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-10800 | Derivative of Derivative is Subset of Derivative in T1 Space | Let $T = \struct {S, \tau}$ be a $T_1$ topological space.
Let $A$ be a subset of $S$.
Then
:$A' ' \subseteq A'$
where
:$A'$ denotes the derivative of $A$ | Let:
:$(1): \quad x \in A' '$
{{AimForCont}} $x \notin A'$.
Then by Characterization of Derivative by Open Sets there exists an open subset $G$ of $T$ such that:
:$(2): \quad x \in G$
and:
:$(3): \quad \lnot \exists y: y \in A \cap G \land x \ne y$
By definition of $T_1$ space:
:$\set x$ is closed.
Then by Open Set min... | Let $T = \struct {S, \tau}$ be a [[Definition:T1 Space|$T_1$ topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then
:$A' ' \subseteq A'$
where
:$A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$ | Let:
:$(1): \quad x \in A' '$
{{AimForCont}} $x \notin A'$.
Then by [[Characterization of Derivative by Open Sets]] there exists an open subset $G$ of $T$ such that:
:$(2): \quad x \in G$
and:
:$(3): \quad \lnot \exists y: y \in A \cap G \land x \ne y$
By definition of [[Definition:T1 Space|$T_1$ space]]:
:$\set x$ ... | Derivative of Derivative is Subset of Derivative in T1 Space | https://proofwiki.org/wiki/Derivative_of_Derivative_is_Subset_of_Derivative_in_T1_Space | https://proofwiki.org/wiki/Derivative_of_Derivative_is_Subset_of_Derivative_in_T1_Space | [
"Set Derivatives",
"T1 Spaces"
] | [
"Definition:T1 Space",
"Definition:Subset",
"Definition:Set Derivative"
] | [
"Characterization of Derivative by Open Sets",
"Definition:T1 Space",
"Definition:Closed Set/Topology",
"Open Set minus Closed Set is Open",
"Definition:Open Set/Topology",
"Characterization of Derivative by Open Sets",
"Definition:Set Intersection",
"Definition:Set Difference",
"Definition:Set Inte... |
proofwiki-10801 | Closure of Derivative is Derivative in T1 Space | Let $T = \struct {S, \tau}$ be a $T_1$ topological space.
Let $A$ be a subset of $S$.
Then
:$A'^- = A'$
where
:$A'$ denotes the derivative of $A$
:$A^-$ denotes the closure of $A$. | {{begin-eqn}}
{{eqn | l = A'^-
| r = A' \cup A' '
| c = Closure Equals Union with Derivative
}}
{{eqn | o = \subseteq
| r = A' \cup A'
| c = $A' ' \subseteq A'$ by Derivative of Derivative is Subset of Derivative in T1 Space
}}
{{eqn | r = A'
| c = Set Union is Idempotent
}}
{{end-eqn}}
So... | Let $T = \struct {S, \tau}$ be a [[Definition:T1 Space|$T_1$ topological space]].
Let $A$ be a [[Definition:Subset|subset]] of $S$.
Then
:$A'^- = A'$
where
:$A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$
:$A^-$ denotes the [[Definition:Closure (Topology)|closure]] of $A$. | {{begin-eqn}}
{{eqn | l = A'^-
| r = A' \cup A' '
| c = [[Closure Equals Union with Derivative]]
}}
{{eqn | o = \subseteq
| r = A' \cup A'
| c = $A' ' \subseteq A'$ by [[Derivative of Derivative is Subset of Derivative in T1 Space]]
}}
{{eqn | r = A'
| c = [[Set Union is Idempotent]]
}}
{{... | Closure of Derivative is Derivative in T1 Space | https://proofwiki.org/wiki/Closure_of_Derivative_is_Derivative_in_T1_Space | https://proofwiki.org/wiki/Closure_of_Derivative_is_Derivative_in_T1_Space | [
"Set Derivatives",
"Set Closures",
"T1 Spaces"
] | [
"Definition:T1 Space",
"Definition:Subset",
"Definition:Set Derivative",
"Definition:Closure (Topology)"
] | [
"Closure Equals Union with Derivative",
"Derivative of Derivative is Subset of Derivative in T1 Space",
"Set Union is Idempotent",
"Definition:Closure (Topology)/Definition 3",
"Definition:Set Equality"
] |
proofwiki-10802 | Union of Derivatives is Subset of Derivative of Union | Let $T = \struct {S, \tau}$ be a topological space.
Let:
:$\FF \subseteq \powerset S$ be a set of subsets of $S$
where $\powerset S$ denotes the power set of $S$.
Then:
:$\ds \bigcup_{A \mathop \in \FF} A' \subseteq \paren {\bigcup_{A \mathop \in \FF} A}'$
where $A'$ denotes the derivative of $A$. | Let $\ds x \in \bigcup_{A \mathop \in \FF} A'$.
Then by definition of union there exists $A \in \FF$ such that:
:$(1): \quad x \in A'$
By Set is Subset of Union:
:$\ds A \subseteq \bigcup_{A \mathop \in \FF} A$
Then by Derivative of Subset is Subset of Derivative:
:$\ds A' \subseteq \paren {\bigcup_{A \mathop \in \FF} ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let:
:$\FF \subseteq \powerset S$ be a [[Definition:Set of Sets|set of subsets]] of $S$
where $\powerset S$ denotes the [[Definition:Power Set|power set]] of $S$.
Then:
:$\ds \bigcup_{A \mathop \in \FF} A' \subseteq \paren {\bigcup_... | Let $\ds x \in \bigcup_{A \mathop \in \FF} A'$.
Then by definition of [[Definition:Union of Family|union]] there exists $A \in \FF$ such that:
:$(1): \quad x \in A'$
By [[Set is Subset of Union/Set of Sets|Set is Subset of Union]]:
:$\ds A \subseteq \bigcup_{A \mathop \in \FF} A$
Then by [[Derivative of Subset is Su... | Union of Derivatives is Subset of Derivative of Union | https://proofwiki.org/wiki/Union_of_Derivatives_is_Subset_of_Derivative_of_Union | https://proofwiki.org/wiki/Union_of_Derivatives_is_Subset_of_Derivative_of_Union | [
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Power Set",
"Definition:Set Derivative"
] | [
"Definition:Set Union/Family of Sets",
"Set is Subset of Union/Set of Sets",
"Derivative of Subset is Subset of Derivative",
"Definition:Subset"
] |
proofwiki-10803 | Point is Isolated iff not Accumulation Point | Let $T = \struct {S, \tau}$ be a topological space.
Let $H \subseteq S$.
Let $x \in H$.
Then:
:$x$ is an isolated point in $H$
{{iff}}:
:$x$ is not an accumulation point of $H$ | === Sufficient Condition ===
Let $x \in H$ be an isolated point in $H$.
Then by definition of isolated point:
:$\exists U \in \tau: H \cap U = \set x$
That is, by definition of uniqueness:
:$\lnot \forall U \in \tau: \paren {x \in U \implies \exists y \in S: \paren {y \in H \cap U \land x \ne y} }$
Hence by Characteriz... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$.
Let $x \in H$.
Then:
:$x$ is an [[Definition:Isolated Point (Topology)|isolated point]] in $H$
{{iff}}:
:$x$ is not an [[Definition:Accumulation Point of Set|accumulation point]] of $H$ | === Sufficient Condition ===
Let $x \in H$ be an [[Definition:Isolated Point (Topology)|isolated point]] in $H$.
Then by definition of [[Definition:Isolated Point (Topology)|isolated point]]:
:$\exists U \in \tau: H \cap U = \set x$
That is, by definition of [[Definition:Unique|uniqueness]]:
:$\lnot \forall U \in \t... | Point is Isolated iff not Accumulation Point | https://proofwiki.org/wiki/Point_is_Isolated_iff_not_Accumulation_Point | https://proofwiki.org/wiki/Point_is_Isolated_iff_not_Accumulation_Point | [
"Isolated Points",
"Accumulation Points"
] | [
"Definition:Topological Space",
"Definition:Isolated Point (Topology)",
"Definition:Accumulation Point/Set"
] | [
"Definition:Isolated Point (Topology)",
"Definition:Isolated Point (Topology)",
"Definition:Unique",
"Characterization of Derivative by Open Sets",
"Definition:Set Derivative",
"Definition:Set Derivative",
"Definition:Accumulation Point/Set",
"Definition:Accumulation Point/Set",
"Definition:Set Deri... |
proofwiki-10804 | Point is Isolated iff belongs to Set less Derivative | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $H \subseteq S$.
Let $x \in S$.
Then:
:$x$ is an isolated point in $H$
{{iff}}:
:$x \in H \setminus H'$
where
:$H'$ denotes the derivative of $H$. | $x$ is an isolated point in $H$
$\iff$ $x \in H$ and $x$ is not an accumulation point of $H$ by Point is Isolated iff not Accumulation Point
$\iff$ $x \in H$ and $x \notin H'$ by definition of derivative
$\iff$ $x \in H \setminus H'$ by definition of set difference.
{{qed}} | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $H \subseteq S$.
Let $x \in S$.
Then:
:$x$ is an [[Definition:Isolated Point (Topology)|isolated point]] in $H$
{{iff}}:
:$x \in H \setminus H'$
where
:$H'$ denotes the [[Definition:Set Derivative|derivative]] of $H$. | $x$ is an [[Definition:Isolated Point (Topology)|isolated point]] in $H$
$\iff$ $x \in H$ and $x$ is not an [[Definition:Accumulation Point of Sequence|accumulation point]] of $H$ by [[Point is Isolated iff not Accumulation Point]]
$\iff$ $x \in H$ and $x \notin H'$ by definition of [[Definition:Set Derivative|deriva... | Point is Isolated iff belongs to Set less Derivative | https://proofwiki.org/wiki/Point_is_Isolated_iff_belongs_to_Set_less_Derivative | https://proofwiki.org/wiki/Point_is_Isolated_iff_belongs_to_Set_less_Derivative | [
"Isolated Points",
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Isolated Point (Topology)",
"Definition:Set Derivative"
] | [
"Definition:Isolated Point (Topology)",
"Definition:Accumulation Point/Sequence",
"Point is Isolated iff not Accumulation Point",
"Definition:Set Derivative",
"Definition:Set Difference"
] |
proofwiki-10805 | Dense-in-itself iff Subset of Derivative | Let $T$ be a topological space.
Let $A \subseteq T$.
Then:
:$A$ is dense-in-itself
{{iff}}:
:$A \subseteq A'$
where $A'$ denotes the derivative of $A$. | {{begin-eqn}}
{{eqn | o =
| c = $A$ is dense-in-itself
}}
{{eqn | ll= \leadstoandfrom
| o =
| c = every $x \in A$ is not an isolated point in $A$
| cc= {{Defof|Dense-in-itself}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| c = every $x \in A$ is an accumulation point of $A$
| cc= Poi... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq T$.
Then:
:$A$ is [[Definition:Dense-in-itself|dense-in-itself]]
{{iff}}:
:$A \subseteq A'$
where $A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$. | {{begin-eqn}}
{{eqn | o =
| c = $A$ is [[Definition:Dense-in-itself|dense-in-itself]]
}}
{{eqn | ll= \leadstoandfrom
| o =
| c = every $x \in A$ is not an [[Definition:Isolated Point (Topology)|isolated point]] in $A$
| cc= {{Defof|Dense-in-itself}}
}}
{{eqn | ll= \leadstoandfrom
| o =
... | Dense-in-itself iff Subset of Derivative | https://proofwiki.org/wiki/Dense-in-itself_iff_Subset_of_Derivative | https://proofwiki.org/wiki/Dense-in-itself_iff_Subset_of_Derivative | [
"Set Derivatives",
"Dense-in-itself"
] | [
"Definition:Topological Space",
"Definition:Dense-in-itself",
"Definition:Set Derivative"
] | [
"Definition:Dense-in-itself",
"Definition:Isolated Point (Topology)",
"Definition:Accumulation Point/Set",
"Point is Isolated iff not Accumulation Point",
"Definition:Element",
"Category:Set Derivatives",
"Category:Dense-in-itself"
] |
proofwiki-10806 | Closure of Dense-in-itself is Dense-in-itself in T1 Space | Let $T$ be a $T_1$ topological space.
Let $A \subseteq T$.
Let $A$ be dense-in-itself.
Then the closure $A^-$ of $A$ is also dense-in-itself. | Let $A$ be dense-in-itself.
Then by Dense-in-itself iff Subset of Derivative:
:$(1): \quad A \subseteq A'$
where $A'$ denotes the derivative of $A$.
By Derivative of Derivative is Subset of Derivative in $T_1$ Space:
:$(2): \quad A' ' \subseteq A'$
By Dense-in-itself iff Subset of Derivative it is sufficient to proof t... | Let $T$ be a [[Definition:T1 Space|$T_1$]] [[Definition:Topological Space|topological space]].
Let $A \subseteq T$.
Let $A$ be [[Definition:Dense-in-itself|dense-in-itself]].
Then the [[Definition:Closure (Topology)|closure]] $A^-$ of $A$ is also [[Definition:Dense-in-itself|dense-in-itself]]. | Let $A$ be [[Definition:Dense-in-itself|dense-in-itself]].
Then by [[Dense-in-itself iff Subset of Derivative]]:
:$(1): \quad A \subseteq A'$
where $A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$.
By [[Derivative of Derivative is Subset of Derivative in T1 Space|Derivative of Derivative is Subset of ... | Closure of Dense-in-itself is Dense-in-itself in T1 Space | https://proofwiki.org/wiki/Closure_of_Dense-in-itself_is_Dense-in-itself_in_T1_Space | https://proofwiki.org/wiki/Closure_of_Dense-in-itself_is_Dense-in-itself_in_T1_Space | [
"Dense-in-itself",
"Set Closures",
"T1 Spaces"
] | [
"Definition:T1 Space",
"Definition:Topological Space",
"Definition:Dense-in-itself",
"Definition:Closure (Topology)",
"Definition:Dense-in-itself"
] | [
"Definition:Dense-in-itself",
"Dense-in-itself iff Subset of Derivative",
"Definition:Set Derivative",
"Derivative of Derivative is Subset of Derivative in T1 Space",
"Dense-in-itself iff Subset of Derivative",
"Closure Equals Union with Derivative",
"Derivative of Union is Union of Derivatives",
"Uni... |
proofwiki-10807 | Union of Set of Dense-in-itself Sets is Dense-in-itself | Let $T$ be a topological space.
Let $\FF \subseteq \powerset T$ such that:
:every element of $\FF$ is dense-in-itself.
Then the union $\bigcup \FF$ is also dense-in-itself. | By Dense-in-itself iff Subset of Derivative:
:$\forall A \in \FF: A \subseteq A'$
where $A'$ denotes the derivative of $A$.
Then by Set Union Preserves Subsets:
:$\ds \bigcup \FF \subseteq \bigcup_{A \mathop \in \FF} A'$
By Union of Derivatives is Subset of Derivative of Union:
:$\ds \bigcup_{A \mathop \in \FF} A' \sub... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $\FF \subseteq \powerset T$ such that:
:every [[Definition:Element|element]] of $\FF$ is [[Definition:Dense-in-itself|dense-in-itself]].
Then the [[Definition:Union of Set of Sets|union]] $\bigcup \FF$ is also [[Definition:Dense-in-itself|dense-in-... | By [[Dense-in-itself iff Subset of Derivative]]:
:$\forall A \in \FF: A \subseteq A'$
where $A'$ denotes the [[Definition:Set Derivative|derivative]] of $A$.
Then by [[Set Union Preserves Subsets]]:
:$\ds \bigcup \FF \subseteq \bigcup_{A \mathop \in \FF} A'$
By [[Union of Derivatives is Subset of Derivative of Union]... | Union of Set of Dense-in-itself Sets is Dense-in-itself | https://proofwiki.org/wiki/Union_of_Set_of_Dense-in-itself_Sets_is_Dense-in-itself | https://proofwiki.org/wiki/Union_of_Set_of_Dense-in-itself_Sets_is_Dense-in-itself | [
"Dense-in-itself",
"Set Derivatives"
] | [
"Definition:Topological Space",
"Definition:Element",
"Definition:Dense-in-itself",
"Definition:Set Union/Set of Sets",
"Definition:Dense-in-itself"
] | [
"Dense-in-itself iff Subset of Derivative",
"Definition:Set Derivative",
"Set Union Preserves Subsets",
"Union of Derivatives is Subset of Derivative of Union",
"Subset Relation is Transitive",
"Dense-in-itself iff Subset of Derivative"
] |
proofwiki-10808 | Equivalence of Definitions of Weight of Topological Space | Let $T$ be a topological space.
Let $\mathbb B$ be the set of all bases of $T$.
The following definitions of the weight of $T$ are equivalent:
=== Definition 1 ===
{{:Definition:Weight of Topological Space/Definition 1}}
=== Definition 2 ===
{{:Definition:Weight of Topological Space/Definition 2}} | By Class of All Cardinals is Subclass of Class of All Ordinals, the set:
:$M = \set {\card \BB: \BB \in \mathbb B}$
is a subclass of the class of all ordinals.
By Class of All Ordinals is Well-Ordered by Subset Relation:
:$M$ is well ordered by the $\subseteq$ relation.
By Class of All Ordinals is Well-Ordered by Subse... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $\mathbb B$ be the [[Definition:Set|set]] of all [[Definition:Analytic Basis|bases]] of $T$.
The following definitions of the [[Definition:Weight of Topological Space|weight of $T$]] are [[Definition:Logical Equivalence|equivalent]]:
=== [[Definit... | By [[Class of All Cardinals is Subclass of Class of All Ordinals]], the [[Definition:Set|set]]:
:$M = \set {\card \BB: \BB \in \mathbb B}$
is a [[Definition:Subclass|subclass]] of the [[Definition:Class of All Ordinals|class of all ordinals]].
By [[Class of All Ordinals is Well-Ordered by Subset Relation]]:
:$M$ is [[... | Equivalence of Definitions of Weight of Topological Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weight_of_Topological_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Weight_of_Topological_Space | [
"Weights of Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Set",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Weight of Topological Space",
"Definition:Logical Equivalence",
"Definition:Weight of Topological Space/Definition 1",
"Definition:Weight of Topological Space/Definition 2"
] | [
"Class of All Cardinals is Subclass of Class of All Ordinals",
"Definition:Set",
"Definition:Subclass",
"Definition:Class of All Ordinals",
"Class of All Ordinals is Well-Ordered by Subset Relation",
"Definition:Well-Ordering",
"Definition:Subset Relation",
"Class of All Ordinals is Well-Ordered by Su... |
proofwiki-10809 | Space is First-Countable iff Character not greater than Aleph 0 | Let $T$ be a topological space.
$T$ is first-countable {{iff}}:
:$\map \chi T \le \aleph_0$
where $\map \chi T$ denotes the character of $T$. | === Sufficient Condition ===
Let $T$ be first-countable.
By definition of first-countable:
:$\forall x \in T: \exists \BB \in \map {\mathbb B} x: \card \BB \le \aleph_0$
where $\map {\mathbb B} x$ denotes the set of all local bases at $x$.
Then by definition of character of a point:
:$\forall x \in T: \map \chi {x, T} ... | Let $T$ be a [[Definition:Topological Space|topological space]].
$T$ is [[Definition:First-Countable Space|first-countable]] {{iff}}:
:$\map \chi T \le \aleph_0$
where $\map \chi T$ denotes the [[Definition:Character of Topological Space|character]] of $T$. | === Sufficient Condition ===
Let $T$ be [[Definition:First-Countable Space|first-countable]].
By definition of [[Definition:First-Countable Space|first-countable]]:
:$\forall x \in T: \exists \BB \in \map {\mathbb B} x: \card \BB \le \aleph_0$
where $\map {\mathbb B} x$ denotes the set of all [[Definition:Local Basi... | Space is First-Countable iff Character not greater than Aleph 0 | https://proofwiki.org/wiki/Space_is_First-Countable_iff_Character_not_greater_than_Aleph_0 | https://proofwiki.org/wiki/Space_is_First-Countable_iff_Character_not_greater_than_Aleph_0 | [
"First-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:First-Countable Space",
"Definition:Character of Topological Space"
] | [
"Definition:First-Countable Space",
"Definition:First-Countable Space",
"Definition:Local Basis",
"Definition:Character of Point in Topological Space",
"Definition:Character of Topological Space",
"Definition:Character of Topological Space",
"Definition:Character of Point in Topological Space",
"Defin... |
proofwiki-10810 | Difference of Two Powers/Examples/Difference of Two Cubes | :$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$ | From Difference of Two Powers:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
The result follows directly by setting $n = 3$.
{{qed}} | :$x^3 - y^3 = \paren {x - y} \paren {x^2 + x y + y^2}$ | From [[Difference of Two Powers]]:
:$\ds a^n - b^n = \paren {a - b} \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} b^j$
The result follows directly by setting $n = 3$.
{{qed}} | Difference of Two Powers/Examples/Difference of Two Cubes | https://proofwiki.org/wiki/Difference_of_Two_Powers/Examples/Difference_of_Two_Cubes | https://proofwiki.org/wiki/Difference_of_Two_Powers/Examples/Difference_of_Two_Cubes | [
"Third Powers",
"Difference of Two Cubes",
"Examples of Use of Difference of Two Powers"
] | [] | [
"Difference of Two Powers"
] |
proofwiki-10811 | Existence of Subfamily of Cardinality not greater than Weight of Space and Unions Equal | Let $T$ be a topological space.
Let $\FF$ be a set of open sets of $T$.
