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-15400 | Parallelism is Equivalence Relation | Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$.
Then $\parallel$ is an equivalence relation on $S$. | Checking in turn each of the criteria for equivalence: | Let $S$ be the [[Definition:Set|set]] of [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is [[Definition:Parallel Lines|parallel]] to $l_2$.
Then $\parallel$ is an [[Definition:Equivalence Relation|equivalence relation... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Parallelism is Equivalence Relation | https://proofwiki.org/wiki/Parallelism_is_Equivalence_Relation | https://proofwiki.org/wiki/Parallelism_is_Equivalence_Relation | [
"Parallel Lines",
"Examples of Equivalence Relations"
] | [
"Definition:Set",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Parallel (Geometry)/Lines",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-15401 | Connected Equivalence Relation is Trivial | Let $S$ be a set.
Let $\RR$ be a relation on $S$ which is both connected and an equivalence relation.
Then $\RR$ is the trivial relation on $S$. | By definition of equivalence relation, $\RR$ is an equivalence relation {{iff}}:
:$\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$
From Relation is Connected iff Union with Inverse and Diagonal is Trivial Relation:
:$\Delta_S \cup \RR^{-1} \cup \RR = S \times S$
Hence the result.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\RR$ be a [[Definition:Endorelation|relation on $S$]] which is both [[Definition:Connected Relation|connected]] and an [[Definition:Equivalence Relation|equivalence relation]].
Then $\RR$ is the [[Definition:Trivial Relation|trivial relation]] on $S$. | By definition of [[Definition:Equivalence Relation|equivalence relation]], $\RR$ is an [[Definition:Equivalence Relation|equivalence relation]] {{iff}}:
:$\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$
From [[Relation is Connected iff Union with Inverse and Diagonal is Trivial Relation]]:
:$\Delta_S \cup \... | Connected Equivalence Relation is Trivial | https://proofwiki.org/wiki/Connected_Equivalence_Relation_is_Trivial | https://proofwiki.org/wiki/Connected_Equivalence_Relation_is_Trivial | [
"Connected Relations",
"Equivalence Relations",
"Trivial Relation",
"Connected Equivalence Relation is Trivial"
] | [
"Definition:Set",
"Definition:Endorelation",
"Definition:Connected Relation",
"Definition:Equivalence Relation",
"Definition:Trivial Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation",
"Relation is Connected iff Union with Inverse and Diagonal is Trivial Relation"
] |
proofwiki-15402 | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3 | Let $\sequence {x_n}$ be the real sequence defined as $x_n = \paren {n + 1}^{1/n}$, using exponentiation.
Then $\sequence {x_n}$ converges with a limit of $1$. | We have the definition of the power to a real number:
:$\paren {n + 1}^{1/n} = \map \exp {\dfrac 1 n \map \ln {n + 1} }$
For $n \ge 1$ then $n + 1 \le 2 n$.
Hence:
{{begin-eqn}}
{{eqn | l = \frac 1 n \map \ln {n + 1}
| o = \le
| r = \frac 1 n \map \ln {2 n}
| c = Logarithm is Strictly Increasing
}}
{{... | Let $\sequence {x_n}$ be the [[Definition:Real Sequence|real sequence]] defined as $x_n = \paren {n + 1}^{1/n}$, using [[Definition:Real Exponential Function|exponentiation]].
Then $\sequence {x_n}$ [[Definition:Convergent Sequence|converges]] with a [[Definition:Limit of Sequence (Number Field)|limit]] of $1$. | We have the definition of the [[Definition:Power to Real Number|power to a real number]]:
:$\paren {n + 1}^{1/n} = \map \exp {\dfrac 1 n \map \ln {n + 1} }$
For $n \ge 1$ then $n + 1 \le 2 n$.
Hence:
{{begin-eqn}}
{{eqn | l = \frac 1 n \map \ln {n + 1}
| o = \le
| r = \frac 1 n \map \ln {2 n}
| c ... | Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition/Lemma_3 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Sufficient_Condition/Lemma_3 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Real Sequence",
"Definition:Exponential Function/Real",
"Definition:Convergent Sequence",
"Definition:Limit of Sequence (Number Field)"
] | [
"Definition:Power (Algebra)/Real Number",
"Logarithm is Strictly Increasing",
"Logarithm on Positive Real Numbers is Group Isomorphism",
"Powers Drown Logarithms",
"Sequence of Powers of Reciprocals is Null Sequence/Corollary",
"Combination Theorem for Sequences/Real/Combined Sum Rule",
"Squeeze Theorem... |
proofwiki-15403 | Characterisation of Non-Archimedean Division Ring Norms/Corollary 1 | $\norm {\,\cdot\,}$ is non-Archimedean {{iff}}:
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0}} = 1$. | By Characterisation of Non-Archimedean Division Ring Norms then:
:$\norm {\,\cdot\,}$ is non-Archimedean {{iff}}:
::$\sup \set {\norm {n \cdot 1_R}: n \in \N_{\gt 0}} \le 1$
By norm of unity then:
:$\norm {1_R} = 1$
The result follows.
{{qed}} | $\norm {\,\cdot\,}$ is [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] {{iff}}:
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0}} = 1$. | By [[Characterisation of Non-Archimedean Division Ring Norms]] then:
:$\norm {\,\cdot\,}$ is [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] {{iff}}:
::$\sup \set {\norm {n \cdot 1_R}: n \in \N_{\gt 0}} \le 1$
By [[Properties of Norm on Division Ring/Norm of Unity|norm of unity]] then:
:$\norm {1_R} ... | Characterisation of Non-Archimedean Division Ring Norms/Corollary 1 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_1 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_1 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Characterisation of Non-Archimedean Division Ring Norms",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Properties of Norm on Division Ring/Norm of Unity"
] |
proofwiki-15404 | Characterisation of Non-Archimedean Division Ring Norms/Corollary 3 | $\norm {\,\cdot\,}$ is Archimedean {{iff}}:
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } = +\infty$ | By Characterisation of Non-Archimedean Division Ring Norms:
:$\norm {\,\cdot\,}$ is Archimedean $\iff \sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1$
By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|2}}:
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1 \iff \sup \set {\norm {n \cdot 1_R}... | $\norm {\,\cdot\,}$ is [[Definition:Archimedean Division Ring Norm|Archimedean]] {{iff}}:
:$\sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } = +\infty$ | By [[Characterisation of Non-Archimedean Division Ring Norms]]:
:$\norm {\,\cdot\,}$ is [[Definition:Archimedean Division Ring Norm|Archimedean]] $\iff \sup \set {\norm {n \cdot 1_R}: n \in \N_{>0} } > 1$
By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|2}}:
:$\sup \set {\norm {n \cdot 1_R}: n \... | Characterisation of Non-Archimedean Division Ring Norms/Corollary 3 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_3 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_3 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)/Archimedean"
] | [
"Characterisation of Non-Archimedean Division Ring Norms",
"Definition:Non-Archimedean/Norm (Division Ring)/Archimedean"
] |
proofwiki-15405 | Characterisation of Non-Archimedean Division Ring Norms/Corollary 2 | Let $\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0} } = C < +\infty$. | {{AimForCont}} $C > 1$.
By Characterizing Property of Supremum of Subset of Real Numbers:
:$\exists m \in \N_{> 0}: \norm {m \cdot 1_R} > 1$
Let
:$x = m \cdot 1_R$
:$y = x^{-1}$
By Norm of Inverse:
:$\norm y < 1$
By Sequence of Powers of Number less than One:
:$\ds \lim_{n \mathop \to \infty} \norm y^n = 0$
By Reci... | Let $\sup \set {\norm {n \cdot 1_R}: n \in \N_{> 0} } = C < +\infty$. | {{AimForCont}} $C > 1$.
By [[Characterizing Property of Supremum of Subset of Real Numbers]]:
:$\exists m \in \N_{> 0}: \norm {m \cdot 1_R} > 1$
Let
:$x = m \cdot 1_R$
:$y = x^{-1}$
By [[Properties of Norm on Division Ring/Norm of Inverse|Norm of Inverse]]:
:$\norm y < 1$
By [[Sequence of Powers of Number le... | Characterisation of Non-Archimedean Division Ring Norms/Corollary 2 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_2 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_2 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [] | [
"Characterizing Property of Supremum of Subset of Real Numbers",
"Properties of Norm on Division Ring/Norm of Inverse",
"Sequence of Powers of Number less than One",
"Reciprocal of Null Sequence",
"Properties of Norm on Division Ring/Norm of Inverse",
"Definition:Contradiction",
"Characterisation of Non... |
proofwiki-15406 | Characterisation of Non-Archimedean Division Ring Norms/Corollary 4 | Let $R$ have characteristic $p > 0$.
Then $\norm {\,\cdot\,}$ is a non_Archimedean norm on $R$. | Because $R$ has characteristic $p > 0$, the set:
:$\set {n \cdot 1_k: n \in \Z}$
has cardinality $p - 1$.
Therefore:
:$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = \max \set {\norm {1 \cdot 1_R}, \norm {2 \cdot 1_R}, \cdots, \norm {\paren {p - 1} \cdot 1_R} } < +\infty$
By {{Corollary|Characterisation of Non-Archimedean... | Let $R$ have [[Definition:Characteristic of Ring|characteristic]] $p > 0$.
Then $\norm {\,\cdot\,}$ is a [[Definition:Non-Archimedean Division Ring Norm|non_Archimedean norm]] on $R$. | Because $R$ has [[Definition:Characteristic of Ring|characteristic]] $p > 0$, the [[Definition:Set|set]]:
:$\set {n \cdot 1_k: n \in \Z}$
has [[Definition:Cardinality|cardinality]] $p - 1$.
Therefore:
:$\sup \set {\norm {n \cdot 1_R}: n \in \Z} = \max \set {\norm {1 \cdot 1_R}, \norm {2 \cdot 1_R}, \cdots, \norm {\p... | Characterisation of Non-Archimedean Division Ring Norms/Corollary 4 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_4 | https://proofwiki.org/wiki/Characterisation_of_Non-Archimedean_Division_Ring_Norms/Corollary_4 | [
"Characterisation of Non-Archimedean Division Ring Norms"
] | [
"Definition:Characteristic of Ring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Characteristic of Ring",
"Definition:Set",
"Definition:Cardinality",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Category:Characterisation of Non-Archimedean Division Ring Norms"
] |
proofwiki-15407 | Norms Equivalent to Absolute Value on Rational Numbers | Let $\alpha \in \R_{> 0}$.
Let $\norm {\,\cdot\,}:\Q \to \R$ be the mapping defined by:
:$\forall x \in \Q: \norm x = \size x^\alpha$
where $\size x$ is the absolute value of $x$ in $\Q$.
Then:
:$\norm {\,\cdot\,}$ be a norm on $\Q$
{{iff}}:
:$\alpha \le 1$ | === Necessary Condition ===
{{:Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition}}{{qed|lemma}} | Let $\alpha \in \R_{> 0}$.
Let $\norm {\,\cdot\,}:\Q \to \R$ be the [[Definition:Mapping|mapping]] defined by:
:$\forall x \in \Q: \norm x = \size x^\alpha$
where $\size x$ is the [[Definition:Absolute Value|absolute value]] of $x$ in $\Q$.
Then:
:$\norm {\,\cdot\,}$ be a [[Definition:Norm on Division Ring|norm]] o... | === [[Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition|Necessary Condition]] ===
{{:Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition}}{{qed|lemma}} | Norms Equivalent to Absolute Value on Rational Numbers | https://proofwiki.org/wiki/Norms_Equivalent_to_Absolute_Value_on_Rational_Numbers | https://proofwiki.org/wiki/Norms_Equivalent_to_Absolute_Value_on_Rational_Numbers | [
"Normed Division Rings"
] | [
"Definition:Mapping",
"Definition:Absolute Value",
"Definition:Norm/Division Ring"
] | [
"Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition"
] |
proofwiki-15408 | Reflexive Relation on Set of Cardinality 2 is Transitive | Let $S$ be a set whose cardinality is equal to $2$:
:$\card S = 2$
Let $\odot \subseteq S \times S$ be a reflexive relation on $S$.
