spherical Triangle|spherical triangle]], $AB$,... | Spherical Law of Cosines/Proof 2 | https://proofwiki.org/wiki/Spherical_Law_of_Cosines | https://proofwiki.org/wiki/Spherical_Law_of_Cosines/Proof_2 | [
"Spherical Law of Cosines",
"Spherical Trigonometry",
"Named Theorems"
] | [
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Center",
"Definition:Spherical Triangle/Side",
"Definition:Subtend",
"Definition:Triangle (Geometry)/Opposite"
] | [
"File:Spherical-Cosine-Formula-2.png",
"Definition:Polygon/Vertex",
"Definition:Spherical Triangle",
"Definition:Sphere/Geometry",
"Definition:Spherical Triangle",
"Definition:Circle/Arc",
"Definition:Great Circle",
"Definition:Great Circle",
"Definition:Circle/Center",
"Definition:Great Circle",
... |
proofwiki-10736 | Spherical Law of Cosines/Angles | :$\cos A = -\cos B \cos C + \sin B \sin C \cos a$ | Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.
From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
:not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
:but also $\tria... | :$\cos A = -\cos B \cos C + \sin B \sin C \cos a$ | Let $\triangle A'B'C'$ be the [[Definition:Polar Triangle|polar triangle]] of $\triangle ABC$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a', b', c'$ of $\triangle A'B'C'$ be [[Definition:Opposite (in Triangle)|opposite]] $A', B', C'$ respectively.
From [[Spherical Triangle is Polar Triangle of its Pol... | Spherical Law of Cosines/Angles | https://proofwiki.org/wiki/Spherical_Law_of_Cosines/Angles | https://proofwiki.org/wiki/Spherical_Law_of_Cosines/Angles | [
"Spherical Law of Cosines",
"Spherical Trigonometry"
] | [] | [
"Definition:Polar Triangle",
"Definition:Spherical Triangle/Side",
"Definition:Triangle (Geometry)/Opposite",
"Spherical Triangle is Polar Triangle of its Polar Triangle",
"Definition:Polar Triangle",
"Definition:Polar Triangle",
"Spherical Law of Cosines",
"Side of Spherical Triangle is Supplement of... |
proofwiki-10737 | Spherical Law of Tangents | :$\dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} } = \dfrac {\tan \frac 1 2 \paren {a + b} } {\tan \frac 1 2 \paren {a - b} }$ | {{begin-eqn}}
{{eqn | l = \tan \dfrac {A + B} 2
| r = \dfrac {\cos \frac {a - b} 2} {\cos \frac {a + b} 2} \cot \dfrac C 2
| c = Napier's Analogies
}}
{{eqn | n = 1
| ll= \leadsto
| l = \tan \frac {A + B} 2 \cos \frac {a + b} 2
| r = \cos \frac {a - b} 2 \cot \frac C 2
| c = more man... | :$\dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} } = \dfrac {\tan \frac 1 2 \paren {a + b} } {\tan \frac 1 2 \paren {a - b} }$ | {{begin-eqn}}
{{eqn | l = \tan \dfrac {A + B} 2
| r = \dfrac {\cos \frac {a - b} 2} {\cos \frac {a + b} 2} \cot \dfrac C 2
| c = [[Napier's Analogies]]
}}
{{eqn | n = 1
| ll= \leadsto
| l = \tan \frac {A + B} 2 \cos \frac {a + b} 2
| r = \cos \frac {a - b} 2 \cot \frac C 2
| c = more... | Spherical Law of Tangents | https://proofwiki.org/wiki/Spherical_Law_of_Tangents | https://proofwiki.org/wiki/Spherical_Law_of_Tangents | [
"Spherical Trigonometry",
"Named Theorems"
] | [] | [
"Napier's Analogies",
"Napier's Analogies"
] |
proofwiki-10738 | Cosine of Half Angle for Spherical Triangles | :$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} }$
where $s = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \cos a
| r = \cos b \cos c + \sin b \sin c \cos A
| c = Spherical Law of Cosines
}}
{{eqn | r = \cos b \cos c + \sin b \sin c \paren {2 \cos^2 \dfrac A 2 - 1}
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | r = \map \cos {b + c} + 2 \sin b \sin c \cos^2 \dfra... | :$\cos \dfrac A 2 = \sqrt {\dfrac {\sin s \, \map \sin {s - a} } {\sin b \sin c} }$
where $s = \dfrac {a + b + c} 2$. | {{begin-eqn}}
{{eqn | l = \cos a
| r = \cos b \cos c + \sin b \sin c \cos A
| c = [[Spherical Law of Cosines]]
}}
{{eqn | r = \cos b \cos c + \sin b \sin c \paren {2 \cos^2 \dfrac A 2 - 1}
| c = {{Corollary|Double Angle Formula for Cosine|1}}
}}
{{eqn | r = \map \cos {b + c} + 2 \sin b \sin c \cos^2 \... | Cosine of Half Angle for Spherical Triangles | https://proofwiki.org/wiki/Cosine_of_Half_Angle_for_Spherical_Triangles | https://proofwiki.org/wiki/Cosine_of_Half_Angle_for_Spherical_Triangles | [
"Half Angle Formulas for Spherical Triangles"
] | [] | [
"Spherical Law of Cosines",
"Cosine of Sum",
"Prosthaphaeresis Formulas/Cosine minus Cosine"
] |
proofwiki-10739 | Cosine of Half Side for Spherical Triangles | :$\cos \dfrac a 2 = \sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} }$
where $S = \dfrac {A + B + C} 2$. | Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.
Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.
From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
:not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
:but also $\tria... | :$\cos \dfrac a 2 = \sqrt {\dfrac {\map \cos {S - B} \, \map \cos {S - C} } {\sin B \sin C} }$
where $S = \dfrac {A + B + C} 2$. | Let $\triangle A'B'C'$ be the [[Definition:Polar Triangle|polar triangle]] of $\triangle ABC$.
Let the [[Definition:Side of Spherical Triangle|sides]] $a', b', c'$ of $\triangle A'B'C'$ be [[Definition:Opposite (in Triangle)|opposite]] $A', B', C'$ respectively.
From [[Spherical Triangle is Polar Triangle of its Pol... | Cosine of Half Side for Spherical Triangles | https://proofwiki.org/wiki/Cosine_of_Half_Side_for_Spherical_Triangles | https://proofwiki.org/wiki/Cosine_of_Half_Side_for_Spherical_Triangles | [
"Half Side Formulas for Spherical Triangles"
] | [] | [
"Definition:Polar Triangle",
"Definition:Spherical Triangle/Side",
"Definition:Triangle (Geometry)/Opposite",
"Spherical Triangle is Polar Triangle of its Polar Triangle",
"Definition:Polar Triangle",
"Definition:Polar Triangle",
"Sine of Half Angle for Spherical Triangles",
"Side of Spherical Triangl... |
proofwiki-10740 | Equation of Circle/Parametric | The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\tuple {a, b}$ can be expressed as a parametric equation:
:$\begin {cases} x = a + R \cos t \\ y = b + R \sin t \end {cases}$ | Let the point $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a + R \cos t
}}
{{eqn | l = y
| r = b + R \sin t
}}
{{end-eqn}}
By the Distance Formula, the distance between $\tuple {x, y}$ and $\tuple {a, b}$ is:
:$\sqrt {\paren {\paren {a + R \cos t} - a}^2 + \paren {\paren {b + R \... | The [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Circle|circle]] embedded in the [[Definition:Cartesian Plane|Cartesian plane]] with [[Definition:Radius of Circle|radius]] $R$ and [[Definition:Center of Circle|center]] $\tuple {a, b}$ can be expressed as a [[Definition:Parametric Equation|para... | Let the [[Definition:Point|point]] $\tuple {x, y}$ satisfy the equations:
{{begin-eqn}}
{{eqn | l = x
| r = a + R \cos t
}}
{{eqn | l = y
| r = b + R \sin t
}}
{{end-eqn}}
By the [[Distance Formula]], the [[Definition:Distance between Points|distance]] between $\tuple {x, y}$ and $\tuple {a, b}$ is:
:$\sq... | Equation of Circle/Parametric | https://proofwiki.org/wiki/Equation_of_Circle/Parametric | https://proofwiki.org/wiki/Equation_of_Circle/Parametric | [
"Equation of Circle"
] | [
"Definition:Equation of Geometric Figure",
"Definition:Circle",
"Definition:Cartesian Plane",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Parametric Equation"
] | [
"Definition:Point",
"Distance Formula",
"Definition:Distance between Points",
"Sum of Squares of Sine and Cosine",
"Definition:Distance between Points",
"Definition:Distance between Points",
"Definition:Point",
"Definition:Circle/Center",
"Definition:Constant",
"Definition:Circle/Radius",
"Defin... |
proofwiki-10741 | Equation of Circle/Polar | :$r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$ | Let the point $\polar {r, \theta}_\text {Polar}$ satisfy the equation:
:$r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$
Let the points $\polar {r, \theta}$ and $\polar {r_0, \varphi}$ be rewritten in Cartesian coordinates:
:$\polar {r, \theta}_\text {Polar} = \tuple {r \cos \theta, r \sin \theta}_\t... | :$r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$ | Let the point $\polar {r, \theta}_\text {Polar}$ satisfy the [[Definition:Equation of Geometric Figure|equation]]:
:$r^2 - 2 r r_0 \map \cos {\theta - \varphi} + \paren {r_0}^2 = R^2$
Let the [[Definition:Point|points]] $\polar {r, \theta}$ and $\polar {r_0, \varphi}$ be rewritten in [[Definition:Cartesian Coordinate... | Equation of Circle/Polar | https://proofwiki.org/wiki/Equation_of_Circle/Polar | https://proofwiki.org/wiki/Equation_of_Circle/Polar | [
"Equation of Circle"
] | [] | [
"Definition:Equation of Geometric Figure",
"Definition:Point",
"Definition:Cartesian Coordinate System",
"Definition:Distance between Points",
"Cosine of Difference",
"Sum of Squares of Sine and Cosine",
"Definition:Circle/Center",
"Definition:Constant",
"Definition:Circle/Radius",
"Category:Equat... |
proofwiki-10742 | Equation of Circle/Cartesian/Formulation 2 | The equation:
:$A \paren {x^2 + y^2} + B x + C y + D = 0$
is the equation of a circle with radius $R$ and center $\tuple {a, b}$, where:
:$R = \dfrac 1 {2 A} \sqrt {B^2 + C^2 - 4 A D}$
:$\tuple {a, b} = \tuple {\dfrac {-B} {2 A}, \dfrac {-C} {2 A} }$
provided:
:$A > 0$
:$B^2 + C^2 \ge 4 A D$ | {{begin-eqn}}
{{eqn | l = A \paren {x^2 + y^2} + B x + C y + D
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x^2 + y^2 + \frac B A x + \frac C A y
| r = - \frac D A
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x^2 + 2 \frac B {2 A} x + \frac {B^2} {4 A^2} + y^2 + 2 \frac C {2 A... | The [[Definition:Equation of Geometric Figure|equation]]:
:$A \paren {x^2 + y^2} + B x + C y + D = 0$
is the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Circle|circle]] with [[Definition:Radius of Circle|radius]] $R$ and [[Definition:Center of Circle|center]] $\tuple {a, b}$, where:
:$R = \df... | {{begin-eqn}}
{{eqn | l = A \paren {x^2 + y^2} + B x + C y + D
| r = 0
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x^2 + y^2 + \frac B A x + \frac C A y
| r = - \frac D A
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x^2 + 2 \frac B {2 A} x + \frac {B^2} {4 A^2} + y^2 + 2 \frac C {2 A... | Equation of Circle/Cartesian/Formulation 2 | https://proofwiki.org/wiki/Equation_of_Circle/Cartesian/Formulation_2 | https://proofwiki.org/wiki/Equation_of_Circle/Cartesian/Formulation_2 | [
"Equation of Circle"
] | [
"Definition:Equation of Geometric Figure",
"Definition:Equation of Geometric Figure",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center"
] | [
"Definition:Positive/Real Number",
"Equation of Circle/Cartesian/Formulation 1",
"Category:Equation of Circle"
] |
proofwiki-10743 | Equation of Circle center Origin | The circle with radius $R$ whose center is at the origin expressed in Cartesian coordinates is given by the equation:
:$x^2 + y^2 = R^2$ | From Equation of Circle in Cartesian Plane, the equation of a circle with radius $R$ and center $\tuple {a, b}$ expressed in Cartesian coordinates is:
:$\paren {x - a}^2 + \paren {y - b}^2 = R^2$
Setting $a = b = 0$ yields the result.
{{qed}} | The [[Definition:Circle|circle]] with [[Definition:Radius of Circle|radius]] $R$ whose [[Definition:Center of Circle|center]] is at the [[Definition:Origin|origin]] expressed in [[Definition:Cartesian Coordinate System|Cartesian coordinates]] is given by the [[Definition:Equation of Geometric Figure|equation]]:
:$x^2 ... | From [[Equation of Circle in Cartesian Plane]], the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Circle|circle]] with [[Definition:Radius of Circle|radius]] $R$ and [[Definition:Center of Circle|center]] $\tuple {a, b}$ expressed in [[Definition:Cartesian Coordinate System|Cartesian coordinate... | Equation of Circle center Origin | https://proofwiki.org/wiki/Equation_of_Circle_center_Origin | https://proofwiki.org/wiki/Equation_of_Circle_center_Origin | [
"Equation of Circle"
] | [
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Coordinate System",
"Definition:Equation of Geometric Figure"
] | [
"Equation of Circle/Cartesian",
"Definition:Equation of Geometric Figure",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Cartesian Coordinate System"
] |
proofwiki-10744 | Distance Formula/3 Dimensions | The distance $d$ between two points $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$ in a Cartesian space of 3 dimensions is:
:$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2 + \paren {z_1 - z_2}^2}$ | :600px
Let $d$ be the distance to be found between $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$.
Let the points $C$ and $D$ be defined as:
:$C = \tuple {x_2, y_1, z_1}$
:$D = \tuple {x_2, y_2, z_1}$
Let $d'$ be the distance between $A$ and $D$.