There exists a subset $\GG \subseteq \FF$ such that:
:$\ds \bigcup \GG = \bigcup \FF$
and:
:$\card \GG \le \map w T$
where:
:$\map w T$ denotes the weight of $T$
:$\card \GG$ denotes the cardinality of $\GG$. | By definition of weight of $T$ there exists a basis $\BB$ of $T$ such that:
:$(1): \quad \card \BB = \map w T$
Let:
:$\BB_1 = \set {W \in \BB: \exists U \in \FF: W \subseteq U}$
By definition of subset:
:$\BB_1 \subseteq \BB$
Then by Subset implies Cardinal Inequality:
:$(2): \quad \card {\BB_1} \le \card \BB$
By defin... | Let $T$ be a [[Definition:Topological Space|topological space]].
Let $\FF$ be a [[Definition:Set of Sets|set]] of [[Definition:Open Set (Topology)|open sets]] of $T$.
There exists a [[Definition:Subset|subset]] $\GG \subseteq \FF$ such that:
:$\ds \bigcup \GG = \bigcup \FF$
and:
:$\card \GG \le \map w T$
where:
:$\... | By definition of [[Definition:Weight of Topological Space|weight]] of $T$ there exists a [[Definition:Analytic Basis|basis]] $\BB$ of $T$ such that:
:$(1): \quad \card \BB = \map w T$
Let:
:$\BB_1 = \set {W \in \BB: \exists U \in \FF: W \subseteq U}$
By definition of [[Definition:Subset|subset]]:
:$\BB_1 \subseteq \B... | Existence of Subfamily of Cardinality not greater than Weight of Space and Unions Equal | https://proofwiki.org/wiki/Existence_of_Subfamily_of_Cardinality_not_greater_than_Weight_of_Space_and_Unions_Equal | https://proofwiki.org/wiki/Existence_of_Subfamily_of_Cardinality_not_greater_than_Weight_of_Space_and_Unions_Equal | [
"Topology"
] | [
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Weight of Topological Space",
"Definition:Cardinality"
] | [
"Definition:Weight of Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Subset",
"Subset implies Cardinal Inequality",
"Axiom:Axiom of Choice",
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:By Hypothesis",
"Union of Subset of Family is... |
proofwiki-10812 | Set of Singletons is Smallest Basis of Discrete Space | Let $T = \struct {S, \tau}$ be a discrete topological space.
Let $\BB = \set {\set x : x \in S}$.
Then $\BB$ is the smallest basis of $T$.
That is:
:$\BB$ is a basis of $T$
and:
:for every basis $\CC$ of $T$, $\BB \subseteq \CC$. | By Basis for Discrete Topology $\BB$ is a basis of $T$.
It remains to be shown that $\BB$ is the smallest basis of $T$.
Let $\CC$ be a basis of $T$.
Let $A \in \BB$.
By definition of the set $\BB$:
:$\exists x \in S: A = \set x$
By definition of basis:
:$\exists B \in \CC: x \in B \subseteq A$
Then by Singleton of Elem... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete]] [[Definition:Topological Space|topological space]].
Let $\BB = \set {\set x : x \in S}$.
Then $\BB$ is the smallest [[Definition:Analytic Basis|basis]] of $T$.
That is:
:$\BB$ is a [[Definition:Analytic Basis|basis]] of $T$
and:
:for every [[De... | By [[Basis for Discrete Topology]] $\BB$ is a [[Definition:Analytic Basis|basis]] of $T$.
It remains to be shown that $\BB$ is the smallest [[Definition:Analytic Basis|basis]] of $T$.
Let $\CC$ be a [[Definition:Analytic Basis|basis]] of $T$.
Let $A \in \BB$.
By definition of the set $\BB$:
:$\exists x \in S: A = ... | Set of Singletons is Smallest Basis of Discrete Space | https://proofwiki.org/wiki/Set_of_Singletons_is_Smallest_Basis_of_Discrete_Space | https://proofwiki.org/wiki/Set_of_Singletons_is_Smallest_Basis_of_Discrete_Space | [
"Discrete Topologies"
] | [
"Definition:Discrete Topology",
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis"
] | [
"Basis for Discrete Topology",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Singleton of Element is Subset",
"Definition:Set Equality"
] |
proofwiki-10813 | Conditional and Inverse are not Equivalent | A conditional statement:
:$p \implies q$
is not logically equivalent to its inverse:
:$\lnot p \implies \lnot q$ | We apply the Method of Truth Tables to the proposition:
:$\paren {p \implies q} \iff \paren {\lnot p \implies \lnot q}$
$\begin{array}{|ccc|c|ccc|} \hline
p & \implies & q) & \iff & (\lnot & p & \implies & \lnot & q) \\
\hline
\F & \T & \F & \T & \T & \F & \T & \T & \F \\
\F & \T & \T & \F & \T & \F & \F & \F & \T \\
\... | A [[Definition:Conditional|conditional statement]]:
:$p \implies q$
is not [[Definition:Logical Equivalence|logically equivalent]] to its [[Definition:Inverse Statement|inverse]]:
:$\lnot p \implies \lnot q$ | We apply the [[Method of Truth Tables]] to the proposition:
:$\paren {p \implies q} \iff \paren {\lnot p \implies \lnot q}$
$\begin{array}{|ccc|c|ccc|} \hline
p & \implies & q) & \iff & (\lnot & p & \implies & \lnot & q) \\
\hline
\F & \T & \F & \T & \T & \F & \T & \T & \F \\
\F & \T & \T & \F & \T & \F & \F & \F & \T... | Conditional and Inverse are not Equivalent | https://proofwiki.org/wiki/Conditional_and_Inverse_are_not_Equivalent | https://proofwiki.org/wiki/Conditional_and_Inverse_are_not_Equivalent | [
"Conditional"
] | [
"Definition:Conditional",
"Definition:Logical Equivalence",
"Definition:Inverse Statement"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-10814 | Weight of Discrete Topology equals Cardinality of Space | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then:
:$\map w T = \size S$
where:
:$\map w T$ denotes the weight of $T$
:$\card S$ denotes the cardinality of $S$. | By Basis for Discrete Topology the set $\BB = \set {\set x: x \in S}$ is a basis of $T$.
By Set of Singletons is Smallest Basis of Discrete Space $\BB$ is smallest basis of $T$:
:for every basis $\CC$ of $T$, $\BB \subseteq \CC$.
Then by Subset implies Cardinal Inequality:
:for every basis $\CC$ of $T$, $\card \BB \le ... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete]] [[Definition:Topological Space|topological space]].
Then:
:$\map w T = \size S$
where:
:$\map w T$ denotes the [[Definition:Weight of Topological Space|weight]] of $T$
:$\card S$ denotes the [[Definition:Cardinality|cardinality]] of $S$. | By [[Basis for Discrete Topology]] the set $\BB = \set {\set x: x \in S}$ is a [[Definition:Analytic Basis|basis]] of $T$.
By [[Set of Singletons is Smallest Basis of Discrete Space]] $\BB$ is smallest [[Definition:Analytic Basis|basis]] of $T$:
:for every [[Definition:Analytic Basis|basis]] $\CC$ of $T$, $\BB \subset... | Weight of Discrete Topology equals Cardinality of Space | https://proofwiki.org/wiki/Weight_of_Discrete_Topology_equals_Cardinality_of_Space | https://proofwiki.org/wiki/Weight_of_Discrete_Topology_equals_Cardinality_of_Space | [
"Discrete Topologies",
"Weights of Topological Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Topological Space",
"Definition:Weight of Topological Space",
"Definition:Cardinality"
] | [
"Basis for Discrete Topology",
"Definition:Basis (Topology)/Analytic Basis",
"Set of Singletons is Smallest Basis of Discrete Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Subset implies Cardinal Inequality",
"Definition:Basis (Topology)/Analytic Ba... |
proofwiki-10815 | Cardinality of Set of Singletons | Let $S$ be a set.
Let $T = \set {\set x: x \in S}$ be the set of all singletons of elements of $S$.
Then:
:$\card T = \card S$
where $\card S$ denotes the cardinality of $S$. | Define a mapping $f: S \to T$:
:$\forall x \in S: \map f x = \set x$
By Singleton Equality:
:$\forall x, y \in S: \map f x = \map f y \implies x = y$
Then, by definition, $f$ is an injection.
By the definition of set $T$:
:$\forall y \in T: \exists x \in S: y = \map f x$
Then, by definition, $f$ is a surjection.
Hence,... | Let $S$ be a [[Definition:Set|set]].
Let $T = \set {\set x: x \in S}$ be the [[Definition:Set|set]] of all [[Definition:Singleton|singletons]] of [[Definition:Element|elements]] of $S$.
Then:
:$\card T = \card S$
where $\card S$ denotes the [[Definition:Cardinality|cardinality]] of $S$. | Define a mapping $f: S \to T$:
:$\forall x \in S: \map f x = \set x$
By [[Singleton Equality]]:
:$\forall x, y \in S: \map f x = \map f y \implies x = y$
Then, by definition, $f$ is an [[Definition:Injection|injection]].
By the definition of set $T$:
:$\forall y \in T: \exists x \in S: y = \map f x$
Then, by defini... | Cardinality of Set of Singletons | https://proofwiki.org/wiki/Cardinality_of_Set_of_Singletons | https://proofwiki.org/wiki/Cardinality_of_Set_of_Singletons | [
"Cardinals"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Singleton",
"Definition:Element",
"Definition:Cardinality"
] | [
"Singleton Equality",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Set Equivalence",
"Definition:Cardinality"
] |
proofwiki-10816 | Basis has Subset Basis of Cardinality equal to Weight of Space | Let $T = \struct {X, \tau}$ be a topological space.
Let $\BB$ be a basis of $T$.
Then there exists a basis $\BB_0$ of $T$ such that
:$\BB_0 \subseteq \BB$ and $\card {\BB_0} = \map w T$
where:
:$\card {\BB_0}$ denotes the cardinality of $\BB_0$
:$\map w T$ denotes the weight of $T$. | There are two cases:
:infinite weight
:finite weight. | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\BB$ be a [[Definition:Analytic Basis|basis]] of $T$.
Then there exists a [[Definition:Analytic Basis|basis]] $\BB_0$ of $T$ such that
:$\BB_0 \subseteq \BB$ and $\card {\BB_0} = \map w T$
where:
:$\card {\BB_0}$ denotes the [... | There are two cases:
:[[Definition:Infinite Set|infinite]] [[Definition:Weight of Topological Space|weight]]
:[[Definition:Finite Set|finite]] [[Definition:Weight of Topological Space|weight]]. | Basis has Subset Basis of Cardinality equal to Weight of Space | https://proofwiki.org/wiki/Basis_has_Subset_Basis_of_Cardinality_equal_to_Weight_of_Space | https://proofwiki.org/wiki/Basis_has_Subset_Basis_of_Cardinality_equal_to_Weight_of_Space | [
"Topology"
] | [
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Cardinality",
"Definition:Weight of Topological Space"
] | [
"Definition:Infinite Set",
"Definition:Weight of Topological Space",
"Definition:Finite Set",
"Definition:Weight of Topological Space",
"Definition:Infinite Set",
"Definition:Weight of Topological Space",
"Definition:Weight of Topological Space",
"Definition:Weight of Topological Space",
"Definition... |
proofwiki-10817 | Cardinality of Union not greater than Product | Let $\FF$ be a set of sets.
Let:
:$\size \FF \le \mathbf m$
where
:$\size \FF$ denotes the cardinality of $\FF$
:$\mathbf m$ is cardinal number (possibly infinite).
Let:
:$\forall A \in \FF: \size A \le \mathbf n$
where
:$\mathbf n$ is cardinal number (possibly infinite).
Then:
:$\ds \size {\bigcup \FF} \le \size {\ma... | $\FF = \O$ or $\FF = \set \O$ or $\O \ne \FF \ne \set \O$.
In case when $\FF = \O$ or $\FF = \set \O$:
{{begin-eqn}}
{{eqn | l = \size {\bigcup \FF}
| r = \size \O
| c = Union of Empty Set, Union of Singleton
}}
{{eqn | o = \le
| r = \size {\mathbf m \times \mathbf n}
| c = Subset implies Cardin... | Let $\FF$ be a [[Definition:Set of Sets|set of sets]].
Let:
:$\size \FF \le \mathbf m$
where
:$\size \FF$ denotes the [[Definition:Cardinality|cardinality]] of $\FF$
:$\mathbf m$ is [[Definition:Cardinal Number|cardinal number]] (possibly [[Definition:Infinite Set|infinite]]).
Let:
:$\forall A \in \FF: \size A \le \... | $\FF = \O$ or $\FF = \set \O$ or $\O \ne \FF \ne \set \O$.
In case when $\FF = \O$ or $\FF = \set \O$:
{{begin-eqn}}
{{eqn | l = \size {\bigcup \FF}
| r = \size \O
| c = [[Union of Empty Set]], [[Union of Singleton]]
}}
{{eqn | o = \le
| r = \size {\mathbf m \times \mathbf n}
| c = [[Subset imp... | Cardinality of Union not greater than Product | https://proofwiki.org/wiki/Cardinality_of_Union_not_greater_than_Product | https://proofwiki.org/wiki/Cardinality_of_Union_not_greater_than_Product | [
"Cardinals",
"Set Union",
"Cardinality"
] | [
"Definition:Set of Sets",
"Definition:Cardinality",
"Definition:Cardinal Number",
"Definition:Infinite Set",
"Definition:Cardinal Number",
"Definition:Infinite Set"
] | [
"Union of Empty Set",
"Union of Singleton",
"Subset implies Cardinal Inequality",
"Empty Set is Subset of All Sets",
"Surjection iff Cardinal Inequality",
"Cardinal of Cardinal Equal to Cardinal",
"Surjection iff Cardinal Inequality",
"Definition:Surjection",
"Definition:Surjection",
"Definition:S... |
proofwiki-10818 | Image of Mapping of Intersections is Smallest Basis | Let $T = \struct {X, \tau}$ be a topological space.
Let $f: X \to \tau$ be a mapping such that:
:$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$.
Then the image $\Img f$ is subset of every basis of $T$. | Let $\BB$ be a basis.
Let $V \in \Img f$.
Then by definition of image there exists a point $b \in X$ such that:
:$V = \map f b$
Then $V$ is open because $\Img f \subseteq \tau$.
By assumption of mapping $f$:
:$b \in V$
Then by definition of basis there exists a subset $U \in \BB$ such that:
:$b \in U \subseteq V$
By de... | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: X \to \tau$ be a mapping such that:
:$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$.
Then the [[Definition:Image of Mapping|image]] $\Img f$ is subset of every [[Def... | Let $\BB$ be a [[Definition:Analytic Basis|basis]].
Let $V \in \Img f$.
Then by definition of [[Definition:Image of Mapping|image]] there exists a [[Definition:Element|point]] $b \in X$ such that:
:$V = \map f b$
Then $V$ is [[Definition:Open Set (Topology)|open]] because $\Img f \subseteq \tau$.
By assumption of m... | Image of Mapping of Intersections is Smallest Basis | https://proofwiki.org/wiki/Image_of_Mapping_of_Intersections_is_Smallest_Basis | https://proofwiki.org/wiki/Image_of_Mapping_of_Intersections_is_Smallest_Basis | [
"Topology"
] | [
"Definition:Topological Space",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Basis (Topology)/Analytic Basis"
] | [
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Element",
"Definition:Open Set/Topology",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Subset",
"Definition:Set Equality"
] |
proofwiki-10819 | Cardinality of Image of Mapping of Intersections is not greater than Weight of Space | Let $T = \struct {X, \tau}$ be a topological space.
Let $f: X \to \tau$ be a mapping such that:
:$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$
Then the cardinality of the image of $f$ is no greater than the weight of $T$:
:$\card {\Img f} \le \map w T$ | By definition of weight, there exists a basis $\BB$ of $T$ such that:
:$\card \BB = \map w T$
By Image of Mapping of Intersections is Smallest Basis:
:$\Img f \subseteq \BB$
Thus by Subset implies Cardinal Inequality:
:$\card {\Img f} \le \card \BB = \map w T$
{{qed}} | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $f: X \to \tau$ be a [[Definition:Mapping|mapping]] such that:
:$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$
Then the [[Definition:Cardinality|cardinality]] of the [[D... | By definition of [[Definition:Weight of Topological Space|weight]], there exists a [[Definition:Analytic Basis|basis]] $\BB$ of $T$ such that:
:$\card \BB = \map w T$
By [[Image of Mapping of Intersections is Smallest Basis]]:
:$\Img f \subseteq \BB$
Thus by [[Subset implies Cardinal Inequality]]:
:$\card {\Img f} \l... | Cardinality of Image of Mapping of Intersections is not greater than Weight of Space | https://proofwiki.org/wiki/Cardinality_of_Image_of_Mapping_of_Intersections_is_not_greater_than_Weight_of_Space | https://proofwiki.org/wiki/Cardinality_of_Image_of_Mapping_of_Intersections_is_not_greater_than_Weight_of_Space | [
"Topology",
"Cardinals"
] | [
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Cardinality",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Weight of Topological Space"
] | [
"Definition:Weight of Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Image of Mapping of Intersections is Smallest Basis",
"Subset implies Cardinal Inequality"
] |
proofwiki-10820 | Finite Weight Space has Basis equal to Image of Mapping of Intersections | Let $T = \struct {X, \tau}$ be a topological space with finite weight.
Then there exist a basis $\BB$ of $T$ and a mapping $f:X \to \tau$ such that:
:$\BB = \Img f$ and
:$\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$
where $\Img f$ denotes the image of $f$. | By definition of weight there exists a basis $\BB$ such that:
:$\card \BB = \map w T$
where:
:$\map w T$ denotes the weight of $T$
:$\card \BB$ denotes the cardinality of $\BB$.
By assumption that weight is finite:
:$\card \BB$ is finite
Then by Cardinality of Set is Finite iff Set is Finite:
:$\BB$ is finite
Define a ... | Let $T = \struct {X, \tau}$ be a [[Definition:Topological Space|topological space]] with [[Definition:Finite Set|finite]] [[Definition:Weight of Topological Space|weight]].
Then there exist a [[Definition:Analytic Basis|basis]] $\BB$ of $T$ and a mapping $f:X \to \tau$ such that:
:$\BB = \Img f$ and
:$\forall x \in X:... | By definition of [[Definition:Weight of Topological Space|weight]] there exists a [[Definition:Analytic Basis|basis]] $\BB$ such that:
:$\card \BB = \map w T$
where:
:$\map w T$ denotes the [[Definition:Weight of Topological Space|weight]] of $T$
:$\card \BB$ denotes the [[Definition:Cardinality|cardinality]] of $\BB$.... | Finite Weight Space has Basis equal to Image of Mapping of Intersections | https://proofwiki.org/wiki/Finite_Weight_Space_has_Basis_equal_to_Image_of_Mapping_of_Intersections | https://proofwiki.org/wiki/Finite_Weight_Space_has_Basis_equal_to_Image_of_Mapping_of_Intersections | [
"Topology"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Weight of Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Image (Set Theory)/Mapping/Mapping"
] | [
"Definition:Weight of Topological Space",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Weight of Topological Space",
"Definition:Cardinality",
"Cardinality of Set is Finite iff Set is Finite",
"Definition:Subset",
"Subset of Finite Set is Finite",
"General Intersection Property of Topolog... |
proofwiki-10821 | Rubik's Cube has 54 Facets | Let $S$ be the set of facets of Rubik's cube.
Then the cardinality of $S$ is given by:
:$\card S = 54$
That is:
:A Rubik's cube has $54$ facets. | A cube, by definition, has $6$ faces.
Each face is subdivided into $9$ facets.
Hence there are $6 \times 9 = 54$ facets in total.
{{qed}} | Let $S$ be the [[Definition:Set|set]] of [[Definition:Facet of Rubik's Cube|facets]] of [[Definition:Rubik's Cube|Rubik's cube]].
Then the [[Definition:Cardinality of Finite Set|cardinality]] of $S$ is given by:
:$\card S = 54$
That is:
:A [[Definition:Rubik's Cube|Rubik's cube]] has $54$ [[Definition:Facet of Rubik... | A [[Definition:Cube (Geometry)|cube]], by definition, has $6$ [[Definition:Face of Polyhedron|faces]].
Each [[Definition:Face of Polyhedron|face]] is subdivided into $9$ [[Definition:Facet of Rubik's Cube|facets]].
Hence there are $6 \times 9 = 54$ [[Definition:Facet of Rubik's Cube|facets]] in total.
{{qed}} | Rubik's Cube has 54 Facets | https://proofwiki.org/wiki/Rubik's_Cube_has_54_Facets | https://proofwiki.org/wiki/Rubik's_Cube_has_54_Facets | [
"Rubik's Cube"
] | [
"Definition:Set",
"Definition:Rubik's Cube/Facet",
"Definition:Rubik's Cube",
"Definition:Cardinality/Finite",
"Definition:Rubik's Cube",
"Definition:Rubik's Cube/Facet"
] | [
"Definition:Cube/Geometry",
"Definition:Polyhedron/Face",
"Definition:Polyhedron/Face",
"Definition:Rubik's Cube/Facet",
"Definition:Rubik's Cube/Facet"
] |
proofwiki-10822 | Equivalence of Definitions of Symmetric Difference/(3) iff (5) | Let $S$ and $T$ be sets.
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | o =
| r = x \in S \symdif T
}}
{{eqn | o = \leadstoandfrom
| r = x \in S \oplus x \in T
| c = {{Defof|Symmetric Difference|index = 5}}
}}
{{eqn | r = \paren {\neg \paren {x \in S} \land \paren {x \in T} } \lor \paren {\paren {x \in S} \land \neg \paren {x \in T} }
| o = \l... | Let $S$ and $T$ be [[Definition:Set|sets]].