Then $\odot$ is also transitive. | {{WLOG}}, let $S = \set {a, b}$.
Let $\odot$ be reflexive.
By definition of reflexive relation:
:$\Delta_S \subseteq \odot$
where $\Delta_S$ is the diagonal relation:
:$\Delta_S = \set {\tuple {x, x}: x \in S}$
That is:
:$\set {\tuple {a, a}, \tuple {b, b} } \subseteq \odot$
Suppose $\set {\tuple {a, a}, \tuple {b, b} ... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is equal to $2$:
:$\card S = 2$
Let $\odot \subseteq S \times S$ be a [[Definition:Reflexive Relation|reflexive relation]] on $S$.
Then $\odot$ is also [[Definition:Transitive Relation|transitive]]. | {{WLOG}}, let $S = \set {a, b}$.
Let $\odot$ be [[Definition:Reflexive Relation|reflexive]].
By definition of [[Definition:Reflexive Relation|reflexive relation]]:
:$\Delta_S \subseteq \odot$
where $\Delta_S$ is the [[Definition:Diagonal Relation|diagonal relation]]:
:$\Delta_S = \set {\tuple {x, x}: x \in S}$
That ... | Reflexive Relation on Set of Cardinality 2 is Transitive | https://proofwiki.org/wiki/Reflexive_Relation_on_Set_of_Cardinality_2_is_Transitive | https://proofwiki.org/wiki/Reflexive_Relation_on_Set_of_Cardinality_2_is_Transitive | [
"Reflexive Relations",
"Transitive Relations"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Reflexive Relation",
"Definition:Transitive Relation"
] | [
"Definition:Reflexive Relation",
"Definition:Reflexive Relation",
"Definition:Diagonal Relation",
"Diagonal Relation is Equivalence",
"Definition:Transitive Relation",
"Definition:Reflexive Relation",
"Definition:Reflexive Relation",
"Definition:Transitive Relation",
"Category:Reflexive Relations",
... |
proofwiki-15409 | Relation on Set of Cardinality 2 cannot be Non-Symmetric and Non-Transitive | Let $S$ be a set whose cardinality is equal to $2$:
:$\card S = 2$
Let $\odot \subseteq S \times S$ be a relation on $S$.
Then it is not possible for $\odot$ to be not symmetric and also not transitive. | {{WLOG}}, let $S = \set {a, b}$.
Let $\odot$ not be symmetric.
{{AimForCont}} $\odot$ is not transitive.
As $\odot$ is not symmetric:
:$\exists \tuple {x, y} \in \odot: \tuple {y, x} \notin \odot$
Thus there are two possibilities:
:$\exists \tuple {a, b} \in \odot: \tuple {b, a} \notin \odot$
:$\exists \tuple {b, a} \i... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is equal to $2$:
:$\card S = 2$
Let $\odot \subseteq S \times S$ be a [[Definition:Endorelation|relation on $S$]].
Then it is not possible for $\odot$ to be not [[Definition:Symmetric Relation|symmetric]] and also not [[Definition:Tran... | {{WLOG}}, let $S = \set {a, b}$.
Let $\odot$ not be [[Definition:Symmetric Relation|symmetric]].
{{AimForCont}} $\odot$ is not [[Definition:Transitive Relation|transitive]].
As $\odot$ is not [[Definition:Symmetric Relation|symmetric]]:
:$\exists \tuple {x, y} \in \odot: \tuple {y, x} \notin \odot$
Thus there are ... | Relation on Set of Cardinality 2 cannot be Non-Symmetric and Non-Transitive | https://proofwiki.org/wiki/Relation_on_Set_of_Cardinality_2_cannot_be_Non-Symmetric_and_Non-Transitive | https://proofwiki.org/wiki/Relation_on_Set_of_Cardinality_2_cannot_be_Non-Symmetric_and_Non-Transitive | [
"Non-Symmetric Relations",
"Non-Transitive Relations"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Endorelation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation"
] | [
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Symmetric Relation",
"Definition:Symmetric Relation",
"Definition:Contradiction",
"Definition:Transitive Relation",
"Definition:Symmetric Relation",
"De... |
proofwiki-15410 | Relations with Combinations of Reflexivity, Symmetry and Transitivity Properties | Let $S$ be a set which has at least $3$ elements.
Then it is possible to set up a relation $\circledcirc$ on $S$ which has any combination of the $3$ properties:
:Reflexivity
:Symmetry
:Transitivity
but this is not possible for a set which has fewer than $3$ elements. | In the following:
:$S_n$ denotes the set $S_n = \set {s_1, s_2, \ldots, s_n}$ of cardinality $n$, where $n \in \Z_{\ge 0}$ is a non-negative integer.
:$\circledcirc$ denotes an arbitrary relation on $S_n$.
Let:
:$\map R \circledcirc$ denote that $\circledcirc$ is reflexive
:$\map S \circledcirc$ denote that $\circledci... | Let $S$ be a [[Definition:Set|set]] which has at least $3$ [[Definition:Element|elements]].
Then it is possible to set up a [[Definition:Endorelation|relation $\circledcirc$ on $S$]] which has any combination of the $3$ properties:
:[[Definition:Reflexive Relation|Reflexivity]]
:[[Definition:Symmetric Relation|Symmetr... | In the following:
:$S_n$ denotes the [[Definition:Set|set]] $S_n = \set {s_1, s_2, \ldots, s_n}$ of [[Definition:Cardinality|cardinality]] $n$, where $n \in \Z_{\ge 0}$ is a [[Definition:Non-Negative Integer|non-negative integer]].
:$\circledcirc$ denotes an arbitrary [[Definition:Endorelation|relation on $S_n$]].
L... | Relations with Combinations of Reflexivity, Symmetry and Transitivity Properties | https://proofwiki.org/wiki/Relations_with_Combinations_of_Reflexivity,_Symmetry_and_Transitivity_Properties | https://proofwiki.org/wiki/Relations_with_Combinations_of_Reflexivity,_Symmetry_and_Transitivity_Properties | [
"Reflexive Relations",
"Symmetric Relations",
"Transitive Relations"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Set",
"Definition:Element"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Positive/Integer",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relatio... |
proofwiki-15411 | Relation on Set of Cardinality 1 is Symmetric and Transitive | Let $S$ be a set whose cardinality is equal to $1$:
:$\card S = 1$
Let $\odot \subseteq S \times S$ be a relation on $S$.
Then $\odot$ is both symmetric and transitive. | {{WLOG}}, let $S = \set a$.
There are $2$ relations on $S$:
$(1): \quad \odot := \O$, which is the null relation on $S$.
From Null Relation is Antireflexive, Symmetric and Transitive, $\odot$ is antireflexive, symmetric and transitive.
Thus in this case $\odot$ is both symmetric and transitive.
$(2): \quad \odot := \se... | Let $S$ be a [[Definition:Set|set]] whose [[Definition:Cardinality|cardinality]] is equal to $1$:
:$\card S = 1$
Let $\odot \subseteq S \times S$ be a [[Definition:Endorelation|relation on $S$]].
Then $\odot$ is both [[Definition:Symmetric Relation|symmetric]] and [[Definition:Transitive Relation|transitive]]. | {{WLOG}}, let $S = \set a$.
There are $2$ [[Definition:Endorelation|relations on $S$]]:
$(1): \quad \odot := \O$, which is the [[Definition:Null Relation|null relation]] on $S$.
From [[Null Relation is Antireflexive, Symmetric and Transitive]], $\odot$ is [[Definition:Antireflexive Relation|antireflexive]], [[Defin... | Relation on Set of Cardinality 1 is Symmetric and Transitive | https://proofwiki.org/wiki/Relation_on_Set_of_Cardinality_1_is_Symmetric_and_Transitive | https://proofwiki.org/wiki/Relation_on_Set_of_Cardinality_1_is_Symmetric_and_Transitive | [
"Symmetric Relations",
"Transitive Relations"
] | [
"Definition:Set",
"Definition:Cardinality",
"Definition:Endorelation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation"
] | [
"Definition:Endorelation",
"Definition:Null Relation",
"Null Relation is Antireflexive, Symmetric and Transitive",
"Definition:Antireflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Tr... |
proofwiki-15412 | Congruence of Triangles is Equivalence Relation | Let $S$ denote the set of all triangles in the plane.
Let $\triangle A \cong \triangle B$ denote the relation that $\triangle A$ is congruent to $\triangle B$.
Then $\cong$ is an equivalence relation on $S$. | Checking in turn each of the criteria for equivalence: | Let $S$ denote the [[Definition:Set|set]] of all [[Definition:Triangle (Geometry)|triangles]] in [[Definition:The Plane|the plane]].
Let $\triangle A \cong \triangle B$ denote the [[Definition:Relation|relation]] that $\triangle A$ is [[Definition:Congruence (Geometry)|congruent]] to $\triangle B$.
Then $\cong$ is an... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Congruence of Triangles is Equivalence Relation | https://proofwiki.org/wiki/Congruence_of_Triangles_is_Equivalence_Relation | https://proofwiki.org/wiki/Congruence_of_Triangles_is_Equivalence_Relation | [
"Examples of Equivalence Relations",
"Triangles"
] | [
"Definition:Set",
"Definition:Triangle (Geometry)",
"Definition:Plane Surface/The Plane",
"Definition:Relation",
"Definition:Congruence (Geometry)",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-15413 | Cardinality of Mapping | Let $S$ be a finite set whose cardinality is $n$:
:$\card S = n$
Let $T$ be a non-empty set
Let $f: S \to T$ be a mapping.
Then:
:$\card f = n$ | First suppose $T = \O$ there are no elements in $f$
From Null Relation is Mapping iff Domain is Empty Set, there are no elements in $f$.
Hence in this case $\card f = 0$, whatever $\card S$ may be,
By definition of mapping, $f$ is a set of ordered pairs $\tuple {s, t}$ where $s \in S$ and $t \in T$, such that:
:$(1): \... | Let $S$ be a [[Definition:Finite Set|finite set]] whose [[Definition:Cardinality|cardinality]] is $n$:
:$\card S = n$
Let $T$ be a [[Definition:Non-Empty|non-empty set]]
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Then:
:$\card f = n$ | First suppose $T = \O$ there are no [[Definition:Element|elements]] in $f$
From [[Null Relation is Mapping iff Domain is Empty Set]], there are no [[Definition:Element|elements]] in $f$.
Hence in this case $\card f = 0$, whatever $\card S$ may be,
By definition of [[Definition:Mapping|mapping]], $f$ is a [[Definiti... | Cardinality of Mapping | https://proofwiki.org/wiki/Cardinality_of_Mapping | https://proofwiki.org/wiki/Cardinality_of_Mapping | [
"Mapping Theory",
"Cardinality"
] | [
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Non-Empty",
"Definition:Mapping"
] | [
"Definition:Element",
"Null Relation is Mapping iff Domain is Empty Set",
"Definition:Element",
"Definition:Mapping",
"Definition:Set",
"Definition:Ordered Pair",
"Dirichlet's Box Principle/Corollary",
"Definition:Mapping",
"Definition:Contradiction"
] |
proofwiki-15414 | Even Integer Plus 5 is Odd | Let $x \in \Z$ be an even integer.
Then $x + 5$ is odd. | Let $x$ be an even integer.
Let $y = 2 n + 5$.
Assume $y = x + 5$ is not an odd integer.
Then:
:$y = x + 5 = 2 n$
where $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = 2 n - 5
| c =
}}
{{eqn | r = \paren {2 n - 6} + 1
| c =
}}
{{eqn | r = 2 \paren {n - 3} + 1
| c =
}}
{{eqn | r = 2 r + 1
... | Let $x \in \Z$ be an [[Definition:Even Integer|even integer]].
Then $x + 5$ is [[Definition:Odd Integer|odd]]. | Let $x$ be an [[Definition:Even Integer|even integer]].