From Distance Formula, it can be seen that:
:$d' = \sqrt {... | The [[Definition:Distance between Points|distance]] $d$ between two [[Definition:Point|points]] $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$ in a [[Definition:Cartesian Space|Cartesian space]] of [[Definition:Dimension (Geometry)|3 dimensions]] is:
:$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}... | :[[File:DistanceFormula3D.png|600px]]
Let $d$ be the [[Definition:Distance between Points|distance]] to be found between $A = \tuple {x_1, y_1, z_1}$ and $B = \tuple {x_2, y_2, z_2}$.
Let the [[Definition:Point|points]] $C$ and $D$ be defined as:
:$C = \tuple {x_2, y_1, z_1}$
:$D = \tuple {x_2, y_2, z_1}$
Let $d'$ ... | Distance Formula/3 Dimensions | https://proofwiki.org/wiki/Distance_Formula/3_Dimensions | https://proofwiki.org/wiki/Distance_Formula/3_Dimensions | [
"Distance Formula",
"Euclidean Geometry",
"Solid Analytic Geometry"
] | [
"Definition:Distance between Points",
"Definition:Point",
"Definition:Cartesian Product/Cartesian Space",
"Definition:Dimension (Geometry)"
] | [
"File:DistanceFormula3D.png",
"Definition:Distance between Points",
"Definition:Point",
"Definition:Distance between Points",
"Distance Formula",
"Definition:Triangle (Geometry)/Right-Angled",
"Pythagoras's Theorem"
] |
proofwiki-10745 | Equation of Sphere/Rectangular Coordinates | :$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$ | Let the point $\tuple {x, y, z}$ satisfy the equation:
:$(1): \quad \paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$
By the Distance Formula in 3 Dimensions, the distance between this $\tuple {x, y, z}$ and $\tuple {a, b, c}$ is:
:$\sqrt {\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2}$
But from eq... | :$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$ | Let the [[Definition:Point|point]] $\tuple {x, y, z}$ satisfy the [[Definition:Equation of Geometric Figure|equation]]:
:$(1): \quad \paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$
By the [[Distance Formula in 3 Dimensions]], the [[Definition:Distance between Points|distance]] between this $\tuple {x, y,... | Equation of Sphere/Rectangular Coordinates | https://proofwiki.org/wiki/Equation_of_Sphere/Rectangular_Coordinates | https://proofwiki.org/wiki/Equation_of_Sphere/Rectangular_Coordinates | [
"Spheres",
"Examples of Surfaces"
] | [] | [
"Definition:Point",
"Definition:Equation of Geometric Figure",
"Distance Formula/3 Dimensions",
"Definition:Distance between Points",
"Definition:Distance between Points",
"Definition:Point",
"Definition:Equation of Geometric Figure",
"Definition:Sphere/Geometry/Center",
"Definition:Constant",
"De... |
proofwiki-10746 | Equation of Sphere/Rectangular Coordinates/Corollary | The equation of a sphere with radius $R$ whose center is at the origin expressed in Cartesian coordinates is:
:$x^2 + y^2 + z^2 = R^2$ | From Equation of Sphere in Rectangular Coordinates, the equation of a sphere with radius $R$ and center $\tuple {a, b, c}$ expressed in Cartesian coordinates is:
:$\paren {x - a}^2 + \paren {y - b}^2 + \paren {z - c}^2 = R^2$
Setting $a = b = c = 0$ yields the result.
{{qed}} | The [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Sphere (Geometry)|sphere]] with [[Definition:Radius of Sphere|radius]] $R$ whose [[Definition:Center of Sphere|center]] is at the [[Definition:Origin|origin]] expressed in [[Definition:Cartesian Coordinate System|Cartesian coordinates]] is:
:$x... | From [[Equation of Sphere in Rectangular Coordinates]], the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Sphere (Geometry)|sphere]] with [[Definition:Radius of Sphere|radius]] $R$ and [[Definition:Center of Sphere|center]] $\tuple {a, b, c}$ expressed in [[Definition:Cartesian Coordinate Syste... | Equation of Sphere/Rectangular Coordinates/Corollary | https://proofwiki.org/wiki/Equation_of_Sphere/Rectangular_Coordinates/Corollary | https://proofwiki.org/wiki/Equation_of_Sphere/Rectangular_Coordinates/Corollary | [
"Spheres",
"Examples of Surfaces"
] | [
"Definition:Equation of Geometric Figure",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius",
"Definition:Sphere/Geometry/Center",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Coordinate System"
] | [
"Equation of Sphere/Rectangular Coordinates",
"Definition:Equation of Geometric Figure",
"Definition:Sphere/Geometry",
"Definition:Sphere/Geometry/Radius",
"Definition:Sphere/Geometry/Center",
"Definition:Cartesian Coordinate System"
] |
proofwiki-10747 | Equation of Conic Section/Cartesian Form | The general conic section can be expressed in Cartesian coordinates in the form:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
for some $a, b, c, f, g, h \in \R$. | By definition, a conic section is the set of points of intersection between a cone and a plane.
Let $P = \tuple {\alpha, \beta, \gamma}$ be the apex of the cone.
Let $Q = \tuple {x, y, z}$ be a point of intersection between the plane and the cone.
From Equation of Right Circular Cone, we have:
:$(1): \quad \paren {x - ... | The general [[Definition:Conic Section|conic section]] can be expressed in [[Definition:Cartesian Coordinate System|Cartesian coordinates]] in the form:
:$a x^2 + 2 h x y + b y^2 + 2 g x + 2 f y + c = 0$
for some $a, b, c, f, g, h \in \R$. | By definition, a [[Definition:Conic Section|conic section]] is the [[Definition:Set|set]] of [[Definition:Point|points]] of [[Definition:Intersection (Geometry)|intersection]] between a [[Definition:Cone (Geometry)|cone]] and a [[Definition:Plane|plane]].
Let $P = \tuple {\alpha, \beta, \gamma}$ be the [[Definition:A... | Equation of Conic Section/Cartesian Form | https://proofwiki.org/wiki/Equation_of_Conic_Section/Cartesian_Form | https://proofwiki.org/wiki/Equation_of_Conic_Section/Cartesian_Form | [
"Equation of Conic Section"
] | [
"Definition:Conic Section",
"Definition:Cartesian Coordinate System"
] | [
"Definition:Conic Section",
"Definition:Set",
"Definition:Point",
"Definition:Intersection (Geometry)",
"Definition:Cone (Geometry)",
"Definition:Plane Surface",
"Definition:Cone (Geometry)/Apex",
"Definition:Cone (Geometry)",
"Definition:Intersection (Geometry)",
"Definition:Plane Surface",
"De... |
proofwiki-10748 | Graph of Quadratic describes Parabola | The locus of the equation defining a quadratic:
:$y = a x^2 + b x + c$
describes a parabola. | Consider the focus-directrix property of a parabola $P$.
Let the focus of $P$ be the point $\tuple {0, f}$ on a Cartesian plane.
Let the directrix of $P$ be the straight line $y = -d$.
Let $\tuple {x, y}$ be an arbitrary point on $P$.
Then by the focus-directrix property:
:$y + d = \sqrt {\paren {x - k}^2 + \tuple {y -... | The [[Definition:Locus|locus]] of the [[Definition:Equation of Geometric Figure|equation]] defining a [[Definition:Quadratic Equation|quadratic]]:
:$y = a x^2 + b x + c$
describes a [[Definition:Parabola|parabola]]. | Consider the [[Definition:Focus-Directrix Property of Parabola|focus-directrix property]] of a [[Definition:Parabola|parabola]] $P$.
Let the [[Definition:Focus of Parabola|focus]] of $P$ be the point $\tuple {0, f}$ on a [[Definition:Cartesian Plane|Cartesian plane]].
Let the [[Definition:Directrix of Parabola|direct... | Graph of Quadratic describes Parabola | https://proofwiki.org/wiki/Graph_of_Quadratic_describes_Parabola | https://proofwiki.org/wiki/Graph_of_Quadratic_describes_Parabola | [
"Parabolas",
"Quadratic Equations",
"Graph of Quadratic describes Parabola"
] | [
"Definition:Locus",
"Definition:Equation of Geometric Figure",
"Definition:Quadratic Equation",
"Definition:Parabola"
] | [
"Definition:Parabola/Focus-Directrix",
"Definition:Parabola",
"Definition:Parabola/Focus",
"Definition:Cartesian Plane",
"Definition:Parabola/Directrix",
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Parabola/Focus-Directrix",
"Definition:Perpendicular Distance between Point and S... |
proofwiki-10749 | Graph of Quadratic describes Parabola/Corollary 1 | The locus of the equation of the square function:
:$y = x^2$
describes a parabola. | This is a particular instance of Graph of Quadratic describes Parabola, where:
:$y = a x^2 + b x + c$
is the equation of a parabola.
The result follows by setting $a = 1, b = 0, c = 0$.
{{qed}} | The [[Definition:Locus|locus]] of the [[Definition:Equation of Geometric Figure|equation]] of the [[Definition:Square (Algebra)|square function]]:
:$y = x^2$
describes a [[Definition:Parabola|parabola]]. | This is a particular instance of [[Graph of Quadratic describes Parabola]], where:
:$y = a x^2 + b x + c$
is the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Parabola|parabola]].
The result follows by setting $a = 1, b = 0, c = 0$.
{{qed}} | Graph of Quadratic describes Parabola/Corollary 1 | https://proofwiki.org/wiki/Graph_of_Quadratic_describes_Parabola/Corollary_1 | https://proofwiki.org/wiki/Graph_of_Quadratic_describes_Parabola/Corollary_1 | [
"Graph of Quadratic describes Parabola"
] | [
"Definition:Locus",
"Definition:Equation of Geometric Figure",
"Definition:Square/Function",
"Definition:Parabola"
] | [
"Graph of Quadratic describes Parabola",
"Definition:Equation of Geometric Figure",
"Definition:Parabola"
] |
proofwiki-10750 | Graph of Quadratic describes Parabola/Corollary 2 | The locus of the equation of the square root function on the non-negative reals:
:$\forall x \in \R_{\ge 0}: \map f x = \sqrt x$
describes half of a parabola. | From {{Corollary|Graph of Quadratic describes Parabola|1}}, where:
:$y = x^2$
is the equation of a parabola.
Let $f: \R \to \R$ be the real function defined as:
:$\map f x = x^2$
From Square of Real Number is Non-Negative, the image of $f$ is $\R_{\ge 0}$.
Also we have from Positive Real Number has Two Square Roots:
:$... | The [[Definition:Locus|locus]] of the [[Definition:Equation of Geometric Figure|equation]] of the [[Definition:Square Root|square root function]] on the [[Definition:Positive Real Number|non-negative reals]]:
:$\forall x \in \R_{\ge 0}: \map f x = \sqrt x$
describes half of a [[Definition:Parabola|parabola]]. | From {{Corollary|Graph of Quadratic describes Parabola|1}}, where:
:$y = x^2$
is the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Parabola|parabola]].
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = x^2$
From [[Square of Real Number is Non-Negati... | Graph of Quadratic describes Parabola/Corollary 2 | https://proofwiki.org/wiki/Graph_of_Quadratic_describes_Parabola/Corollary_2 | https://proofwiki.org/wiki/Graph_of_Quadratic_describes_Parabola/Corollary_2 | [
"Graph of Quadratic describes Parabola"
] | [
"Definition:Locus",
"Definition:Equation of Geometric Figure",
"Definition:Square Root",
"Definition:Positive/Real Number",
"Definition:Parabola"
] | [
"Definition:Equation of Geometric Figure",
"Definition:Parabola",
"Definition:Real Function",
"Square of Real Number is Non-Negative",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Positive Real Number has Two Square Roots",
"Definition:Bijective Restriction",
"Definition:Bijective Restriction",
... |
proofwiki-10751 | Natural Numbers under Addition form Commutative Semigroup | The algebraic structure $\struct {\N, +}$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative semigroup. | By Naturally Ordered Semigroup Exists, there exists a naturally ordered semigroup.
Consider the natural numbers $\N$ defined as the naturally ordered semigroup.
From the definition of the naturally ordered semigroup, $\struct {\N, +}$ is {{apriori}} a semigroup.
By Naturally Ordered Semigroup Axioms imply Commutativity... | The [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {\N, +}$ consisting of the [[Definition:Set|set]] of [[Definition:Natural Numbers|natural numbers]] $\N$ under [[Definition:Natural Number Addition|addition]] $+$ is a [[Definition:Commutative Semigroup|commutative semigroup]]. | By [[Naturally Ordered Semigroup Exists]], there exists a [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
Consider the [[Definition:Natural Numbers|natural numbers]] $\N$ defined as the [[Definition:Naturally Ordered Semigroup|naturally ordered semigroup]].
From the definition of the [[Definit... | Natural Numbers under Addition form Commutative Semigroup | https://proofwiki.org/wiki/Natural_Numbers_under_Addition_form_Commutative_Semigroup | https://proofwiki.org/wiki/Natural_Numbers_under_Addition_form_Commutative_Semigroup | [
"Natural Number Addition",
"Examples of Commutative Semigroups"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Addition/Natural Numbers",
"Definition:Commutative Semigroup"
] | [
"Naturally Ordered Semigroup Exists",
"Definition:Naturally Ordered Semigroup",
"Definition:Natural Numbers",
"Definition:Naturally Ordered Semigroup",
"Definition:Naturally Ordered Semigroup",
"Definition:Semigroup",
"Naturally Ordered Semigroup Axioms imply Commutativity",
"Definition:Commutative Se... |
proofwiki-10752 | Gödel's Incompleteness Theorems/First/Corollary | If $T$ is both consistent and complete, it does not contain minimal arithmetic. | This is simply the contrapositive of Gödel's First Incompleteness Theorem.
{{qed}}
{{Namedfor|Kurt Friedrich Gödel}} | If $T$ is both [[Definition:Consistent (Logic)|consistent]] and [[Definition:Complete Theory|complete]], it does not contain [[Definition:Minimal Arithmetic|minimal arithmetic]]. | This is simply the [[Definition:Contrapositive Statement|contrapositive]] of [[Gödel's First Incompleteness Theorem]].