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | o =
| r = x \in S \symdif T
}}
{{eqn | o = \leadstoandfrom
| r = x \in S \oplus x \in T
| c = {{Defof|Symmetric Difference|index = 5}}
}}
{{eqn | r = \paren {\neg \paren {x \in S} \land \paren {x \in T} } \lor \paren {\paren {x \in S} \land \neg \paren {x \in T} }
| o = \l... | Equivalence of Definitions of Symmetric Difference/(3) iff (5) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(3)_iff_(5) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(3)_iff_(5) | [
"Equivalence of Definitions of Symmetric Difference"
] | [
"Definition:Set"
] | [
"Non-Equivalence as Disjunction of Conjunctions",
"Union is Commutative",
"Definition:Set Equality"
] |
proofwiki-10823 | Equivalence of Definitions of Symmetric Difference/(2) iff (5) | Let $S$ and $T$ be sets.
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | o =
| r = x \in S \symdif T
}}
{{eqn | o = \leadstoandfrom
| r = x \in S \oplus x \in T
| c = {{Defof|Symmetric Difference|index = 5}}
}}
{{eqn | r = \paren {x \in S \lor x \in T} \land \neg \paren {x \in S \land x \in T}
| o = \leadstoandfrom
| c = {{Defof|Exclusiv... | Let $S$ and $T$ be [[Definition:Set|sets]].
{{TFAENocat|def = Symmetric Difference|view = symmetric difference $S \symdif T$ between $S$ and $T$}} | {{begin-eqn}}
{{eqn | o =
| r = x \in S \symdif T
}}
{{eqn | o = \leadstoandfrom
| r = x \in S \oplus x \in T
| c = {{Defof|Symmetric Difference|index = 5}}
}}
{{eqn | r = \paren {x \in S \lor x \in T} \land \neg \paren {x \in S \land x \in T}
| o = \leadstoandfrom
| c = {{Defof|Exclusiv... | Equivalence of Definitions of Symmetric Difference/(2) iff (5) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(2)_iff_(5) | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Symmetric_Difference/(2)_iff_(5) | [
"Equivalence of Definitions of Symmetric Difference"
] | [
"Definition:Set"
] | [
"Definition:Set Equality"
] |
proofwiki-10824 | Partition of Facets of Rubik's Cube | Let $S$ denote the set of the facets of Rukik's cube.
Then $S$ can be partitioned as follows:
:$S = \set {S_C \mid S_E \mid S_Z}$
where:
:$S_C$ denotes the set of corner facets
:$S_E$ denotes the set of edge facets
:$S_Z$ denotes the set of center facets. | From the definition of the facets, each face is divided into $9$ facets.
:600px
A facet is either:
:on the corner of a face, for example $flu$, $fru$
:on the edge of a face, for example $fu$, $fr$
:in the center of a face, for example $F$.
:$(1):\quad$ Each facet can be either in $S_C$ or $S_E$ or $S_Z$ and can not be ... | Let $S$ denote the [[Definition:Set|set]] of the [[Definition:Facet of Rubik's Cube|facets]] of [[Definition:Rubik's Cube|Rukik's cube]].
Then $S$ can be [[Definition:Set Partition|partitioned]] as follows:
:$S = \set {S_C \mid S_E \mid S_Z}$
where:
:$S_C$ denotes the [[Definition:Set|set]] of [[Definition:Corner Face... | From the definition of the [[Definition:Facet of Rubik's Cube|facets]], each [[Definition:Face of Rubik's Cube|face]] is divided into $9$ [[Definition:Facet of Rubik's Cube|facets]].
:[[File:RubiksCubeFacets.png|600px]]
A [[Definition:Facet of Rubik's Cube|facet]] is either:
:on the corner of a [[Definition:Face of R... | Partition of Facets of Rubik's Cube | https://proofwiki.org/wiki/Partition_of_Facets_of_Rubik's_Cube | https://proofwiki.org/wiki/Partition_of_Facets_of_Rubik's_Cube | [
"Rubik's Cube"
] | [
"Definition:Set",
"Definition:Rubik's Cube/Facet",
"Definition:Rubik's Cube",
"Definition:Set Partition",
"Definition:Set",
"Definition:Rubik's Cube/Corner Facet",
"Definition:Set",
"Definition:Rubik's Cube/Edge Facet",
"Definition:Set",
"Definition:Rubik's Cube/Center Facet"
] | [
"Definition:Rubik's Cube/Facet",
"Definition:Rubik's Cube/Face",
"Definition:Rubik's Cube/Facet",
"File:RubiksCubeFacets.png",
"Definition:Rubik's Cube/Facet",
"Definition:Rubik's Cube/Face",
"Definition:Rubik's Cube/Face",
"Definition:Rubik's Cube/Face",
"Definition:Rubik's Cube/Facet",
"Definiti... |
proofwiki-10825 | Even and Odd Integers form Partition of Integers | The odd integers and even integers form a partition of the integers. | Let $n \in \Z$ be an integer.
Let $\Bbb O$ be the set of odd integers and $\Bbb E$ be the set of even integers.
By the Division Theorem:
:$\forall n \in \Z: \exists! q, r \in \Z: n = 2 q + r, 0 \le r < 2$
from which it follows that either:
:$n = 2 q \in \Bbb E$
or:
:$n = 2 q + 1 \in \Bbb O$
Thus:
:$(1): \quad$ each ele... | The [[Definition:Odd Integer|odd integers]] and [[Definition:Even Integer|even integers]] form a [[Definition:Set Partition|partition]] of the [[Definition:Integer|integers]]. | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Let $\Bbb O$ be the set of [[Definition:Odd Integer|odd integers]] and $\Bbb E$ be the set of [[Definition:Even Integer|even integers]].
By the [[Division Theorem]]:
:$\forall n \in \Z: \exists! q, r \in \Z: n = 2 q + r, 0 \le r < 2$
from which it follows that eit... | Even and Odd Integers form Partition of Integers | https://proofwiki.org/wiki/Even_and_Odd_Integers_form_Partition_of_Integers | https://proofwiki.org/wiki/Even_and_Odd_Integers_form_Partition_of_Integers | [
"Set Partitions",
"Odd Integers",
"Even Integers",
"Integers",
"2"
] | [
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Set Partition",
"Definition:Integer"
] | [
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Even Integer",
"Division Theorem",
"Definition:Element",
"Definition:Element",
"Definition:Empty Set",
"Definition:Set Partition"
] |
proofwiki-10826 | Analog between Logic and Set Theory | The concepts of set theory have directly corresponding concepts in logic:
:{| border = "1"
|-
! style="padding: 2px 10px" | Set Theory
! style="padding: 2px 10px" | Logic
|-
| align="left" style="padding: 2px 10px"| Set: $S, T$
| align="left" style="padding: 2px 10px"| Statement: $p, q$
|-
| align="left" style="padding... | Let $P$ and $Q$ be propositional functions.
Let $S$ and $T$ be subsets of a universe $\Bbb U$ such that:
:$S = \set {x \in \Bbb U: \map P x}$
:$T = \set {x \in \Bbb U: \map Q x}$
By the following definitions:
{{begin-axiom}}
{{axiom | n = 1
| lc= Intersection:
| ml= S \cap T
| mo= :=
| m... | The concepts of [[Definition:Set Theory|set theory]] have directly corresponding concepts in [[Definition:Logic|logic]]:
:{| border = "1"
|-
! style="padding: 2px 10px" | Set Theory
! style="padding: 2px 10px" | Logic
|-
| align="left" style="padding: 2px 10px"| [[Definition:Set|Set]]: $S, T$
| align="left" style="pad... | Let $P$ and $Q$ be [[Definition:Propositional Function|propositional functions]].
Let $S$ and $T$ be [[Definition:Subset|subsets]] of a [[Definition:Universal Set|universe]] $\Bbb U$ such that:
:$S = \set {x \in \Bbb U: \map P x}$
:$T = \set {x \in \Bbb U: \map Q x}$
By the following definitions:
{{begin-axiom}}
{{... | Analog between Logic and Set Theory | https://proofwiki.org/wiki/Analog_between_Logic_and_Set_Theory | https://proofwiki.org/wiki/Analog_between_Logic_and_Set_Theory | [
"Set Theory",
"Logic"
] | [
"Definition:Set Theory",
"Definition:Logic",
"Definition:Set",
"Definition:Statement",
"Definition:Set Union",
"Definition:Disjunction",
"Definition:Set Intersection",
"Definition:Conjunction",
"Definition:Subset",
"Definition:Conditional",
"Definition:Symmetric Difference",
"Definition:Exclus... | [
"Definition:Propositional Function",
"Definition:Subset",
"Definition:Universal Set",
"Definition:Set Intersection",
"Definition:Set Union",
"Definition:Subset",
"Definition:Symmetric Difference",
"Definition:Set Complement",
"Definition:Set Equality"
] |
proofwiki-10827 | Euler Phi Function of 1 | :$\map \phi 1 = 1$ | The only (strictly) positive integer less than or equal to $1$ is $1$ itself.
By Integer is Coprime to 1, $1$ is coprime to itself.
Hence, by definition, there is exactly $1$ integer less than or equal to $1$ which is coprime with $1$.
Hence the result.
{{qed}} | :$\map \phi 1 = 1$ | The only [[Definition:Strictly Positive Integer|(strictly) positive integer]] less than or equal to $1$ is $1$ itself.
By [[Integer is Coprime to 1]], $1$ is [[Definition:Coprime Integers|coprime]] to itself.
Hence, by definition, there is exactly $1$ [[Definition:Integer|integer]] less than or equal to $1$ which is ... | Euler Phi Function of 1 | https://proofwiki.org/wiki/Euler_Phi_Function_of_1 | https://proofwiki.org/wiki/Euler_Phi_Function_of_1 | [
"Examples of Euler Phi Function",
"1"
] | [] | [
"Definition:Strictly Positive/Integer",
"Integer is Coprime to 1",
"Definition:Coprime/Integers",
"Definition:Integer",
"Definition:Coprime/Integers"
] |
proofwiki-10828 | Cardinality of Set is Finite iff Set is Finite | Let $A$ be a set.
:$\card A$ is finite
{{iff}}:
:$A$ is finite
where $\card A$ denotes the cardinality of $A$. | Definition of cardinal:
:$(1): \quad \card A \sim A$
:$\card A$ is finite
{{iff}}:
:$\exists n \in \N: \card A \sim \N_n$ by definition of finite set
{{iff}}:
:$\exists n \in \N: A \sim \N_n$ by $(1)$ and Set Equivalence behaves like Equivalence Relation
{{iff}}:
:$A$ is finite by definition of finite set.
{{qed}} | Let $A$ be a [[Definition:Set|set]].
:$\card A$ is [[Definition:Finite Set|finite]]
{{iff}}:
:$A$ is [[Definition:Finite Set|finite]]
where $\card A$ denotes the [[Definition:Cardinality|cardinality]] of $A$. | Definition of [[Definition:Cardinal|cardinal]]:
:$(1): \quad \card A \sim A$
:$\card A$ is [[Definition:Finite Set|finite]]
{{iff}}:
:$\exists n \in \N: \card A \sim \N_n$ by definition of [[Definition:Finite Set|finite set]]
{{iff}}:
:$\exists n \in \N: A \sim \N_n$ by $(1)$ and [[Set Equivalence behaves like Equiv... | Cardinality of Set is Finite iff Set is Finite | https://proofwiki.org/wiki/Cardinality_of_Set_is_Finite_iff_Set_is_Finite | https://proofwiki.org/wiki/Cardinality_of_Set_is_Finite_iff_Set_is_Finite | [
"Cardinals"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Cardinality"
] | [
"Definition:Cardinal",
"Definition:Finite Set",
"Definition:Finite Set",
"Set Equivalence behaves like Equivalence Relation",
"Definition:Finite Set",
"Definition:Finite Set"
] |
proofwiki-10829 | Multiplication using Parabola | :500pxrightthumb
Let the parabola $P$ defined as $y = x^2$ be plotted on the Cartesian plane.
Let $A = \tuple {x_a, y_a}$ and $B = \tuple {x_b, y_b}$ be points on the curve $\map f x$ so that $x_a < x_b$.
Then the line segment joining $A B$ will cross the $y$-axis at $-x_a x_b$.
Thus $P$ can be used as a nomogram to ca... | Let $\map f x = x^2$.
Then:
:$\map f {x_a} = x_a^2$
and:
:$\map f {x_b} = x_b^2$
Then the slope $m$ of the line segment joining $A B$ will be:
{{begin-eqn}}
{{eqn | l = m
| r = \frac {x_b^2 - x_a^2} {x_b - x_a}
| c = Equation of Straight Line in Plane: Point-Slope Form
}}
{{eqn | r = \frac {\paren {x_b - x... | :[[File:Multiplication-using-Parabola.png|500px|right|thumb]]
Let the [[Definition:Parabola|parabola]] $P$ defined as $y = x^2$ be plotted on the [[Definition:Cartesian Plane|Cartesian plane]].
Let $A = \tuple {x_a, y_a}$ and $B = \tuple {x_b, y_b}$ be [[Definition:Point|points]] on the curve $\map f x$ so that $x_a... | Let $\map f x = x^2$.
Then:
:$\map f {x_a} = x_a^2$
and:
:$\map f {x_b} = x_b^2$
Then the [[Definition:Slope of Straight Line|slope]] $m$ of the [[Definition:Line Segment|line segment]] joining $A B$ will be:
{{begin-eqn}}
{{eqn | l = m
| r = \frac {x_b^2 - x_a^2} {x_b - x_a}
| c = [[Equation of Str... | Multiplication using Parabola | https://proofwiki.org/wiki/Multiplication_using_Parabola | https://proofwiki.org/wiki/Multiplication_using_Parabola | [
"Multiplication",
"Parabolas",
"Nomograms"
] | [
"File:Multiplication-using-Parabola.png",
"Definition:Parabola",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Line/Segment",
"Definition:Axis/Y-Axis",
"Definition:Nomogram",
"Definition:Multiplication/Real Numbers",
"Definition:Real Number",
"Definition:Axis/X-Axis",
"Definition... | [
"Definition:Slope/Straight Line",
"Definition:Line/Segment",
"Equation of Straight Line in Plane/Point-Slope Form",
"Difference of Two Squares",
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Intercept",
"Definition:Coordinate System/Coordinate"
] |
proofwiki-10830 | Equivalence of Definitions of Countably Infinite Set | Let $S$ be a set.
{{TFAE|def = Countably Infinite Set}} | From Integers are Countably Infinite there is a bijection between $\Z$, the set of integers, and $\N$, the set of natural numbers.
Let $h: \N \to \Z$ be such a bijection.
Let $f: S \to \N$ be a bijection.
From Composite of Bijections is Bijection:
:$h \circ f: S \to \Z$ is a bijection.
Similarly, let $g: S \to \Z$ be a... | Let $S$ be a [[Definition:Set|set]].
{{TFAE|def = Countably Infinite Set}} | From [[Integers are Countably Infinite]] there is a [[Definition:Bijection|bijection]] between $\Z$, the [[Definition:Integer|set of integers]], and $\N$, the [[Definition:Natural Numbers|set of natural numbers]].
Let $h: \N \to \Z$ be such a [[Definition:Bijection|bijection]].
Let $f: S \to \N$ be a [[Definition:Bi... | Equivalence of Definitions of Countably Infinite Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Countably_Infinite_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Countably_Infinite_Set | [
"Countable Sets"
] | [
"Definition:Set"
] | [
"Integers are Countably Infinite",
"Definition:Bijection",
"Definition:Integer",
"Definition:Natural Numbers",
"Definition:Bijection",
"Definition:Bijection",
"Composite of Bijections is Bijection",
"Definition:Bijection",
"Definition:Bijection",
"Inverse of Bijection is Bijection",
"Definition:... |
proofwiki-10831 | Set of Odd Integers is Countably Infinite | Let $\Bbb O$ be the set of odd integers.
Then $\Bbb O$ is countably infinite. | Let $f: \Bbb O \to \Z$ be the mapping defined as:
:$\forall x \in \Bbb O: \map f x = \dfrac {x + 1} 2$
$f$ is well-defined as $x + 1$ is even and so $\dfrac {x + 1} 2 \in \Z$.
Let $x, y \in \Bbb O$ such that $\map f x = \map f y$.
Then:
{{begin-eqn}}
{{eqn | l = \map f x
| r = \map f y
| c =
}}
{{eqn | ll=... | Let $\Bbb O$ be the [[Definition:Set|set]] of [[Definition:Odd Integer|odd integers]].
Then $\Bbb O$ is [[Definition:Countably Infinite Set|countably infinite]]. | Let $f: \Bbb O \to \Z$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x \in \Bbb O: \map f x = \dfrac {x + 1} 2$
$f$ is [[Definition:Well-Defined Mapping|well-defined]] as $x + 1$ is [[Definition:Even Integer|even]] and so $\dfrac {x + 1} 2 \in \Z$.
Let $x, y \in \Bbb O$ such that $\map f x = \map f y$.
... | Set of Odd Integers is Countably Infinite | https://proofwiki.org/wiki/Set_of_Odd_Integers_is_Countably_Infinite | https://proofwiki.org/wiki/Set_of_Odd_Integers_is_Countably_Infinite | [
"Countable Sets",
"Odd Integers"
] | [
"Definition:Set",
"Definition:Odd Integer",
"Definition:Countably Infinite/Set"
] | [
"Definition:Mapping",
"Definition:Well-Defined/Mapping",
"Definition:Even Integer",
"Definition:Injection",
"Definition:Inverse of Mapping",
"Definition:Well-Defined/Mapping",
"Definition:odd Integer",
"Definition:Mapping",
"Definition:Injection",
"Cantor-Bernstein-Schröder Theorem",
"Definition... |
proofwiki-10832 | Unique Readability for Prefix Notation | Let $\AA$ be an alphabet.
Then prefix notation for $\AA$ has the unique readability property. | Let $\phi$ be a WFF of prefix notation for $\AA$.
Apply the Principle of Mathematical Induction on the length of $\phi$ to prove:
:$(1): \quad$ No prefix of $\phi$ is a WFF, except $\phi$ itself;
:$(2): \quad$ If the first symbol of $\phi$ has arity $n$, then there exist unique WFFs $\phi_1, \ldots, \phi_n$ such that $... | Let $\AA$ be an [[Definition:Alphabet of Formal Language|alphabet]].
Then [[Definition:Prefix Notation|prefix notation]] for $\AA$ has the [[Definition:Unique Readability Property|unique readability property]]. | Let $\phi$ be a [[Definition:WFF|WFF]] of [[Definition:Prefix Notation|prefix notation]] for $\AA$.
Apply the [[Principle of Mathematical Induction]] on the [[Definition:Length of Sequence|length]] of $\phi$ to prove:
:$(1): \quad$ No [[Definition:Prefix|prefix]] of $\phi$ is a [[Definition:WFF|WFF]], except $\phi$ i... | Unique Readability for Prefix Notation | https://proofwiki.org/wiki/Unique_Readability_for_Prefix_Notation | https://proofwiki.org/wiki/Unique_Readability_for_Prefix_Notation | [
"Prefix Notation",
"Collations"
] | [
"Definition:Formal Language/Alphabet",
"Definition:Operation/Binary Operation/Prefix Notation",
"Definition:Collation/Unique Readability"
] | [
"Symbols:Abbreviations/W/WFF",
"Definition:Operation/Binary Operation/Prefix Notation",
"Principle of Mathematical Induction",
"Definition:Length of Sequence",
"Definition:Prefix",
"Symbols:Abbreviations/W/WFF",
"Definition:Symbol",
"Definition:Operation/Arity",
"Definition:Unique",
"Symbols:Abbre... |
proofwiki-10833 | Identity Matrix is Permutation Matrix | An identity matrix is an example of a permutation matrix. | An identity matrix, by definition, has instances of $1$ on the main diagonal and $0$ elsewhere.
Each diagonal element is by definition on one row and one column of the matrix.
Also by definition, each diagonal element is on a different row and
column from each other diagonal element.
The result follows by definition o... | An [[Definition:Identity Matrix|identity matrix]] is an example of a [[Definition:Permutation Matrix|permutation matrix]]. | An [[Definition:Identity Matrix|identity matrix]], by definition, has instances of $1$ on the [[Definition:Main Diagonal|main diagonal]] and $0$ elsewhere.
Each [[Definition:Diagonal Element|diagonal element]] is by definition on one [[Definition:Row of Matrix|row]] and one [[Definition:Column of Matrix|column]] of th... | Identity Matrix is Permutation Matrix | https://proofwiki.org/wiki/Identity_Matrix_is_Permutation_Matrix | https://proofwiki.org/wiki/Identity_Matrix_is_Permutation_Matrix | [
"Unit Matrices",
"Permutation Matrices"
] | [
"Definition:Unit Matrix",
"Definition:Permutation Matrix"
] | [
"Definition:Unit Matrix",
"Definition:Matrix/Diagonal/Main",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Matrix/Square Matrix",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Def... |
proofwiki-10834 | Full Rook Matrix is Nonsingular | A full rook matrix is nonsingular. | Let $\mathbf A$ be a full rook matrix.
By definition, $\mathbf A$ is an instance of a permutation matrix.
By Determinant of Permutation Matrix, it follows that $\det \mathbf A = \pm 1$.
By Matrix is Nonsingular iff Determinant has Multiplicative Inverse:
:$\mathbf A$ is nonsingular.
{{qed}} | A [[Definition:Full Rook Matrix|full rook matrix]] is [[Definition:Nonsingular Matrix|nonsingular]]. | Let $\mathbf A$ be a [[Definition:Full Rook Matrix|full rook matrix]].