Let $y = 2 n + 5$.
Assume $y = x + 5$ is not an [[Definition:Odd Integer|odd integer]].
Then:
:$y = x + 5 = 2 n$
where $n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = 2 n - 5
| c =
}}
{{eqn | r = \paren {2 n - 6} + 1
| c =
}}
{{eqn | ... | Even Integer Plus 5 is Odd/Indirect Proof | https://proofwiki.org/wiki/Even_Integer_Plus_5_is_Odd | https://proofwiki.org/wiki/Even_Integer_Plus_5_is_Odd/Indirect_Proof | [
"Even Integers",
"Odd Integers",
"Even Integer Plus 5 is Odd"
] | [
"Definition:Even Integer",
"Definition:Odd Integer"
] | [
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:False",
"Definition:Even Integer",
"Rule of Transposition",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-15415 | Even Integer Plus 5 is Odd | Let $x \in \Z$ be an even integer.
Then $x + 5$ is odd. | Let $x$ be an even integer.
Then by definition:
:$x = 2 n$
for some integer $n$.
{{AimForCont}} $y = x + 5 = 2 m$ for some integer $m$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = 2 m - 5
| c =
}}
{{eqn | r = \paren {2 m - 6} + 1
| c =
}}
{{eqn | r = 2 \paren {m - 3} + 1
| c =
}}
{{eqn | r = 2 r ... | Let $x \in \Z$ be an [[Definition:Even Integer|even integer]].
Then $x + 5$ is [[Definition:Odd Integer|odd]]. | Let $x$ be an [[Definition:Even Integer|even integer]].
Then by definition:
:$x = 2 n$
for some [[Definition:Integer|integer]] $n$.
{{AimForCont}} $y = x + 5 = 2 m$ for some [[Definition:Integer|integer]] $m$.
Then:
{{begin-eqn}}
{{eqn | l = x
| r = 2 m - 5
| c =
}}
{{eqn | r = \paren {2 m - 6} + 1
... | Even Integer Plus 5 is Odd/Proof by Contradiction | https://proofwiki.org/wiki/Even_Integer_Plus_5_is_Odd | https://proofwiki.org/wiki/Even_Integer_Plus_5_is_Odd/Proof_by_Contradiction | [
"Even Integers",
"Odd Integers",
"Even Integer Plus 5 is Odd"
] | [
"Definition:Even Integer",
"Definition:Odd Integer"
] | [
"Definition:Even Integer",
"Definition:Integer",
"Definition:Integer",
"Definition:Odd Integer",
"Definition:Contradiction",
"Definition:Premise",
"Definition:Even Integer",
"Proof by Contradiction"
] |
proofwiki-15416 | Arcsine as Integral | :$\ds \map \arcsin x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$ | === Lemma 1 ===
{{:Arcsine as Integral/Lemma 1}}{{qed|lemma}} | :$\ds \map \arcsin x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$ | === [[Arcsine as Integral/Lemma 1|Lemma 1]] ===
{{:Arcsine as Integral/Lemma 1}}{{qed|lemma}} | Arcsine as Integral | https://proofwiki.org/wiki/Arcsine_as_Integral | https://proofwiki.org/wiki/Arcsine_as_Integral | [
"Arcsine Function",
"Primitive of Reciprocal of Root of a squared minus x squared",
"Arcsine as Integral"
] | [] | [
"Arcsine as Integral/Lemma 1"
] |
proofwiki-15417 | Sum of Even Sequence of Products of Consecutive Fibonacci Numbers | :$\ds \sum_{j \mathop = 1}^{2 n} F_j F_{j + 1} = {F_{2 n + 1} }^2 - 1$ | From Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers:
:$(1): \quad \ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^{2 n} F_j F_{j + 1}
| r = \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} + F_{2 n} F_{2 n + 1}
| c =
}}
{{... | :$\ds \sum_{j \mathop = 1}^{2 n} F_j F_{j + 1} = {F_{2 n + 1} }^2 - 1$ | From [[Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers]]:
:$(1): \quad \ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^{2 n} F_j F_{j + 1}
| r = \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} + F_{2 n} F_{2 n + 1}
| c = ... | Sum of Even Sequence of Products of Consecutive Fibonacci Numbers | https://proofwiki.org/wiki/Sum_of_Even_Sequence_of_Products_of_Consecutive_Fibonacci_Numbers | https://proofwiki.org/wiki/Sum_of_Even_Sequence_of_Products_of_Consecutive_Fibonacci_Numbers | [
"Sums of Sequences",
"Fibonacci Numbers"
] | [] | [
"Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers",
"Cassini's Identity"
] |
proofwiki-15418 | Lucas Number as Element of Recursive Sequence | Let $L_k$ be the $k$th Lucas number, defined as the sum of two Fibonacci numbers:
:$L_n = F_{n - 1} + F_{n + 1}$
Then $L_n$ can be defined as the $n$th element of the recursive sequence:
:$L_n = \begin{cases}
2 & : n = 0 \\
1 & : n = 1 \\
L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$ | Proof by induction:
Let $L_n$ be the Lucas number defined as the sum of two Fibonacci numbers:
:$L_n = F_{n - 1} + F_{n + 1}$
For all $n \in \N$, let $\map P n$ be the proposition:
:$L_n = \begin{cases}
2 & : n = 0 \\
1 & : n = 1 \\
L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$ | Let $L_k$ be the $k$th [[Definition:Lucas Number/Definition 2|Lucas number]], defined as the sum of two [[Definition:Fibonacci Number|Fibonacci numbers]]:
:$L_n = F_{n - 1} + F_{n + 1}$
Then $L_n$ can be defined as the $n$th [[Definition:Element|element]] of the [[Definition:Recursive Sequence|recursive sequence]]:
:... | Proof by [[Second Principle of Mathematical Induction|induction]]:
Let $L_n$ be the [[Definition:Lucas Number/Definition 2|Lucas number]] defined as the sum of two [[Definition:Fibonacci Number|Fibonacci numbers]]:
:$L_n = F_{n - 1} + F_{n + 1}$
For all $n \in \N$, let $\map P n$ be the [[Definition:Proposition|prop... | Lucas Number as Element of Recursive Sequence | https://proofwiki.org/wiki/Lucas_Number_as_Element_of_Recursive_Sequence | https://proofwiki.org/wiki/Lucas_Number_as_Element_of_Recursive_Sequence | [
"Lucas Numbers"
] | [
"Definition:Lucas Number/Definition 2",
"Definition:Fibonacci Number",
"Definition:Element",
"Definition:Recursive Sequence"
] | [
"Second Principle of Mathematical Induction",
"Definition:Lucas Number/Definition 2",
"Definition:Fibonacci Number",
"Definition:Proposition",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction",
"Second Principle of ... |
proofwiki-15419 | Product of nth Lucas and Fibonacci Numbers | Let $L_k$ be the $k$th Lucas number.
Let $F_k$ be the $k$th Fibonacci number.
Then:
:$\forall n \in \N_{>0}: F_n L_n = F_{2 n}$ | By definition of Lucas numbers:
:$L_n = F_{n - 1} + F_{n + 1}$
Hence:
:$F_n L_n = F_n \paren {F_{n - 1} + F_{n + 1} }$
From Honsberger's Identity:
:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
The result follows by setting $m = n$.
{{qed}} | Let $L_k$ be the $k$th [[Definition:Lucas Number|Lucas number]].
Let $F_k$ be the $k$th [[Definition:Fibonacci Number|Fibonacci number]].
Then:
:$\forall n \in \N_{>0}: F_n L_n = F_{2 n}$ | By definition of [[Definition:Lucas Number/Definition 2|Lucas numbers]]:
:$L_n = F_{n - 1} + F_{n + 1}$
Hence:
:$F_n L_n = F_n \paren {F_{n - 1} + F_{n + 1} }$
From [[Honsberger's Identity]]:
:$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$
The result follows by setting $m = n$.
{{qed}} | Product of nth Lucas and Fibonacci Numbers | https://proofwiki.org/wiki/Product_of_nth_Lucas_and_Fibonacci_Numbers | https://proofwiki.org/wiki/Product_of_nth_Lucas_and_Fibonacci_Numbers | [
"Lucas Numbers",
"Fibonacci Numbers"
] | [
"Definition:Lucas Number",
"Definition:Fibonacci Number"
] | [
"Definition:Lucas Number/Definition 2",
"Honsberger's Identity"
] |
proofwiki-15420 | Representation of Integers in Balanced Ternary | Let $n \in \Z$ be an integer.
$n$ can be represented uniquely in balanced ternary:
:$\ds n = \sum_{j \mathop = 0}^m r_j 3^j$
:$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}$
such that:
where:
:$m \in \Z_{>0}$ is a strictly positive integer such that $3^m < \size {2 n} < 3^{m + 1}$
:all the $r_j$ are such that $r_j \in \set... | Let $n \in \Z$.
Let $m \in \Z_{\ge 0}$ be such that:
:$3^m + 1 \le \size {2 n} \le 3^{m + 1} - 1$
where $\size {2 n}$ denotes the absolute value of $2 n$.
As $2 n$ is even, this is always possible, because $3^r$ is always an odd integer for non-negative $r$.
Let $d = \dfrac {3^{m + 1} - 1} 2$.
Let $k = n + d$.
We have ... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
$n$ can be represented [[Definition:Unique|uniquely]] in [[Definition:Balanced Ternary Representation|balanced ternary]]:
:$\ds n = \sum_{j \mathop = 0}^m r_j 3^j$
:$\sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}$
such that:
where:
:$m \in \Z_{>0}$ is a [[Definition:... | Let $n \in \Z$.
Let $m \in \Z_{\ge 0}$ be such that:
:$3^m + 1 \le \size {2 n} \le 3^{m + 1} - 1$
where $\size {2 n}$ denotes the [[Definition:Absolute Value|absolute value]] of $2 n$.
As $2 n$ is [[Definition:Even Integer|even]], this is always possible, because $3^r$ is always an [[Definition:Odd Integer|odd intege... | Representation of Integers in Balanced Ternary | https://proofwiki.org/wiki/Representation_of_Integers_in_Balanced_Ternary | https://proofwiki.org/wiki/Representation_of_Integers_in_Balanced_Ternary | [
"Balanced Ternary Representation"
] | [
"Definition:Integer",
"Definition:Unique",
"Definition:Balanced Ternary Representation",
"Definition:Strictly Positive/Integer"
] | [
"Definition:Absolute Value",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Positive/Integer",
"Definition:Ternary Notation",
"Basis Representation Theorem",
"Definition:Unique",
"Sum of Geometric Sequence",
"Definition:Balanced Ternary Representation",
"Definition:Ternary Notati... |
proofwiki-15421 | Absolute Value is Norm | The absolute value is a norm on the set of real numbers $\R$. | By Complex Modulus is Norm then the complex modulus satisfies the norm axioms on the set of complex numbers $\C$.
Since the real numbers $\R$ is a subset of the complex numbers $\C$ then the complex modulus satisfies the norm axioms on the real numbers $\R$.
By Complex Modulus of Real Number equals Absolute Value then ... | The [[Definition:Absolute Value|absolute value]] is a [[Definition:Norm on Division Ring|norm]] on the [[Definition:Real Number|set of real numbers]] $\R$. | By [[Complex Modulus is Norm]] then the [[Definition:Complex Modulus|complex modulus]] satisfies the [[Definition:Norm on Division Ring|norm axioms]] on the [[Definition:Complex Number|set of complex numbers]] $\C$.