{{qed}}
{{Namedfor|Kurt Friedrich Gödel}} | Gödel's Incompleteness Theorems/First/Corollary | https://proofwiki.org/wiki/Gödel's_Incompleteness_Theorems/First/Corollary | https://proofwiki.org/wiki/Gödel's_Incompleteness_Theorems/First/Corollary | [
"Gödel's Incompleteness Theorems"
] | [
"Definition:Consistent (Logic)",
"Definition:Complete Theory",
"Definition:Minimal Arithmetic"
] | [
"Definition:Contrapositive Statement",
"Gödel's Incompleteness Theorems/First"
] |
proofwiki-10753 | Monomorphism that is Split Epimorphism is Split Monomorphism | Let $\mathbf C$ be a metacategory.
Let $C$ and $D$ be objects of $\mathbf C$.
Let $f: C \to D$ be a morphism in $\mathbf C$ such that $f$ is a monomorphism and a split epimorphism.
Then $f: C \to D$ is a split monomorphism. | Let $g: D \to C$ be the right inverse of $f$:
:$f \circ g = \operatorname{id}_D$
which is guaranteed to exist by definition of split epimorphism.
Therefore:
:$f \circ g \circ f = \operatorname{id}_D \circ f = f \circ \operatorname{id}_C$
by the property of the identity morphism.
Since $f$ is left cancellable, by the de... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $C$ and $D$ be [[Definition:Object (Category Theory)|objects]] of $\mathbf C$.
Let $f: C \to D$ be a [[Definition:Morphism|morphism]] in $\mathbf C$ such that $f$ is a [[Definition:Monomorphism (Category Theory)|monomorphism]] and a [[Definition:Split... | Let $g: D \to C$ be the [[Definition:Right Inverse (Category Theory)|right inverse]] of $f$:
:$f \circ g = \operatorname{id}_D$
which is guaranteed to exist by definition of [[Definition:Split Epimorphism|split epimorphism]].
Therefore:
:$f \circ g \circ f = \operatorname{id}_D \circ f = f \circ \operatorname{id}_C... | Monomorphism that is Split Epimorphism is Split Monomorphism | https://proofwiki.org/wiki/Monomorphism_that_is_Split_Epimorphism_is_Split_Monomorphism | https://proofwiki.org/wiki/Monomorphism_that_is_Split_Epimorphism_is_Split_Monomorphism | [
"Morphisms"
] | [
"Definition:Metacategory",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Monomorphism (Category Theory)",
"Definition:Split Epimorphism",
"Definition:Split Monomorphism"
] | [
"Definition:Section (Category Theory)",
"Definition:Split Epimorphism",
"Definition:Identity Morphism",
"Definition:Cancellable Element/Left Cancellable",
"Definition:Monomorphism (Category Theory)",
"Definition:Split Monomorphism",
"Definition:Retraction (Category Theory)",
"Category:Morphisms"
] |
proofwiki-10754 | Epimorphism that is Split Monomorphism is Split Epimorphism | Let $\mathbf C$ be a metacategory.
Let $f: C \to D$ be a epimorphism and a split monomorphism.
Then $f: C \to D$ is a split epimorphism. | {{explain|What is a Dual proof}}
Dual proof of Monomorphism that is Split Epimorphism is Split Monomorphism.
{{qed}}
Category:Epimorphisms
Category:Monomorphisms (Category Theory)
k41tqj6wiaco8bl898w5naxuwmhw8kz | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $f: C \to D$ be a [[Definition:Epimorphism (Category Theory)|epimorphism]] and a [[Definition:Split Monomorphism|split monomorphism]].
Then $f: C \to D$ is a [[Definition:Split Epimorphism|split epimorphism]]. | {{explain|What is a Dual proof}}
Dual proof of [[Monomorphism that is Split Epimorphism is Split Monomorphism]].
{{qed}}
[[Category:Epimorphisms]]
[[Category:Monomorphisms (Category Theory)]]
k41tqj6wiaco8bl898w5naxuwmhw8kz | Epimorphism that is Split Monomorphism is Split Epimorphism | https://proofwiki.org/wiki/Epimorphism_that_is_Split_Monomorphism_is_Split_Epimorphism | https://proofwiki.org/wiki/Epimorphism_that_is_Split_Monomorphism_is_Split_Epimorphism | [
"Epimorphisms",
"Monomorphisms (Category Theory)"
] | [
"Definition:Metacategory",
"Definition:Epimorphism (Category Theory)",
"Definition:Split Monomorphism",
"Definition:Split Epimorphism"
] | [
"Monomorphism that is Split Epimorphism is Split Monomorphism",
"Category:Epimorphisms",
"Category:Monomorphisms (Category Theory)"
] |
proofwiki-10755 | Westwood's Puzzle | :500px
Take any rectangle $ABCD$ and draw the diagonal $AC$.
Inscribe a circle $GFJ$ in one of the resulting triangles $\triangle ABC$.
Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle.
Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABC... | Construct the perpendicular from $E$ to $AC$, and call its foot $G$.
Let $K$ be the intersection of $IE$ and $AC$.
Let $L$ be the intersection of $EH$ and $AC$.
:500px
First we have:
{{begin-eqn}}
{{eqn | n = 1
| l = \angle CKI
| r = \angle EKG
| c = Two Straight Lines make Equal Opposite Angles
}}
{{... | :[[File:WestwoodsPuzzle.png|500px]]
Take any [[Definition:Rectangle|rectangle]] $ABCD$ and draw the [[Definition:Diagonal of Quadrilateral|diagonal]] $AC$.
[[Definition:Incircle of Triangle|Inscribe]] a [[Definition:Circle|circle]] $GFJ$ in one of the resulting [[Definition:Triangle (Geometry)|triangles]] $\triangle ... | [[Perpendicular through Given Point|Construct the perpendicular]] from $E$ to $AC$, and call its [[Definition:Foot of Perpendicular|foot]] $G$.
Let $K$ be the [[Definition:Intersection (Geometry)|intersection]] of $IE$ and $AC$.
Let $L$ be the [[Definition:Intersection (Geometry)|intersection]] of $EH$ and $AC$.
:[[... | Westwood's Puzzle/Proof 1 | https://proofwiki.org/wiki/Westwood's_Puzzle | https://proofwiki.org/wiki/Westwood's_Puzzle/Proof_1 | [
"Westwood's Puzzle",
"Incircles of Triangles",
"Euclidean Geometry"
] | [
"File:WestwoodsPuzzle.png",
"Definition:Quadrilateral/Rectangle",
"Definition:Diameter of Quadrilateral",
"Definition:Incircle of Triangle",
"Definition:Circle",
"Definition:Triangle (Geometry)",
"Definition:Right Angle/Perpendicular",
"Definition:Incircle of Triangle/Incenter",
"Definition:Polygon/... | [
"Perpendicular through Given Point",
"Definition:Right Angle/Perpendicular/Foot",
"Definition:Intersection (Geometry)",
"Definition:Intersection (Geometry)",
"File:Westwood's Puzzle Proof.png",
"Two Straight Lines make Equal Opposite Angles",
"Radius at Right Angle to Tangent",
"Opposite Sides and Ang... |
proofwiki-10756 | Westwood's Puzzle | :500px
Take any rectangle $ABCD$ and draw the diagonal $AC$.
Inscribe a circle $GFJ$ in one of the resulting triangles $\triangle ABC$.
Drop perpendiculars $IEF$ and $HEJ$ from the center of this incircle $E$ to the sides of the rectangle.
Then the area of the rectangle $DHEI$ equals half the area of the rectangle $ABC... | The crucial geometric truth to note is that:
:$CJ = CG, AG = AF, BF = BJ$
This follows from the fact that:
:$\triangle CEJ \cong \triangle CEG$, $\triangle AEF \cong \triangle AEG$ and $\triangle BEF \cong \triangle BEJ$
This is a direct consequence of the point $E$ being the center of the incircle of $\triangle ABC$.
... | :[[File:WestwoodsPuzzle.png|500px]]
Take any [[Definition:Rectangle|rectangle]] $ABCD$ and draw the [[Definition:Diagonal of Quadrilateral|diagonal]] $AC$.
[[Definition:Incircle of Triangle|Inscribe]] a [[Definition:Circle|circle]] $GFJ$ in one of the resulting [[Definition:Triangle (Geometry)|triangles]] $\triangle ... | The crucial geometric truth to note is that:
:$CJ = CG, AG = AF, BF = BJ$
This follows from the fact that:
:$\triangle CEJ \cong \triangle CEG$, $\triangle AEF \cong \triangle AEG$ and $\triangle BEF \cong \triangle BEJ$
This is a direct consequence of the point $E$ being the [[Definition:Center of Circle|center]] of... | Westwood's Puzzle/Proof 2 | https://proofwiki.org/wiki/Westwood's_Puzzle | https://proofwiki.org/wiki/Westwood's_Puzzle/Proof_2 | [
"Westwood's Puzzle",
"Incircles of Triangles",
"Euclidean Geometry"
] | [
"File:WestwoodsPuzzle.png",
"Definition:Quadrilateral/Rectangle",
"Definition:Diameter of Quadrilateral",
"Definition:Incircle of Triangle",
"Definition:Circle",
"Definition:Triangle (Geometry)",
"Definition:Right Angle/Perpendicular",
"Definition:Incircle of Triangle/Incenter",
"Definition:Polygon/... | [
"Definition:Circle/Center",
"Definition:Incircle of Triangle",
"Pythagoras's Theorem"
] |
proofwiki-10757 | Vector Cross Product Operator is Bilinear | Let $\mathbf u$, $\mathbf v$ and $\mathbf w$ be vectors in a vector space $\mathbf V$ of $3$ dimensions:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = u_i \mathbf i + u_j \mathbf j + u_k \mathbf k
}}
{{eqn | l = \mathbf v
| r = v_i \mathbf i + v_j \mathbf j + v_k \mathbf k
}}
{{eqn | l = \mathbf w
| r = w... | {{begin-eqn}}
{{eqn | l = \left({c \mathbf u + \mathbf v}\right) \times \mathbf w
| r = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ c u_i + v_i & c u_j + v_j & c u_k + v_k \\ w_i & w_j & w_k \end{vmatrix}
| c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = \begin{vmatrix} \mathbf i & \mathbf j & \m... | Let $\mathbf u$, $\mathbf v$ and $\mathbf w$ be [[Definition:Vector (Linear Algebra)|vectors]] in a [[Definition:Vector Space|vector space]] $\mathbf V$ of [[Definition:Dimension of Vector Space|$3$ dimensions]]:
{{begin-eqn}}
{{eqn | l = \mathbf u
| r = u_i \mathbf i + u_j \mathbf j + u_k \mathbf k
}}
{{eqn | l... | {{begin-eqn}}
{{eqn | l = \left({c \mathbf u + \mathbf v}\right) \times \mathbf w
| r = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ c u_i + v_i & c u_j + v_j & c u_k + v_k \\ w_i & w_j & w_k \end{vmatrix}
| c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = \begin{vmatrix} \mathbf i & \mathbf j & \m... | Vector Cross Product Operator is Bilinear | https://proofwiki.org/wiki/Vector_Cross_Product_Operator_is_Bilinear | https://proofwiki.org/wiki/Vector_Cross_Product_Operator_is_Bilinear | [
"Vector Cross Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Real Number"
] | [
"Determinant as Sum of Determinants",
"Determinant with Row Multiplied by Constant",
"Category:Vector Cross Product"
] |
proofwiki-10758 | Natural Numbers have No Proper Zero Divisors | Let $\N$ be the natural numbers.
Then for all $m, n \in \N$:
:$m \times n = 0 \iff m = 0 \lor n = 0$
That is, $\N$ has no proper zero divisors. | === Necessary Condition ===
Suppose that $n = 0$ or $m = 0$.
Then from Zero is Zero Element for Natural Number Multiplication:
:$m \times n = 0$
{{qed|lemma}} | Let $\N$ be the [[Definition:Natural Numbers|natural numbers]].
Then for all $m, n \in \N$:
:$m \times n = 0 \iff m = 0 \lor n = 0$
That is, $\N$ has no [[Definition:Proper Zero Divisor|proper zero divisors]]. | === Necessary Condition ===
Suppose that $n = 0$ or $m = 0$.
Then from [[Zero is Zero Element for Natural Number Multiplication]]:
:$m \times n = 0$
{{qed|lemma}} | Natural Numbers have No Proper Zero Divisors | https://proofwiki.org/wiki/Natural_Numbers_have_No_Proper_Zero_Divisors | https://proofwiki.org/wiki/Natural_Numbers_have_No_Proper_Zero_Divisors | [
"Natural Numbers"
] | [
"Definition:Natural Numbers",
"Definition:Proper Zero Divisor"
] | [
"Zero is Zero Element for Natural Number Multiplication"
] |
proofwiki-10759 | Diagonals of Rhombus Bisect Angles | Let $OABC$ be a rhombus.
Then:
:$(1): \quad OB$ bisects $\angle AOC$ and $\angle ABC$
:$(2): \quad AC$ bisects $\angle OAB$ and $\angle OCB$
400px | {{WLOG}}, we will only prove $OB$ bisects $\angle AOC$.
We have:
{{begin-eqn}}
{{eqn | l = OA
| r = OC
| c = {{Defof|Rhombus}}
}}
{{eqn | l = BA
| r = BC
| c = {{Defof|Rhombus}}
}}
{{eqn | l = OB
| r = OB
| c = Common Side
}}
{{eqn | ll = \leadsto
| l = \triangle OAB
| o ... | Let $OABC$ be a [[Definition:Rhombus|rhombus]].
Then:
:$(1): \quad OB$ [[Definition:Angle Bisector|bisects]] $\angle AOC$ and $\angle ABC$
:$(2): \quad AC$ [[Definition:Angle Bisector|bisects]] $\angle OAB$ and $\angle OCB$
[[File:RhombusBisectAngles.png|400px]] | {{WLOG}}, we will only prove $OB$ [[Definition:Angle Bisector|bisects]] $\angle AOC$.