By definition, $\mathbf A$ is an instance of a [[Definition:Permutation Matrix|permutation matrix]].
By [[Determinant of Permutation Matrix]], it follows that $\det \mathbf A = \pm 1$.
By [[Matrix is Nonsingular iff Determinant has Multiplicative... | Full Rook Matrix is Nonsingular | https://proofwiki.org/wiki/Full_Rook_Matrix_is_Nonsingular | https://proofwiki.org/wiki/Full_Rook_Matrix_is_Nonsingular | [
"Full Rook Matrices",
"Nonsingular Matrices"
] | [
"Definition:Full Rook Matrix",
"Definition:Nonsingular Matrix"
] | [
"Definition:Full Rook Matrix",
"Definition:Permutation Matrix",
"Determinant of Permutation Matrix",
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Definition:Nonsingular Matrix"
] |
proofwiki-10835 | Product of Rook Matrices is Rook Matrix | Let $\mathbf A$ and $\mathbf B$ be rook matrices.
Their product $\mathbf {A B}$ is also a rook matrix. | An element $a b_{ij}$ of $\mathbf {A B}$ is formed by multiplying each element of row $i$ of $\mathbf A$ by its corresponding element of column $j$ of $\mathbf B$.
No more than $1$ element of row $i$ equals $1$, and the rest equal $0$.
No more than $1$ column $k$ of $\mathbf B$ contains $1$ in its $i$th element, and th... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Rook Matrix|rook matrices]].
Their [[Definition:Matrix Product (Conventional)|product]] $\mathbf {A B}$ is also a [[Definition:Rook Matrix|rook matrix]]. | An [[Definition:Element of Matrix|element]] $a b_{ij}$ of $\mathbf {A B}$ is formed by [[Definition:Real Multiplication|multiplying]] each [[Definition:Element of Matrix|element]] of [[Definition:Row of Matrix|row]] $i$ of $\mathbf A$ by its corresponding [[Definition:Element of Matrix|element]] of [[Definition:Column ... | Product of Rook Matrices is Rook Matrix | https://proofwiki.org/wiki/Product_of_Rook_Matrices_is_Rook_Matrix | https://proofwiki.org/wiki/Product_of_Rook_Matrices_is_Rook_Matrix | [
"Rook Matrices"
] | [
"Definition:Rook Matrix",
"Definition:Matrix Product (Conventional)",
"Definition:Rook Matrix"
] | [
"Definition:Matrix/Element",
"Definition:Multiplication/Real Numbers",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"Definition:Matrix/Element",
"Definition:Matrix/Column",
"Definition:Matrix/Element",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Matrix/Element",
"De... |
proofwiki-10836 | Topology Defined by Basis | Let $S$ be a set.
Let $\BB$ be a set of subsets of $S$.
Suppose that
:$(\text B1): \quad \forall A_1, A_2 \in \BB: \forall x \in A_1 \cap A_2: \exists A \in \BB: x \in A \subseteq A_1 \cap A_2$
:$(\text B2): \quad \forall x \in X: \exists A \in \BB: x \in A$
::$\tau = \set {\bigcup \GG: \GG \subseteq \BB}$
Then:
:$T = ... | We have to prove Open Set Axioms: | Let $S$ be a [[Definition:Set|set]].
Let $\BB$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $S$.
Suppose that
:$(\text B1): \quad \forall A_1, A_2 \in \BB: \forall x \in A_1 \cap A_2: \exists A \in \BB: x \in A \subseteq A_1 \cap A_2$
:$(\text B2): \quad \forall x \in X: \exists A \in \BB: ... | We have to prove [[Axiom:Open Set Axioms|Open Set Axioms]]: | Topology Defined by Basis | https://proofwiki.org/wiki/Topology_Defined_by_Basis | https://proofwiki.org/wiki/Topology_Defined_by_Basis | [
"Topological Bases"
] | [
"Definition:Set",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Topological Space",
"Definition:Basis (Topology)/Analytic Basis"
] | [
"Axiom:Open Set Axioms",
"Axiom:Open Set Axioms"
] |
proofwiki-10837 | Equivalence of Definitions of Singular Matrix | {{TFAE|def = Singular Matrix}}
Let $\struct {R, +, \circ}$ be a ring with unity.
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\mathbf A$ be an element of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$. | Follows directly from Matrix is Nonsingular iff Determinant has Multiplicative Inverse.
{{qed}}
Category:Singular Matrices
avq5fidzrgypqx4rm6rl12cyoniwy22 | {{TFAE|def = Singular Matrix}}
Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]].
Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\mathbf A$ be an [[Definition:Element|element]] of the [[Definition:Ring of Square Matrices|ring of squa... | Follows directly from [[Matrix is Nonsingular iff Determinant has Multiplicative Inverse]].
{{qed}}
[[Category:Singular Matrices]]
avq5fidzrgypqx4rm6rl12cyoniwy22 | Equivalence of Definitions of Singular Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Singular_Matrix | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Singular_Matrix | [
"Singular Matrices"
] | [
"Definition:Ring with Unity",
"Definition:Strictly Positive/Integer",
"Definition:Element",
"Definition:Ring of Square Matrices"
] | [
"Matrix is Nonsingular iff Determinant has Multiplicative Inverse",
"Category:Singular Matrices"
] |
proofwiki-10838 | Matrix is Singular iff Product with Non-Zero Vector is Zero | Let $\mathbb F$ be a field.
Let $\mathbf A$ be a square matrix of order $n$ over $\mathbb F$.
Then $\mathbf A$ is singular {{iff}} there exists a vector $\mathbf v \in \mathbb F^n$:
:$\mathbf v \ne \mathbf 0$
:$\mathbf A \mathbf v = \mathbf 0$
where $\mathbf 0$ is the zero vector. | === Sufficient Case ===
Aiming for a Proof by Contraposition, suppose that
:$\neg \exists \mathbf v \in \mathbb F^n : \mathbf v \neq \mathbf 0 \wedge \mathbf A \mathbf v = \mathbf 0$
Then, for any two vectors $\mathbf p, \mathbf q \in \mathbb F^n$,
{{begin-eqn}}
{{eqn | l = \mathbf A \mathbf p
| r = \mathbf A \ma... | Let $\mathbb F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Square Matrix|order $n$]] over $\mathbb F$.
Then $\mathbf A$ is [[Definition:Singular Matrix|singular]] {{iff}} there exists a [[Definition:Vector (Linear Algebr... | === Sufficient Case ===
Aiming for a [[Proof by Contraposition]], suppose that
:$\neg \exists \mathbf v \in \mathbb F^n : \mathbf v \neq \mathbf 0 \wedge \mathbf A \mathbf v = \mathbf 0$
Then, for any two [[Definition:Vector|vectors]] $\mathbf p, \mathbf q \in \mathbb F^n$,
{{begin-eqn}}
{{eqn | l = \mathbf A \mathb... | Matrix is Singular iff Product with Non-Zero Vector is Zero | https://proofwiki.org/wiki/Matrix_is_Singular_iff_Product_with_Non-Zero_Vector_is_Zero | https://proofwiki.org/wiki/Matrix_is_Singular_iff_Product_with_Non-Zero_Vector_is_Zero | [
"Singular Matrices",
"Proofs by Contraposition"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Singular Matrix",
"Definition:Vector/Linear Algebra",
"Definition:Zero Vector"
] | [
"Proof by Contraposition",
"Definition:Vector",
"Matrix Multiplication Distributes over Matrix Addition",
"Definition:Matrix Product",
"Definition:Injection",
"Definition:Block Multiplication",
"Definition:Matrix Product",
"Definition:Matrix/Column",
"Definition:Vector Quantity/Component",
"Defini... |
proofwiki-10839 | Equivalence of Definitions of Integer Congruence | Let $m \in \Z_{> 0}$.
{{TFAE|def = Congruence Modulo Integer|view = congruence modulo $m$}} | Let $x_1, x_2, z \in \Z$.
Let $x_1 \equiv x_2 \pmod z$ as defined by the equal remainder after division:
:$\RR_z = \set {\tuple {x, y} \in \Z \times \Z: \exists k \in \Z: x = y + k z}$
Let $\tuple {x_1, x_2} \in \RR_z$.
Then by definition:
:$\exists k \in \Z: x_1 = x_2 + k z$
So, by definition of the modulo operation:
... | Let $m \in \Z_{> 0}$.
{{TFAE|def = Congruence Modulo Integer|view = congruence modulo $m$}} | Let $x_1, x_2, z \in \Z$.
Let $x_1 \equiv x_2 \pmod z$ as defined by the [[Definition:Congruence (Number Theory)/Integers/Remainder after Division|equal remainder after division]]:
:$\RR_z = \set {\tuple {x, y} \in \Z \times \Z: \exists k \in \Z: x = y + k z}$
Let $\tuple {x_1, x_2} \in \RR_z$.
Then by definition:
... | Equivalence of Definitions of Integer Congruence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Integer_Congruence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Integer_Congruence | [
"Modulo Arithmetic"
] | [] | [
"Definition:Congruence (Number Theory)/Integers/Remainder after Division",
"Definition:Modulo Operation",
"Definition:Congruence (Number Theory)/Integers/Modulo Operation",
"Definition:Congruence (Number Theory)/Integers/Modulo Operation",
"Definition:Congruence (Number Theory)/Integers/Integer Multiple",
... |
proofwiki-10840 | Occurrence in Prefix Notation has Unique Scope | Let $\FF$ be a formal language in prefix notation.
Let $\mathbf A$ be a well-formed formula of $\FF$.
Let $a$ be an occurrence in $\mathbf A$.
Then $a$ has a unique scope. | From the formal definition of prefix notation, it follows that $a$ must be introduced by the rule of formation:
:$a \mathbf A_1 \cdots \mathbf A_n$
for some well-formed formulas $\mathbf A_1, \ldots, \mathbf A_n$.
By Unique Readability for Prefix Notation, the $\mathbf A_i$ are uniquely determined.
Then $\mathbf A' = a... | Let $\FF$ be a [[Definition:Formal Language|formal language]] in [[Definition:Prefix Notation|prefix notation]].
Let $\mathbf A$ be a [[Definition:Well-Formed Formula|well-formed formula]] of $\FF$.
Let $a$ be an [[Definition:Occurrence (Formal Systems)|occurrence]] in $\mathbf A$.
Then $a$ has a unique [[Definitio... | From the [[Definition:Prefix Notation/Formal Definition|formal definition of prefix notation]], it follows that $a$ must be introduced by the [[Definition:Rule of Formation|rule of formation]]:
:$a \mathbf A_1 \cdots \mathbf A_n$
for some [[Definition:Well-Formed Formula|well-formed formulas]] $\mathbf A_1, \ldots, \... | Occurrence in Prefix Notation has Unique Scope | https://proofwiki.org/wiki/Occurrence_in_Prefix_Notation_has_Unique_Scope | https://proofwiki.org/wiki/Occurrence_in_Prefix_Notation_has_Unique_Scope | [
"Prefix Notation",
"Formal Languages"
] | [
"Definition:Formal Language",
"Definition:Operation/Binary Operation/Prefix Notation",
"Definition:Well-Formed Formula",
"Definition:Occurrence (Formal Systems)",
"Definition:Scope of Occurrence"
] | [
"Definition:Prefix Notation/Formal Definition",
"Definition:Rule of Formation",
"Definition:Well-Formed Formula",
"Unique Readability for Prefix Notation",
"Definition:Well-Formed Part",
"Definition:Well-Formed Part",
"Definition:Scope of Occurrence"
] |
proofwiki-10841 | Krull's Theorem | Let $R$ be a non-null ring with unity.
Then $R$ has a maximal ideal. | Let $\struct {P, \subseteq}$ be the ordered set consisting of all proper ideals of $R$, ordered by inclusion.
The theorem is proved by applying Zorn's Lemma to $P$.
First, we check that the conditions for Zorn's Lemma are met: $P$ must be non-empty, and every non-empty chain in $P$ must have an upper bound. | Let $R$ be a [[Definition:Non-Null Ring|non-null]] [[Definition:Ring with Unity|ring with unity]].
Then $R$ has a [[Definition:Maximal Ideal of Ring|maximal ideal]]. | Let $\struct {P, \subseteq}$ be the [[Definition:Ordered Set|ordered set]] consisting of all [[Definition:Proper Ideal of Ring|proper ideals]] of $R$, ordered by [[Definition:Subset|inclusion]].
The theorem is proved by applying [[Zorn's Lemma]] to $P$.
First, we check that the conditions for [[Zorn's Lemma]] are met... | Krull's Theorem | https://proofwiki.org/wiki/Krull's_Theorem | https://proofwiki.org/wiki/Krull's_Theorem | [
"Ideal Theory",
"Maximal Ideals of Rings"
] | [
"Definition:Non-Null Ring",
"Definition:Ring with Unity",
"Definition:Maximal Ideal of Ring"
] | [
"Definition:Ordered Set",
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Subset",
"Zorn's Lemma",
"Zorn's Lemma",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Chain (Order Theory)/Subset Relation",
"Definition:Upper Bound of Set",
"Definition:Non-Empty Set",
"Definit... |
proofwiki-10842 | Exclusive Or as Conjunction of Disjunctions | :$p \oplus q \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$ | {{begin-eqn}}
{{eqn | l = p \oplus q
| o = \dashv \vdash
| r = \paren {p \lor q} \land \neg \paren {p \land q}
| c = {{Defof|Exclusive Or}}
}}
{{eqn | o = \dashv \vdash
| r = \paren {p \lor q} \land \paren {\neg p \lor \neg q}
| c = De Morgan's Laws: Disjunction of Negations
}}
{{end-eqn}}... | :$p \oplus q \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$ | {{begin-eqn}}
{{eqn | l = p \oplus q
| o = \dashv \vdash
| r = \paren {p \lor q} \land \neg \paren {p \land q}
| c = {{Defof|Exclusive Or}}
}}
{{eqn | o = \dashv \vdash
| r = \paren {p \lor q} \land \paren {\neg p \lor \neg q}
| c = [[De Morgan's Laws (Logic)/Disjunction of Negations|De Mo... | Exclusive Or as Conjunction of Disjunctions/Proof 1 | https://proofwiki.org/wiki/Exclusive_Or_as_Conjunction_of_Disjunctions | https://proofwiki.org/wiki/Exclusive_Or_as_Conjunction_of_Disjunctions/Proof_1 | [
"Exclusive Or as Conjunction of Disjunctions",
"Exclusive Or",
"Disjunction",
"Conjunction"
] | [] | [
"De Morgan's Laws (Logic)/Disjunction of Negations"
] |
proofwiki-10843 | Exclusive Or as Conjunction of Disjunctions | :$p \oplus q \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$ | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
:<nowiki>$\begin {array} {|ccc||ccccccccc|} \hline
p & \oplus & q & (p & \lor & q) & \land & (\neg & p & \lor & \neg & q) \\
\hline
\F & \F & \F & \F & \F & \F & \F & \T ... | :$p \oplus q \dashv \vdash \paren {p \lor q} \land \paren {\neg p \lor \neg q}$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin {array} {|ccc||ccccccccc|} \h... | Exclusive Or as Conjunction of Disjunctions/Proof by Truth Table | https://proofwiki.org/wiki/Exclusive_Or_as_Conjunction_of_Disjunctions | https://proofwiki.org/wiki/Exclusive_Or_as_Conjunction_of_Disjunctions/Proof_by_Truth_Table | [
"Exclusive Or as Conjunction of Disjunctions",
"Exclusive Or",
"Disjunction",
"Conjunction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-10844 | NAND as Disjunction of Negations | :$p \uparrow q \dashv \vdash \neg p \lor \neg q$ | {{begin-eqn}}
{{eqn | l = p \uparrow q
| o = \dashv \vdash
| r = \map \neg {p \land q}
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \vdash
| r = \neg p \lor \neg q
| c = De Morgan's Laws: Disjunction of Negations
}}
{{end-eqn}}
{{qed}} | :$p \uparrow q \dashv \vdash \neg p \lor \neg q$ | {{begin-eqn}}
{{eqn | l = p \uparrow q
| o = \dashv \vdash
| r = \map \neg {p \land q}
| c = {{Defof|Logical NAND}}
}}
{{eqn | o = \dashv \vdash
| r = \neg p \lor \neg q
| c = [[De Morgan's Laws (Logic)/Disjunction of Negations|De Morgan's Laws: Disjunction of Negations]]
}}
{{end-eqn}}
{{... | NAND as Disjunction of Negations/Proof 1 | https://proofwiki.org/wiki/NAND_as_Disjunction_of_Negations | https://proofwiki.org/wiki/NAND_as_Disjunction_of_Negations/Proof_1 | [
"Logical NAND",
"Disjunction",
"NAND as Disjunction of Negations"
] | [] | [
"De Morgan's Laws (Logic)/Disjunction of Negations"
] |
proofwiki-10845 | NAND as Disjunction of Negations | :$p \uparrow q \dashv \vdash \neg p \lor \neg q$ | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|ccc||ccccc|} \hline
p & \uparrow & q & \neg & p & \lor & \neg & q \\
\hline
\F & \T & \F & \T & \F & \T & \T & \F \\
\F & \T & \T & \T & \F & \T & \F & \T... | :$p \uparrow q \dashv \vdash \neg p \lor \neg q$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|ccc||ccccc|} \hline
p & \upar... | NAND as Disjunction of Negations/Proof by Truth Table | https://proofwiki.org/wiki/NAND_as_Disjunction_of_Negations | https://proofwiki.org/wiki/NAND_as_Disjunction_of_Negations/Proof_by_Truth_Table | [
"Logical NAND",
"Disjunction",
"NAND as Disjunction of Negations"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-10846 | Rule of Idempotence/Disjunction/Formulation 2/Forward Implication | : $\vdash p \implies \paren {p \lor p}$ | {{BeginTableau|p \implies \paren {p \lor p} }}
{{Assumption|1|p}}
{{Addition|2|1|p \lor p|1|1}}
{{Implication|3||p \implies \paren {p \lor p}|1|2}}
{{EndTableau}}
{{qed}}
Category:Rule of Idempotence
9466a8vtv1pw8m79feaeim5qqcdowfs | : $\vdash p \implies \paren {p \lor p}$ | {{BeginTableau|p \implies \paren {p \lor p} }}
{{Assumption|1|p}}
{{Addition|2|1|p \lor p|1|1}}
{{Implication|3||p \implies \paren {p \lor p}|1|2}}
{{EndTableau}}
{{qed}}
[[Category:Rule of Idempotence]]
9466a8vtv1pw8m79feaeim5qqcdowfs | Rule of Idempotence/Disjunction/Formulation 2/Forward Implication | https://proofwiki.org/wiki/Rule_of_Idempotence/Disjunction/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Rule_of_Idempotence/Disjunction/Formulation_2/Forward_Implication | [
"Rule of Idempotence"
] | [] | [
"Category:Rule of Idempotence"
] |
proofwiki-10847 | Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication | : $\vdash \left({p \lor p}\right) \implies p$ | {{BeginTableau|\left({p \lor p}\right) \implies p}}
{{Premise|1|p \lor p}}
{{Assumption|2|p}}
{{ProofByCases|3|1|p|1|2|2|2|2}}
{{Implication|4||\left({p \lor p}\right) \implies p|1|3}}
{{EndTableau}}
{{qed}} | : $\vdash \left({p \lor p}\right) \implies p$ | {{BeginTableau|\left({p \lor p}\right) \implies p}}
{{Premise|1|p \lor p}}
{{Assumption|2|p}}
{{ProofByCases|3|1|p|1|2|2|2|2}}
{{Implication|4||\left({p \lor p}\right) \implies p|1|3}}
{{EndTableau}}
{{qed}} | Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication | https://proofwiki.org/wiki/Rule_of_Idempotence/Disjunction/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/Rule_of_Idempotence/Disjunction/Formulation_2/Reverse_Implication | [
"Rule of Idempotence"
] | [] | [] |
proofwiki-10848 | Rule of Addition/Sequent Form/Formulation 2/Form 1 | :$\vdash p \implies \paren {p \lor q}$ | {{BeginTableau|p \implies \paren {p \lor q} }}
{{Premise|1|p}}
{{Addition|2|1|p \lor q|1|1}}
{{Implication|3||p \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | :$\vdash p \implies \paren {p \lor q}$ | {{BeginTableau|p \implies \paren {p \lor q} }}
{{Premise|1|p}}
{{Addition|2|1|p \lor q|1|1}}
{{Implication|3||p \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 1 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_1 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_1/Proof_1 | [
"Rule of Addition"
] | [] | [] |
proofwiki-10849 | Rule of Addition/Sequent Form/Formulation 2/Form 1 | :$\vdash p \implies \paren {p \lor q}$ | {{BeginTableau|p \implies \paren {p \lor q}|Instance 2 of the Hilbert-style systems}}
{{TableauLine
| n = 1
| f = q \implies \paren {p \lor q}
| rlnk = Definition:Hilbert Proof System/Instance 2
| rtxt = Axiom $\text A 2$
}}
{{TableauLine
| n = 2
| f = p \implies \paren {q \lor p}
| rlnk = Definition:Hilbert Pro... | :$\vdash p \implies \paren {p \lor q}$ | {{BeginTableau|p \implies \paren {p \lor q}|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
{{TableauLine
| n = 1
| f = q \implies \paren {p \lor q}
| rlnk = Definition:Hilbert Proof System/Instance 2
| rtxt = Axiom $\text A 2$
}}
{{TableauLine
| n = 2
| f = p \implies \pa... | Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_1 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_1/Proof_2 | [
"Rule of Addition"
] | [] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-10850 | Rule of Addition/Sequent Form/Formulation 2/Form 2 | :$\vdash q \implies \left({p \lor q}\right)$ | {{BeginTableau|q \implies \paren {p \lor q} }}
{{Premise|1|q}}
{{Addition|2|1|p \lor q|1|2}}
{{Implication|3||q \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | :$\vdash q \implies \left({p \lor q}\right)$ | {{BeginTableau|q \implies \paren {p \lor q} }}
{{Premise|1|q}}
{{Addition|2|1|p \lor q|1|2}}
{{Implication|3||q \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | Rule of Addition/Sequent Form/Formulation 2/Form 2/Proof 1 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_2/Proof_1 | [
"Rule of Addition"
] | [] | [] |
proofwiki-10851 | Rule of Addition/Sequent Form/Formulation 2 | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | {{BeginTableau|p \implies \paren {p \lor q} }}
{{Premise|1|p}}
{{Addition|2|1|p \lor q|1|1}}
{{Implication|3||p \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | {{BeginTableau|p \implies \paren {p \lor q} }}
{{Premise|1|p}}
{{Addition|2|1|p \lor q|1|1}}
{{Implication|3||p \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 1 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_1/Proof_1 | [
"Rule of Addition"
] | [] | [] |
proofwiki-10852 | Rule of Addition/Sequent Form/Formulation 2 | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | {{BeginTableau|p \implies \paren {p \lor q}|Instance 2 of the Hilbert-style systems}}
{{TableauLine
| n = 1
| f = q \implies \paren {p \lor q}
| rlnk = Definition:Hilbert Proof System/Instance 2
| rtxt = Axiom $\text A 2$
}}
{{TableauLine
| n = 2
| f = p \implies \paren {q \lor p}
| rlnk = Definition:Hilbert Pro... | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | {{BeginTableau|p \implies \paren {p \lor q}|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
{{TableauLine
| n = 1
| f = q \implies \paren {p \lor q}
| rlnk = Definition:Hilbert Proof System/Instance 2
| rtxt = Axiom $\text A 2$
}}
{{TableauLine
| n = 2
| f = p \implies \pa... | Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_1/Proof_2 | [
"Rule of Addition"
] | [] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-10853 | Rule of Addition/Sequent Form/Formulation 2 | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | {{BeginTableau|q \implies \paren {p \lor q} }}
{{Premise|1|q}}
{{Addition|2|1|p \lor q|1|2}}
{{Implication|3||q \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | {{BeginTableau|q \implies \paren {p \lor q} }}
{{Premise|1|q}}
{{Addition|2|1|p \lor q|1|2}}
{{Implication|3||q \implies \paren {p \lor q}|1|3}}
{{EndTableau}}
{{Qed}} | Rule of Addition/Sequent Form/Formulation 2/Form 2/Proof 1 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Form_2/Proof_1 | [
"Rule of Addition"
] | [] | [] |
proofwiki-10854 | Rule of Addition/Sequent Form/Formulation 2 | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | === Form 1 ===
{{:Rule of Addition/Sequent Form/Formulation 2/Proof 1/Form 1}}
=== Form 2 ===
{{:Rule of Addition/Sequent Form/Formulation 2/Proof 1/Form 2}} | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | === [[Rule of Addition/Sequent Form/Formulation 2/Proof 1/Form 1|Form 1]] ===
{{:Rule of Addition/Sequent Form/Formulation 2/Proof 1/Form 1}}
=== [[Rule of Addition/Sequent Form/Formulation 2/Proof 1/Form 2|Form 2]] ===
{{:Rule of Addition/Sequent Form/Formulation 2/Proof 1/Form 2}} | Rule of Addition/Sequent Form/Formulation 2/Proof 1 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Proof_1 | [
"Rule of Addition"
] | [] | [
"Rule of Addition/Sequent Form/Formulation 2/Form 1/Proof 1",
"Rule of Addition/Sequent Form/Formulation 2/Form 2/Proof 1"
] |
proofwiki-10855 | Rule of Addition/Sequent Form/Formulation 2 | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives are $T$ for all boolean interpretations.