Since the [[Definition:Real Number|real numbers]] $\R$ is a [[Definition:Subset|subset]] of the [[Defin... | Absolute Value is Norm | https://proofwiki.org/wiki/Absolute_Value_is_Norm | https://proofwiki.org/wiki/Absolute_Value_is_Norm | [
"Examples of Norms",
"Absolute Value Function"
] | [
"Definition:Absolute Value",
"Definition:Norm/Division Ring",
"Definition:Real Number"
] | [
"Complex Modulus is Norm",
"Definition:Complex Modulus",
"Definition:Norm/Division Ring",
"Definition:Complex Number",
"Definition:Real Number",
"Definition:Subset",
"Definition:Complex Number",
"Definition:Complex Modulus",
"Definition:Norm/Division Ring",
"Definition:Real Number",
"Complex Mod... |
proofwiki-15422 | Bounds for Integer Expressed in Base k | Let $n \in \Z$ be an integer.
Let $k \in \Z$ such that $k \ge 2$.
Let $n$ be expressed in base $k$ notation:
:$n = \ds \sum_{j \mathop = 1}^s a_j k^j$
where each of the $a_j$ are such that $a_j \in \set {0, 1, \ldots, k - 1}$.
Then:
:$0 \le n < k^{s + 1}$ | As none of the coefficients $a_j$ in $\ds \sum_{j \mathop = 1}^s a_j k^j$ is (strictly) negative, the summation itself likewise cannot be negative
Thus:
:$0 \le n$
The equality is satisfied when $a_j = 0$ for all $j$.
We then have:
{{begin-eqn}}
{{eqn | l = n
| r = \sum_{j \mathop = 1}^s a_j k^j
| c =
}}
{... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Let $k \in \Z$ such that $k \ge 2$.
Let $n$ be expressed in [[Definition:Number Base|base $k$ notation]]:
:$n = \ds \sum_{j \mathop = 1}^s a_j k^j$
where each of the $a_j$ are such that $a_j \in \set {0, 1, \ldots, k - 1}$.
Then:
:$0 \le n < k^{s + 1}$ | As none of the [[Definition:Coefficient of Polynomial|coefficients]] $a_j$ in $\ds \sum_{j \mathop = 1}^s a_j k^j$ is [[Definition:Strictly Negative Integer|(strictly) negative]], the [[Definition:Summation|summation]] itself likewise cannot be [[Definition:Strictly Negative Integer|negative]]
Thus:
:$0 \le n$
The eq... | Bounds for Integer Expressed in Base k | https://proofwiki.org/wiki/Bounds_for_Integer_Expressed_in_Base_k | https://proofwiki.org/wiki/Bounds_for_Integer_Expressed_in_Base_k | [
"Number Bases"
] | [
"Definition:Integer",
"Definition:Number Base"
] | [
"Definition:Coefficient of Polynomial",
"Definition:Strictly Negative/Integer",
"Definition:Summation",
"Definition:Strictly Negative/Integer",
"Sum of Geometric Sequence"
] |
proofwiki-15423 | Different Representations to Number Base represent Different Integers | Let $k \in \Z$ such that $k \ge 2$.
Let $a$ and $b$ be representations of integers in base $k$ notation:
:$a = \ds \sum_{j \mathop = 0}^r a_j k^j$
:$b = \ds \sum_{j \mathop = 0}^s b_j k^j$
such that either:
:$r \ne s$
or:
:$\exists j \in \set {0, 1, \ldots, r}: a_j \ne b_j$
Then $a$ and $b$ represent different integers... | First suppose that $r \ne s$.
{{WLOG}}, suppose $r > s$.
Then from Bounds for Integer Expressed in Base k:
{{begin-eqn}}
{{eqn | l = a_r k^r
| o = >
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = a
| o = >
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = a
| o = \ne
... | Let $k \in \Z$ such that $k \ge 2$.
Let $a$ and $b$ be representations of [[Definition:Integer|integers]] in [[Definition:Number Base|base $k$ notation]]:
:$a = \ds \sum_{j \mathop = 0}^r a_j k^j$
:$b = \ds \sum_{j \mathop = 0}^s b_j k^j$
such that either:
:$r \ne s$
or:
:$\exists j \in \set {0, 1, \ldots, r}: a_j \... | First suppose that $r \ne s$.
{{WLOG}}, suppose $r > s$.
Then from [[Bounds for Integer Expressed in Base k]]:
{{begin-eqn}}
{{eqn | l = a_r k^r
| o = >
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = a
| o = >
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = a
| o = \n... | Different Representations to Number Base represent Different Integers | https://proofwiki.org/wiki/Different_Representations_to_Number_Base_represent_Different_Integers | https://proofwiki.org/wiki/Different_Representations_to_Number_Base_represent_Different_Integers | [
"Number Bases"
] | [
"Definition:Integer",
"Definition:Number Base",
"Definition:Integer"
] | [
"Bounds for Integer Expressed in Base k",
"Bounds for Integer Expressed in Base k"
] |
proofwiki-15424 | Existence of q for which j - qk is Positive | Let $j, k \in \Z$ be integers such that $k > 0$.
Then there exist $q \in \Z$ such that $j - q k > 0$. | Let $q = -\size j - 1$.
Then:
{{begin-eqn}}
{{eqn | l = j - q k
| r = j - \paren {-\size j - 1} k
| c =
}}
{{eqn | r = j + \size j + k
| c =
}}
{{end-eqn}}
We have that:
:$\forall j \le 0: j + \size j = 0$
and:
:$\forall j > 0: j + \size j = 2 j$
So:
:$j - q k \ge k$
and as $k > 0$ the result follow... | Let $j, k \in \Z$ be [[Definition:Integer|integers]] such that $k > 0$.
Then there exist $q \in \Z$ such that $j - q k > 0$. | Let $q = -\size j - 1$.
Then:
{{begin-eqn}}
{{eqn | l = j - q k
| r = j - \paren {-\size j - 1} k
| c =
}}
{{eqn | r = j + \size j + k
| c =
}}
{{end-eqn}}
We have that:
:$\forall j \le 0: j + \size j = 0$
and:
:$\forall j > 0: j + \size j = 2 j$
So:
:$j - q k \ge k$
and as $k > 0$ the result f... | Existence of q for which j - qk is Positive | https://proofwiki.org/wiki/Existence_of_q_for_which_j_-_qk_is_Positive | https://proofwiki.org/wiki/Existence_of_q_for_which_j_-_qk_is_Positive | [
"Number Theory"
] | [
"Definition:Integer"
] | [] |
proofwiki-15425 | Power Function on Base between Zero and One is Strictly Decreasing/Real Number | Let $a \in \R$ be a real number such that $0 \lt a \lt 1$.
Let $f: \R \to \R$ be the real function defined as:
:$\map f x = a^x$
where $a^x$ denotes $a$ to the power of $x$.
Then $f$ is strictly decreasing. | Let $x, y \in \R$ be such that $x < y$.
Since $0 < a < 1$, we have that:
:$\dfrac 1 a > 1$
Then we have that:
{{begin-eqn}}
{{eqn | l = \paren {\dfrac 1 a}^x
| o = <
| r = \paren {\dfrac 1 a}^y
| c = Real Power Function on Base Greater than One is Strictly Increasing
}}
{{eqn | ll= \leadstoandfrom
... | Let $a \in \R$ be a [[Definition:Real Number|real number]] such that $0 \lt a \lt 1$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map f x = a^x$
where $a^x$ denotes [[Definition:Power to Real Number|$a$ to the power of $x$]].
Then $f$ is [[Definition:Strictly Decreasing Real F... | Let $x, y \in \R$ be such that $x < y$.
Since $0 < a < 1$, we have that:
:$\dfrac 1 a > 1$
Then we have that:
{{begin-eqn}}
{{eqn | l = \paren {\dfrac 1 a}^x
| o = <
| r = \paren {\dfrac 1 a}^y
| c = [[Real Power Function on Base Greater than One is Strictly Increasing]]
}}
{{eqn | ll= \leadstoand... | Power Function on Base between Zero and One is Strictly Decreasing/Real Number | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Real_Number | https://proofwiki.org/wiki/Power_Function_on_Base_between_Zero_and_One_is_Strictly_Decreasing/Real_Number | [
"Power Function on Base between Zero and One is Strictly Decreasing"
] | [
"Definition:Real Number",
"Definition:Real Function",
"Definition:Power (Algebra)/Real Number",
"Definition:Strictly Decreasing/Real Function"
] | [
"Power Function on Base Greater than One is Strictly Increasing/Real Number",
"Reciprocal Function is Strictly Decreasing",
"Category:Power Function on Base between Zero and One is Strictly Decreasing"
] |
proofwiki-15426 | Integral Ideal is Ideal of Ring | Let $J$ be a non-empty subset of the set of integers $\Z$.
Then:
:$J$ is an integral ideal
{{iff}}:
:$J$ is an ideal of the ring of integers $\struct {\Z, +, \times}$. | Let $J \subseteq \Z$ fulfil the conditions of an integral ideal:
:$(1): \quad n, m \in J \implies m + n \in J, m - n \in J$
:$(2): \quad n \in J, r \in \Z \implies r n \in J$
First note that $J$ is non-empty by definition.
Then from $(1)$ we have in particular:
:$n, m \in J \implies m - n \in J$
Thus by the One-Step Su... | Let $J$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of the [[Definition:Integer|set of integers]] $\Z$.
Then:
:$J$ is an [[Definition:Integral Ideal|integral ideal]]
{{iff}}:
:$J$ is an [[Definition:Ideal of Ring|ideal]] of the [[Definition:Ring of Integers|ring of integers]] $\struct {\Z... | Let $J \subseteq \Z$ fulfil the conditions of an [[Definition:Integral Ideal|integral ideal]]:
:$(1): \quad n, m \in J \implies m + n \in J, m - n \in J$
:$(2): \quad n \in J, r \in \Z \implies r n \in J$
First note that $J$ is [[Definition:Non-Empty Set|non-empty]] by definition.
Then from $(1)$ we have in parti... | Integral Ideal is Ideal of Ring | https://proofwiki.org/wiki/Integral_Ideal_is_Ideal_of_Ring | https://proofwiki.org/wiki/Integral_Ideal_is_Ideal_of_Ring | [
"Integral Ideals",
"Ideal Theory"
] | [
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Integer",
"Definition:Integral Ideal",
"Definition:Ideal of Ring",
"Definition:Ring of Integers"
] | [
"Definition:Integral Ideal",
"Definition:Non-Empty Set",
"One-Step Subgroup Test",
"Definition:Subgroup",
"Definition:Additive Group of Integers",
"Integer Multiplication is Commutative",
"Definition:Ideal of Ring",
"Definition:Ring of Integers",
"Definition:Subgroup",
"One-Step Subgroup Test",
... |
proofwiki-15427 | Set of Integer Multiples is Integral Ideal | Let $m \in \Z$ be an integer.
Let $m \Z$ denote the set of integer multiples of $m$.
Then $m \Z$ is an integral ideal. | First note that $m \times 0 \in m \Z$ whatever $m$ may be.
Thus $m \Z \ne \O$.
Let $a, b \in m \Z$.
Then:
{{begin-eqn}}
{{eqn | l = a + b
| r = m j + m k
| c = for some $j, k \in \Z$ by definition of $m \Z$
}}
{{eqn | r = m \paren {j + k}
| c =
}}
{{eqn | o = \in
| r = m \Z
| c =
}}
{{en... | Let $m \in \Z$ be an [[Definition:Integer|integer]].
Let $m \Z$ denote the [[Definition:Set of Integer Multiples|set of integer multiples]] of $m$.
Then $m \Z$ is an [[Definition:Integral Ideal|integral ideal]]. | First note that $m \times 0 \in m \Z$ whatever $m$ may be.
Thus $m \Z \ne \O$.
Let $a, b \in m \Z$.
Then:
{{begin-eqn}}
{{eqn | l = a + b
| r = m j + m k
| c = for some $j, k \in \Z$ by definition of $m \Z$
}}
{{eqn | r = m \paren {j + k}
| c =
}}
{{eqn | o = \in
| r = m \Z
| c =
}}... | Set of Integer Multiples is Integral Ideal | https://proofwiki.org/wiki/Set_of_Integer_Multiples_is_Integral_Ideal | https://proofwiki.org/wiki/Set_of_Integer_Multiples_is_Integral_Ideal | [
"Integral Ideals",
"Sets of Integer Multiples"
] | [
"Definition:Integer",
"Definition:Set of Integer Multiples",
"Definition:Integral Ideal"
] | [
"Definition:Integral Ideal"
] |
proofwiki-15428 | Minimal Smooth Surface Spanned by Contour | Let $\map z {x, y}: \R^2 \to \R$ be a real-valued function.