We have:
{{begin-eqn}}
{{eqn | l = OA
| r = OC
| c = {{Defof|Rhombus}}
}}
{{eqn | l = BA
| r = BC
| c = {{Defof|Rhombus}}
}}
{{eqn | l = OB
| r = OB
| c = Common Side
}}
{{eqn | ll = \leadsto
... | Diagonals of Rhombus Bisect Angles/Proof 1 | https://proofwiki.org/wiki/Diagonals_of_Rhombus_Bisect_Angles | https://proofwiki.org/wiki/Diagonals_of_Rhombus_Bisect_Angles/Proof_1 | [
"Euclidean Geometry",
"Vector Algebra",
"Parallelograms",
"Diagonals of Rhombus Bisect Angles"
] | [
"Definition:Quadrilateral/Rhombus",
"Definition:Angle Bisector",
"Definition:Angle Bisector",
"File:RhombusBisectAngles.png"
] | [
"Definition:Angle Bisector",
"Triangle Side-Side-Side Congruence",
"Definition:Angle Bisector"
] |
proofwiki-10760 | Diagonals of Rhombus Bisect Angles | Let $OABC$ be a rhombus.
Then:
:$(1): \quad OB$ bisects $\angle AOC$ and $\angle ABC$
:$(2): \quad AC$ bisects $\angle OAB$ and $\angle OCB$
400px | {{WLOG}}, we will only prove $OB$ bisects $\angle AOC$.
Let the position vectors of $A$, $B$ and $C$ with respect to $O$ be $\mathbf a$, $\mathbf b$ and $\mathbf c$ respectively.
By definition of rhombus, we have:
{{begin-eqn}}
{{eqn | n = a
| l = \mathbf a + \mathbf c
| r = \mathbf b
| c = Parallelog... | Let $OABC$ be a [[Definition:Rhombus|rhombus]].
Then:
:$(1): \quad OB$ [[Definition:Angle Bisector|bisects]] $\angle AOC$ and $\angle ABC$
:$(2): \quad AC$ [[Definition:Angle Bisector|bisects]] $\angle OAB$ and $\angle OCB$
[[File:RhombusBisectAngles.png|400px]] | {{WLOG}}, we will only prove $OB$ [[Definition:Angle Bisector|bisects]] $\angle AOC$.
Let the [[Definition:Position Vector|position vectors]] of $A$, $B$ and $C$ with respect to $O$ be $\mathbf a$, $\mathbf b$ and $\mathbf c$ respectively.
By definition of [[Definition:Rhombus|rhombus]], we have:
{{begin-eqn}}
{{eqn ... | Diagonals of Rhombus Bisect Angles/Proof 2 | https://proofwiki.org/wiki/Diagonals_of_Rhombus_Bisect_Angles | https://proofwiki.org/wiki/Diagonals_of_Rhombus_Bisect_Angles/Proof_2 | [
"Euclidean Geometry",
"Vector Algebra",
"Parallelograms",
"Diagonals of Rhombus Bisect Angles"
] | [
"Definition:Quadrilateral/Rhombus",
"Definition:Angle Bisector",
"Definition:Angle Bisector",
"File:RhombusBisectAngles.png"
] | [
"Definition:Angle Bisector",
"Definition:Position Vector",
"Definition:Quadrilateral/Rhombus",
"Parallelogram Law",
"Dot Product Distributes over Addition",
"Dot Product of Vector with Itself",
"Dot Product of Vector with Itself",
"Dot Product Distributes over Addition",
"Definition:Dot Product/Real... |
proofwiki-10761 | Summation Formula for Polygonal Numbers | Let $\map P {k, n}$ be the $n$th $k$-gonal number.
Then:
:$\ds \map P {k, n} = \sum_{j \mathop = 1}^n \paren {\paren {k - 2} \paren {j - 1} + 1}$ | We have that:
$\map P {k, n} = \begin{cases}
0 & : n = 0 \\
\map P {k, n - 1} + \paren {k - 2} \paren {n - 1} + 1 & : n > 0
\end{cases}$
Proof by induction:
For all $n \in \N_{>0}$, let $\map \Pi n$ be the proposition:
:$\ds \map P {k, n} = \sum_{j \mathop = 1}^n \paren {\paren {k - 2} \paren {j - 1} + 1}$ | Let $\map P {k, n}$ be the $n$th [[Definition:Polygonal Number|$k$-gonal number]].
Then:
:$\ds \map P {k, n} = \sum_{j \mathop = 1}^n \paren {\paren {k - 2} \paren {j - 1} + 1}$ | We have that:
$\map P {k, n} = \begin{cases}
0 & : n = 0 \\
\map P {k, n - 1} + \paren {k - 2} \paren {n - 1} + 1 & : n > 0
\end{cases}$
Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map \Pi n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map P {k, n} = \sum_{j ... | Summation Formula for Polygonal Numbers | https://proofwiki.org/wiki/Summation_Formula_for_Polygonal_Numbers | https://proofwiki.org/wiki/Summation_Formula_for_Polygonal_Numbers | [
"Polygonal Numbers",
"Proofs by Induction"
] | [
"Definition:Polygonal Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10762 | Relative Prime Modulo Tensor is Zero | Let $p \in \Z_{>0}$ and $q \in \Z_{>0}$ be positive coprime integers.
Let $\Z / p \Z$ and $\Z / q \Z$ be $\Z$-modules.
{{explain|It is not a good idea to use the same notation for both a ring and a module. Either $\Z / p \Z$ is a ring or it is a module. Please consider taking the advice in the explain template at the b... | By Bézout's Identity there exists $a, b \in \Z$ such that $a p + b q = 1$.
Then for $s \otimes_\Z t \in \Z / p \Z \otimes \Z / q \Z$:
{{begin-eqn}}
{{eqn | l = s \otimes t
| r = (s \left({a p + b q}\right)) \otimes t
| c = $s = s \cdot 1$
}}
{{eqn | r = (s a p + s b q) \otimes t
| c = {{Module-axiom|2... | Let $p \in \Z_{>0}$ and $q \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|positive]] [[Definition:Coprime Integers|coprime integers]].
Let [[Definition:Ring of Integers Modulo m|$\Z / p \Z$]] and $\Z / q \Z$ be [[Definition:Module over Ring|$\Z$-modules]].
{{explain|It is not a good idea to use the same notat... | By [[Bézout's Identity]] there exists $a, b \in \Z$ such that $a p + b q = 1$.
Then for $s \otimes_\Z t \in \Z / p \Z \otimes \Z / q \Z$:
{{begin-eqn}}
{{eqn | l = s \otimes t
| r = (s \left({a p + b q}\right)) \otimes t
| c = $s = s \cdot 1$
}}
{{eqn | r = (s a p + s b q) \otimes t
| c = {{Module-... | Relative Prime Modulo Tensor is Zero | https://proofwiki.org/wiki/Relative_Prime_Modulo_Tensor_is_Zero | https://proofwiki.org/wiki/Relative_Prime_Modulo_Tensor_is_Zero | [
"Tensor Algebra"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Ring of Integers Modulo m",
"Definition:Module over Ring",
"Definition:Tensor Product of Modules"
] | [
"Bézout's Identity",
"Definition:Tensor Product of Modules",
"Definition:Tensor Product of Modules",
"Tensor with Zero Element is Zero in Tensor"
] |
proofwiki-10763 | Tensor Product is Module | Let $R$ be a ring.
Let $M$ be a $R$-right module.
Let $N$ be a $R$-left module.
Then:
:$\ds T = \bigoplus_{s \mathop \in M \times N} R s$
is a left module. | === Axiom 1 ===
Let $x, y \in T$ with $x = \family {s_i}_{i \mathop \in I}$ and $y = (t_i)_{i \mathop \in I}$.
Let $\lambda\in R$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \circ \paren {x + y}
| r = \lambda \circ (\family {s_i}_{i \mathop \in I} + \family {t_i}_{i \mathop \in I})
| c = Definition of elements... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $M$ be a $R$-[[Definition:Right Module over Ring|right module]].
Let $N$ be a $R$-[[Definition:Left Module over Ring|left module]].
Then:
:$\ds T = \bigoplus_{s \mathop \in M \times N} R s$
is a [[Definition:Left Module over Ring|left module]]. | === Axiom 1 ===
Let $x, y \in T$ with [[Definition:Module Direct Product|$x = \family {s_i}_{i \mathop \in I}$ and $y = (t_i)_{i \mathop \in I}$.]]
Let $\lambda\in R$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \circ \paren {x + y}
| r = \lambda \circ (\family {s_i}_{i \mathop \in I} + \family {t_i}_{i \mathop \... | Tensor Product is Module | https://proofwiki.org/wiki/Tensor_Product_is_Module | https://proofwiki.org/wiki/Tensor_Product_is_Module | [
"Tensor Algebra",
"Module Theory"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Right Module over Ring",
"Definition:Left Module over Ring",
"Definition:Left Module over Ring"
] | [
"Definition:Module Direct Product",
"Definition:Direct Sum of Modules",
"Definition:Direct Sum of Modules",
"Definition:Direct Sum of Modules",
"Definition:Direct Sum of Modules",
"Definition:Direct Sum of Modules",
"Definition:Direct Sum of Modules",
"Definition:Direct Sum of Modules"
] |
proofwiki-10764 | Supremum of Subset of Real Numbers is Arbitrarily Close | Let $A \subseteq \R$ be a subset of the real numbers.
Let $b$ be a supremum of $A$.
Let $\epsilon \in \R_{>0}$.
Then:
:$\exists x \in A: b − x < \epsilon$ | Note that $A$ is non-empty as the empty set does not admit a supremum (in $\R$).
Suppose $\epsilon \in \R_{>0}$ such that:
:$\forall x \in A: b − x \ge \epsilon$
Then:
:$\forall x \in A: b − \epsilon \ge x$
and so $b − \epsilon$ would be an upper bound of $A$ which is less than $b$.
But since $b$ is a supremum of $A$ t... | Let $A \subseteq \R$ be a [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]].
Let $b$ be a [[Definition:Supremum of Subset of Real Numbers|supremum]] of $A$.
Let $\epsilon \in \R_{>0}$.
Then:
:$\exists x \in A: b − x < \epsilon$ | Note that $A$ is [[Definition:Non-Empty Set|non-empty]] as the [[Definition:Empty Set|empty set]] does not admit a [[Definition:Supremum of Subset of Real Numbers|supremum]] (in $\R$).
Suppose $\epsilon \in \R_{>0}$ such that:
:$\forall x \in A: b − x \ge \epsilon$
Then:
:$\forall x \in A: b − \epsilon \ge x$
and so... | Supremum of Subset of Real Numbers is Arbitrarily Close | https://proofwiki.org/wiki/Supremum_of_Subset_of_Real_Numbers_is_Arbitrarily_Close | https://proofwiki.org/wiki/Supremum_of_Subset_of_Real_Numbers_is_Arbitrarily_Close | [
"Real Analysis"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers"
] | [
"Definition:Non-Empty Set",
"Definition:Empty Set",
"Definition:Supremum of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Proof by Contradiction"
] |
proofwiki-10765 | Tensor with Zero Element is Zero in Tensor | Let $R$ be a ring.
Let $M$ be a right $R$-module.
Let $N$ be a left $R$-module.
Let $M \otimes_R N$ denote their tensor product.
Then:
:$0\otimes_R n = m \otimes_R 0 = 0 \otimes_R 0$
is the zero in $M \otimes_R N$. | Let $m \in M$ and $n \in N$
Then
{{begin-eqn}}
{{eqn | l = m \otimes_R n
| r = \paren {m + 0} \otimes_R n
| c = {{Group-axiom|2}}
}}
{{eqn | r = m \otimes_R n + 0 \otimes_R n
| c = {{Defof|Tensor Equality}}
}}
{{eqn | r = m \otimes_R \paren {n + 0}
| c = {{Group-axiom|2}}
}}
{{eqn | r = m \otime... | Let $R$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $M$ be a [[Definition:Right Module over Ring|right $R$-module]].
Let $N$ be a [[Definition:Left Module over Ring|left $R$-module]].
Let $M \otimes_R N$ denote their [[Definition:Tensor Product of Modules|tensor product]].
Then:
:$0\otimes_R n = m \otim... | Let $m \in M$ and $n \in N$
Then
{{begin-eqn}}
{{eqn | l = m \otimes_R n
| r = \paren {m + 0} \otimes_R n
| c = {{Group-axiom|2}}
}}
{{eqn | r = m \otimes_R n + 0 \otimes_R n
| c = {{Defof|Tensor Equality}}
}}
{{eqn | r = m \otimes_R \paren {n + 0}
| c = {{Group-axiom|2}}
}}
{{eqn | r = m \oti... | Tensor with Zero Element is Zero in Tensor | https://proofwiki.org/wiki/Tensor_with_Zero_Element_is_Zero_in_Tensor | https://proofwiki.org/wiki/Tensor_with_Zero_Element_is_Zero_in_Tensor | [
"Tensor Algebra",
"Homological Algebra"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Right Module over Ring",
"Definition:Left Module over Ring",
"Definition:Tensor Product of Modules",
"Definition:Zero of Tensor Product"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Left Module over Ring",
"Category:Tensor Algebra",
"Category:Homological Algebra"
] |
proofwiki-10766 | Primitive of Arcsecant of x over a/Formulation 1 | :$\ds \int \arcsec \frac x a \rd x = \begin {cases}
x \arcsec \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
x \arcsec \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end {cases}$ | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begin {cases} \dfrac a {x \sqrt {x^2 - a^... | :$\ds \int \arcsec \frac x a \rd x = \begin {cases}
x \arcsec \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\
x \arcsec \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\
\end {cases}$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \begi... | Primitive of Arcsecant of x over a/Formulation 1 | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a/Formulation_1 | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a/Formulation_1 | [
"Primitives involving Inverse Secant Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Constant",
"Definition:Real Interval/Open",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form",
"Definition:... |
proofwiki-10767 | Primitive of Arcsecant of x over a/Formulation 2 | :$\ds \int \arcsec \frac x a \rd x = x \arcsec \frac x a - a \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $x^2 > 1$.
$\arcsec \dfrac x a$ is undefined on the real numbers for $x^2 < 1$. | With a view to expressing the primitive in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \dfrac a {\size x \sqrt {x^2 - a^2} }
... | :$\ds \int \arcsec \frac x a \rd x = x \arcsec \frac x a - a \ln \size {x + \sqrt {x^2 - a^2} } + C$
for $x^2 > 1$.