:<nowiki>$\begin{array}{|c|c|ccccc|ccccc|} \hline
p & q & p & \implies & (p & \lor & q) & q & \implies & (p & \lor & q) \\
\hline
\F & \F & \F & \T & \F & \F & \F & \F ... | {{begin-eqn}}
{{eqn | n = 1
| l = \vdash p
| o = \implies
| r = \paren {p \lor q}
}}
{{eqn | n = 2
| l = \vdash q
| o = \implies
| r = \paren {p \lor q}
}}
{{end-eqn}} | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] are $T$ for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|c|c|ccccc|ccccc|} \... | Rule of Addition/Sequent Form/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2 | https://proofwiki.org/wiki/Rule_of_Addition/Sequent_Form/Formulation_2/Proof_by_Truth_Table | [
"Rule of Addition"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-10856 | Hilbert Proof System Instance 2 is Consistent | Instance 2 of the Hilbert proof systems $\mathscr H_2$ is consistent. | Consider Instance 1 of a constructed semantics, denoted $\mathscr C_1$.
Note that $\neg p$ is not a tautology for $\mathscr C_1$.
We will establish that every $\mathscr H_2$-theorem is a $\mathscr C_1$-tautology.
That is, that $\mathscr H_2$ is sound for $\mathscr C_1$.
Starting with the axioms:
{{begin-axiom}}
{{axiom... | [[Definition:Hilbert Proof System/Instance 2|Instance 2]] of the [[Definition:Hilbert Proof System|Hilbert proof systems]] $\mathscr H_2$ is [[Definition:Consistent Proof System|consistent]]. | Consider [[Definition:Constructed Semantics/Instance 1|Instance 1]] of a [[Definition:Constructed Semantics|constructed semantics]], denoted $\mathscr C_1$.
Note that $\neg p$ is not a [[Definition:Tautology (Formal Semantics)|tautology]] for $\mathscr C_1$.
We will establish that every $\mathscr H_2$-[[Definition:Th... | Hilbert Proof System Instance 2 is Consistent | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_is_Consistent | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_is_Consistent | [
"Hilbert Proof System Instance 2"
] | [
"Definition:Hilbert Proof System/Instance 2",
"Definition:Hilbert Proof System",
"Definition:Consistent (Logic)/Proof System"
] | [
"Definition:Constructed Semantics/Instance 1",
"Definition:Constructed Semantics",
"Definition:Tautology/Formal Semantics",
"Definition:Theorem/Formal System",
"Definition:Tautology/Formal Semantics",
"Definition:Sound Proof System",
"Definition:Axiom/Formal Systems",
"Rule of Idempotence/Disjunction/... |
proofwiki-10857 | Set of Local Minimum is Countable | Let $X$ be a subset of $\R$.
The set:
:$\leftset {x \in X: x}$ is local minimum in $\rightset X$
is countable. | Define:
:$Y := \leftset {x \in X: x}$ is local minimum in $\rightset X$
By definition of $Y$ and definition of local minimum in set:
:$\forall x \in Y: \exists y \in \R: y < x \land \openint y x \cap X = \O$
By the Axiom of Choice, define a mapping $f: Y \to \powerset \R$ as:
:$\forall x \in Y: \exists y \in \R: \map f... | Let $X$ be a [[Definition:Subset|subset]] of $\R$.
The set:
:$\leftset {x \in X: x}$ is [[Definition:Local Minimum in Set of Reals|local minimum in]] $\rightset X$
is [[Definition:Countable Set|countable]]. | Define:
:$Y := \leftset {x \in X: x}$ is [[Definition:Local Minimum in Set of Reals|local minimum in]] $\rightset X$
By definition of $Y$ and definition of [[Definition:Local Minimum in Set of Reals|local minimum in set]]:
:$\forall x \in Y: \exists y \in \R: y < x \land \openint y x \cap X = \O$
By the [[Axiom:Axiom... | Set of Local Minimum is Countable | https://proofwiki.org/wiki/Set_of_Local_Minimum_is_Countable | https://proofwiki.org/wiki/Set_of_Local_Minimum_is_Countable | [
"Real Analysis",
"Countable Sets"
] | [
"Definition:Subset",
"Definition:Local Minimum in Set of Reals",
"Definition:Countable Set"
] | [
"Definition:Local Minimum in Set of Reals",
"Definition:Local Minimum in Set of Reals",
"Axiom:Axiom of Choice",
"Definition:Mapping",
"Definition:Injection",
"Definition:Injection",
"Cardinality of Image of Injection",
"Definition:Cardinality",
"Definition:Image (Set Theory)/Mapping/Subset",
"Def... |
proofwiki-10858 | Set of Pairwise Disjoint Intervals is Countable | Let $X$ be a subset of $\powerset \R$ such that:
:$(1): \quad X$ is pairwise disjoint:
::::$\forall A, B \in X: A \ne B \implies A \cap B = \O$.
:$(2): \quad$ every element of $X$ contains an open interval:
::::$\forall A \in X: \exists x, y \in \R: x < y \land \openint x y \subseteq A$.
Then $X$ is countable. | By Between two Real Numbers exists Rational Number:
:$\forall A \in X: \exists x, y \in \R, q \in \Q: x < y \land q \in \openint x y \subseteq A$
By the Axiom of Choice define a mapping $f: X \to \Q$:
:$\forall A \in X: \map f A \in A$
First it needs to be shown that $f$ is an injection by definition.
Let $A, B \in X$ ... | Let $X$ be a [[Definition:Subset|subset]] of $\powerset \R$ such that:
:$(1): \quad X$ is [[Definition:Pairwise Disjoint|pairwise disjoint]]:
::::$\forall A, B \in X: A \ne B \implies A \cap B = \O$.
:$(2): \quad$ every [[Definition:Element|element]] of $X$ [[Definition:Superset|contains]] an [[Definition:Open Real Int... | By [[Between two Real Numbers exists Rational Number]]:
:$\forall A \in X: \exists x, y \in \R, q \in \Q: x < y \land q \in \openint x y \subseteq A$
By the [[Axiom:Axiom of Choice|Axiom of Choice]] define a [[Definition:Mapping|mapping]] $f: X \to \Q$:
:$\forall A \in X: \map f A \in A$
First it needs to be shown t... | Set of Pairwise Disjoint Intervals is Countable | https://proofwiki.org/wiki/Set_of_Pairwise_Disjoint_Intervals_is_Countable | https://proofwiki.org/wiki/Set_of_Pairwise_Disjoint_Intervals_is_Countable | [
"Countable Sets"
] | [
"Definition:Subset",
"Definition:Pairwise Disjoint",
"Definition:Element",
"Definition:Subset/Superset",
"Definition:Real Interval/Open",
"Definition:Countable Set"
] | [
"Between two Real Numbers exists Rational Number",
"Axiom:Axiom of Choice",
"Definition:Mapping",
"Definition:Injection",
"Definition:Set Intersection",
"Definition:Empty Set",
"Definition:Pairwise Disjoint",
"Definition:Injection",
"Set is Subset of Itself",
"Definition:Subset",
"Cardinality of... |
proofwiki-10859 | Set is Countable if Cardinality equals Cardinality of Countable Set | Let $X, Y$ be sets.
Let:
:$\card X = \card Y$
where $\card X$ denotes the cardinality of $X$.
If $X$ is countable then $Y$ is countable. | Assume that $X$ is countable.
By definition of countable set there exists an injection:L
:$f: X \to \N$
By definition of cardinality the sets $Y$ and $X$ are equivalent:
:$Y \sim X$
Then by definition of set equivalence there exists a bijection:
:$g: Y \to X$
By definition of bijection:
:$g$ is an injection.
Hence by C... | Let $X, Y$ be [[Definition:Set|sets]].
Let:
:$\card X = \card Y$
where $\card X$ denotes the [[Definition:Cardinality|cardinality]] of $X$.
If $X$ is [[Definition:Countable Set|countable]] then $Y$ is [[Definition:Countable Set|countable]]. | Assume that $X$ is [[Definition:Countable Set|countable]].
By definition of [[Definition:Countable Set|countable set]] there exists an [[Definition:Injection|injection]]:L
:$f: X \to \N$
By definition of [[Definition:Cardinality|cardinality]] the sets $Y$ and $X$ are [[Definition:Set Equivalence|equivalent]]:
:$Y \si... | Set is Countable if Cardinality equals Cardinality of Countable Set | https://proofwiki.org/wiki/Set_is_Countable_if_Cardinality_equals_Cardinality_of_Countable_Set | https://proofwiki.org/wiki/Set_is_Countable_if_Cardinality_equals_Cardinality_of_Countable_Set | [
"Countable Sets"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Countable Set",
"Definition:Countable Set"
] | [
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Injection",
"Definition:Cardinality",
"Definition:Set Equivalence",
"Definition:Set Equivalence",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Injection",
"Composite of Injections is Injection",
"Definition:Inject... |
proofwiki-10860 | Double Negation/Double Negation Introduction/Sequent Form | {{:Double Negation/Double Negation Introduction/Sequent Form/Formulation 1}} | {{BeginTableau|p \vdash \neg \neg p}}
{{Premise|1|p}}
{{Assumption|2|\neg p}}
{{NonContradiction|3|1, 2|1|2}}
{{Contradiction|4|1|\neg \neg p|2|3}}
{{EndTableau|qed}} | {{:Double Negation/Double Negation Introduction/Sequent Form/Formulation 1}} | {{BeginTableau|p \vdash \neg \neg p}}
{{Premise|1|p}}
{{Assumption|2|\neg p}}
{{NonContradiction|3|1, 2|1|2}}
{{Contradiction|4|1|\neg \neg p|2|3}}
{{EndTableau|qed}} | Double Negation/Double Negation Introduction/Sequent Form/Formulation 1/Proof | https://proofwiki.org/wiki/Double_Negation/Double_Negation_Introduction/Sequent_Form | https://proofwiki.org/wiki/Double_Negation/Double_Negation_Introduction/Sequent_Form/Formulation_1/Proof | [
"Double Negation Introduction"
] | [] | [] |
proofwiki-10861 | Principle of Non-Contradiction/Sequent Form | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | {{BeginTableau|p, \neg p \vdash \bot}}
{{Premise|1|p}}
{{Premise|2|\neg p}}
{{NonContradiction|3|1, 2|1|2}}
{{EndTableau}}
{{Qed}} | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | {{BeginTableau|p, \neg p \vdash \bot}}
{{Premise|1|p}}
{{Premise|2|\neg p}}
{{NonContradiction|3|1, 2|1|2}}
{{EndTableau}}
{{Qed}} | Principle of Non-Contradiction/Sequent Form/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_1/Proof_1 | [
"Principle of Non-Contradiction"
] | [] | [] |
proofwiki-10862 | Principle of Non-Contradiction/Sequent Form | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | We apply the Method of Truth Tables.
:<nowiki>$\begin {array} {|cccc||c|} \hline
p & \land & \neg & p & \bot \\
\hline
\F & \F & \T & \F & \F \\
\T & \F & \F & \T & \F \\
\hline
\end {array}$</nowiki>
As can be seen by inspection, the truth value of the main connective, that is $\land$, is $F$ for each boolean interpre... | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | We apply the [[Method of Truth Tables]].
:<nowiki>$\begin {array} {|cccc||c|} \hline
p & \land & \neg & p & \bot \\
\hline
\F & \F & \T & \F & \F \\
\T & \F & \F & \T & \F \\
\hline
\end {array}$</nowiki>
As can be seen by inspection, the [[Definition:Truth Value|truth value]] of the [[Definition:Main Connective (Pro... | Principle of Non-Contradiction/Sequent Form/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_1/Proof_by_Truth_Table | [
"Principle of Non-Contradiction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-10863 | Principle of Non-Contradiction/Sequent Form | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | {{BeginTableau|\vdash \neg \left({p \land \neg p}\right)}}
{{Assumption|1|p \land \neg p}}
{{Simplification|2|1|p|1|1}}
{{Simplification|3|1|\neg p|1|2}}
{{NonContradiction|4|1|2|3}}
{{Contradiction|5||\neg \left({p \land \neg p}\right)|1|4}}
{{EndTableau|qed}} | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | {{BeginTableau|\vdash \neg \left({p \land \neg p}\right)}}
{{Assumption|1|p \land \neg p}}
{{Simplification|2|1|p|1|1}}
{{Simplification|3|1|\neg p|1|2}}
{{NonContradiction|4|1|2|3}}
{{Contradiction|5||\neg \left({p \land \neg p}\right)|1|4}}
{{EndTableau|qed}} | Principle of Non-Contradiction/Sequent Form/Formulation 2/Proof 1 | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_2/Proof_1 | [
"Principle of Non-Contradiction"
] | [] | [] |
proofwiki-10864 | Principle of Non-Contradiction/Sequent Form | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | We apply the Method of Truth Tables to the proposition $\neg \paren {p \land \neg p}$.
As can be seen by inspection, the truth value of the main connective, that is $\neg$, is $T$ for each boolean interpretation for $p$.
:<nowiki>$\begin {array} {|ccccc|} \hline
\neg & (p & \land & \neg & p)\\
\hline
\T & \F & \F & \T ... | {{:Principle of Non-Contradiction/Sequent Form/Formulation 1}} | We apply the [[Method of Truth Tables]] to the proposition $\neg \paren {p \land \neg p}$.
As can be seen by inspection, the [[Definition:Truth Value|truth value]] of the [[Definition:Main Connective (Propositional Logic)|main connective]], that is $\neg$, is $T$ for each [[Definition:Boolean Interpretation|boolean in... | Principle of Non-Contradiction/Sequent Form/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_2/Proof_by_Truth_Table | [
"Principle of Non-Contradiction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-10865 | Factorization of Natural Numbers within 4 n + 1 not Unique | Let:
:$S = \set {4 n + 1: n \in \N} = \set {1, 5, 9, 13, 17, \ldots}$
be the set of natural numbers of the form $4 n + 1$.
Then not all elements of $S$ have a complete factorization by other elements of $S$ which is unique. | Proof by Counterexample:
Consider the number:
:$m = 693 = 3^2 \times 7 \times 11$
Thus:
:$m = 9 \times 77 = 21 \times 33$
We have that:
{{begin-eqn}}
{{eqn | l = 9
| r = 4 \times 2 + 1
| rr= \in S
}}
{{eqn | l = 77
| r = 4 \times 19 + 1
| rr= \in S
}}
{{eqn | l = 21
| r = 4 \times 5 + 1
... | Let:
:$S = \set {4 n + 1: n \in \N} = \set {1, 5, 9, 13, 17, \ldots}$
be the [[Definition:Set|set]] of [[Definition:Natural Number|natural numbers]] of the form $4 n + 1$.