Let $\Gamma$ be a closed contour in $3$-dimensional Euclidean space.
Then the smooth surface of least area spanned by the contour $\Gamma$ has to satisfy the following Euler's equation:
:$r \paren {1 + q^2} - 2 s p q + t \paren {1 + p^2} = 0$
where:
{{begin-eq... | The surface area for a smooth surface embedded in $3$-dimensional Euclidean space is given by:
:$\ds A \sqbrk z = \iint_\Gamma \sqrt {1 + z_x^2 + z_y^2} \rd x \rd y$
{{MissingLinks|We need a link to the above}}
It follows that:
{{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \frac \partial {\partial z_x} \sqrt {1 + z_x^2 + ... | Let $\map z {x, y}: \R^2 \to \R$ be a [[Definition:Real-Valued Function|real-valued function]].
Let $\Gamma$ be a [[Definition:Closed Contour|closed contour]] in $3$-[[Definition:Dimension of Vector Space|dimensional]] [[Definition:Real Euclidean Space|Euclidean space]].
Then the [[Definition:Smooth Real Function|smo... | The [[Definition:Surface|surface]] [[Definition:Area|area]] for a [[Definition:Smooth Real Function|smooth]] [[Definition:Surface|surface]] embedded in $3$-[[Definition:Dimension of Vector Space|dimensional]] [[Definition:Real Euclidean Space|Euclidean space]] is given by:
:$\ds A \sqbrk z = \iint_\Gamma \sqrt {1 + z_... | Minimal Smooth Surface Spanned by Contour | https://proofwiki.org/wiki/Minimal_Smooth_Surface_Spanned_by_Contour | https://proofwiki.org/wiki/Minimal_Smooth_Surface_Spanned_by_Contour | [
"Calculus of Variations"
] | [
"Definition:Real-Valued Function",
"Definition:Contour/Closed",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Definition:Smooth Real Function",
"Necessary Condition for Integral Functional to have Extremum/Two Variables",
"Definition:Partial Derivative",
"Definition:Mean ... | [
"Definition:Surface",
"Definition:Area",
"Definition:Smooth Real Function",
"Definition:Surface",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space/Real",
"Necessary Condition for Integral Functional to have Extremum/Two Variables",
"Definition:Smooth Real Function",
"Definition:Su... |
proofwiki-15429 | Integral Ideal is Set of Integer Multiples | Let $J$ be an integral ideal.
Then $J$ is in the form of a set of integer multiples $m \Z$ for some $m \in \Z$. | By definition, $J$ satisfies the following conditions:
:$(1): \quad n, m \in J \implies m + n \in J, m - n \in J$
:$(2): \quad n \in J, r \in \Z \implies r n \in J$
First note that the null ideal $\set 0$ is an integral ideal.
This is of the form $0 \Z$.
Let $J \ne \set 0$.
Then $\exists a \in J: a \ne 0$.
As $0 \in \Z... | Let $J$ be an [[Definition:Integral Ideal|integral ideal]].
Then $J$ is in the form of a [[Definition:Set of Integer Multiples|set of integer multiples]] $m \Z$ for some $m \in \Z$. | By definition, $J$ satisfies the following conditions:
:$(1): \quad n, m \in J \implies m + n \in J, m - n \in J$
:$(2): \quad n \in J, r \in \Z \implies r n \in J$
First note that the [[Definition:Null Ideal|null ideal]] $\set 0$ is an [[Definition:Integral Ideal|integral ideal]].
This is of the form $0 \Z$.
Le... | Integral Ideal is Set of Integer Multiples | https://proofwiki.org/wiki/Integral_Ideal_is_Set_of_Integer_Multiples | https://proofwiki.org/wiki/Integral_Ideal_is_Set_of_Integer_Multiples | [
"Integral Ideals",
"Sets of Integer Multiples"
] | [
"Definition:Integral Ideal",
"Definition:Set of Integer Multiples"
] | [
"Definition:Null Ideal",
"Definition:Integral Ideal",
"Definition:Bounded Below Set",
"Definition:Empty Set",
"Set of Integers Bounded Below has Smallest Element",
"Definition:Smallest Element",
"Definition:Subset",
"Definition:Contradiction",
"Definition:Smallest Element",
"Definition:Strictly Po... |
proofwiki-15430 | Difference between Odd Squares is Divisible by 8 | Let $a$ and $b$ be odd integers.
Then $a^2 - b^2$ is divisible by $8$. | Let $a = 2 m + 1$, $b = 2 n + 1$.
Then:
{{begin-eqn}}
{{eqn | l = a^2 - b^2
| r = \paren {2 m + 1}^2 - \paren {2 n + 1}^2
| c =
}}
{{eqn | r = \paren {4 m^2 + 4 m + 1} - \paren {4 n^2 + 4 n - 1}
| c =
}}
{{eqn | r = 4 \paren {m^2 - n^2} + 4 \paren {m - n}
| c =
}}
{{eqn | r = 4 \paren {m + n... | Let $a$ and $b$ be [[Definition:Odd Integer|odd integers]].
Then $a^2 - b^2$ is [[Definition:Divisor of Integer|divisible]] by $8$. | Let $a = 2 m + 1$, $b = 2 n + 1$.
Then:
{{begin-eqn}}
{{eqn | l = a^2 - b^2
| r = \paren {2 m + 1}^2 - \paren {2 n + 1}^2
| c =
}}
{{eqn | r = \paren {4 m^2 + 4 m + 1} - \paren {4 n^2 + 4 n - 1}
| c =
}}
{{eqn | r = 4 \paren {m^2 - n^2} + 4 \paren {m - n}
| c =
}}
{{eqn | r = 4 \paren {m + ... | Difference between Odd Squares is Divisible by 8 | https://proofwiki.org/wiki/Difference_between_Odd_Squares_is_Divisible_by_8 | https://proofwiki.org/wiki/Difference_between_Odd_Squares_is_Divisible_by_8 | [
"Odd Squares"
] | [
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Difference of Two Squares",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-15431 | Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition | Let $\alpha \in \R_{> 0}$.
Let $\norm {\,\cdot\,}:\Q \to \R$ be the mapping defined by:
:$\forall x \in \Q: \norm x = \size x^\alpha$
where $\size x$ is the absolute value of $x$ in $\Q$.
Let $\norm {\,\cdot\,}$ be a norm on $\Q$.
Then:
:$\alpha \le 1$ | The contrapositive is proved.
Let $\alpha > 1$.
The {{Norm-axiom-mult|3}} is not satisfied:
{{begin-eqn}}
{{eqn | l = \norm {1 + 1}
| r = \size {1 + 1}^\alpha
}}
{{eqn | r = 2^\alpha
}}
{{eqn | o = >
| r = 2
| c = Power Function on Base Greater than One is Strictly Increasing
}}
{{eqn | r = \size 1^\a... | Let $\alpha \in \R_{> 0}$.
Let $\norm {\,\cdot\,}:\Q \to \R$ be the [[Definition:Mapping|mapping]] defined by:
:$\forall x \in \Q: \norm x = \size x^\alpha$
where $\size x$ is the [[Definition:Absolute Value|absolute value]] of $x$ in $\Q$.
Let $\norm {\,\cdot\,}$ be a [[Definition:Norm on Division Ring|norm]] on $\... | The [[Definition:Contrapositive|contrapositive]] is proved.
Let $\alpha > 1$.
The {{Norm-axiom-mult|3}} is not satisfied:
{{begin-eqn}}
{{eqn | l = \norm {1 + 1}
| r = \size {1 + 1}^\alpha
}}
{{eqn | r = 2^\alpha
}}
{{eqn | o = >
| r = 2
| c = [[Power Function on Base Greater than One is Strictly I... | Norms Equivalent to Absolute Value on Rational Numbers/Necessary Condition | https://proofwiki.org/wiki/Norms_Equivalent_to_Absolute_Value_on_Rational_Numbers/Necessary_Condition | https://proofwiki.org/wiki/Norms_Equivalent_to_Absolute_Value_on_Rational_Numbers/Necessary_Condition | [
"Normed Division Rings"
] | [
"Definition:Mapping",
"Definition:Absolute Value",
"Definition:Norm/Division Ring"
] | [
"Definition:Contrapositive Statement",
"Power Function on Base Greater than One is Strictly Increasing",
"Rule of Transposition"
] |
proofwiki-15432 | Norms Equivalent to Absolute Value on Rational Numbers/Sufficient Condition | Let $\alpha \in \R_{> 0}$.
Let $\norm {\,\cdot\,}: \Q \to \R$ be the mapping defined by:
:$\forall x \in \Q: \norm x = \size x^\alpha$
where $\size x$ is the absolute value of $x$ in $\Q$.
Then:
:$\alpha \le 1 \implies \norm {\,\cdot\,}$ is a norm on $\Q$ | Suppose $\alpha \le 1$.
It is shown that $\norm {\,\cdot\,}$ satisfies the norm axioms $(\text N 1)$-$(\text N 3)$.
=== {{Norm-axiom-mult|1|nolink}} ===
Let $x \in \Q$.
{{begin-eqn}}
{{eqn | l = \norm x = 0
| o = \leadstoandfrom
| r = \size x^\alpha = 0
| c = Definition of $\norm {\,\cdot\,}$
}}
{{eqn... | Let $\alpha \in \R_{> 0}$.
Let $\norm {\,\cdot\,}: \Q \to \R$ be the [[Definition:Mapping|mapping]] defined by:
:$\forall x \in \Q: \norm x = \size x^\alpha$
where $\size x$ is the [[Definition:Absolute Value|absolute value]] of $x$ in $\Q$.
Then:
:$\alpha \le 1 \implies \norm {\,\cdot\,}$ is a [[Definition:Norm on ... | Suppose $\alpha \le 1$.
It is shown that $\norm {\,\cdot\,}$ satisfies the [[Axiom:Multiplicative Norm Axioms|norm axioms $(\text N 1)$-$(\text N 3)$]].
=== {{Norm-axiom-mult|1|nolink}} ===
Let $x \in \Q$.
{{begin-eqn}}
{{eqn | l = \norm x = 0
| o = \leadstoandfrom
| r = \size x^\alpha = 0
| c = ... | Norms Equivalent to Absolute Value on Rational Numbers/Sufficient Condition | https://proofwiki.org/wiki/Norms_Equivalent_to_Absolute_Value_on_Rational_Numbers/Sufficient_Condition | https://proofwiki.org/wiki/Norms_Equivalent_to_Absolute_Value_on_Rational_Numbers/Sufficient_Condition | [
"Normed Division Rings"
] | [
"Definition:Mapping",
"Definition:Absolute Value",
"Definition:Norm/Division Ring"
] | [
"Axiom:Multiplicative Norm Axioms",
"Definition:Power (Algebra)",
"Absolute Value is Norm",
"Absolute Value is Norm",
"Exponent Combination Laws/Power of Product",
"Power Function on Base Greater than One is Strictly Increasing",
"Power Function on Base between Zero and One is Strictly Decreasing"
] |
proofwiki-15433 | Lowest Common Multiple of Integers with Common Divisor | Let $b, d \in \Z_{>0}$ be (strictly) positive integers
Then:
:$\lcm \set {a b, a d} = a \lcm \set {b, d}$
where:
:$a \in \Z_{>0}$
:$\lcm \set {b, d}$ denotes the lowest common multiple of $m$ and $n$. | We have that:
{{begin-eqn}}
{{eqn | l = b
| o = \divides
| r = \lcm \set {b, d}
| c = {{Defof|Lowest Common Multiple of Integers}}
}}
{{eqn | lo= \land
| l = d
| o = \divides
| r = \lcm \set {b, d}
| c =
}}
{{eqn | ll= \leadsto
| l = r b
| r = \lcm \set {b, d}
... | Let $b, d \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]]
Then:
:$\lcm \set {a b, a d} = a \lcm \set {b, d}$
where:
:$a \in \Z_{>0}$
:$\lcm \set {b, d}$ denotes the [[Definition:Lowest Common Multiple of Integers|lowest common multiple]] of $m$ and $n$. | We have that:
{{begin-eqn}}
{{eqn | l = b
| o = \divides
| r = \lcm \set {b, d}
| c = {{Defof|Lowest Common Multiple of Integers}}
}}
{{eqn | lo= \land
| l = d
| o = \divides
| r = \lcm \set {b, d}
| c =
}}
{{eqn | ll= \leadsto
| l = r b
| r = \lcm \set {b, d}
... | Lowest Common Multiple of Integers with Common Divisor | https://proofwiki.org/wiki/Lowest_Common_Multiple_of_Integers_with_Common_Divisor | https://proofwiki.org/wiki/Lowest_Common_Multiple_of_Integers_with_Common_Divisor | [
"Lowest Common Multiple"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Lowest Common Multiple/Integers"
] | [
"Definition:By Hypothesis",
"LCM Divides Common Multiple",
"LCM iff Divides All Common Multiples"
] |
proofwiki-15434 | Addition of Fractions | Let $a, b, c, d \in \Z$ such that $b d \ne 0$.