$\arcsec \dfrac x a$ is undefined on the [[Definition:Real Numbers|real numbers]] for $x^2 < 1$. | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
{{begin-eqn}}
{{eqn | l = u
| r = \arcsec \frac x a
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d u} {\d x}
| r = \dfra... | Primitive of Arcsecant of x over a/Formulation 2 | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a/Formulation_2 | https://proofwiki.org/wiki/Primitive_of_Arcsecant_of_x_over_a/Formulation_2 | [
"Primitives involving Inverse Secant Function"
] | [
"Definition:Real Number"
] | [
"Definition:Primitive (Calculus)",
"Derivative of Arcsecant Function/Corollary 1",
"Primitive of Constant",
"Integration by Parts",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form",
"Definition:Natural Logarithm",
"Definition:S... |
proofwiki-10768 | Cross Product of Vector with Itself is Zero | Let $\mathbf x$ be a vector in a vector space of $3$ dimensions:
:$\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$
Then:
:$\mathbf x \times \mathbf x = \mathbf 0$
where $\times$ denotes vector cross product. | {{begin-eqn}}
{{eqn | l = \mathbf x \times \mathbf x
| r = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ x_i & x_j & x_k \\ x_i & x_j & x_k \end {vmatrix}
| c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = \mathbf 0
| c = Square Matrix with Duplicate Rows has Zero Determinant
}}
{{end-eqn}}
{... | Let $\mathbf x$ be a [[Definition:Vector (Linear Algebra)|vector]] in a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]]:
:$\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$
Then:
:$\mathbf x \times \mathbf x = \mathbf 0$
where $\times$ denotes [[Definition:... | {{begin-eqn}}
{{eqn | l = \mathbf x \times \mathbf x
| r = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ x_i & x_j & x_k \\ x_i & x_j & x_k \end {vmatrix}
| c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = \mathbf 0
| c = [[Square Matrix with Duplicate Rows has Zero Determinant]]
}}
{{end-eqn... | Cross Product of Vector with Itself is Zero/Proof 1 | https://proofwiki.org/wiki/Cross_Product_of_Vector_with_Itself_is_Zero | https://proofwiki.org/wiki/Cross_Product_of_Vector_with_Itself_is_Zero/Proof_1 | [
"Cross Product of Vector with Itself is Zero",
"Vector Cross Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Cross Product"
] | [
"Square Matrix with Duplicate Rows has Zero Determinant"
] |
proofwiki-10769 | Cross Product of Vector with Itself is Zero | Let $\mathbf x$ be a vector in a vector space of $3$ dimensions:
:$\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$
Then:
:$\mathbf x \times \mathbf x = \mathbf 0$
where $\times$ denotes vector cross product. | By definition, a vector is parallel to itself.
The result follows from Cross Product of Parallel Vectors.
{{qed}} | Let $\mathbf x$ be a [[Definition:Vector (Linear Algebra)|vector]] in a [[Definition:Vector Space|vector space]] of [[Definition:Dimension of Vector Space|$3$ dimensions]]:
:$\mathbf x = x_i \mathbf i + x_j \mathbf j + x_k \mathbf k$
Then:
:$\mathbf x \times \mathbf x = \mathbf 0$
where $\times$ denotes [[Definition:... | By definition, a [[Definition:Vector (Linear Algebra)|vector]] is [[Definition:Parallel Lines|parallel]] to itself.
The result follows from [[Cross Product of Parallel Vectors]].
{{qed}} | Cross Product of Vector with Itself is Zero/Proof 2 | https://proofwiki.org/wiki/Cross_Product_of_Vector_with_Itself_is_Zero | https://proofwiki.org/wiki/Cross_Product_of_Vector_with_Itself_is_Zero/Proof_2 | [
"Cross Product of Vector with Itself is Zero",
"Vector Cross Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Vector Cross Product"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Parallel (Geometry)/Lines",
"Cross Product of Parallel Vectors"
] |
proofwiki-10770 | Reciprocal of Riemann Zeta Function | For $\map \Re z > 1$:
:$\ds \frac 1 {\map \zeta z} = \sum_{k \mathop = 1}^\infty \frac{\mu \left({k}\right)} {k^z}$
where:
:$\zeta$ is the Riemann zeta function
:$\mu$ is the Möbius function. | By Sum of Reciprocals of Powers as Euler Product:
{{begin-eqn}}
{{eqn | l = \frac 1 {\map \zeta z}
| r = \prod_{\text {$p$ prime} } \paren {1 - p^{-z} }
| c =
}}
{{eqn | r = \paren {1 - \frac 1 {2^z} } \paren {1 - \frac 1 {3^z} } \paren {1 - \frac 1 {5^z} } \paren {1 - \frac 1 {7^z} } \paren {1 - \frac 1 {... | For $\map \Re z > 1$:
:$\ds \frac 1 {\map \zeta z} = \sum_{k \mathop = 1}^\infty \frac{\mu \left({k}\right)} {k^z}$
where:
:$\zeta$ is the [[Definition:Riemann Zeta Function|Riemann zeta function]]
:$\mu$ is the [[Definition:Möbius Function|Möbius function]]. | By [[Sum of Reciprocals of Powers as Euler Product]]:
{{begin-eqn}}
{{eqn | l = \frac 1 {\map \zeta z}
| r = \prod_{\text {$p$ prime} } \paren {1 - p^{-z} }
| c =
}}
{{eqn | r = \paren {1 - \frac 1 {2^z} } \paren {1 - \frac 1 {3^z} } \paren {1 - \frac 1 {5^z} } \paren {1 - \frac 1 {7^z} } \paren {1 - \fra... | Reciprocal of Riemann Zeta Function | https://proofwiki.org/wiki/Reciprocal_of_Riemann_Zeta_Function | https://proofwiki.org/wiki/Reciprocal_of_Riemann_Zeta_Function | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Möbius Function"
] | [
"Sum of Reciprocals of Powers as Euler Product",
"Dirichlet Series of Inverse of Arithmetic Function",
"Category:Riemann Zeta Function"
] |
proofwiki-10771 | Condition for Agreement of Family of Mappings | Let $\family {A_i}_{i \mathop \in I}, \family {B_i}_{i \mathop \in I}$ be families of non empty sets.
Let $\family {f_i}_{i \mathop \in I}$ be a family of mappings such that:
:$\forall i \in I: f_i \in \map \FF {A_i, B_i}$
{{explain|Clarify: what is $\map \FF {A_i, B_i}$? From the context it can be understood as being ... | === Sufficient Condition ===
Let:
:$\ds \bigcup_{i \mathop \in I} f_i \in \map \FF {\bigcup_{i \mathop \in I} A_i, \bigcup_{i \mathop \in I} B_i}$
Let $i, j \in I$ be such that:
:$\Dom {f_i} \cap \Dom {f_j} \ne \O$
Let $a \in \paren {\Dom {f_i} \cap \Dom {f_j} }$.
Let $\ds b \in \bigcup_{i \mathop \in I} B_i$ be such t... | Let $\family {A_i}_{i \mathop \in I}, \family {B_i}_{i \mathop \in I}$ be [[Definition:Indexed Family of Sets|families]] of [[Definition:Non-Empty Set|non empty sets]].
Let $\family {f_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Mapping|mappings]] such that:
:$\forall i \in I: f_i \... | === Sufficient Condition ===
Let:
:$\ds \bigcup_{i \mathop \in I} f_i \in \map \FF {\bigcup_{i \mathop \in I} A_i, \bigcup_{i \mathop \in I} B_i}$
Let $i, j \in I$ be such that:
:$\Dom {f_i} \cap \Dom {f_j} \ne \O$
Let $a \in \paren {\Dom {f_i} \cap \Dom {f_j} }$.
Let $\ds b \in \bigcup_{i \mathop \in I} B_i$ be su... | Condition for Agreement of Family of Mappings | https://proofwiki.org/wiki/Condition_for_Agreement_of_Family_of_Mappings | https://proofwiki.org/wiki/Condition_for_Agreement_of_Family_of_Mappings | [
"Mapping Theory",
"Set Union"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Non-Empty Set",
"Definition:Indexing Set/Family",
"Definition:Mapping",
"Definition:Mapping"
] | [
"Definition:Mapping",
"Definition:Contradiction",
"Definition:Contradiction",
"Definition:Mapping"
] |
proofwiki-10772 | Supremum of Set of Real Numbers is at least Supremum of Subset | Let $S$ be a set of real numbers.
Let $S$ have a supremum.
Let $T$ be a non-empty subset of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | The number $\sup S$ is an upper bound for $S$.
Therefore, $\sup S$ is an upper bound for $T$ as $T$ is a non-empty subset of $S$.
Accordingly, $T$ has a supremum by the Continuum Property.
The number $\sup S$ is an upper bound for $T$.
Therefore, $\sup S$ is greater than or equal to $\sup T$ as $\sup T$ is the least up... | Let $S$ be a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $S$ have a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Let $T$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | The [[Definition:Real Number|number]] $\sup S$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for $S$.
Therefore, $\sup S$ is an [[Definition:Upper Bound of Subset of Real Numbers|upper bound]] for $T$ as $T$ is a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Acco... | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 1 | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_1 | [
"Real Analysis",
"Supremum of Set of Real Numbers is at least Supremum of Subset"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Subset"
] | [
"Definition:Real Number",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Supremum of Set/Real Numbers",
"Continuum Property",
"Definition:Real Number",
"Definition:Upper Bound of Set/Real Number... |
proofwiki-10773 | Supremum of Set of Real Numbers is at least Supremum of Subset | Let $S$ be a set of real numbers.
Let $S$ have a supremum.
Let $T$ be a non-empty subset of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | By the Continuum Property, $T$ admits a supremum.
It follows from Supremum of Subset that $\sup T \le \sup S$.
{{qed}} | Let $S$ be a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $S$ have a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Let $T$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | By the [[Continuum Property]], $T$ admits a [[Definition:Supremum of Subset of Real Numbers|supremum]].
It follows from [[Supremum of Subset]] that $\sup T \le \sup S$.
{{qed}} | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 2 | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_2 | [
"Real Analysis",
"Supremum of Set of Real Numbers is at least Supremum of Subset"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Subset"
] | [
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"Supremum of Subset"
] |
proofwiki-10774 | Supremum of Set of Real Numbers is at least Supremum of Subset | Let $S$ be a set of real numbers.
Let $S$ have a supremum.
Let $T$ be a non-empty subset of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | $S$ is bounded above as $S$ has a supremum.
Therefore, $T$ is bounded above as $T$ is a subset of $S$.
Accordingly, $T$ admits a supremum by the Continuum Property as $T$ is non-empty.
We know that $\sup T$ and $\sup S$ exist.
Therefore by Suprema of two Real Sets:
:$\forall \epsilon \in \R_{>0}: \forall t \in T: \exis... | Let $S$ be a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $S$ have a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Let $T$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | $S$ is [[Definition:Bounded Above Set|bounded above]] as $S$ has a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Therefore, $T$ is [[Definition:Bounded Above Set|bounded above]] as $T$ is a [[Definition:Subset|subset]] of $S$.
Accordingly, $T$ admits a [[Definition:Supremum of Subset of Real Numbers|sup... | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 3 | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_3 | [
"Real Analysis",
"Supremum of Set of Real Numbers is at least Supremum of Subset"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Subset"
] | [
"Definition:Bounded Above Set",
"Definition:Supremum of Set/Real Numbers",
"Definition:Bounded Above Set",
"Definition:Subset",
"Definition:Supremum of Set/Real Numbers",
"Continuum Property",
"Definition:Non-Empty Set",
"Suprema of two Real Sets"
] |
proofwiki-10775 | Supremum of Set of Real Numbers is at least Supremum of Subset | Let $S$ be a set of real numbers.
Let $S$ have a supremum.
Let $T$ be a non-empty subset of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | By definition $\sup S$ is an upper bound for $S$.
Thus:
:$\forall x \in S: x \le \sup S$
As $T \subseteq S$ we have by definition of subset that:
:$\forall x \in T: x \in S$
Hence:
:$\forall x \in T: x \le \sup S$
So by definition $\sup S$ is an upper bound for $T$.
So $\sup S$ is at least as big as the smallest upper ... | Let $S$ be a [[Definition:Set|set]] of [[Definition:Real Number|real numbers]].
Let $S$ have a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Let $T$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $S$.
Then $\sup T$ exists and:
:$\sup T \le \sup S$ | By definition $\sup S$ is an [[Definition:Upper Bound|upper bound]] for $S$.
Thus:
:$\forall x \in S: x \le \sup S$
As $T \subseteq S$ we have by definition of [[Definition:Subset|subset]] that:
:$\forall x \in T: x \in S$
Hence:
:$\forall x \in T: x \le \sup S$
So by definition $\sup S$ is an [[Definition:Upper Bo... | Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 4 | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset | https://proofwiki.org/wiki/Supremum_of_Set_of_Real_Numbers_is_at_least_Supremum_of_Subset/Proof_4 | [
"Real Analysis",
"Supremum of Set of Real Numbers is at least Supremum of Subset"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Subset"
] | [
"Definition:Upper Bound",
"Definition:Subset",
"Definition:Upper Bound",
"Definition:Smallest Element",
"Definition:Upper Bound",
"Definition:Supremum"
] |
proofwiki-10776 | Supremum of Subset of Union Equals Supremum of Union | Let $S$ be a non-empty real set.
Let $S$ have a supremum.
Let $\set {S_i: i \in \set {1, 2, \ldots, n} }$, $n \in \N_{>0}$, be a set of non-empty subsets of $S$.
Let $\bigcup S_i = S$.
Then there exists a $j$ in $\set {1, 2, \ldots, n}$ such that:
:$\sup S_j = \sup S$ | If $S$ equals $S_j$ for a $j$ in $\set {1, 2, \ldots, n}$, it is trivially true that $\sup S = \sup S_j$.
Now assume that $S$ is unequal to $S_i$ for every $i$ in $\left\{{1, 2, \ldots, n}\right\}$.
By Supremum of Set of Real Numbers is at least Supremum of Subset, $\sup S \ge \sup S_i$ for every $i$ in $\set{1, 2, \ld... | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Real Number|real set]].
Let $S$ have a [[Definition:Supremum of Subset of Real Numbers|supremum]].
Let $\set {S_i: i \in \set {1, 2, \ldots, n} }$, $n \in \N_{>0}$, be a set of [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subsets]] of $S$.... | If $S$ equals $S_j$ for a $j$ in $\set {1, 2, \ldots, n}$, it is trivially true that $\sup S = \sup S_j$.