Then not all [[Definition:Element|elements]] of $S$ have a [[Definition:Complete Factorization|complete factorization]] by other [[Definition:Elem... | [[Proof by Counterexample]]:
Consider the [[Definition:Natural Number|number]]:
:$m = 693 = 3^2 \times 7 \times 11$
Thus:
:$m = 9 \times 77 = 21 \times 33$
We have that:
{{begin-eqn}}
{{eqn | l = 9
| r = 4 \times 2 + 1
| rr= \in S
}}
{{eqn | l = 77
| r = 4 \times 19 + 1
| rr= \in S
}}
{{eqn... | Factorization of Natural Numbers within 4 n + 1 not Unique | https://proofwiki.org/wiki/Factorization_of_Natural_Numbers_within_4_n_+_1_not_Unique | https://proofwiki.org/wiki/Factorization_of_Natural_Numbers_within_4_n_+_1_not_Unique | [
"Number Theory"
] | [
"Definition:Set",
"Definition:Natural Numbers",
"Definition:Element",
"Definition:Complete Factorization",
"Definition:Element",
"Definition:Unique"
] | [
"Proof by Counterexample",
"Definition:Natural Numbers",
"Definition:Divisor (Algebra)/Integer",
"Definition:Complete Factorization",
"Definition:Element"
] |
proofwiki-10866 | Solutions of Pythagorean Equation/Primitive | The set of all primitive Pythagorean triples is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
:$m, n \in \Z_{>0}$ are (strictly) positive integers
:$m \perp n$, that is, $m$ and $n$ are coprime
:$m$ and $n$ are of opposite parity
:$m > n$ | First we show that $\tuple {2 m n, m^2 - n^2, m^2 + n^2}$ is a Pythagorean triple:
{{begin-eqn}}
{{eqn | l = \paren {2 m n}^2 + \paren {m^2 - n^2}^2
| r = 4 m^2 n^2 + m^4 - 2 m^2 n^2 + n^4
| c =
}}
{{eqn | r = m^4 + 2 m^2 n^2 + n^4
| c =
}}
{{eqn | r = \paren {m^2 + n^2}^2
| c =
}}
{{end-eqn}}
S... | The [[Definition:Set|set]] of all [[Definition:Primitive Pythagorean Triple|primitive Pythagorean triples]] is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
:$m, n \in \Z_{>0}$ are [[Definition:Strictly Positive Integer|(strictly) positive integers]]
:$m \perp n$, that is, $m$ and $n$ are [[Definition:C... | First we show that $\tuple {2 m n, m^2 - n^2, m^2 + n^2}$ is a [[Definition:Pythagorean Triple|Pythagorean triple]]:
{{begin-eqn}}
{{eqn | l = \paren {2 m n}^2 + \paren {m^2 - n^2}^2
| r = 4 m^2 n^2 + m^4 - 2 m^2 n^2 + n^4
| c =
}}
{{eqn | r = m^4 + 2 m^2 n^2 + n^4
| c =
}}
{{eqn | r = \paren {m^2 + ... | Solutions of Pythagorean Equation/Primitive/Proof 1 | https://proofwiki.org/wiki/Solutions_of_Pythagorean_Equation/Primitive | https://proofwiki.org/wiki/Solutions_of_Pythagorean_Equation/Primitive/Proof_1 | [
"Solutions of Pythagorean Equation"
] | [
"Definition:Set",
"Definition:Pythagorean Triple/Primitive",
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Parity of Integer"
] | [
"Definition:Pythagorean Triple",
"Definition:Pythagorean Triple",
"Definition:Pythagorean Triple/Primitive",
"Definition:Pythagorean Triple/Primitive",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Prime Divides Power",
"Common Divisor Divides Integer Combination",
"Prime Divid... |
proofwiki-10867 | Solutions of Pythagorean Equation/Primitive | The set of all primitive Pythagorean triples is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
:$m, n \in \Z_{>0}$ are (strictly) positive integers
:$m \perp n$, that is, $m$ and $n$ are coprime
:$m$ and $n$ are of opposite parity
:$m > n$ | Let $\tuple {A, B, C}$ be a Pythagorean Triple:
:$A^2 + B^2 = C^2$
By the Pythagorean theorem, this equation describes the sides of a right triangle:
:400px
By the definitions of sine and cosine:
{{begin-eqn}}
{{eqn | l = \sin \theta
| r = \frac A C
}}
{{eqn | l = \cos \theta
| r = \frac B C
}}
{{end-eqn}}
... | The [[Definition:Set|set]] of all [[Definition:Primitive Pythagorean Triple|primitive Pythagorean triples]] is generated by:
:$\tuple {2 m n, m^2 - n^2, m^2 + n^2}$
where:
:$m, n \in \Z_{>0}$ are [[Definition:Strictly Positive Integer|(strictly) positive integers]]
:$m \perp n$, that is, $m$ and $n$ are [[Definition:C... | Let $\tuple {A, B, C}$ be a [[Definition:Pythagorean Triple|Pythagorean Triple]]:
:$A^2 + B^2 = C^2$
By the [[Pythagoras's Theorem|Pythagorean theorem]], this equation describes the sides of a [[Definition:Right Triangle|right triangle]]:
:[[File:RightTriangleWithTheta.png|400px]]
By the [[Definition:Sine of Angle|... | Solutions of Pythagorean Equation/Primitive/Proof 2 | https://proofwiki.org/wiki/Solutions_of_Pythagorean_Equation/Primitive | https://proofwiki.org/wiki/Solutions_of_Pythagorean_Equation/Primitive/Proof_2 | [
"Solutions of Pythagorean Equation"
] | [
"Definition:Set",
"Definition:Pythagorean Triple/Primitive",
"Definition:Strictly Positive/Integer",
"Definition:Coprime/Integers",
"Definition:Parity of Integer"
] | [
"Definition:Pythagorean Triple",
"Pythagoras's Theorem",
"Definition:Triangle (Geometry)/Right-Angled",
"File:RightTriangleWithTheta.png",
"Definition:Sine/Definition from Triangle",
"Definition:Cosine/Definition from Triangle",
"Equiangular Triangles are Similar",
"Proportion is Equivalence Relation"... |
proofwiki-10868 | Solutions of Pythagorean Equation/General | Let $x, y, z$ be a solution to the Pythagorean equation.
Then $x = k x', y = k y', z = k z'$, where:
:$\tuple {x', y', z'}$ is a primitive Pythagorean triple
:$k \in \Z: k \ge 1$ | Let $\tuple {x, y, z}$ be non-primitive solution to the Pythagorean equation.
Let:
:$\exists k \in \Z: k \ge 2, k \divides x, k \divides y$
such that $x \perp y$.
Then we can express $x$ and $y$ as $x = k x', y = k y'$.
Thus:
:$z^2 = k^2 x'^2 + k^2 y'^2 = k^2 z'^2$
for some $z' \in \Z$.
Let:
:$\exists k \in \Z: k \ge 2... | Let $x, y, z$ be a solution to the [[Definition:Pythagorean Equation|Pythagorean equation]].
Then $x = k x', y = k y', z = k z'$, where:
:$\tuple {x', y', z'}$ is a [[Definition:Primitive Pythagorean Triple|primitive Pythagorean triple]]
:$k \in \Z: k \ge 1$ | Let $\tuple {x, y, z}$ be non-[[Definition:Primitive Pythagorean Triple|primitive solution]] to the [[Definition:Pythagorean Equation|Pythagorean equation]].
Let:
:$\exists k \in \Z: k \ge 2, k \divides x, k \divides y$
such that $x \perp y$.
Then we can express $x$ and $y$ as $x = k x', y = k y'$.
Thus:
:$z^2 = k^... | Solutions of Pythagorean Equation/General | https://proofwiki.org/wiki/Solutions_of_Pythagorean_Equation/General | https://proofwiki.org/wiki/Solutions_of_Pythagorean_Equation/General | [
"Solutions of Pythagorean Equation"
] | [
"Definition:Pythagorean Equation",
"Definition:Pythagorean Triple/Primitive"
] | [
"Definition:Pythagorean Triple/Primitive",
"Definition:Pythagorean Equation",
"Definition:Common Divisor/Integers",
"Definition:Common Divisor/Integers",
"Definition:Common Divisor/Integers",
"Definition:Pythagorean Triple/Primitive",
"Definition:Pythagorean Equation",
"Definition:Pythagorean Triple/P... |
proofwiki-10869 | Goldbach Conjecture implies Goldbach's Marginal Conjecture | Suppose the Goldbach Conjecture holds:
:Every even integer greater than $2$ is the sum of two primes.
Then Goldbach's Marginal Conjecture follows:
:Every integer greater than $5$ can be written as the sum of three primes. | Suppose the Goldbach Conjecture holds.
Let $n \in \Z$ such that $n > 5$.
Let $n$ be an odd integer.
Then $n - 3$ is an even integer greater than $2$.
By the Goldbach Conjecture:
:$n - 3 = p_1 + p_2$
where $p_1$ and $p_2$ are both primes.
Then:
:$n = p_1 + p_2 + 3$
As $3$ is prime, the result follows.
Let $n$ be an even... | Suppose the [[Goldbach Conjecture]] holds:
:Every [[Definition:Even Integer|even integer]] greater than $2$ is the sum of two [[Definition:Prime Number|primes]].
Then [[Goldbach's Marginal Conjecture]] follows:
:Every [[Definition:Integer|integer]] greater than $5$ can be written as the sum of three [[Definition:Prim... | Suppose the [[Goldbach Conjecture]] holds.
Let $n \in \Z$ such that $n > 5$.
Let $n$ be an [[Definition:Odd Integer|odd integer]].
Then $n - 3$ is an [[Definition:Even Integer|even integer]] greater than $2$.
By the [[Goldbach Conjecture]]:
:$n - 3 = p_1 + p_2$
where $p_1$ and $p_2$ are both [[Definition:Prime Num... | Goldbach Conjecture implies Goldbach's Marginal Conjecture | https://proofwiki.org/wiki/Goldbach_Conjecture_implies_Goldbach's_Marginal_Conjecture | https://proofwiki.org/wiki/Goldbach_Conjecture_implies_Goldbach's_Marginal_Conjecture | [
"Prime Numbers",
"Goldbach Conjecture"
] | [
"Goldbach Conjecture",
"Definition:Even Integer",
"Definition:Prime Number",
"Goldbach Conjecture/Marginal",
"Definition:Integer",
"Definition:Prime Number"
] | [
"Goldbach Conjecture",
"Definition:Odd Integer",
"Definition:Even Integer",
"Goldbach Conjecture",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Even Integer",
"Definition:Even Integer",
"Goldbach Conjecture",
"Definition:Prime Number",
"Definition:Prime Number",
"Category:... |
proofwiki-10870 | Congruent Integers are of same Quadratic Character | Let $p$ be an odd prime.
Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$.
Let $a \equiv b \pmod p$.
Then $a$ and $b$ have the same quadratic character. | Let $a \equiv b \pmod p$.
Then by Congruence of Powers:
:$a^2 \equiv b^2 \pmod p$
Hence:
:$x^2 \equiv a \pmod p$ has a solution {{iff}} $x^2 \equiv b \pmod p$.
Hence the result.
{{qed}}
Category:Quadratic Residues
rqg7tzzvmszbgwmoy21zvr5o7qdyw94 | Let $p$ be an [[Definition:Odd Prime|odd prime]].
Let $a \in \Z$ be an [[Definition:Integer|integer]] such that $a \not \equiv 0 \pmod p$.
Let $a \equiv b \pmod p$.
Then $a$ and $b$ have the same [[Definition:Quadratic Character|quadratic character]]. | Let $a \equiv b \pmod p$.
Then by [[Congruence of Powers]]:
:$a^2 \equiv b^2 \pmod p$
Hence:
:$x^2 \equiv a \pmod p$ has a solution {{iff}} $x^2 \equiv b \pmod p$.
Hence the result.
{{qed}}
[[Category:Quadratic Residues]]
rqg7tzzvmszbgwmoy21zvr5o7qdyw94 | Congruent Integers are of same Quadratic Character | https://proofwiki.org/wiki/Congruent_Integers_are_of_same_Quadratic_Character | https://proofwiki.org/wiki/Congruent_Integers_are_of_same_Quadratic_Character | [
"Quadratic Residues"
] | [
"Definition:Odd Prime",
"Definition:Integer",
"Definition:Quadratic Residue/Character"
] | [
"Congruence of Powers",
"Category:Quadratic Residues"
] |
proofwiki-10871 | Weight of Sorgenfrey Line is Continuum | Let $T = \struct {\R, \tau}$ be the Sorgenfrey line.
Then $\map w T = \mathfrak c$
where
:$\map w T$ denotes the weight of $T$
:$\mathfrak c$ denotes continuum, the cardinality of real numbers. | By definition of Sorgenfrey line, the set:
:$\BB = \set {\hointr x y: x, y \in \R \land x < y}$
is a basis of $T$.
By definition of weight:
:$\map w T \le \card \BB$
where $\card \BB$ denotes the cardinality of $\BB$.
By Cardinality of Basis of Sorgenfrey Line not greater than Continuum:
:$\card \BB \le \mathfrak c$
Th... | Let $T = \struct {\R, \tau}$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Then $\map w T = \mathfrak c$
where
:$\map w T$ denotes the [[Definition:Weight of Topological Space|weight]] of $T$
:$\mathfrak c$ denotes [[Definition:Cardinality of Continuum|continuum]], the [[Definition:Cardinality|cardinality]] ... | By definition of [[Definition:Sorgenfrey Line|Sorgenfrey line]], the [[Definition:Set|set]]:
:$\BB = \set {\hointr x y: x, y \in \R \land x < y}$
is a [[Definition:Analytic Basis|basis]] of $T$.
By definition of [[Definition:Weight of Topological Space|weight]]:
:$\map w T \le \card \BB$
where $\card \BB$ denotes the ... | Weight of Sorgenfrey Line is Continuum | https://proofwiki.org/wiki/Weight_of_Sorgenfrey_Line_is_Continuum | https://proofwiki.org/wiki/Weight_of_Sorgenfrey_Line_is_Continuum | [
"Sorgenfrey Line"
] | [
"Definition:Sorgenfrey Line",
"Definition:Weight of Topological Space",
"Definition:Cardinality of Continuum",
"Definition:Cardinality",
"Definition:Real Number"
] | [
"Definition:Sorgenfrey Line",
"Definition:Set",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Weight of Topological Space",
"Definition:Cardinality",
"Cardinality of Basis of Sorgenfrey Line not greater than Continuum",
"Definition:Weight of Topological Space",
"Definition:Basis (Topology)... |
proofwiki-10872 | Construction of Regular Heptadecagon | It is possible to construct a regular hepadecagon (that is, a regular polygon with $17$ sides) using a compass and straightedge construction. | It remains to be demonstrated that the line segment $NM$ is the side of a regular hepadecagon inscribed in circle $ACB$.
This will be done by demonstrating that $\angle NOM$ is equal to $\dfrac {2 \pi} {17}$ radians, that is, $\dfrac 1 {17}$ of the full circle $ACB$.
For convenience, let the radius $OA$ be equal to $4 ... | It is possible to construct a [[Definition:Regular Heptadecagon|regular hepadecagon]] (that is, a [[Definition:Regular Polygon|regular polygon]] with $17$ [[Definition:Side of Polygon|sides]]) using a [[Definition:Compass and Straightedge Construction|compass and straightedge construction]]. | It remains to be demonstrated that the [[Definition:Line Segment|line segment]] $NM$ is the [[Definition:Side of Polygon|side]] of a [[Definition:Regular Heptadecagon|regular hepadecagon]] [[Definition:Polygon Inscribed in Circle|inscribed]] in [[Definition:Circle|circle]] $ACB$.
This will be done by demonstrating tha... | Construction of Regular Heptadecagon | https://proofwiki.org/wiki/Construction_of_Regular_Heptadecagon | https://proofwiki.org/wiki/Construction_of_Regular_Heptadecagon | [
"Compass and Straightedge Constructions",
"Regular Polygons",
"17"
] | [
"Definition:Heptadecagon/Regular",
"Definition:Polygon/Regular",
"Definition:Polygon/Side",
"Definition:Compass and Straightedge Construction"
] | [
"Definition:Line/Segment",
"Definition:Polygon/Side",
"Definition:Heptadecagon/Regular",
"Definition:Inscribe/Polygon in Circle",
"Definition:Circle",
"Definition:Angular Measure/Radian",
"Definition:Circle",
"Definition:Circle/Radius",
"Pythagoras's Theorem",
"Definition:Tangent Function/Definiti... |
proofwiki-10873 | Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure | Let $\BB$ be a set of subsets of $\R$, the set of all real numbers.
Let:
:$\card \BB < \mathfrak c$
where
:$\card \BB$ denotes the cardinality of $\BB$
:$\mathfrak c = \card \R$ denotes continuum.
Let $\FF = \set {\bigcup \GG: \GG \subseteq \BB}$.
Then:
:$\exists x, y \in \R: x < y \land \hointr x y \notin \FF$ | Define:
:$ Z = \leftset {x \in \R: \exists U \in \FF: x}$ is local minimum in $\rightset U$
By Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum:
:$\card Z < \mathfrak c$
Then by Cardinalities form Inequality implies Difference is Nonempty:
:... | Let $\BB$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $\R$, the [[Definition:Set|set]] of all [[Definition:Real Number|real numbers]].
Let:
:$\card \BB < \mathfrak c$
where
:$\card \BB$ denotes the [[Definition:Cardinality|cardinality]] of $\BB$
:$\mathfrak c = \card \R$ denotes [[Definitio... | Define:
:$ Z = \leftset {x \in \R: \exists U \in \FF: x}$ is local minimum in $\rightset U$
By [[Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum]]:
:$\card Z < \mathfrak c$
Then by [[Cardinalities form Inequality implies Difference is Non... | Set of Subset of Reals with Cardinality less than Continuum has not Interval in Union Closure | https://proofwiki.org/wiki/Set_of_Subset_of_Reals_with_Cardinality_less_than_Continuum_has_not_Interval_in_Union_Closure | https://proofwiki.org/wiki/Set_of_Subset_of_Reals_with_Cardinality_less_than_Continuum_has_not_Interval_in_Union_Closure | [
"Infinite Sets"
] | [
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Set",
"Definition:Real Number",
"Definition:Cardinality",
"Definition:Cardinality of Continuum"
] | [
"Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum",
"Cardinalities form Inequality implies Difference is Nonempty",
"Definition:Empty Set",
"Definition:Set Difference",
"Definition:Local Minimum in Set of Reals",
"Definition:Lo... |
proofwiki-10874 | Cardinality of Basis of Sorgenfrey Line not greater than Continuum | Let $T = \struct {\R, \tau}$ be the Sorgenfrey line.
Let
:$\BB = \set {\hointr x y: x, y \in \R \land x < y}$
be the basis of $T$.
Then $\card \BB \le \mathfrak c$
where
:$\card \BB$ denotes the cardinality of $\BB$
:$\mathfrak c = \card \R$ denotes the continuum. | Define a mapping $f: \BB \to \R \times \R$:
:$\forall I \in \BB: \map f I = \tuple {\min I, \sup I}$
That is:
:$\map f {\hointr x y} = \tuple {x, y} \forall x, y \in \R: x < y$
We will show that $f$ is an injection by definition.
Let $I_1, I_2 \in \BB$ such that:
:$\map f {I_1} = \map f {I_2}$
{{begin-eqn}}
{{eqn | l =... | Let $T = \struct {\R, \tau}$ be the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
Let
:$\BB = \set {\hointr x y: x, y \in \R \land x < y}$
be the [[Definition:Analytic Basis|basis]] of $T$.
Then $\card \BB \le \mathfrak c$
where
:$\card \BB$ denotes the [[Definition:Cardinality|cardinality]] of $\BB$
:$\mathfrak ... | Define a [[Definition:Mapping|mapping]] $f: \BB \to \R \times \R$:
:$\forall I \in \BB: \map f I = \tuple {\min I, \sup I}$
That is:
:$\map f {\hointr x y} = \tuple {x, y} \forall x, y \in \R: x < y$
We will show that $f$ is an [[Definition:Injection|injection]] by definition.
Let $I_1, I_2 \in \BB$ such that:
:$\m... | Cardinality of Basis of Sorgenfrey Line not greater than Continuum | https://proofwiki.org/wiki/Cardinality_of_Basis_of_Sorgenfrey_Line_not_greater_than_Continuum | https://proofwiki.org/wiki/Cardinality_of_Basis_of_Sorgenfrey_Line_not_greater_than_Continuum | [
"Sorgenfrey Line"
] | [
"Definition:Sorgenfrey Line",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Cardinality",
"Definition:Cardinality of Continuum"
] | [
"Definition:Mapping",
"Definition:Injection",
"Definition:Injection",
"Injection implies Cardinal Inequality",
"Cardinal Product Equal to Maximum"
] |
proofwiki-10875 | Cardinalities form Inequality implies Difference is Nonempty | Let $X, Y$ be sets.
Let
:$\card X < \card Y$
where $\card X$ denotes the cardinality of $X$.
Then:
:$Y \setminus X \ne \O$ | {{AimForCont}} that:
:$Y \setminus X = \O$
Then by Set Difference with Superset is Empty Set:
:$Y \subseteq X$
Hence by Subset implies Cardinal Inequality:
:$\card Y \le \card X$
This contradicts:
:$\card X < \card Y$
Hence the result.
{{qed}} | Let $X, Y$ be [[Definition:Set|sets]].
Let
:$\card X < \card Y$
where $\card X$ denotes the [[Definition:Cardinality|cardinality]] of $X$.
Then:
:$Y \setminus X \ne \O$ | {{AimForCont}} that:
:$Y \setminus X = \O$
Then by [[Set Difference with Superset is Empty Set]]:
:$Y \subseteq X$
Hence by [[Subset implies Cardinal Inequality]]:
:$\card Y \le \card X$
This [[Definition:Contradiction|contradicts]]:
:$\card X < \card Y$
Hence the result.
{{qed}} | Cardinalities form Inequality implies Difference is Nonempty | https://proofwiki.org/wiki/Cardinalities_form_Inequality_implies_Difference_is_Nonempty | https://proofwiki.org/wiki/Cardinalities_form_Inequality_implies_Difference_is_Nonempty | [
"Cardinals"
] | [
"Definition:Set",
"Definition:Cardinality"
] | [
"Set Difference with Superset is Empty Set",
"Subset implies Cardinal Inequality",
"Definition:Contradiction"
] |
proofwiki-10876 | Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum | Let $\BB$ be a set of subsets of $\R$.
Let:
:$\size \BB < \mathfrak c$
where
:$\size \BB$ denotes the cardinality of $\BB$
:$\mathfrak c = \size \R$ denotes the cardinality of the continuum.
Let
:$X = \leftset {x \in \R: \exists U \in \set {\bigcup \GG: \GG \subseteq \BB}: x}$ is a local minimum in $\rightset U$
Then:
... | We will prove that:
:$(1): \quad \size \BB \aleph_0 < \mathfrak c$
where $\aleph_0 = \size \N$ by Aleph Zero equals Cardinality of Naturals.
In the case when $\size \BB = \mathbf 0$ we have by Zero of Cardinal Product is Zero:
:$\size \BB \aleph_0 = \mathbf 0 < \mathfrak c$
In the case when $\mathbf 0 < \size \BB < \al... | Let $\BB$ be a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $\R$.
Let:
:$\size \BB < \mathfrak c$
where
:$\size \BB$ denotes the [[Definition:Cardinality|cardinality]] of $\BB$
:$\mathfrak c = \size \R$ denotes the [[Definition:Cardinality of Continuum|cardinality of the continuum]].
Let
:$X = \... | We will prove that:
:$(1): \quad \size \BB \aleph_0 < \mathfrak c$
where $\aleph_0 = \size \N$ by [[Aleph Zero equals Cardinality of Naturals]].
In the case when $\size \BB = \mathbf 0$ we have by [[Zero of Cardinal Product is Zero]]:
:$\size \BB \aleph_0 = \mathbf 0 < \mathfrak c$
In the case when $\mathbf 0 < \s... | Set of Subsets of Reals with Cardinality less than Continuum Cardinality of Local Minimums of Union Closure less than Continuum | https://proofwiki.org/wiki/Set_of_Subsets_of_Reals_with_Cardinality_less_than_Continuum_Cardinality_of_Local_Minimums_of_Union_Closure_less_than_Continuum | https://proofwiki.org/wiki/Set_of_Subsets_of_Reals_with_Cardinality_less_than_Continuum_Cardinality_of_Local_Minimums_of_Union_Closure_less_than_Continuum | [
"Infinite Sets",
"Cardinality of Continuum",
"Real Numbers"
] | [
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Cardinality",
"Definition:Cardinality of Continuum",
"Definition:Local Minimum in Set of Reals"
] | [
"Aleph Zero equals Cardinality of Naturals",
"Zero of Cardinal Product is Zero",
"Product of Cardinals is Commutative",
"Cardinal Product Equal to Maximum",
"Aleph Zero is less than Cardinality of Continuum",
"Cardinal Product Equal to Maximum",
"Definition:Local Minimum in Set of Reals",
"Definition:... |
proofwiki-10877 | Slope of Secant | Let $f: \R \to \R$ be a real function.