Then:
:$\dfrac a b + \dfrac c d = \dfrac {a D + B c} {\lcm \set {b, d} }$
where:
:$B = \dfrac b {\gcd \set {b, d} }$
:$D = \dfrac d {\gcd \set {b, d} }$
:$\lcm$ denotes lowest common multiple
:$\gcd$ denotes greatest common divisor. | {{begin-eqn}}
{{eqn | l = \dfrac a b + \dfrac c d
| r = \dfrac {a d} {b d} + \dfrac {b c} {b d}
| c =
}}
{{eqn | r = \dfrac {a d + b c} {b d}
| c =
}}
{{eqn | r = \dfrac {a d + b c} {\gcd \set {b, d} \lcm \set {b, d} }
| c = Product of GCD and LCM
}}
{{eqn | r = \dfrac {a D \gcd \set {b, d} + ... | Let $a, b, c, d \in \Z$ such that $b d \ne 0$.
Then:
:$\dfrac a b + \dfrac c d = \dfrac {a D + B c} {\lcm \set {b, d} }$
where:
:$B = \dfrac b {\gcd \set {b, d} }$
:$D = \dfrac d {\gcd \set {b, d} }$
:$\lcm$ denotes [[Definition:Lowest Common Multiple of Integers|lowest common multiple]]
:$\gcd$ denotes [[Definit... | {{begin-eqn}}
{{eqn | l = \dfrac a b + \dfrac c d
| r = \dfrac {a d} {b d} + \dfrac {b c} {b d}
| c =
}}
{{eqn | r = \dfrac {a d + b c} {b d}
| c =
}}
{{eqn | r = \dfrac {a d + b c} {\gcd \set {b, d} \lcm \set {b, d} }
| c = [[Product of GCD and LCM]]
}}
{{eqn | r = \dfrac {a D \gcd \set {b, d... | Addition of Fractions | https://proofwiki.org/wiki/Addition_of_Fractions | https://proofwiki.org/wiki/Addition_of_Fractions | [
"Addition of Fractions",
"Fractions",
"Addition",
"Lowest Common Multiple",
"Greatest Common Divisor"
] | [
"Definition:Lowest Common Multiple/Integers",
"Definition:Greatest Common Divisor/Integers"
] | [
"Product of GCD and LCM",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-15435 | GCD of Sum and Difference of Integers | :$\gcd \set {a + b, a - b} \ge \gcd \set {a, b}$ | Let $d = \gcd \set {a, b}$.
Then by definition of greatest common divisor:
:$d \divides a \land d \divides b$
From Common Divisor Divides Integer Combination:
:$d \divides \paren {a + b} \land d \divides \paren {a - b}$
By definition of common divisor:
:$d \divides \gcd \set {a + b, a - b}$
Hence from Absolute Value of... | :$\gcd \set {a + b, a - b} \ge \gcd \set {a, b}$ | Let $d = \gcd \set {a, b}$.
Then by definition of [[Definition:Greatest Common Divisor of Integers|greatest common divisor]]:
:$d \divides a \land d \divides b$
From [[Common Divisor Divides Integer Combination]]:
:$d \divides \paren {a + b} \land d \divides \paren {a - b}$
By definition of [[Definition:Common Divis... | GCD of Sum and Difference of Integers | https://proofwiki.org/wiki/GCD_of_Sum_and_Difference_of_Integers | https://proofwiki.org/wiki/GCD_of_Sum_and_Difference_of_Integers | [
"Greatest Common Divisor"
] | [] | [
"Definition:Greatest Common Divisor/Integers",
"Common Divisor Divides Integer Combination",
"Definition:Common Divisor/Integers",
"Absolute Value of Integer is not less than Divisors"
] |
proofwiki-15436 | Greatest Common Divisor divides Lowest Common Multiple | Let $a, b \in \Z$ such that $a b \ne 0$.
Then:
:$\gcd \set {a, b} \divides \lcm \set {a, b}$
where:
:$\lcm$ denotes lowest common multiple
:$\gcd$ denotes greatest common divisor.
:$\divides$ denotes divisibility. | We have that:
:$\gcd \set {a, b} \divides a$
and:
:$a \divides \lcm \set {a, b}$
The result follows from Divisor Relation is Transitive.
{{qed}} | Let $a, b \in \Z$ such that $a b \ne 0$.
Then:
:$\gcd \set {a, b} \divides \lcm \set {a, b}$
where:
:$\lcm$ denotes [[Definition:Lowest Common Multiple of Integers|lowest common multiple]]
:$\gcd$ denotes [[Definition:Greatest Common Divisor of Integers|greatest common divisor]].
:$\divides$ denotes [[Definition:D... | We have that:
:$\gcd \set {a, b} \divides a$
and:
:$a \divides \lcm \set {a, b}$
The result follows from [[Divisor Relation is Transitive]].
{{qed}} | Greatest Common Divisor divides Lowest Common Multiple | https://proofwiki.org/wiki/Greatest_Common_Divisor_divides_Lowest_Common_Multiple | https://proofwiki.org/wiki/Greatest_Common_Divisor_divides_Lowest_Common_Multiple | [
"Greatest Common Divisor",
"Lowest Common Multiple"
] | [
"Definition:Lowest Common Multiple/Integers",
"Definition:Greatest Common Divisor/Integers",
"Definition:Divisor (Algebra)/Integer"
] | [
"Divisor Relation is Transitive"
] |
proofwiki-15437 | Intersection of Sets of Integer Multiples | Let $m, n \in \Z$ such that $m n \ne 0$.
Let $m \Z$ denote the set of integer multiples of $m$.
Then:
:$m \Z \cap n \Z = \lcm \set {m, n} \Z$
where $\lcm$ denotes lowest common multiple. | Let $x \in m \Z \cap n \Z$.
Then by definition of set intersection:
:$m \divides x$ and $n \divides x$
So from LCM Divides Common Multiple:
:$\lcm \set {m, n} \divides x$
and so $x \in \lcm \set {m, n} \Z$
That is:
:$m \Z \cap n \Z \subseteq \lcm \set {m, n} \Z$
{{qed|lemma}}
Now suppose $x \in \lcm \set {m, n} \Z$.
Th... | Let $m, n \in \Z$ such that $m n \ne 0$.
Let $m \Z$ denote the [[Definition:Set of Integer Multiples|set of integer multiples of $m$]].
Then:
:$m \Z \cap n \Z = \lcm \set {m, n} \Z$
where $\lcm$ denotes [[Definition:Lowest Common Multiple of Integers|lowest common multiple]]. | Let $x \in m \Z \cap n \Z$.
Then by definition of [[Definition:Set Intersection|set intersection]]:
:$m \divides x$ and $n \divides x$
So from [[LCM Divides Common Multiple]]:
:$\lcm \set {m, n} \divides x$
and so $x \in \lcm \set {m, n} \Z$
That is:
:$m \Z \cap n \Z \subseteq \lcm \set {m, n} \Z$
{{qed|lemma}}
No... | Intersection of Sets of Integer Multiples | https://proofwiki.org/wiki/Intersection_of_Sets_of_Integer_Multiples | https://proofwiki.org/wiki/Intersection_of_Sets_of_Integer_Multiples | [
"Sets of Integer Multiples",
"Lowest Common Multiple",
"Intersection of Sets of Integer Multiples"
] | [
"Definition:Set of Integer Multiples",
"Definition:Lowest Common Multiple/Integers"
] | [
"Definition:Set Intersection",
"LCM Divides Common Multiple",
"Definition:Lowest Common Multiple/Integers",
"Definition:Set Equality"
] |
proofwiki-15438 | Set of Integer Multiples of GCD | Let $m, n \in \Z$.
Let $m \Z$ denote the set of integer multiples of $m$
Then:
:$m \Z \cup n \Z \subseteq \gcd \set {m, n} \Z$
where $\gcd$ denotes greatest common divisor. | Let $x \in m \Z \cup n \Z$.
Then either:
:$m \divides x$
or:
:$n \divides x$
In both cases:
:$\gcd \set {m, n} \divides x$
and so:
:$x \in \gcd \set {m, n} \Z$
Hence by definition of subset:
:$m \Z \cup n \Z \subseteq \gcd \set {m, n} \Z$
{{qed}} | Let $m, n \in \Z$.
Let $m \Z$ denote the [[Definition:Set of Integer Multiples|set of integer multiples of $m$]]
Then:
:$m \Z \cup n \Z \subseteq \gcd \set {m, n} \Z$
where $\gcd$ denotes [[Definition:Greatest Common Divisor of Integers|greatest common divisor]]. | Let $x \in m \Z \cup n \Z$.
Then either:
:$m \divides x$
or:
:$n \divides x$
In both cases:
:$\gcd \set {m, n} \divides x$
and so:
:$x \in \gcd \set {m, n} \Z$
Hence by definition of [[Definition:Subset|subset]]:
:$m \Z \cup n \Z \subseteq \gcd \set {m, n} \Z$
{{qed}} | Set of Integer Multiples of GCD | https://proofwiki.org/wiki/Set_of_Integer_Multiples_of_GCD | https://proofwiki.org/wiki/Set_of_Integer_Multiples_of_GCD | [
"Sets of Integer Multiples",
"Greatest Common Divisor"
] | [
"Definition:Set of Integer Multiples",
"Definition:Greatest Common Divisor/Integers"
] | [
"Definition:Subset"
] |
proofwiki-15439 | Join of Sets of Integer Multiples is Set of Integer Multiples of GCD | Let $m, n \in \Z$.
Let $m \Z$ denote the set of integer multiples of $m$
Let $r \in \Z$ such that:
:$m \Z \subseteq r \Z$
and:
:$n \Z \subseteq r \Z$
Then:
:$\gcd \set {m, n} \Z \subseteq r \Z$
where $\gcd$ denotes greatest common divisor. | From Set of Integer Multiples is Integral Ideal, each of $m \Z$, $n \Z$, $r \Z$ and $\gcd \set {m, n} \Z$ are integral ideals.
Let $c \in \gcd \set {m, n} \Z$.
By definition of integral ideal:
:$\gcd \set {m, n} \divides c$
By Set of Integer Combinations equals Set of Multiples of GCD:
:$\exists x, y \in \Z: c = x m + ... | Let $m, n \in \Z$.