Now assume that $S$ is unequal to $S_i$ for every $i$ in $\left\{{1, 2, \ldots, n}\right\}$.
By [[Supremum of Set of Real Numbers is at least Supremum of Subset]], $\sup S \ge \sup S_i$ for every $i$ in $\set{1,... | Supremum of Subset of Union Equals Supremum of Union | https://proofwiki.org/wiki/Supremum_of_Subset_of_Union_Equals_Supremum_of_Union | https://proofwiki.org/wiki/Supremum_of_Subset_of_Union_Equals_Supremum_of_Union | [
"Real Analysis"
] | [
"Definition:Non-Empty Set",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Subset"
] | [
"Supremum of Set of Real Numbers is at least Supremum of Subset",
"Supremum of Subset of Real Numbers is Arbitrarily Close",
"Definition:Element",
"Definition:Element",
"Category:Real Analysis"
] |
proofwiki-10777 | Condition for Ideal to be Total Ring | Let $\struct {A, +, \circ}$ be a commutative ring with unity.
Let $I$ be an ideal of $A$ such that the quotient ring $A / I$ is a field.
Let $J$ be an ideal of $A$ such that $I \subsetneq J$.
Then:
:$A = J$ | Let $A$ be a commutative ring with unity.
Let $I$ be an ideal of $A$ such that the quotient ring $A / I$ is a field.
Let $J$ be an ideal of $A$ such that $I \subsetneq J$.
From Ideal is Subring:
:$J \subseteq A$
It remains to be proved that that $A \subseteq J$.
Let $a \in A$.
As $I \subsetneq J$, it follows from defin... | Let $\struct {A, +, \circ}$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $I$ be an [[Definition:Ideal of Ring|ideal]] of $A$ such that the [[Definition:Quotient Ring|quotient ring]] $A / I$ is a [[Definition:Field (Abstract Algebra)|field]].
Let $J$ be an [[Definition:Ideal of Ri... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $I$ be an [[Definition:Ideal of Ring|ideal]] of $A$ such that the [[Definition:Quotient Ring|quotient ring]] $A / I$ is a [[Definition:Field (Abstract Algebra)|field]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $A$ suc... | Condition for Ideal to be Total Ring | https://proofwiki.org/wiki/Condition_for_Ideal_to_be_Total_Ring | https://proofwiki.org/wiki/Condition_for_Ideal_to_be_Total_Ring | [
"Ideal Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Ideal of Ring"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Ideal of Ring",
"Ideal is Subring",
"Definition:Proper Subset",
"Definition:Coset",
"Definition:Field (Abstract Algebra)",
"Definition:Subset",
"D... |
proofwiki-10778 | Area between Two Non-Intersecting Chords | Let $AB$ and $CD$ be two chords of a circle whose center is at $O$ and whose radius is $r$.
:400px
:400px
Let $\alpha$ and $\theta$ be respectively the measures in radians of the angles $\angle COD$ and $\angle AOB$.
Then the area $\AA$ between the two chords is given by:
:$\AA = \dfrac {r^2} 2 \paren {\theta - \sin \t... | Let $\SS_\alpha$ be the area of the segment whose base subtends $\alpha$.
Let $\SS_\theta$ be the area of the segment whose base subtends $\theta$. | Let $AB$ and $CD$ be two [[Definition:Chord of Circle|chords]] of a [[Definition:Circle|circle]] whose [[Definition:Center of Circle|center]] is at $O$ and whose [[Definition:Radius of Circle|radius]] is $r$.
:[[File:Circle with chords and area.png|400px]]
:[[File:Circle with chords and area 3.png|400px]]
Let $\alpha... | Let $\SS_\alpha$ be the [[Definition:Area|area]] of the [[Definition:Segment of Circle|segment]] whose [[Definition:Base of Segment|base]] [[Definition:Subtend|subtends]] $\alpha$.
Let $\SS_\theta$ be the [[Definition:Area|area]] of the [[Definition:Segment of Circle|segment]] whose [[Definition:Base of Segment|base]]... | Area between Two Non-Intersecting Chords | https://proofwiki.org/wiki/Area_between_Two_Non-Intersecting_Chords | https://proofwiki.org/wiki/Area_between_Two_Non-Intersecting_Chords | [
"Circles"
] | [
"Definition:Circle/Chord",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"File:Circle with chords and area.png",
"File:Circle with chords and area 3.png",
"Definition:Angular Measure/Radian",
"Definition:Angle",
"Definition:Area",
"Definition:Circle/Chord",
"Defini... | [
"Definition:Area",
"Definition:Segment of Circle",
"Definition:Segment of Circle/Base",
"Definition:Subtend",
"Definition:Area",
"Definition:Segment of Circle",
"Definition:Segment of Circle/Base",
"Definition:Subtend",
"Definition:Area",
"Definition:Area",
"Definition:Area",
"Definition:Area"... |
proofwiki-10779 | Supremum of Absolute Value of Difference equals Supremum of Difference | Let $S$ be a non-empty real set.
Let $\ds \sup_{x, y \mathop \in S} \paren {x - y}$ exist.
Then $\ds \sup_{x, y \mathop \in S} \size {x - y}$ exists and:
:$\ds \sup_{x, y \mathop \in S} \size {x - y} = \sup_{x, y \mathop \in S} \paren {x - y}$ | Consider the set $\set {x - y: x, y \in S, x - y \le 0}$.
There is a number $x'$ in $S$ as $S$ is non-empty.
Therefore, $0 \in \set {x - y: x, y \in S, x - y \le 0}$ as $x = y = x'$ implies that $x - y = 0$, $x, y \in S$, and $x - y \le 0$.
Also, $0$ is an upper bound for $\set {x - y: x, y \in S, x - y \le 0}$ by defi... | Let $S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Real Number|real set]].
Let $\ds \sup_{x, y \mathop \in S} \paren {x - y}$ exist.
Then $\ds \sup_{x, y \mathop \in S} \size {x - y}$ exists and:
:$\ds \sup_{x, y \mathop \in S} \size {x - y} = \sup_{x, y \mathop \in S} \paren {x - y}$ | Consider the [[Definition:Set|set]] $\set {x - y: x, y \in S, x - y \le 0}$.
There is a [[Definition:Real Number|number]] $x'$ in $S$ as $S$ is [[Definition:Non-Empty Set|non-empty]].
Therefore, $0 \in \set {x - y: x, y \in S, x - y \le 0}$ as $x = y = x'$ implies that $x - y = 0$, $x, y \in S$, and $x - y \le 0$.
A... | Supremum of Absolute Value of Difference equals Supremum of Difference | https://proofwiki.org/wiki/Supremum_of_Absolute_Value_of_Difference_equals_Supremum_of_Difference | https://proofwiki.org/wiki/Supremum_of_Absolute_Value_of_Difference_equals_Supremum_of_Difference | [
"Suprema",
"Absolute Value Function"
] | [
"Definition:Non-Empty Set",
"Definition:Real Number"
] | [
"Definition:Set",
"Definition:Real Number",
"Definition:Non-Empty Set",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Non-Empty Set",
"Supremum of Set Equals Maximum of Suprema of Subsets",
"Supremum of Set Equals Maximum of Suprema of Subsets"... |
proofwiki-10780 | Supremum of Sum equals Sum of Suprema | Let $A$ and $B$ be non-empty sets of real numbers.
Let $A + B$ be $\set {x + y: x \in A, y \in B}$.
Let either $A$ and $B$ have suprema or $A + B$ have a supremum.
Then all $\sup A$, $\sup B$, and $\sup \paren {A + B}$ exist and:
:$\sup \paren {A + B} = \sup A + \sup B$ | Assume first that $A$ and $B$ have suprema.
We have:
:$x \le \sup A$ for an arbitrary $x$ in $A$
:$y \le \sup B$ for an arbitrary $y$ in $B$
Adding these inequalities, we get:
:$x + y \le \sup A + \sup B$
The number $x + y$ is an arbitrary element of $A + B$ as $x$ and $y$ are arbitrary elements of $A$ and $B$ respecti... | Let $A$ and $B$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]] of [[Definition:Real Number|real numbers]].
Let $A + B$ be $\set {x + y: x \in A, y \in B}$.
Let either $A$ and $B$ have [[Definition:Supremum of Subset of Real Numbers|suprema]] or $A + B$ have a [[Definition:Supremum of Subset of Real... | Assume first that $A$ and $B$ have [[Definition:Supremum of Subset of Real Numbers|suprema]].
We have:
:$x \le \sup A$ for an arbitrary $x$ in $A$
:$y \le \sup B$ for an arbitrary $y$ in $B$
Adding these inequalities, we get:
:$x + y \le \sup A + \sup B$
The number $x + y$ is an arbitrary [[Definition:Element|ele... | Supremum of Sum equals Sum of Suprema | https://proofwiki.org/wiki/Supremum_of_Sum_equals_Sum_of_Suprema | https://proofwiki.org/wiki/Supremum_of_Sum_equals_Sum_of_Suprema | [
"Real Analysis"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Real Number",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers"
] | [
"Definition:Supremum of Set/Real Numbers",
"Definition:Element",
"Definition:Element",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Supremum of Set/Real Numbers",
"Continuum Property",
"Definition:Supremum of Set/Real Numbers",
"... |
proofwiki-10781 | Edge is Bridge iff in All Spanning Trees | Let $G$ be a simple graph.
Let $e$ be an edge of $G$.
Then $e$ is a bridge in $G$ {{iff}} $e$ belongs to every spanning tree for $G$. | === Necessary Condition ===
Let $e$ be a bridge.
That is, suppose the edge deletion $G - e$ is disconnected.
Let $T$ be an arbitrary spanning tree for $G$.
By definition $T$ is a connected subgraph of $G$.
If $T$ did not contain $e$, then it would also be a subgraph of $G - e$.
This contradicts the fact that $G - e$ is... | Let $G$ be a [[Definition:Simple Graph|simple graph]].
Let $e$ be an [[Definition:Edge of Graph|edge]] of $G$.
Then $e$ is a [[Definition:Bridge (Graph Theory)|bridge]] in $G$ {{iff}} $e$ belongs to every [[Definition:Spanning Tree|spanning tree]] for $G$. | === Necessary Condition ===
Let $e$ be a [[Definition:Bridge (Graph Theory)|bridge]].
That is, suppose the [[Definition:Edge Deletion|edge deletion]] $G - e$ is [[Definition:Disconnected Graph|disconnected]].
Let $T$ be an arbitrary [[Definition:Spanning Tree|spanning tree]] for $G$.
By definition $T$ is a [[Defin... | Edge is Bridge iff in All Spanning Trees | https://proofwiki.org/wiki/Edge_is_Bridge_iff_in_All_Spanning_Trees | https://proofwiki.org/wiki/Edge_is_Bridge_iff_in_All_Spanning_Trees | [
"Graph Theory"
] | [
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Bridge (Graph Theory)",
"Definition:Spanning Tree"
] | [
"Definition:Bridge (Graph Theory)",
"Definition:Edge Deletion",
"Definition:Connected (Graph Theory)/Graph/Disconnected",
"Definition:Spanning Tree",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Subgraph",
"Definition:Subgraph",
"Definition:Connected (Graph Theory)/Graph/Disconnected",
"... |
proofwiki-10782 | Edge is Minimum Weight Bridge iff in All Minimum Spanning Trees | Let $G$ be an undirected network.
Let every edge of $G$ have a unique weight.
Let $e$ be an edge of $G$.
Then $e$ is a bridge of minimum weight in $G$ {{iff}} $e$ belongs to every minimum spanning tree of $G$. | === Necessary Condition ===
{{AimForCont}} $e$ is a bridge of minimum weight that does not belong to some minimum spanning tree $Q$.
Let $e$ be added to $Q$ to make $Q'$.
Then $e$ forms part of a unique cycle $C$ in $Q$.
Thus there exists an edge $f \in C$ such that $\map w Q < \map w {Q + e - f}$.
This contradicts the... | Let $G$ be an [[Definition:Undirected Network|undirected network]].
Let every [[Definition:Edge of Graph|edge]] of $G$ have a [[Definition:Unique|unique]] [[Definition:Weight (Network Theory)|weight]].
Let $e$ be an [[Definition:Edge of Graph|edge]] of $G$.
Then $e$ is a [[Definition:Bridge (Graph Theory)|bridge]] ... | === Necessary Condition ===
{{AimForCont}} $e$ is a [[Definition:Bridge (Graph Theory)|bridge]] of minimum [[Definition:Weight (Network Theory)|weight]] that does not belong to some [[Definition:Minimum Spanning Tree|minimum spanning tree]] $Q$.
Let $e$ be added to $Q$ to make $Q'$.
Then $e$ forms part of a unique c... | Edge is Minimum Weight Bridge iff in All Minimum Spanning Trees | https://proofwiki.org/wiki/Edge_is_Minimum_Weight_Bridge_iff_in_All_Minimum_Spanning_Trees | https://proofwiki.org/wiki/Edge_is_Minimum_Weight_Bridge_iff_in_All_Minimum_Spanning_Trees | [
"Network Theory"
] | [
"Definition:Network (Graph Theory)/Undirected",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Unique",
"Definition:Network (Graph Theory)/Weight",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Bridge (Graph Theory)",
"Definition:Network (Graph Theory)/Weight",
"Definition:Minimum Spanning ... | [
"Definition:Bridge (Graph Theory)",
"Definition:Network (Graph Theory)/Weight",
"Definition:Minimum Spanning Tree",
"Definition:Graph (Graph Theory)/Edge",
"Proof by Contradiction"
] |
proofwiki-10783 | Maximum Weight Edge in all Minimum Spanning Trees is Bridge | Let $G$ be an undirected network.
Let every edge of $G$ have a unique weight.
Let $e$ be an edge of $G$ that belongs to every minimum spanning tree of $G$.
Let $e$ have maximum weight in $G$.
Then $e$ is a bridge in $G$. | {{proof wanted}}
Category:Network Theory
qsmxtomolnooyfurqaijieuu9q39qc0 | Let $G$ be an [[Definition:Undirected Network|undirected network]].
Let every [[Definition:Edge of Graph|edge]] of $G$ have a [[Definition:Unique|unique]] [[Definition:Weight (Network Theory)|weight]].