Let the graph of $f$ be depicted on a Cartesian plane.
:400px
Let $AB$ be a secant of $f$ where:
:$A = \tuple {x, \map f x}$
:$A = \tuple {x + h, \map f {x + h} }$
Then the slope of $AB$ is given by:
:$\dfrac {\map f {x + h} - \map f x} h$ | The slope of $AB$ is defined as the change in $y$ divided by the change in $x$.
Between $A$ and $B$:
:the change in $x$ is $\paren {x + h} - x = h$
:the change in $y$ is $\map f {x + h} - \map f x$.
Hence the result.
{{qed}} | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let the [[Definition:Graph of Mapping|graph]] of $f$ be depicted on a [[Definition:Cartesian Plane|Cartesian plane]].
:[[File:SecantToCurve.png|400px]]
Let $AB$ be a [[Definition:Secant of Curve|secant]] of $f$ where:
:$A = \tuple {x, \map f x}$
:... | The [[Definition:Slope of Straight Line|slope]] of $AB$ is defined as the change in $y$ divided by the change in $x$.
Between $A$ and $B$:
:the change in $x$ is $\paren {x + h} - x = h$
:the change in $y$ is $\map f {x + h} - \map f x$.
Hence the result.
{{qed}} | Slope of Secant | https://proofwiki.org/wiki/Slope_of_Secant | https://proofwiki.org/wiki/Slope_of_Secant | [
"Analytic Geometry"
] | [
"Definition:Real Function",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"File:SecantToCurve.png",
"Definition:Secant Line",
"Definition:Slope/Straight Line"
] | [
"Definition:Slope/Straight Line"
] |
proofwiki-10878 | Derivative of Curve at Point | Let $f: \R \to \R$ be a real function.
Let the graph $G$ of $f$ be depicted on a Cartesian plane.
Then the derivative of $f$ at $x = \xi$ is equal to the tangent to $G$ at $x = \xi$. | Let $f: \R \to \R$ be a real function.
:400px
Let the graph $G$ of $f$ be depicted on a Cartesian plane.
Let $A = \tuple {\xi, \map f \xi}$ be a point on $G$.
Consider the secant $AB$ to $G$ where $B = \tuple {\xi + h, \map f {\xi + h} }$.
From Slope of Secant, the slope of $AB$ is given by:
:$\dfrac {\map f {x + h} - ... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
Let the [[Definition:Graph of Mapping|graph]] $G$ of $f$ be depicted on a [[Definition:Cartesian Plane|Cartesian plane]].
Then the [[Definition:Derivative of Real Function at Point|derivative]] of $f$ at $x = \xi$ is equal to the [[Definition:Tangen... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]].
:[[File:DerivativeOfCurve.png|400px]]
Let the [[Definition:Graph of Mapping|graph]] $G$ of $f$ be depicted on a [[Definition:Cartesian Plane|Cartesian plane]].
Let $A = \tuple {\xi, \map f \xi}$ be a [[Definition:Point|point]] on $G$.
Consider ... | Derivative of Curve at Point | https://proofwiki.org/wiki/Derivative_of_Curve_at_Point | https://proofwiki.org/wiki/Derivative_of_Curve_at_Point | [
"Derivatives",
"Differential Calculus",
"Analytic Geometry"
] | [
"Definition:Real Function",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"Definition:Derivative/Real Function/Derivative at Point",
"Definition:Tangent Line"
] | [
"Definition:Real Function",
"File:DerivativeOfCurve.png",
"Definition:Graph of Mapping",
"Definition:Cartesian Plane",
"Definition:Point",
"Definition:Secant Line",
"Slope of Secant",
"Definition:Slope/Straight Line",
"Definition:Secant Line",
"Definition:Tangent Line",
"Definition:Limit of Real... |
proofwiki-10879 | Derivative of Square Function | Let $f: \R \to \R$ be the square function:
:$\forall x \in \R: \map f x = x^2$
Then the derivative of $f$ is given by:
:$\map {f'} x = 2 x$ | {{begin-eqn}}
{{eqn | l = \map {f'} x
| r = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\paren {x + h}^2 - x^2} h
| c =
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {x^2 + 2 x h + h^2 - x^2... | Let $f: \R \to \R$ be the [[Definition:Square (Algebra)|square function]]:
:$\forall x \in \R: \map f x = x^2$
Then the [[Definition:Derivative|derivative]] of $f$ is given by:
:$\map {f'} x = 2 x$ | {{begin-eqn}}
{{eqn | l = \map {f'} x
| r = \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h
| c = {{Defof|Derivative of Real Function at Point}}
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {\paren {x + h}^2 - x^2} h
| c =
}}
{{eqn | r = \lim_{h \mathop \to 0} \frac {x^2 + 2 x h + h^2 - x^2... | Derivative of Square Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Square_Function | https://proofwiki.org/wiki/Derivative_of_Square_Function/Proof_1 | [
"Derivative of Square Function",
"Derivatives",
"Square Function"
] | [
"Definition:Square/Function",
"Definition:Derivative"
] | [] |
proofwiki-10880 | Derivative of Square Function | Let $f: \R \to \R$ be the square function:
:$\forall x \in \R: \map f x = x^2$
Then the derivative of $f$ is given by:
:$\map {f'} x = 2 x$ | From Power Rule for Derivatives:
:$\map {\dfrac \d {\d x} } {x^n} = n x^{n - 1}$
The result follows by setting $n = 2$.
{{qed}} | Let $f: \R \to \R$ be the [[Definition:Square (Algebra)|square function]]:
:$\forall x \in \R: \map f x = x^2$
Then the [[Definition:Derivative|derivative]] of $f$ is given by:
:$\map {f'} x = 2 x$ | From [[Power Rule for Derivatives]]:
:$\map {\dfrac \d {\d x} } {x^n} = n x^{n - 1}$
The result follows by setting $n = 2$.
{{qed}} | Derivative of Square Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Square_Function | https://proofwiki.org/wiki/Derivative_of_Square_Function/Proof_2 | [
"Derivative of Square Function",
"Derivatives",
"Square Function"
] | [
"Definition:Square/Function",
"Definition:Derivative"
] | [
"Power Rule for Derivatives"
] |
proofwiki-10881 | Countable iff Cardinality not greater than Aleph Zero | Let $X$ be set.
$X$ is countable {{iff}}:
$\card X \le \aleph_0$
where:
:$\card X$ denotes the cardinality of $X$
:$\aleph_0 = \card \N$ by Aleph Zero equals Cardinality of Naturals. | :$X$ is countable
{{iff}}:
:there exists an injection $f: X \to \N$ by definition of countable set
{{iff}}:
:$\card X \le \card \N$ by Injection iff Cardinal Inequality
{{iff}}:
:$\card X \le \aleph_0$
{{qed}} | Let $X$ be [[Definition:Set|set]].
$X$ is [[Definition:Countable Set|countable]] {{iff}}:
$\card X \le \aleph_0$
where:
:$\card X$ denotes the [[Definition:Cardinality|cardinality]] of $X$
:$\aleph_0 = \card \N$ by [[Aleph Zero equals Cardinality of Naturals]]. | :$X$ is [[Definition:Countable Set|countable]]
{{iff}}:
:there exists an [[Definition:Injection|injection]] $f: X \to \N$ by definition of [[Definition:Countable Set|countable set]]
{{iff}}:
:$\card X \le \card \N$ by [[Injection iff Cardinal Inequality]]
{{iff}}:
:$\card X \le \aleph_0$
{{qed}} | Countable iff Cardinality not greater than Aleph Zero | https://proofwiki.org/wiki/Countable_iff_Cardinality_not_greater_than_Aleph_Zero | https://proofwiki.org/wiki/Countable_iff_Cardinality_not_greater_than_Aleph_Zero | [
"Countable Sets",
"Aleph Mapping"
] | [
"Definition:Set",
"Definition:Countable Set",
"Definition:Cardinality",
"Aleph Zero equals Cardinality of Naturals"
] | [
"Definition:Countable Set",
"Definition:Injection",
"Definition:Countable Set",
"Injection iff Cardinal Inequality"
] |
proofwiki-10882 | Aleph Zero equals Cardinality of Naturals | $\aleph_0 = \card \N$
where
:$\aleph$ denotes the aleph mapping,
:$\card \N$ denotes the cardinality of $\N$. | {{begin-eqn}}
{{eqn | l = \aleph_0
| r = \card {\aleph_0}
| c = Cardinal of Cardinal Equal to Cardinal
}}
{{eqn | r = \card {\omega}
| c = {{Defof|Aleph Mapping}}
}}
{{eqn | r = \card {\N}
| c = {{Defof|Natural Numbers|subdef = Construction}}
}}
{{end-eqn}}
{{qed}} | $\aleph_0 = \card \N$
where
:$\aleph$ denotes the [[Definition:Aleph Mapping|aleph mapping]],
:$\card \N$ denotes the [[Definition:Cardinality|cardinality]] of $\N$. | {{begin-eqn}}
{{eqn | l = \aleph_0
| r = \card {\aleph_0}
| c = [[Cardinal of Cardinal Equal to Cardinal]]
}}
{{eqn | r = \card {\omega}
| c = {{Defof|Aleph Mapping}}
}}
{{eqn | r = \card {\N}
| c = {{Defof|Natural Numbers|subdef = Construction}}
}}
{{end-eqn}}
{{qed}} | Aleph Zero equals Cardinality of Naturals | https://proofwiki.org/wiki/Aleph_Zero_equals_Cardinality_of_Naturals | https://proofwiki.org/wiki/Aleph_Zero_equals_Cardinality_of_Naturals | [
"Aleph Mapping"
] | [
"Definition:Aleph Mapping",
"Definition:Cardinality"
] | [
"Cardinal of Cardinal Equal to Cardinal"
] |
proofwiki-10883 | Distance Moved by Body from Rest under Constant Acceleration | Let a body $B$ be stationary.
Let $B$ be subject to a constant acceleration.
Then the distance travelled by $B$ is proportional to the square of the length of time $B$ is under the acceleration. | From Equations of Motion with Constant Acceleration: Distance after Time:
:$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$
where:
:$\mathbf s$ is the displacement of $B$ at time $t$ from its initial position at time $t$
:$\mathbf u$ is the velocity at time $t = 0$
:$\mathbf a$ is the constant acceleration $t$
In t... | Let a [[Definition:Body|body]] $B$ be [[Definition:Stationary|stationary]].
Let $B$ be subject to a [[Definition:Constant|constant]] [[Definition:Acceleration|acceleration]].
Then the [[Definition:Distance (Geometry)|distance]] travelled by $B$ is [[Definition:Proportion|proportional]] to the [[Definition:Square (Alg... | From [[Equations of Motion with Constant Acceleration/Distance after Time|Equations of Motion with Constant Acceleration: Distance after Time]]:
:$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$
where:
:$\mathbf s$ is the [[Definition:Displacement|displacement]] of $B$ at [[Definition:Time|time]] $t$ from its initi... | Distance Moved by Body from Rest under Constant Acceleration | https://proofwiki.org/wiki/Distance_Moved_by_Body_from_Rest_under_Constant_Acceleration | https://proofwiki.org/wiki/Distance_Moved_by_Body_from_Rest_under_Constant_Acceleration | [
"Equations of Motion with Constant Acceleration",
"Mechanics"
] | [
"Definition:Body",
"Definition:Stationary",
"Definition:Constant",
"Definition:Acceleration",
"Definition:Distance (Geometry)",
"Definition:Proportion",
"Definition:Square/Function",
"Definition:Time/Length",
"Definition:Acceleration"
] | [
"Equations of Motion with Constant Acceleration/Distance after Time",
"Definition:Displacement",
"Definition:Time",
"Definition:Time",
"Definition:Velocity",
"Definition:Time",
"Definition:Constant",
"Definition:Acceleration",
"Definition:Magnitude",
"Definition:Vector Quantity",
"Definition:Pro... |
proofwiki-10884 | Equations of Motion with Constant Acceleration/Velocity after Time | :$\mathbf v = \mathbf u + \mathbf a t$ | By definition of acceleration:
:$\dfrac {\d \mathbf v} {\d t} = \mathbf a$
By Solution to Linear First Order Ordinary Differential Equation:
:$\mathbf v = \mathbf c + \mathbf a t$
where $\mathbf c$ is a constant vector.
We are given the initial condition:
:$\bigvalueat {\mathbf v} {t \mathop = 0} = \mathbf u$
from whic... | :$\mathbf v = \mathbf u + \mathbf a t$ | By definition of [[Definition:Acceleration|acceleration]]:
:$\dfrac {\d \mathbf v} {\d t} = \mathbf a$
By [[Solution to Linear First Order Ordinary Differential Equation]]:
:$\mathbf v = \mathbf c + \mathbf a t$
where $\mathbf c$ is a [[Definition:Constant|constant]] [[Definition:Vector|vector]].
We are given the [[D... | Equations of Motion with Constant Acceleration/Velocity after Time | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Velocity_after_Time | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Velocity_after_Time | [
"Equations of Motion with Constant Acceleration"
] | [] | [
"Definition:Acceleration",
"Solution to Linear First Order Ordinary Differential Equation",
"Definition:Constant",
"Definition:Vector",
"Definition:Initial Condition"
] |
proofwiki-10885 | Equations of Motion with Constant Acceleration/Distance after Time | :$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$ | From Equations of Motion with Constant Acceleration: Velocity after Time:
:$\mathbf v = \mathbf u + \mathbf a t$
By definition of velocity, this can be expressed as:
:$\dfrac {\d \mathbf s} {\d t} = \mathbf u + \mathbf a t$
where both $\mathbf u$ and $\mathbf a$ are constant.
By Solution to Linear First Order Ordinary ... | :$\mathbf s = \mathbf u t + \dfrac {\mathbf a t^2} 2$ | From [[Equations of Motion with Constant Acceleration/Velocity after Time|Equations of Motion with Constant Acceleration: Velocity after Time]]:
:$\mathbf v = \mathbf u + \mathbf a t$
By definition of [[Definition:Velocity|velocity]], this can be expressed as:
:$\dfrac {\d \mathbf s} {\d t} = \mathbf u + \mathbf a t$
... | Equations of Motion with Constant Acceleration/Distance after Time | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Distance_after_Time | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Distance_after_Time | [
"Equations of Motion with Constant Acceleration"
] | [] | [
"Equations of Motion with Constant Acceleration/Velocity after Time",
"Definition:Velocity",
"Definition:Constant",
"Solution to Linear First Order Ordinary Differential Equation",
"Definition:Constant",
"Definition:Vector",
"Definition:Initial Condition"
] |
proofwiki-10886 | Equations of Motion with Constant Acceleration/Velocity after Distance | :$\mathbf v \cdot \mathbf v = \mathbf u \cdot \mathbf u + 2 \mathbf a \cdot \mathbf s$ | From Equations of Motion with Constant Acceleration: Velocity after Time
:$\mathbf v = \mathbf u + \mathbf a t$
Then:
{{begin-eqn}}
{{eqn | l = \mathbf v \cdot \mathbf v
| r = \paren {\mathbf u + \mathbf a t} \cdot \paren {\mathbf u + \mathbf a t}
| c =
}}
{{eqn | r = \mathbf u \cdot \mathbf u + \mathbf u ... | :$\mathbf v \cdot \mathbf v = \mathbf u \cdot \mathbf u + 2 \mathbf a \cdot \mathbf s$ | From [[Equations of Motion with Constant Acceleration/Velocity after Time|Equations of Motion with Constant Acceleration: Velocity after Time]]
:$\mathbf v = \mathbf u + \mathbf a t$
Then:
{{begin-eqn}}
{{eqn | l = \mathbf v \cdot \mathbf v
| r = \paren {\mathbf u + \mathbf a t} \cdot \paren {\mathbf u + \mathbf... | Equations of Motion with Constant Acceleration/Velocity after Distance | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Velocity_after_Distance | https://proofwiki.org/wiki/Equations_of_Motion_with_Constant_Acceleration/Velocity_after_Distance | [
"Equations of Motion with Constant Acceleration"
] | [] | [
"Equations of Motion with Constant Acceleration/Velocity after Time",
"Dot Product Distributes over Addition",
"Dot Product Operator is Commutative",
"Dot Product Associates with Scalar Multiplication",
"Dot Product Distributes over Addition",
"Equations of Motion with Constant Acceleration/Distance after... |
proofwiki-10887 | Aleph Zero is less than Cardinality of Continuum | $\aleph_0 < \mathfrak c$
where
:$\aleph$ denotes the aleph mapping,
:$\mathfrak c$ denotes the cardinality of the continuum. | By Power Set of Natural Numbers has Cardinality of Continuum:
:$\mathfrak c = \card {\powerset \N}$
where:
:$\powerset \N$ denotes the power set of $\N$
:$\card {\powerset \N}$ denotes the cardinality of $\powerset \N$.
By Cardinality of Set less than Cardinality of Power Set:
:$\card \N < \card {\powerset \N}$
Thus b... | $\aleph_0 < \mathfrak c$
where
:$\aleph$ denotes the [[Definition:Aleph Mapping|aleph mapping]],
:$\mathfrak c$ denotes the [[Definition:Cardinality of Continuum|cardinality of the continuum]]. | By [[Power Set of Natural Numbers has Cardinality of Continuum]]:
:$\mathfrak c = \card {\powerset \N}$
where:
:$\powerset \N$ denotes the [[Definition:Power Set|power set]] of $\N$
:$\card {\powerset \N}$ denotes the [[Definition:Cardinality|cardinality]] of $\powerset \N$.
By [[Cardinality of Set less than Cardinali... | Aleph Zero is less than Cardinality of Continuum | https://proofwiki.org/wiki/Aleph_Zero_is_less_than_Cardinality_of_Continuum | https://proofwiki.org/wiki/Aleph_Zero_is_less_than_Cardinality_of_Continuum | [
"Infinite Sets",
"Aleph Mapping",
"Cardinality of Continuum"
] | [
"Definition:Aleph Mapping",
"Definition:Cardinality of Continuum"
] | [
"Power Set of Natural Numbers has Cardinality of Continuum",
"Definition:Power Set",
"Definition:Cardinality",
"Cardinality of Set less than Cardinality of Power Set",
"Aleph Zero equals Cardinality of Naturals"
] |
proofwiki-10888 | Cardinality of Set less than Cardinality of Power Set | Let $X$ be a set.
Then:
:$\card X < \card {\powerset X}$
where
:$\card X$ denotes the cardinality of $X$,
:$\powerset X$ denotes the power set of $X$. | By No Bijection from Set to its Power Set:
: there exist no bijections $X \to \powerset X$
Then by definition of set equivalence:
:$X \not\sim \powerset X$
Hence by definition of cardinality:
:$(1): \quad \card X \ne \card {\powerset X}$
By Cardinality of Set of Singletons:
:$(2): \quad \card {\set {\set {x}: x \in X} ... | Let $X$ be a [[Definition:Set|set]].
Then:
:$\card X < \card {\powerset X}$
where
:$\card X$ denotes the [[Definition:Cardinality|cardinality]] of $X$,
:$\powerset X$ denotes the [[Definition:Power Set|power set]] of $X$. | By [[No Bijection from Set to its Power Set]]:
: there exist no [[Definition:Bijection|bijections]] $X \to \powerset X$
Then by definition of [[Definition:Set Equivalence|set equivalence]]:
:$X \not\sim \powerset X$
Hence by definition of [[Definition:Cardinality|cardinality]]:
:$(1): \quad \card X \ne \card {\powers... | Cardinality of Set less than Cardinality of Power Set | https://proofwiki.org/wiki/Cardinality_of_Set_less_than_Cardinality_of_Power_Set | https://proofwiki.org/wiki/Cardinality_of_Set_less_than_Cardinality_of_Power_Set | [
"Cardinals",
"Power Set",
"Cardinality"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Power Set"
] | [
"No Bijection from Set to its Power Set",
"Definition:Bijection",
"Definition:Set Equivalence",
"Definition:Cardinality",
"Cardinality of Set of Singletons",
"Definition:Subset",
"Definition:Power Set",
"Definition:Subset",
"Subset implies Cardinal Inequality"
] |
proofwiki-10889 | Volume of Solid of Revolution | Let $f: \R \to \R$ be a real function which is integrable on the interval $\closedint a b$.
Let the points be defined:
:$A = \tuple {a, \map f a}$
:$B = \tuple {b, \map f b}$
:$C = \tuple {b, 0}$
:$D = \tuple {a, 0}$
Let the figure $ABCD$ be defined as being bounded by the straight lines $y = 0$, $x = a$, $x = b$ and t... | :500px
Consider a rectangle bounded by the lines:
:$y = 0$
:$x = \xi$
:$x = \xi + \delta x$
:$y = \map f x$
Consider the right circular cylinder generated by revolving it about the $x$-axis.
By Volume of Right Circular Cylinder, the volume of this cylinder is:
:$V_\xi = \pi \paren {\map f x}^2 \delta x$
{{finish|Needs ... | Let $f: \R \to \R$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|interval]] $\closedint a b$.