Let $m \Z$ denote the [[Definition:Set of Integer Multiples|set of integer multiples of $m$]]
Let $r \in \Z$ such that:
:$m \Z \subseteq r \Z$
and:
:$n \Z \subseteq r \Z$
Then:
:$\gcd \set {m, n} \Z \subseteq r \Z$
where $\gcd$ denotes [[Definition:Greatest Common Divisor of Integers|greatest co... | From [[Set of Integer Multiples is Integral Ideal]], each of $m \Z$, $n \Z$, $r \Z$ and $\gcd \set {m, n} \Z$ are [[Definition:Integral Ideal|integral ideals]].
Let $c \in \gcd \set {m, n} \Z$.
By definition of [[Definition:Integral Ideal|integral ideal]]:
:$\gcd \set {m, n} \divides c$
By [[Set of Integer Combinat... | Join of Sets of Integer Multiples is Set of Integer Multiples of GCD | https://proofwiki.org/wiki/Join_of_Sets_of_Integer_Multiples_is_Set_of_Integer_Multiples_of_GCD | https://proofwiki.org/wiki/Join_of_Sets_of_Integer_Multiples_is_Set_of_Integer_Multiples_of_GCD | [
"Sets of Integer Multiples",
"Greatest Common Divisor"
] | [
"Definition:Set of Integer Multiples",
"Definition:Greatest Common Divisor/Integers"
] | [
"Set of Integer Multiples is Integral Ideal",
"Definition:Integral Ideal",
"Definition:Integral Ideal",
"Set of Integer Combinations equals Set of Multiples of GCD",
"Definition:Integral Ideal",
"Definition:Subset"
] |
proofwiki-15440 | GCD of Generators of General Fibonacci Sequence is Divisor of All Terms | Let $\FF = \sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$:
:$a_n = \begin{cases}
r & : n = 0 \\
s & : n = 1 \\
t a_{n - 2} + u a_{n - 1} & : n > 1
\end{cases}$
Let:
:$d = \gcd \set {r, s}$
where $\gcd$ denotes greatest common divisor.
Then:
:$\forall n \in \Z_{>0}: d \divides ... | From the construction of a general Fibonacci sequence, $a_n$ is an integer combination of $r$ and $s$.
From Set of Integer Combinations equals Set of Multiples of GCD, $a_n$ is divisible by $\gcd \set {r, s}$.
Hence the result.
{{qed}} | Let $\FF = \sequence {a_n}$ be a [[Definition:General Fibonacci Sequence|general Fibonacci sequence]] generated by the parameters $r, s, t, u$:
:$a_n = \begin{cases}
r & : n = 0 \\
s & : n = 1 \\
t a_{n - 2} + u a_{n - 1} & : n > 1
\end{cases}$
Let:
:$d = \gcd \set {r, s}$
where $\gcd$ denotes [[Definition:Greatest C... | From the construction of a [[Definition:General Fibonacci Sequence|general Fibonacci sequence]], $a_n$ is an [[Definition:Integer Combination|integer combination]] of $r$ and $s$.
From [[Set of Integer Combinations equals Set of Multiples of GCD]], $a_n$ is [[Definition:Divisor of Integer|divisible]] by $\gcd \set {r,... | GCD of Generators of General Fibonacci Sequence is Divisor of All Terms | https://proofwiki.org/wiki/GCD_of_Generators_of_General_Fibonacci_Sequence_is_Divisor_of_All_Terms | https://proofwiki.org/wiki/GCD_of_Generators_of_General_Fibonacci_Sequence_is_Divisor_of_All_Terms | [
"Greatest Common Divisor",
"Fibonacci Numbers"
] | [
"Definition:General Fibonacci Sequence",
"Definition:Greatest Common Divisor/Integers"
] | [
"Definition:General Fibonacci Sequence",
"Definition:Integer Combination",
"Set of Integer Combinations equals Set of Multiples of GCD",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-15441 | GCD of Consecutive Integers of General Fibonacci Sequence | Let $\FF = \sequence {a_n}$ be a general Fibonacci sequence generated by the parameters $r, s, t, u$:
:$a_n = \begin{cases}
r & : n = 0 \\
s & : n = 1 \\
t a_{n - 2} + u a_{n - 1} & : n > 1
\end{cases}$
Let:
:$d = \gcd \set {r, s}$
where $\gcd$ denotes greatest common divisor.
Let $f = \gcd \set {a_m, a_{m - 1} }$ for ... | Proof by induction:
Let $\map P m$ be the proposition:
:$\gcd \set {f_m, t} = 1 \implies f_m = d$
where $f_m = \gcd \set {a_m, a_{m - 1} }$.
For clarity, we have indexed $f$. | Let $\FF = \sequence {a_n}$ be a [[Definition:General Fibonacci Sequence|general Fibonacci sequence]] generated by the parameters $r, s, t, u$:
:$a_n = \begin{cases}
r & : n = 0 \\
s & : n = 1 \\
t a_{n - 2} + u a_{n - 1} & : n > 1
\end{cases}$
Let:
:$d = \gcd \set {r, s}$
where $\gcd$ denotes [[Definition:Greatest C... | Proof by [[Principle of Mathematical Induction|induction]]:
Let $\map P m$ be the proposition:
:$\gcd \set {f_m, t} = 1 \implies f_m = d$
where $f_m = \gcd \set {a_m, a_{m - 1} }$.
For clarity, we have indexed $f$. | GCD of Consecutive Integers of General Fibonacci Sequence | https://proofwiki.org/wiki/GCD_of_Consecutive_Integers_of_General_Fibonacci_Sequence | https://proofwiki.org/wiki/GCD_of_Consecutive_Integers_of_General_Fibonacci_Sequence | [
"Greatest Common Divisor",
"Fibonacci Numbers"
] | [
"Definition:General Fibonacci Sequence",
"Definition:Greatest Common Divisor/Integers"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-15442 | Three Points in Ultrametric Space have Two Equal Distances/Corollary 3 | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,
Let $x, y \in R$ and $\norm x \lt \norm y$.
Then:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \norm y$ | By {{Corollary|Three Points in Ultrametric Space have Two Equal Distances|2}}:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y} = \norm y$
{{qed}} | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm{\,\cdot\,}$,
Let $x, y \in R$ and $\norm x \lt \norm y$.
Then:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \norm y$ | By {{Corollary|Three Points in Ultrametric Space have Two Equal Distances|2}}:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y} = \norm y$
{{qed}} | Three Points in Ultrametric Space have Two Equal Distances/Corollary 3 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_3 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_3 | [
"Three Points in Ultrametric Space have Two Equal Distances"
] | [
"Definition:Normed Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [] |
proofwiki-15443 | Three Points in Ultrametric Space have Two Equal Distances/Corollary 4 | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$,
Let $x, y \in R$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {x + y} < \norm y
| o = \implies
| r = \norm x = \norm y
}}
{{eqn | l = \norm {x - y} < \norm y
| o = \implies
| r = \norm x = \no... | The contrapositive statements are proved.
Let $\norm x \ne \norm y$
By {{Corollary|Three Points in Ultrametric Space have Two Equal Distances|2}}:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y} \ge \norm y$
The result follows.
{{qed}} | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean norm]] $\norm{\,\cdot\,}$,
Let $x, y \in R$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {x + y} < \norm y
| o = \implies
| r = \norm x = \norm... | The [[Definition:Contrapositive Statement|contrapositive statements]] are proved.
Let $\norm x \ne \norm y$
By {{Corollary|Three Points in Ultrametric Space have Two Equal Distances|2}}:
:$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y} \ge \norm y$
The result follows.
{{qed}} | Three Points in Ultrametric Space have Two Equal Distances/Corollary 4 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_4 | https://proofwiki.org/wiki/Three_Points_in_Ultrametric_Space_have_Two_Equal_Distances/Corollary_4 | [
"Three Points in Ultrametric Space have Two Equal Distances"
] | [
"Definition:Normed Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)"
] | [
"Definition:Contrapositive Statement"
] |
proofwiki-15444 | Equivalent Norms are both Non-Archimedean or both Archimedean | Let $R$ be a division ring with unity $1_R$.
Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be equivalent norms on $R$.
Then $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are either both non-Archimedean or both Archimedean. | By Characterisation of Non-Archimedean Division Ring Norms then:
:$\norm {\,\cdot\,}_1$ is non-Archimedean $\iff \forall n \in \N_{>0}: \norm{n \cdot 1_R}_1 \le 1$.
By the definition of norm equivalence then:
:$\forall n \in \N: \norm {n \cdot 1_R}_1 \le 1 \iff \norm {n \cdot 1_R}_2 \le 1$
Similarly, by Characterisatio... | Let $R$ be a [[Definition:Division Ring|division ring]] with [[Definition:Unity of Ring|unity]] $1_R$.
Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be [[Definition:Equivalent Division Ring Norms|equivalent]] [[Definition:Norm on Division Ring|norms]] on $R$.
Then $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2... | By [[Characterisation of Non-Archimedean Division Ring Norms]] then:
:$\norm {\,\cdot\,}_1$ is [[Definition:Non-Archimedean Division Ring Norm|non-Archimedean]] $\iff \forall n \in \N_{>0}: \norm{n \cdot 1_R}_1 \le 1$.
By the definition of [[Definition:Equivalent Division Ring Norms by Cauchy Sequence|norm equivalence... | Equivalent Norms are both Non-Archimedean or both Archimedean | https://proofwiki.org/wiki/Equivalent_Norms_are_both_Non-Archimedean_or_both_Archimedean | https://proofwiki.org/wiki/Equivalent_Norms_are_both_Non-Archimedean_or_both_Archimedean | [
"Normed Division Rings",
"Non-Archimedean Norms"
] | [
"Definition:Division Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Equivalent Division Ring Norms",
"Definition:Norm/Division Ring",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Non-Archimedean/Norm (Division Ring)/Archimedean"
] | [
"Characterisation of Non-Archimedean Division Ring Norms",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Definition:Equivalent Division Ring Norms/Cauchy Sequence Equivalent",
"Characterisation of Non-Archimedean Division Ring Norms",
"Definition:Non-Archimedean/Norm (Division Ring)"
] |
proofwiki-15445 | Minimum Area of Triangle whose Vertices are Lattice Points | Let $T$ be a triangle embedded in a cartesian plane.
Let the vertices of $T$ be lattice points which are not all on the same straight line.
Then the area of $T$ is such that:
:$\map \Area T \ge \dfrac 1 2$ | {{WLOG}} let one of the vertices of $T$ be at $\tuple {0, 0}$.
Let the other $2$ vertices be at $\tuple {a, b}$ and $\tuple {x, y}$.
By Area of Triangle in Determinant Form with Vertex at Origin:
:$\map \Area T = \dfrac {\size {b y - a x} } 2$
As the vertices of $T$ are non-collinear, $\map \Area T \ge 0$.
Thus $\size ... | Let $T$ be a [[Definition:Triangle (Geometry)|triangle]] embedded in a [[Definition:Cartesian Plane|cartesian plane]].
Let the [[Definition:Vertex of Polygon|vertices]] of $T$ be [[Definition:Lattice Point of Cartesian Coordinate System|lattice points]] which are not all on the same [[Definition:Straight Line|straight... | {{WLOG}} let one of the [[Definition:Vertex of Polygon|vertices]] of $T$ be at $\tuple {0, 0}$.
Let the other $2$ [[Definition:Vertex of Polygon|vertices]] be at $\tuple {a, b}$ and $\tuple {x, y}$.