Let $e$ be an [[Definition:Edge of Graph|edge]] of $G$ that belongs to every [[Definition:Minimum Spanning Tree|mini... | {{proof wanted}}
[[Category:Network Theory]]
qsmxtomolnooyfurqaijieuu9q39qc0 | Maximum Weight Edge in all Minimum Spanning Trees is Bridge | https://proofwiki.org/wiki/Maximum_Weight_Edge_in_all_Minimum_Spanning_Trees_is_Bridge | https://proofwiki.org/wiki/Maximum_Weight_Edge_in_all_Minimum_Spanning_Trees_is_Bridge | [
"Network Theory"
] | [
"Definition:Network (Graph Theory)/Undirected",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Unique",
"Definition:Network (Graph Theory)/Weight",
"Definition:Graph (Graph Theory)/Edge",
"Definition:Minimum Spanning Tree",
"Definition:Network (Graph Theory)/Weight",
"Definition:Bridge (Graph The... | [
"Category:Network Theory"
] |
proofwiki-10784 | Scaling Property of Dirac Delta Function | Let $\map \delta t$ be the Dirac delta function.
Let $a$ be a non zero constant real number.
Then:
:$\map \delta {a t} = \dfrac {\map \delta t} {\size a}$ | The equation can be rearranged as:
:$\size a \map \delta {a t} = \map \delta t$
We will check the definition of Dirac delta function in turn.
Definition of Dirac delta function:
{{begin-eqn}}
{{eqn | n = 1
| l = \map \delta t
| r = \begin{cases} +\infty & : t = 0 \\ 0 & : \text{otherwise} \end{cases}
... | Let $\map \delta t$ be the [[Definition:Dirac Delta Function|Dirac delta function]].
Let $a$ be a non zero [[Definition:Constant|constant]] [[Definition:Real Numbers|real number]].
Then:
:$\map \delta {a t} = \dfrac {\map \delta t} {\size a}$ | The equation can be rearranged as:
:$\size a \map \delta {a t} = \map \delta t$
We will check the [[Definition:Dirac Delta Function|definition]] of Dirac delta function in turn.
Definition of Dirac delta function:
{{begin-eqn}}
{{eqn | n = 1
| l = \map \delta t
| r = \begin{cases} +\infty & : t = 0 \\ 0 ... | Scaling Property of Dirac Delta Function | https://proofwiki.org/wiki/Scaling_Property_of_Dirac_Delta_Function | https://proofwiki.org/wiki/Scaling_Property_of_Dirac_Delta_Function | [
"Dirac Delta Function"
] | [
"Definition:Dirac Delta Function",
"Definition:Constant",
"Definition:Real Number"
] | [
"Definition:Dirac Delta Function",
"Definition:Positive/Real Number",
"Definition:Negative/Real Number",
"Reversal of Limits of Definite Integral",
"Category:Dirac Delta Function"
] |
proofwiki-10785 | Supremum of Function is less than Supremum of Greater Function | Let $f$ and $g$ be real functions.
Let $S$ be a subset of $\Dom f \cap \Dom g$.
Let $\map f x \le \map g x$ for every $x \in S$.
Let $\ds \sup_{x \mathop \in S} \map g x$ exist.
Then $\ds \sup_{x \mathop \in S} \map f x$ exists and:
:$\ds \sup_{x \mathop \in S} \map f x \le \sup_{x \mathop \in S} \map g x$. | We have:
{{begin-eqn}}
{{eqn | l = \sup g
| r = \map \sup {f + \paren {g - f} }
}}
{{eqn | r = \sup f + \sup \left({g - f}\right)
| c = Supremum of Sum equals Sum of Suprema
}}
{{end-eqn}}
Supremum of Sum equals Sum of Suprema also gives that $\sup f$ and $\sup \paren {g - f}$ exist.
We have:
{{begin-eqn}}
... | Let $f$ and $g$ be [[Definition:Real Function|real functions]].
Let $S$ be a [[Definition:Subset|subset]] of $\Dom f \cap \Dom g$.
Let $\map f x \le \map g x$ for every $x \in S$.
Let $\ds \sup_{x \mathop \in S} \map g x$ exist.
Then $\ds \sup_{x \mathop \in S} \map f x$ exists and:
:$\ds \sup_{x \mathop \in S} \... | We have:
{{begin-eqn}}
{{eqn | l = \sup g
| r = \map \sup {f + \paren {g - f} }
}}
{{eqn | r = \sup f + \sup \left({g - f}\right)
| c = [[Supremum of Sum equals Sum of Suprema]]
}}
{{end-eqn}}
[[Supremum of Sum equals Sum of Suprema]] also gives that $\sup f$ and $\sup \paren {g - f}$ exist.
We have:
{{... | Supremum of Function is less than Supremum of Greater Function | https://proofwiki.org/wiki/Supremum_of_Function_is_less_than_Supremum_of_Greater_Function | https://proofwiki.org/wiki/Supremum_of_Function_is_less_than_Supremum_of_Greater_Function | [
"Real Analysis",
"Suprema"
] | [
"Definition:Real Function",
"Definition:Subset"
] | [
"Supremum of Sum equals Sum of Suprema",
"Supremum of Sum equals Sum of Suprema",
"Definition:Upper Bound of Set/Real Numbers",
"Category:Real Analysis",
"Category:Suprema"
] |
proofwiki-10786 | Characterization of Boundary by Open Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $T$.
Let $x$ be a point of $T$.
Then $x \in \partial A$ {{iff}}:
:for every open set $U$ of $T$:
::if $x \in U$
::then $A \cap U \ne \O$ and $\relcomp S A \cap U \ne \O$
where:
:$\relcomp S A = S \setminus A$ denotes the complement of $A$ in $S$... | === Sufficient Condition ===
Let $x \in \partial A$.
Then by Boundary is Intersection of Closure with Closure of Complement:
:$x \in \paren {\relcomp S A}^-$ and $x \in A^-$
where $A^-$ denotes the closure of $A$.
Hence by Condition for Point being in Closure, for every open set $U$ of $T$:
:$x \in U \implies A \cap U ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $T$.
Let $x$ be a point of $T$.
Then $x \in \partial A$ {{iff}}:
:for every [[Definition:Open Set (Topology)|open set]] $U$ of $T$:
::if $x \in U$
::then $A \cap U \ne \O$ and $\relcomp ... | === Sufficient Condition ===
Let $x \in \partial A$.
Then by [[Boundary is Intersection of Closure with Closure of Complement]]:
:$x \in \paren {\relcomp S A}^-$ and $x \in A^-$
where $A^-$ denotes the [[Definition:Closure (Topology)|closure]] of $A$.
Hence by [[Condition for Point being in Closure]], for every [[D... | Characterization of Boundary by Open Sets | https://proofwiki.org/wiki/Characterization_of_Boundary_by_Open_Sets | https://proofwiki.org/wiki/Characterization_of_Boundary_by_Open_Sets | [
"Set Boundaries"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Boundary (Topology)"
] | [
"Boundary is Intersection of Closure with Closure of Complement",
"Definition:Closure (Topology)",
"Condition for Point being in Closure",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Condition for Point being in Closure",
"Boundary is Intersection of Closure with Closure of Compleme... |
proofwiki-10787 | Characterization of Closure by Open Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $S$.
Let $x$ be a point of $T$.
Let $A^-$ denote the closure of $A$.
Then $x \in A^-$ {{iff}}:
:for every open set $U$ of $T$:
::$x \in U \implies A \cap U \ne \O$ | === Sufficient Condition ===
Let $x \in A^-$.
{{AimForCont}} there exists an open set $U$ of $T$ such that:
:$x \in U$ and $A \cap U = \O$
We have that $U$ is open in $T$.
So by definition of closed set, $\relcomp S U$ is closed in $T$.
Then:
{{begin-eqn}}
{{eqn | l = A \cap U
| r = \O
| c =
}}
{{eqn | ll=... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Let $x$ be a point of $T$.
Let $A^-$ denote the [[Definition:Closure (Topology)|closure]] of $A$.
Then $x \in A^-$ {{iff}}:
:for every [[Definition:Open Set (Topology)|open set]] $... | === Sufficient Condition ===
Let $x \in A^-$.
{{AimForCont}} there exists an [[Definition:Open Set (Topology)|open set]] $U$ of $T$ such that:
:$x \in U$ and $A \cap U = \O$
We have that $U$ is [[Definition:Open Set (Topology)|open]] in $T$.
So by definition of [[Definition:Closed Set (Topology)|closed set]], $\re... | Characterization of Closure by Open Sets | https://proofwiki.org/wiki/Characterization_of_Closure_by_Open_Sets | https://proofwiki.org/wiki/Characterization_of_Closure_by_Open_Sets | [
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Closure (Topology)",
"Definition:Open Set/Topology"
] | [
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Empty Intersection iff Subset of Complement",
"Definition:Closed Set/Topology",
"Empty Intersection iff Subset of Complement",
"Definition:Set Intersection",
"Definiti... |
proofwiki-10788 | Summation is Linear/Sum of Summations | :$\ds \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$ | The proof proceeds by mathematical induction.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$ | :$\ds \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$ | The proof proceeds by [[Principle of Mathematical Induction|mathematical induction]].
For all $n \in \N_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n \paren {x_i + y_i}$ | Summation is Linear/Sum of Summations | https://proofwiki.org/wiki/Summation_is_Linear/Sum_of_Summations | https://proofwiki.org/wiki/Summation_is_Linear/Sum_of_Summations | [
"Numbers",
"Proofs by Induction"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-10789 | Summation is Linear/Scaling of Summations | :$\ds \lambda \sum_{i \mathop = 1}^n x_i = \sum_{i \mathop = 1}^n \lambda x_i$ | For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \lambda \sum_{i \mathop = 1}^n x_i = \sum_{i \mathop = 1}^n \lambda x_i$ | :$\ds \lambda \sum_{i \mathop = 1}^n x_i = \sum_{i \mathop = 1}^n \lambda x_i$ | For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \lambda \sum_{i \mathop = 1}^n x_i = \sum_{i \mathop = 1}^n \lambda x_i$ | Summation is Linear/Scaling of Summations | https://proofwiki.org/wiki/Summation_is_Linear/Scaling_of_Summations | https://proofwiki.org/wiki/Summation_is_Linear/Scaling_of_Summations | [
"Numbers",
"Proofs by Induction"
] | [] | [
"Definition:Proposition"
] |
proofwiki-10790 | Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum | Let $f$ be a real function.
Let $S$ be a subset of the domain of $f$.
Let $\ds \sup_{x \mathop \in S} \set {\map f x}$ and $\ds \inf_{x \mathop \in S} \set {\map f x}$ exist.
Then $\ds \sup_{x, y \mathop \in S} \set {\size {\map f x - \map f y} }$ exists and:
:$\ds \sup_{x, y \mathop \in S} \set {\size {\map f x - \map... | {{begin-eqn}}
{{eqn | l = \sup_{x \mathop \in S} \set {\map f x} - \inf_{x \mathop \in S} \set {\map f x}
| r = \sup_{x \mathop \in S} \set {\map f x} + \sup_{x \mathop \in S} \set {-\map f x}
| c = Negative of Infimum is Supremum of Negatives
}}
{{eqn | r = \sup_{x, y \mathop \in S} \set {\map f x + \paren... | Let $f$ be a [[Definition:Real Function|real function]].
Let $S$ be a [[Definition:Subset|subset]] of the [[Definition:Domain of Mapping|domain]] of $f$.
Let $\ds \sup_{x \mathop \in S} \set {\map f x}$ and $\ds \inf_{x \mathop \in S} \set {\map f x}$ exist.
Then $\ds \sup_{x, y \mathop \in S} \set {\size {\map f x... | {{begin-eqn}}
{{eqn | l = \sup_{x \mathop \in S} \set {\map f x} - \inf_{x \mathop \in S} \set {\map f x}
| r = \sup_{x \mathop \in S} \set {\map f x} + \sup_{x \mathop \in S} \set {-\map f x}
| c = [[Negative of Infimum is Supremum of Negatives]]
}}
{{eqn | r = \sup_{x, y \mathop \in S} \set {\map f x + \p... | Supremum of Absolute Value of Difference equals Difference between Supremum and Infimum | https://proofwiki.org/wiki/Supremum_of_Absolute_Value_of_Difference_equals_Difference_between_Supremum_and_Infimum | https://proofwiki.org/wiki/Supremum_of_Absolute_Value_of_Difference_equals_Difference_between_Supremum_and_Infimum | [
"Real Analysis"
] | [
"Definition:Real Function",
"Definition:Subset",
"Definition:Domain (Set Theory)/Mapping"
] | [
"Negative of Infimum is Supremum of Negatives",
"Supremum of Sum equals Sum of Suprema",
"Supremum of Absolute Value of Difference equals Supremum of Difference",
"Category:Real Analysis"
] |
proofwiki-10791 | Diaconescu-Goodman-Myhill Theorem | The {{Axiom-link|Choice}} implies the law of excluded middle. | Let $\mathbb B = \set {0, 1}$.
Let $p$ be a proposition.
Let the following two sets be defined:
:$A = \set {x \in \mathbb B: x = 0 \lor p}$
:$B = \set {x \in \mathbb B: x = 1 \lor p}$
where $\lor$ denotes the disjunction operator.
We have that:
:$0 \in A$
and:
:$1 \in B$
so both $A$ and $B$ are non-empty
Then the set:... | The {{Axiom-link|Choice}} implies the [[Law of Excluded Middle|law of excluded middle]]. | Let $\mathbb B = \set {0, 1}$.
Let $p$ be a [[Definition:Proposition|proposition]].
Let the following two [[Definition:Set|sets]] be defined:
:$A = \set {x \in \mathbb B: x = 0 \lor p}$
:$B = \set {x \in \mathbb B: x = 1 \lor p}$
where $\lor$ denotes the [[Definition:Disjunction|disjunction]] operator.
We have tha... | Diaconescu-Goodman-Myhill Theorem | https://proofwiki.org/wiki/Diaconescu-Goodman-Myhill_Theorem | https://proofwiki.org/wiki/Diaconescu-Goodman-Myhill_Theorem | [
"Diaconescu-Goodman-Myhill Theorem",
"Axiom of Choice",
"Law of Excluded Middle"
] | [
"Law of Excluded Middle"
] | [
"Definition:Proposition",
"Definition:Set",
"Definition:Disjunction",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Choice Function",
"Modus Tollendo Ponens",
"Definition:Element",
"Definition:Contradiction",
"Proof by Cases",
"Law of... |
proofwiki-10792 | Characterization of Boundary by Basis | Let $T = \struct {S, \tau}$ be a topological space.