Let the [[Definition:Point|points]] be defined:
:$A = \tuple {a, \map f a}$
:$B = \tuple {b, \map f b}$
:$C = \tuple {b, 0}$
:$... | :[[File:VolumeOfSolidOfRevolution.png|500px]]
Consider a [[Definition:Rectangle|rectangle]] bounded by the lines:
:$y = 0$
:$x = \xi$
:$x = \xi + \delta x$
:$y = \map f x$
Consider the [[Definition:Right Circular Cylinder|right circular cylinder]] generated by revolving it about the [[Definition:X-Axis|$x$-axis]].
... | Volume of Solid of Revolution | https://proofwiki.org/wiki/Volume_of_Solid_of_Revolution | https://proofwiki.org/wiki/Volume_of_Solid_of_Revolution | [
"Integral Calculus",
"Solids of Revolution"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Point",
"Definition:Geometric Figure/Plane Figure",
"Definition:Line/Straight Line",
"Definition:Line/Curve",
"Definition:Solid of Revolution",
"Definition:Axis/X-Axis",
"Definition:Volume"... | [
"File:VolumeOfSolidOfRevolution.png",
"Definition:Quadrilateral/Rectangle",
"Definition:Right Circular Cylinder",
"Definition:Axis/X-Axis",
"Volume of Right Circular Cylinder",
"Definition:Volume",
"Definition:Right Circular Cylinder"
] |
proofwiki-10890 | Acceleration is Second Derivative of Displacement with respect to Time | The '''acceleration''' $\mathbf a$ of a body $M$ is the second derivative of the displacement $\mathbf s$ of $M$ from a given point of reference {{WRT|Differentiation}} time $t$:
:$\mathbf a = \dfrac {\d^2 \mathbf s} {\d t^2}$ | By definition, the acceleration of a body $M$ is defined as the first derivative of the velocity $\mathbf v$ of $M$ relative to a given point of reference {{WRT|Differentiation}} time:
:$\mathbf a = \dfrac {\d \mathbf v} {\d t}$
Also by definition, the velocity of $M$ is defined as the first derivative of the displacem... | The '''[[Definition:Acceleration|acceleration]]''' $\mathbf a$ of a [[Definition:Body|body]] $M$ is the [[Definition:Second Derivative|second derivative]] of the [[Definition:Displacement|displacement]] $\mathbf s$ of $M$ from a given [[Definition:Point of Reference|point of reference]] {{WRT|Differentiation}} [[Defini... | By definition, the [[Definition:Acceleration|acceleration]] of a [[Definition:Body|body]] $M$ is defined as the [[Definition:Derivative|first derivative]] of the [[Definition:Velocity|velocity]] $\mathbf v$ of $M$ [[Definition:Relative Velocity|relative to]] a given [[Definition:Point of Reference|point of reference]] ... | Acceleration is Second Derivative of Displacement with respect to Time | https://proofwiki.org/wiki/Acceleration_is_Second_Derivative_of_Displacement_with_respect_to_Time | https://proofwiki.org/wiki/Acceleration_is_Second_Derivative_of_Displacement_with_respect_to_Time | [
"Mechanics",
"Acceleration"
] | [
"Definition:Acceleration",
"Definition:Body",
"Definition:Derivative/Higher Derivatives/Second Derivative",
"Definition:Displacement",
"Definition:Frame of Reference/Point of Reference",
"Definition:Time"
] | [
"Definition:Acceleration",
"Definition:Body",
"Definition:Derivative",
"Definition:Velocity",
"Definition:Relative Velocity",
"Definition:Frame of Reference/Point of Reference",
"Definition:Time",
"Definition:Velocity",
"Definition:Derivative",
"Definition:Displacement",
"Definition:Frame of Ref... |
proofwiki-10891 | Equation of Catenary/Cartesian/Formulation 1 | The '''catenary''' is described by the equation:
:$y = \dfrac {e^{a x} + e^{-a x} } {2 a} = \dfrac {\cosh a x} a$
where $a$ is a constant.
The lowest point of the catenary is at $\tuple {0, \dfrac 1 a}$. | Let $\tuple {x, y}$ be an arbitrary point on the chain.
Let $s$ be the length along the arc of the chain from the lowest point to $\tuple {x, y}$.
Let $w_0$ be the linear mass density of the chain, that is, its weight per unit length.
The section of chain between the lowest point and $\tuple {x, y}$ is in static equili... | The '''[[Definition:Catenary|catenary]]''' is described by the equation:
:$y = \dfrac {e^{a x} + e^{-a x} } {2 a} = \dfrac {\cosh a x} a$
where $a$ is a [[Definition:Constant|constant]].
The lowest point of the [[Definition:Catenary|catenary]] is at $\tuple {0, \dfrac 1 a}$. | Let $\tuple {x, y}$ be an arbitrary [[Definition:Point|point]] on the [[Definition:Chain (Physics)|chain]].
Let $s$ be the [[Definition:Linear Measure|length]] along the [[Definition:Arc of Curve|arc]] of the [[Definition:Chain (Physics)|chain]] from the lowest point to $\tuple {x, y}$.
Let $w_0$ be the [[Definition:... | Equation of Catenary/Cartesian/Formulation 1/Proof | https://proofwiki.org/wiki/Equation_of_Catenary/Cartesian/Formulation_1 | https://proofwiki.org/wiki/Equation_of_Catenary/Cartesian/Formulation_1/Proof | [
"Catenary"
] | [
"Definition:Catenary",
"Definition:Constant",
"Definition:Catenary"
] | [
"Definition:Point",
"Definition:Chain (Physics)",
"Definition:Linear Measure",
"Definition:Curve/Arc",
"Definition:Chain (Physics)",
"Definition:Mass Density/Linear",
"Definition:Chain (Physics)",
"Definition:Weight (Physics)",
"Definition:Linear Measure",
"Definition:Chain (Physics)",
"Definiti... |
proofwiki-10892 | Equation of Catenary/Cartesian/Formulation 2 | The '''catenary''' is described by the equation:
:$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$
where $a$ is a constant.
The lowest point of the chain is at $\tuple {0, a}$. | Take the equation of the catenary according to Formulation 1:
:$y = \dfrac {e^{ax} + e^{-ax}} {2 a}$
Put this in a form which uses the hyperbolic cosine:
:$y = \dfrac {\cosh a x} a$
Replace $a$ with $\dfrac 1 a$:
:$y = a \cosh \dfrac x a$
Hence the result.
{{qed}} | The '''[[Definition:Catenary|catenary]]''' is described by the [[Definition:Equation|equation]]:
:$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$
where $a$ is a [[Definition:Constant|constant]].
The lowest point of the [[Definition:Chain (Physics)|chain]] is at $\tuple {0, a}$. | Take the equation of the [[Equation of Catenary/Cartesian/Formulation 1|catenary according to Formulation 1]]:
:$y = \dfrac {e^{ax} + e^{-ax}} {2 a}$
Put this in a form which uses the [[Definition:Hyperbolic Cosine|hyperbolic cosine]]:
:$y = \dfrac {\cosh a x} a$
Replace $a$ with $\dfrac 1 a$:
:$y = a \cosh \dfrac x ... | Equation of Catenary/Cartesian/Formulation 2/Proof | https://proofwiki.org/wiki/Equation_of_Catenary/Cartesian/Formulation_2 | https://proofwiki.org/wiki/Equation_of_Catenary/Cartesian/Formulation_2/Proof | [
"Catenary"
] | [
"Definition:Catenary",
"Definition:Equation",
"Definition:Constant",
"Definition:Chain (Physics)"
] | [
"Equation of Catenary/Cartesian/Formulation 1",
"Definition:Hyperbolic Cosine"
] |
proofwiki-10893 | Cardinality of Power Set is Invariant | Let $X, Y$ be sets.
Let $\card X = \card Y$
where $\card X$ denotes the cardinality of $X$.
Then:
:$\card {\powerset X} = \card {\powerset Y}$
where $\powerset X$ denotes the power set of $X$. | By definition of cardinality:
:$X \sim Y$
where $\sim$ denotes the set equivalence.
Then by definition of set equivalence:
: there exists a bijection $f: X \to Y$
By definition of bijection
:$f$ is an injection and a surjection.
By Mapping is Injection iff Direct Image Mapping is Injection:
:the direct image mapping $\... | Let $X, Y$ be [[Definition:Set|sets]].
Let $\card X = \card Y$
where $\card X$ denotes the [[Definition:Cardinality|cardinality]] of $X$.
Then:
:$\card {\powerset X} = \card {\powerset Y}$
where $\powerset X$ denotes the [[Definition:Power Set|power set]] of $X$. | By definition of [[Definition:Cardinality|cardinality]]:
:$X \sim Y$
where $\sim$ denotes the [[Definition:Set Equivalence|set equivalence]].
Then by definition of [[Definition:Set Equivalence|set equivalence]]:
: there exists a [[Definition:Bijection|bijection]] $f: X \to Y$
By definition of [[Definition:Bijection|b... | Cardinality of Power Set is Invariant | https://proofwiki.org/wiki/Cardinality_of_Power_Set_is_Invariant | https://proofwiki.org/wiki/Cardinality_of_Power_Set_is_Invariant | [
"Cardinals",
"Power Set"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Power Set"
] | [
"Definition:Cardinality",
"Definition:Set Equivalence",
"Definition:Set Equivalence",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Injection",
"Definition:Surjection",
"Mapping is Injection iff Direct Image Mapping is Injection",
"Definition:Direct Image Mapping/Mapping",
"Definitio... |
proofwiki-10894 | Reals are Isomorphic to Dedekind Cuts | Let $\mathscr D$ be the set of all Dedekind cuts of the totally ordered set $\struct {\Q, \le}$.
Define a mapping $f: \R \to \mathscr D$ as:
:$\forall x \in \R: \map f x = \set {y \in \Q: y < x}$
Then $f$ is a bijection. | First, we will prove that:
:$\forall x \in \R: \map f x \in \mathscr D$
Let $x \in \R$.
It is to be proved that $\map f x$ is a proper subset of $\Q$ such that:
:$(1): \quad \forall z \in \map f x: \forall y \in \Q: y < z \implies y \in \map f x$
:$(2): \quad \forall z \in \map f x: \exists y \in \map f x: z < y$
We ha... | Let $\mathscr D$ be the [[Definition:Set of Sets|set]] of all [[Definition:Dedekind Cut|Dedekind cuts]] of the [[Definition:Totally Ordered Set|totally ordered set]] $\struct {\Q, \le}$.
Define a [[Definition:Mapping|mapping]] $f: \R \to \mathscr D$ as:
:$\forall x \in \R: \map f x = \set {y \in \Q: y < x}$
Then $f$ ... | First, we will prove that:
:$\forall x \in \R: \map f x \in \mathscr D$
Let $x \in \R$.
It is to be proved that $\map f x$ is a [[Definition:Proper Subset|proper subset]] of $\Q$ such that:
:$(1): \quad \forall z \in \map f x: \forall y \in \Q: y < z \implies y \in \map f x$
:$(2): \quad \forall z \in \map f x: \exis... | Reals are Isomorphic to Dedekind Cuts | https://proofwiki.org/wiki/Reals_are_Isomorphic_to_Dedekind_Cuts | https://proofwiki.org/wiki/Reals_are_Isomorphic_to_Dedekind_Cuts | [
"Real Numbers",
"Dedekind Cuts"
] | [
"Definition:Set of Sets",
"Definition:Dedekind Cut",
"Definition:Totally Ordered Set",
"Definition:Mapping",
"Definition:Bijection"
] | [
"Definition:Proper Subset",
"Definition:Proper Subset",
"Between two Real Numbers exists Rational Number",
"Definition:Bijection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Injection",
"Between two Real Numbers exists Rational Number",
"Definition:Surjection",
"Definition:Dedeki... |
proofwiki-10895 | Slope of Orthogonal Curves | Let $C_1$ and $C_2$ be curves in a cartesian plane.
Let $C_1$ and $C_2$ intersect each other at $P$.
Let the slope of $C_1$ and $C_2$ at $P$ be $m_1$ and $m_2$.
Then $C_1$ and $C_2$ are orthogonal {{iff}}:
:$m_1 = -\dfrac 1 {m_2}$ | Let the slopes of $C_1$ and $C_2$ at $P$ be defined by the vectors $\mathbf v_1$ and $\mathbf v_2$ represented as column matrices:
:$\mathbf v_1 = \begin{bmatrix} x_1 \\ y_1 \end{bmatrix} , \mathbf v_2 = \begin{bmatrix} x_2 \\ y_2 \end{bmatrix}$
By Non-Zero Vectors are Orthogonal iff Perpendicular:
:$\mathbf v_1 \cdot ... | Let $C_1$ and $C_2$ be [[Definition:Curve|curves]] in a [[Definition:Cartesian Plane|cartesian plane]].
Let $C_1$ and $C_2$ [[Definition:Intersection (Geometry)|intersect]] each other at $P$.
Let the [[Definition:Slope|slope]] of $C_1$ and $C_2$ at $P$ be $m_1$ and $m_2$.
Then $C_1$ and $C_2$ are [[Definition:Ortho... | Let the [[Definition:Slope|slopes]] of $C_1$ and $C_2$ at $P$ be defined by the [[Definition:Vector|vectors]] $\mathbf v_1$ and $\mathbf v_2$ represented as [[Definition:Column Matrix|column matrices]]:
:$\mathbf v_1 = \begin{bmatrix} x_1 \\ y_1 \end{bmatrix} , \mathbf v_2 = \begin{bmatrix} x_2 \\ y_2 \end{bmatrix}$
B... | Slope of Orthogonal Curves | https://proofwiki.org/wiki/Slope_of_Orthogonal_Curves | https://proofwiki.org/wiki/Slope_of_Orthogonal_Curves | [
"Analytic Geometry"
] | [
"Definition:Line/Curve",
"Definition:Cartesian Plane",
"Definition:Intersection (Geometry)",
"Definition:Slope",
"Definition:Orthogonal Curves"
] | [
"Definition:Slope",
"Definition:Vector",
"Definition:Column Matrix",
"Non-Zero Vectors are Orthogonal iff Perpendicular",
"Definition:Orthogonal Curves",
"Definition:Dot Product",
"Category:Analytic Geometry"
] |
proofwiki-10896 | Orthogonal Trajectories/Examples/Concentric Circles | Consider the one-parameter family of curves:
:$(1): \quad x^2 + y^2 = c$
Its family of orthogonal trajectories is given by the equation:
:$y = c x$
:500px | We use the technique of formation of ordinary differential equation by elimination.
Differentiating $(1)$ {{WRT|Differentiation}} $x$ gives:
:$2 x + 2 y \dfrac {\d y} {\d x} = 0$
from which:
:$\dfrac {\d y} {\d x} = -\dfrac x y$
Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogon... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]]:
:$(1): \quad x^2 + y^2 = c$
Its [[Definition:Orthogonal Trajectories|family of orthogonal trajectories]] is given by the equation:
:$y = c x$
:[[File:ConcentricCirclesOrthogonalTrajectories.png|500px]] | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $x$ gives:
:$2 x + 2 y \dfrac {\d y} {\d x} = 0$
from which:
:$\dfrac {\d y} {\d x}... | Orthogonal Trajectories/Examples/Concentric Circles | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Concentric_Circles | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Concentric_Circles | [
"Examples of Orthogonal Trajectories",
"Circles"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Orthogonal Trajectories",
"File:ConcentricCirclesOrthogonalTrajectories.png"
] | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories",
"Solution to Separable Differential Equation",
"Primitive of Reciprocal"
] |
proofwiki-10897 | Angle of Tangent to Radius in Polar Coordinates | Let $C$ be a curve embedded in a plane defined by polar coordinates.
Let $P$ be the point at $\polar {r, \theta}$.
Then the angle $\psi$ made by the tangent to $C$ at $P$ with the radial coordinate is given by:
:$\tan \psi = r \dfrac {\d \theta} {\d r}$ | :400px
{{ProofWanted}} | Let $C$ be a [[Definition:Curve|curve]] embedded in a [[Definition:Plane|plane]] defined by [[Definition:Polar Coordinates|polar coordinates]].
Let $P$ be the [[Definition:Point|point]] at $\polar {r, \theta}$.
Then the [[Definition:Angle|angle]] $\psi$ made by the [[Definition:Tangent Line|tangent]] to $C$ at $P$ wi... | :[[File:TangentToRadiusPolar.png|400px]]
{{ProofWanted}} | Angle of Tangent to Radius in Polar Coordinates | https://proofwiki.org/wiki/Angle_of_Tangent_to_Radius_in_Polar_Coordinates | https://proofwiki.org/wiki/Angle_of_Tangent_to_Radius_in_Polar_Coordinates | [
"Analytic Geometry",
"Polar Coordinates",
"Tangents"
] | [
"Definition:Line/Curve",
"Definition:Plane Surface",
"Definition:Polar Coordinates",
"Definition:Point",
"Definition:Angle",
"Definition:Tangent Line",
"Definition:Polar Coordinates/Radial Coordinate"
] | [
"File:TangentToRadiusPolar.png"
] |
proofwiki-10898 | Orthogonal Trajectories/Examples/Circles Tangent to Y Axis | Consider the one-parameter family of curves:
:$(1): \quad x^2 + y^2 = 2 c x$
which describes the loci of circles tangent to the $y$-axis at the origin.
Its family of orthogonal trajectories is given by the equation:
:$x^2 + y^2 = 2 c y$
which describes the loci of circles tangent to the $x$-axis at the origin.
:600px | We use the technique of formation of ordinary differential equation by elimination.
Differentiating $(1)$ {{WRT|Differentiation}} $x$ gives:
:$2 x + 2 y \dfrac {\d y} {\d x} = 2 c$
from which:
:$\dfrac {\d y} {\d x} = \dfrac {y^2 - x^2} {2 x y}$
Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the f... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]]:
:$(1): \quad x^2 + y^2 = 2 c x$
which describes the [[Definition:Locus|loci]] of [[Definition:Circle|circles]] [[Definition:Tangent Line|tangent]] to the [[Definition:Y-Axis|$y$-axis]] at the [[Definition:Origin|origin]].
Its ... | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
[[Definition:Differentiation|Differentiating]] $(1)$ {{WRT|Differentiation}} $x$ gives:
:$2 x + 2 y \dfrac {\d y} {\d x} = 2 c$
from which:
:$\dfrac {\d y} {\d ... | Orthogonal Trajectories/Examples/Circles Tangent to Y Axis/Proof 1 | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Circles_Tangent_to_Y_Axis | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Circles_Tangent_to_Y_Axis/Proof_1 | [
"Examples of Orthogonal Trajectories",
"Circles",
"Circles Tangent to Y Axis"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Locus",
"Definition:Circle",
"Definition:Tangent Line",
"Definition:Axis/Y-Axis",
"Definition:Coordinate System/Origin",
"Definition:Orthogonal Trajectories",
"Definition:Locus",
"Definition:Circle",
"Definition:Tangent Line",
"Definition:... | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories",
"Definition:Homogeneous Function/Real Space",
"Definition:Homogeneous Function/Real Space/Degree",
"De... |
proofwiki-10899 | Orthogonal Trajectories/Examples/Circles Tangent to Y Axis | Consider the one-parameter family of curves:
:$(1): \quad x^2 + y^2 = 2 c x$
which describes the loci of circles tangent to the $y$-axis at the origin.
Its family of orthogonal trajectories is given by the equation:
:$x^2 + y^2 = 2 c y$
which describes the loci of circles tangent to the $x$-axis at the origin.
:600px | We use the technique of formation of ordinary differential equation by elimination.
Expressing $(1)$ in polar coordinates, we have:
:$(2): \quad r = 2 c \cos \theta$
Differentiating $(1)$ {{WRT|Differentiation}} $\theta$ gives:
:$(3): \quad \dfrac {\d r} {\d \theta} = -2 c \sin \theta$
Eliminating $c$ from $(2)$ and $(... | Consider the [[Definition:One-Parameter Family of Curves|one-parameter family of curves]]:
:$(1): \quad x^2 + y^2 = 2 c x$
which describes the [[Definition:Locus|loci]] of [[Definition:Circle|circles]] [[Definition:Tangent Line|tangent]] to the [[Definition:Y-Axis|$y$-axis]] at the [[Definition:Origin|origin]].
Its ... | We use the technique of [[Definition:Formation of Ordinary Differential Equation by Elimination|formation of ordinary differential equation by elimination]].
Expressing $(1)$ in [[Definition:Polar Coordinates|polar coordinates]], we have:
:$(2): \quad r = 2 c \cos \theta$
[[Definition:Differentiation|Differentiating]... | Orthogonal Trajectories/Examples/Circles Tangent to Y Axis/Proof 2 | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Circles_Tangent_to_Y_Axis | https://proofwiki.org/wiki/Orthogonal_Trajectories/Examples/Circles_Tangent_to_Y_Axis/Proof_2 | [
"Examples of Orthogonal Trajectories",
"Circles",
"Circles Tangent to Y Axis"
] | [
"Definition:Family of Curves/One-Parameter",
"Definition:Locus",
"Definition:Circle",
"Definition:Tangent Line",
"Definition:Axis/Y-Axis",
"Definition:Coordinate System/Origin",
"Definition:Orthogonal Trajectories",
"Definition:Locus",
"Definition:Circle",
"Definition:Tangent Line",
"Definition:... | [
"Definition:Formation of Ordinary Differential Equation by Elimination",
"Definition:Polar Coordinates",
"Definition:Differentiation",
"Orthogonal Trajectories of One-Parameter Family of Curves",
"Definition:Orthogonal Trajectories",
"Solution to Separable Differential Equation",
"Primitive of Reciproca... |
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