Let $\BB \subseteq \tau$ be a basis.
Let $A$ be a subset of $T$.
Let $x$ be a point of $T$.
Then $x \in \partial A$ {{iff}}:
:for every $U \in \BB$:
::if $x \in U$
::then $A \cap U \ne \O$ and $\relcomp S A \cap U \ne \O$
where:
:$\relcomp S A = S \setminus A$ denotes... | === Sufficient Condition ===
Let $x \in \partial A$.
Let $U \in \BB$.
By definition of basis, $U$ is an open set of $T$.
Thus from Characterization of Boundary by Open Sets:
:if $x \in U$
::then $A \cap U \ne \O$ and $\relcomp S A \cap U \ne \O$.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[definition:Topological Space|topological space]].
Let $\BB \subseteq \tau$ be a [[Definition:Analytic Basis|basis]].
Let $A$ be a [[Definition:Subset|subset]] of $T$.
Let $x$ be a point of $T$.
Then $x \in \partial A$ {{iff}}:
:for every $U \in \BB$:
::if $x \in U$
::then $A \cap... | === Sufficient Condition ===
Let $x \in \partial A$.
Let $U \in \BB$.
By definition of [[Definition:Analytic Basis|basis]], $U$ is an [[Definition:Open Set (Topology)|open set of $T$]].
Thus from [[Characterization of Boundary by Open Sets]]:
:if $x \in U$
::then $A \cap U \ne \O$ and $\relcomp S A \cap U \ne \O$.
... | Characterization of Boundary by Basis | https://proofwiki.org/wiki/Characterization_of_Boundary_by_Basis | https://proofwiki.org/wiki/Characterization_of_Boundary_by_Basis | [
"Set Boundaries"
] | [
"definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Subset",
"Definition:Relative Complement",
"Definition:Boundary (Topology)"
] | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Open Set/Topology",
"Characterization of Boundary by Open Sets",
"Characterization of Boundary by Open Sets",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Basis (Topology)/Analytic Basis"
] |
proofwiki-10793 | Union of Interiors and Boundary Equals Whole Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $T$.
Then:
:$S = \Int A \cup \partial A \cup \Int {A'}$
where:
:$A' = S \setminus A$ denotes the complement of $A$ relative to $S$
:$\Int A$ denotes the interior of $A$
:$\partial A$ denotes the boundary of $A$. | {{begin-eqn}}
{{eqn | l = \Int A \cup \partial A \cup \Int {A'}
| r = \Int A \cup \Int {A'} \cup \partial A
| c = Union is Associative, Union is Commutative
}}
{{eqn | r = \Int A \cup \Int {A'} \cup \paren {\map \cl A \cap \map \cl {A'} }
| c = Boundary is Intersection of Closure with Closure of Compl... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $T$.
Then:
:$S = \Int A \cup \partial A \cup \Int {A'}$
where:
:$A' = S \setminus A$ denotes the [[Definition:Relative Complement|complement of $A$ relative to $S$]]
:$\Int A$ denotes th... | {{begin-eqn}}
{{eqn | l = \Int A \cup \partial A \cup \Int {A'}
| r = \Int A \cup \Int {A'} \cup \partial A
| c = [[Union is Associative]], [[Union is Commutative]]
}}
{{eqn | r = \Int A \cup \Int {A'} \cup \paren {\map \cl A \cap \map \cl {A'} }
| c = [[Boundary is Intersection of Closure with Closur... | Union of Interiors and Boundary Equals Whole Space | https://proofwiki.org/wiki/Union_of_Interiors_and_Boundary_Equals_Whole_Space | https://proofwiki.org/wiki/Union_of_Interiors_and_Boundary_Equals_Whole_Space | [
"Set Boundaries",
"Set Interiors"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Relative Complement",
"Definition:Interior (Topology)",
"Definition:Boundary (Topology)"
] | [
"Union is Associative",
"Union is Commutative",
"Boundary is Intersection of Closure with Closure of Complement",
"Intersection Distributes over Union",
"Complement of Interior equals Closure of Complement",
"Union is Associative",
"Union with Relative Complement",
"Complement of Interior equals Closu... |
proofwiki-10794 | Characterization of Derivative by Open Sets | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $T$.
Let $x$ be a point of $T$.
Then
:$x \in A'$
{{iff}}:
:for every open set $U$ of $T$:
::if $x \in U$
::then there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \ne y$
where
:$A'$ denotes the derivative of $A$. | === Sufficient Condition ===
Let $x \in A'$.
Then $x$ is an accumulation point of $A$ by Definition:Set Derivative.
Then by definition of accumulation point:
:$(1): \quad x \in \paren {A \setminus \set x}^-$
where $A^-$ denotes the closure of $A$.
Let $U$ be an open set of $T$.
Let $x \in U$.
Then by $(1)$ and Conditio... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $T$.
Let $x$ be a point of $T$.
Then
:$x \in A'$
{{iff}}:
:for every [[Definition:Open Set (Topology)|open set]] $U$ of $T$:
::if $x \in U$
::then there exists a point $y$ of $T$ such th... | === Sufficient Condition ===
Let $x \in A'$.
Then $x$ is an [[Definition:Accumulation Point of Set|accumulation point]] of $A$ by [[Definition:Set Derivative]].
Then by definition of [[Definition:Accumulation Point of Set|accumulation point]]:
:$(1): \quad x \in \paren {A \setminus \set x}^-$
where $A^-$ denotes the... | Characterization of Derivative by Open Sets | https://proofwiki.org/wiki/Characterization_of_Derivative_by_Open_Sets | https://proofwiki.org/wiki/Characterization_of_Derivative_by_Open_Sets | [
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Set Derivative"
] | [
"Definition:Accumulation Point/Set",
"Definition:Set Derivative",
"Definition:Accumulation Point/Set",
"Definition:Closure (Topology)",
"Definition:Open Set/Topology",
"Condition for Point being in Closure",
"Definition:Set Difference",
"Definition:Set Intersection",
"Definition:Singleton",
"Defin... |
proofwiki-10795 | Characterization of Derivative by Local Basis | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $S$.
Let $x$ be a point of $T$.
Let $\BB \subseteq \tau$ be a local basis at $x$.
Then
:$x \in A'$
{{iff}}:
:for every $U \in \BB$, there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \ne y$
where:
:$A'$ denotes the derivative of $... | === Sufficient Condition ===
Let $x \in A'$.
By Characterization of Derivative by Open Sets:
For every open set $U$ of $T$:
:if $x \in U$
:then there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \ne y$
As the elements of $\BB$ are all open sets, it follows that:
For every open set $U \in \BB$:
:if $x \in... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Let $x$ be a point of $T$.
Let $\BB \subseteq \tau$ be a [[Definition:Local Basis|local basis]] at $x$.
Then
:$x \in A'$
{{iff}}:
:for every $U \in \BB$, there exists a point $y$ o... | === Sufficient Condition ===
Let $x \in A'$.
By [[Characterization of Derivative by Open Sets]]:
For every [[Definition:Open Set (Topology)|open set]] $U$ of $T$:
:if $x \in U$
:then there exists a point $y$ of $T$ such that $y \in A \cap U$ and $x \ne y$
As the [[Definition:Element|elements]] of $\BB$ are all [[D... | Characterization of Derivative by Local Basis | https://proofwiki.org/wiki/Characterization_of_Derivative_by_Local_Basis | https://proofwiki.org/wiki/Characterization_of_Derivative_by_Local_Basis | [
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Local Basis",
"Definition:Set Derivative"
] | [
"Characterization of Derivative by Open Sets",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Characterization of Derivative by Open Sets",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-10796 | Derivative is Included in Closure | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $S$.
Then
:$A' \subseteq A^-$
where
:$A'$ denotes the derivative of $A$
:$A^-$ denotes the closure of $A$. | Let $x \in A'$.
By Condition for Point being in Closure it is enough to prove that:
:for every open set $G$ of $T$:
:: if $x \in G$
:: then $A \cap G \ne \O$.
Let $G$ be an open set of $T$.
Let $x \in G$.
Then by Characterization of Derivative by Open Sets:
: there exists a point $\exists y \in S: y \in A \cap G \land ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then
:$A' \subseteq A^-$
where
:$A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$
:$A^-$ denotes the [[Definition:Closure (Topology)|closure]] of $A$. | Let $x \in A'$.
By [[Condition for Point being in Closure]] it is enough to prove that:
:for every [[Definition:Open Set (Topology)|open set]] $G$ of $T$:
:: if $x \in G$
:: then $A \cap G \ne \O$.
Let $G$ be an [[Definition:Open Set (Topology)|open set]] of $T$.
Let $x \in G$.
Then by [[Characterization of Deriva... | Derivative is Included in Closure | https://proofwiki.org/wiki/Derivative_is_Included_in_Closure | https://proofwiki.org/wiki/Derivative_is_Included_in_Closure | [
"Set Derivatives",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Set Derivative",
"Definition:Closure (Topology)"
] | [
"Condition for Point being in Closure",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Characterization of Derivative by Open Sets"
] |
proofwiki-10797 | Closure Equals Union with Derivative | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a subset of $S$.
Then:
:$A^- = A \cup A'$
where
:$A'$ denotes the derivative of $A$
:$A^-$ denotes the closure of $A$. | === Closure Subset of Union ===
It is to be proved that:
:$A^- \subseteq A \cup A'$
Let $x \in A^-$.
In the case where $x \in A$ then $x \in A \cup A'$ by definition of set union.
Let:
:$(1): \quad x \notin A$
From Characterization of Derivative by Open Sets, to prove $x \in A'$ it is enough to show that:
:for every op... | Let $T = \struct {S, \tau}$ be a [[definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$A^- = A \cup A'$
where
:$A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$
:$A^-$ denotes the [[Definition:Closure (Topology)|closure]] of $A$. | === Closure Subset of Union ===
It is to be proved that:
:$A^- \subseteq A \cup A'$
Let $x \in A^-$.
In the case where $x \in A$ then $x \in A \cup A'$ by definition of [[Definition:Set Union|set union]].
Let:
:$(1): \quad x \notin A$
From [[Characterization of Derivative by Open Sets]], to prove $x \in A'$ it is ... | Closure Equals Union with Derivative | https://proofwiki.org/wiki/Closure_Equals_Union_with_Derivative | https://proofwiki.org/wiki/Closure_Equals_Union_with_Derivative | [
"Set Derivatives",
"Set Closures"
] | [
"definition:Topological Space",
"Definition:Subset",
"Definition:Set Derivative",
"Definition:Closure (Topology)"
] | [
"Definition:Set Union",
"Characterization of Derivative by Open Sets",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Condition for Point being in Closure",
"Definition:Empty Set",
"Definition:Set Intersection",
"Definition:Set Intersection",
"Definition:Set Union"
] |
proofwiki-10798 | Derivative of Subset is Subset of Derivative | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$, $B$ be subsets of $S$.
Then
:$A \subseteq B \implies A' \subseteq B'$
where $A'$ denotes the derivative of $A$ in $T$. | Let $A \subseteq B$.
Let $x \in A'$.
By Characterization of Derivative by Open Sets it is enough to prove that:
:for every open set $G$ of $T$:
::if $x \in G$
::then there exists $y$ such that $y \in B \cap G$ and $x \ne y$.
Let $G$ be an open set of $T$.
Let $x \in G$.
Then by Characterization of Derivative by Open Se... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$, $B$ be [[Definition:Subset|subsets]] of $S$.
Then
:$A \subseteq B \implies A' \subseteq B'$
where $A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$ in $T$. | Let $A \subseteq B$.
Let $x \in A'$.
By [[Characterization of Derivative by Open Sets]] it is enough to prove that:
:for every [[Definition:Open Set (Topology)|open set]] $G$ of $T$:
::if $x \in G$
::then there exists $y$ such that $y \in B \cap G$ and $x \ne y$.
Let $G$ be an [[Definition:Open Set (Topology)|open ... | Derivative of Subset is Subset of Derivative | https://proofwiki.org/wiki/Derivative_of_Subset_is_Subset_of_Derivative | https://proofwiki.org/wiki/Derivative_of_Subset_is_Subset_of_Derivative | [
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Set Derivative"
] | [
"Characterization of Derivative by Open Sets",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Characterization of Derivative by Open Sets",
"Definition:Subset"
] |
proofwiki-10799 | Derivative of Union is Union of Derivatives | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$, $B$ be subsets of $S$.
Then
:$\paren {A \cup B}' = A' \cup B\,'$
where
:$A'$ denotes the derivative of $A$. | === Derivative of Union subset of Union of Derivatives ===
It will be shown that:
:$\paren {A \cup B}' \subseteq A' \cup B\,'$
Let $x \in \paren {A \cup B}'$.
By definition of set derivative:
:$x$ is an accumulation point of $A \cup B$.
Then by definition of accumulation point of set:
:$(1): \quad x \in \paren {\paren ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$, $B$ be [[Definition:Subset|subsets]] of $S$.
Then
:$\paren {A \cup B}' = A' \cup B\,'$
where
:$A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$. | === Derivative of Union subset of Union of Derivatives ===
It will be shown that:
:$\paren {A \cup B}' \subseteq A' \cup B\,'$
Let $x \in \paren {A \cup B}'$.
By definition of [[Definition:Set Derivative|set derivative]]:
:$x$ is an [[Definition:Accumulation Point of Set|accumulation point]] of $A \cup B$.
Then... | Derivative of Union is Union of Derivatives | https://proofwiki.org/wiki/Derivative_of_Union_is_Union_of_Derivatives | https://proofwiki.org/wiki/Derivative_of_Union_is_Union_of_Derivatives | [
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Subset",
"Definition:Set Derivative"
] | [
"Definition:Set Derivative",
"Definition:Accumulation Point/Set",
"Definition:Accumulation Point/Set",
"Definition:Closure (Topology)",
"Set Difference is Right Distributive over Union",
"Closure of Finite Union equals Union of Closures",
"Definition:Set Union",
"Definition:Accumulation Point/Set",
... |
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