By [[Area of Triangle in Determinant Form with Vertex at Origin]]:
:$\map \Area T = \dfrac {\size {b y - a x} } 2$
As ... | Minimum Area of Triangle whose Vertices are Lattice Points | https://proofwiki.org/wiki/Minimum_Area_of_Triangle_whose_Vertices_are_Lattice_Points | https://proofwiki.org/wiki/Minimum_Area_of_Triangle_whose_Vertices_are_Lattice_Points | [
"Areas of Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Cartesian Plane",
"Definition:Polygon/Vertex",
"Definition:Lattice Point/Cartesian Coordinate System",
"Definition:Line/Straight Line",
"Definition:Area"
] | [
"Definition:Polygon/Vertex",
"Definition:Polygon/Vertex",
"Area of Triangle in Determinant Form with Vertex at Origin",
"Definition:Polygon/Vertex",
"Definition:Collinear/Points",
"Definition:Lattice Point/Cartesian Coordinate System"
] |
proofwiki-15446 | Perpendicular Distance from Straight Line in Plane to Origin | Let $L$ be the straight line embedded in the cartesian plane whose equation is given as:
:$a x + b y = c$
Then the perpendicular distance $d$ between $L$ and $\tuple {0, 0}$ is given by:
:$d = \size {\dfrac c {\sqrt {a^2 + b^2} } }$ | From Perpendicular Distance from Straight Line in Plane to Point, the perpendicular distance $d$ between $L$ and the point $\tuple {x_0, y_0}$ is given by:
:$d = \dfrac {\size {a x_0 + b y_0 + c} } {\sqrt {a^2 + b^2} }$
The result follows by setting $x_0 = 0$ and $y_0 = 0$.
{{qed}} | Let $L$ be the [[Definition:Straight Line|straight line]] embedded in the [[Definition:Cartesian Plane|cartesian plane]] whose [[Equation of Straight Line in Plane|equation]] is given as:
:$a x + b y = c$
Then the [[Definition:Perpendicular Distance between Point and Straight Line|perpendicular distance]] $d$ between ... | From [[Perpendicular Distance from Straight Line in Plane to Point/General Form|Perpendicular Distance from Straight Line in Plane to Point]], the [[Definition:Perpendicular Distance between Point and Straight Line|perpendicular distance]] $d$ between $L$ and the [[Definition:Point|point]] $\tuple {x_0, y_0}$ is given ... | Perpendicular Distance from Straight Line in Plane to Origin | https://proofwiki.org/wiki/Perpendicular_Distance_from_Straight_Line_in_Plane_to_Origin | https://proofwiki.org/wiki/Perpendicular_Distance_from_Straight_Line_in_Plane_to_Origin | [
"Perpendicular Distance from Straight Line in Plane to Point"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane",
"Definition:Perpendicular Distance between Point and Straight Line"
] | [
"Perpendicular Distance from Straight Line in Plane to Point/General Form",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Point"
] |
proofwiki-15447 | Necessary Condition for Integral Functional to have Extremum/Two Variables | Let $D \subset \R^2$.
Let $\Gamma$ be the boundary of $D$.
Let $S$ be a set of real mappings such that:
:$S = \set {\map z {x, y}: \paren {z: S_1 \subseteq \R^2 \to S_2 \subseteq \R}, \paren {\map z {x, y} \in \map {C^2}D}, \paren {\map z \Gamma = 0} }$
Let $J \sqbrk z: S \to S_3 \subseteq \R$ be a functional of the fo... | From Condition for Differentiable Functional to have Extremum we have
:$\bigvalueat {\delta J \sqbrk {z; h} } {z \mathop = \hat z} = 0$
The variation exists if $J$ is a differentiable functional.
Since $\map z \Gamma = 0$, $\map h {x, y}$ vanishes on the boundary $\Gamma$:
:$\bigvalueat h \Gamma = 0$
From the definitio... | Let $D \subset \R^2$.
Let $\Gamma$ be the [[Definition:Boundary (Geometry)|boundary]] of $D$.
Let $S$ be a [[Definition:Set|set]] of [[Definition:Real Function|real mappings]] such that:
:$S = \set {\map z {x, y}: \paren {z: S_1 \subseteq \R^2 \to S_2 \subseteq \R}, \paren {\map z {x, y} \in \map {C^2}D}, \paren {\m... | From [[Condition for Differentiable Functional to have Extremum]] we have
:$\bigvalueat {\delta J \sqbrk {z; h} } {z \mathop = \hat z} = 0$
The variation exists if $J$ is a [[Definition:Differentiable Functional|differentiable functional]].
Since $\map z \Gamma = 0$, $\map h {x, y}$ vanishes on the [[Definition:Boun... | Necessary Condition for Integral Functional to have Extremum/Two Variables | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum/Two_Variables | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum/Two_Variables | [
"Calculus of Variations"
] | [
"Definition:Boundary (Geometry)",
"Definition:Set",
"Definition:Real Function",
"Definition:Functional/Real",
"Definition:Conditional/Necessary Condition",
"Definition:Extremum/Functional",
"Definition:Euler's Equation for Vanishing Variation"
] | [
"Condition for Differentiable Functional to have Extremum",
"Definition:Differentiable Functional",
"Definition:Boundary (Geometry)",
"Definition:Differentiable Functional",
"Definition:Functional/Real",
"Definition:Definite Integral",
"Taylor's Theorem",
"Definition:Ordered Tuple",
"Definition:Defi... |
proofwiki-15448 | Line in Plane is Straight iff Slope is Constant | Let $\LL$ be a curve which can be embedded in the plane.
Then $\LL$ is a straight line {{iff}} it is of constant slope. | Let $L$ be embedded in the cartesian plane.
The slope of $\LL$ at a point $p = \tuple {x, y}$ is defined as being its derivative at $p$ {{WRT|Differentiation}} $x$:
:$\grad p = \dfrac {\d y} {\d x}$
:500px
Let $\LL$ be a straight line.
Let $\triangle ABC$ and $\triangle DEF$ be right triangles constructed so that:
:$A,... | Let $\LL$ be a [[Definition:Curve|curve]] which can be embedded in [[Definition:The Plane|the plane]].
Then $\LL$ is a [[Definition:Straight Line|straight line]] {{iff}} it is of [[Definition:Constant|constant]] [[Definition:Slope|slope]]. | Let $L$ be embedded in the [[Definition:Cartesian Plane|cartesian plane]].
The [[Definition:Slope|slope]] of $\LL$ at a point $p = \tuple {x, y}$ is defined as being its [[Definition:Derivative|derivative]] at $p$ {{WRT|Differentiation}} $x$:
:$\grad p = \dfrac {\d y} {\d x}$
:[[File:Gradient-of-Straight-Line.png|5... | Line in Plane is Straight iff Slope is Constant | https://proofwiki.org/wiki/Line_in_Plane_is_Straight_iff_Slope_is_Constant | https://proofwiki.org/wiki/Line_in_Plane_is_Straight_iff_Slope_is_Constant | [
"Straight Lines"
] | [
"Definition:Line/Curve",
"Definition:Plane Surface/The Plane",
"Definition:Line/Straight Line",
"Definition:Constant",
"Definition:Slope"
] | [
"Definition:Cartesian Plane",
"Definition:Slope",
"Definition:Derivative",
"File:Gradient-of-Straight-Line.png",
"Definition:Line/Straight Line",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Parallel (Geometry)/Lines",
"Definition:Axis/X-Axis",
"Definition:Parallel (Geometry)/Lines",
... |
proofwiki-15449 | Equation of Straight Line in Plane/General Equation | A straight line $\LL$ is the set of all $\tuple {x, y} \in \R^2$, where:
:$\alpha_1 x + \alpha_2 y = \beta$
where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero. | Let $y = \map f x$ be the equation of a straight line $\LL$.
From Line in Plane is Straight iff Slope is Constant, $\LL$ has constant slope.
Thus the derivative of $y$ {{WRT|Differentiation}} $x$ will be of the form:
:$y' = c$
Thus:
{{begin-eqn}}
{{eqn | l = y
| r = \int c \rd x
| c = Fundamental Theorem of... | A [[Definition:Straight Line|straight line]] $\LL$ is the [[Definition:Set|set]] of all $\tuple {x, y} \in \R^2$, where:
:$\alpha_1 x + \alpha_2 y = \beta$
where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are [[Definition:Zero (Number)|zero]]. | Let $y = \map f x$ be the [[Definition:Equation of Geometric Figure|equation]] of a [[Definition:Straight Line|straight line]] $\LL$.
From [[Line in Plane is Straight iff Slope is Constant]], $\LL$ has [[Definition:Constant|constant]] [[Definition:Slope of Straight Line|slope]].
Thus the [[Definition:Derivative|deriv... | Equation of Straight Line in Plane/General Equation | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/General_Equation | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/General_Equation | [
"Equation of Straight Line in Plane/General Equation",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Set",
"Definition:Zero (Number)"
] | [
"Definition:Equation of Geometric Figure",
"Definition:Line/Straight Line",
"Line in Plane is Straight iff Slope is Constant",
"Definition:Constant",
"Definition:Slope/Straight Line",
"Definition:Derivative",
"Fundamental Theorem of Calculus",
"Primitive of Constant",
"Definition:Primitive (Calculus... |
proofwiki-15450 | Slope of Straight Line joining Points in Cartesian Plane | Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.
Let $\LL$ be the straight line passing through $p_1$ and $p_2$.
Then the slope of $\LL$ is given by:
:$\tan \theta = \dfrac {y_2 - y_1} {x_2 - x_1}$
where $\theta$ is the angle made by $\LL$ with the $x$-axis. | :500px
The slope of a straight line is defined as the change in $y$ divided by the change in $x$.
The change in $y$ from $p_1$ to $p_2$ is $y_2 - y_1$.
The change in $x$ from $p_1$ to $p_2$ is $x_2 - x_1$.
By definition of tangent of $\theta$:
:$\tan \theta = \dfrac {y_2 - y_1} {x_2 - x_1}$
Hence the result.
{{qed}} | Let $p_1 := \tuple {x_1, y_1}$ and $p_2 := \tuple {x_2, y_2}$ be [[Definition:Point|points]] in a [[Definition:Cartesian Plane|cartesian plane]].
Let $\LL$ be the [[Definition:Straight Line|straight line]] passing through $p_1$ and $p_2$.
Then the [[Definition:Slope of Straight Line|slope]] of $\LL$ is given by:
:$... | :[[File:Slope-of-Line-between-Points.png|500px]]
The [[Definition:Slope of Straight Line|slope of a straight line]] is defined as the change in $y$ divided by the change in $x$.
The change in $y$ from $p_1$ to $p_2$ is $y_2 - y_1$.
The change in $x$ from $p_1$ to $p_2$ is $x_2 - x_1$.
By definition of [[Definition:... | Slope of Straight Line joining Points in Cartesian Plane | https://proofwiki.org/wiki/Slope_of_Straight_Line_joining_Points_in_Cartesian_Plane | https://proofwiki.org/wiki/Slope_of_Straight_Line_joining_Points_in_Cartesian_Plane | [
"Straight Lines",
"Slope"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Definition:Slope/Straight Line",
"Definition:Angle",
"Definition:Axis/X-Axis"
] | [
"File:Slope-of-Line-between-Points.png",
"Definition:Slope/Straight Line",
"Definition:Tangent Function/Definition from Triangle"
] |
proofwiki-15451 | Equation of Straight Line in Plane/Two-Point Form | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.
Let $\LL$ be the straight line passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
or:
:$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} ... | From the slope-intercept form of the equation of the straight line:
:$(1): \quad y = m x + c$
which is to be satisfied by both $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.
We express $m$ and $c$ in terms of $\paren {x_1, y_1}$ and $\paren {x_2, y_2}$:
{{begin-eqn}}
{{eqn | l = y_1
| r = m x_1 + c
| c =
}}
... | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be [[Definition:Point|points]] in a [[Definition:Cartesian Plane|cartesian plane]].
Let $\LL$ be the [[Definition:Straight Line|straight line]] passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \d... | From the [[Equation of Straight Line in Plane/Slope-Intercept Form|slope-intercept form]] of the equation of the [[Definition:Straight Line|straight line]]:
:$(1): \quad y = m x + c$
which is to be satisfied by both $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.
We express $m$ and $c$ in terms of $\paren {x_1, y_1}$ ... | Equation of Straight Line in Plane/Two-Point Form/Proof 1 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Proof_1 | [
"Two-Point Form of Equation of Straight Line in Plane",
"Equations of Straight Lines in Plane"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/Two-Point Form"
] | [
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Line/Straight Line"
] |
proofwiki-15452 | Equation of Straight Line in Plane/Two-Point Form | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.
Let $\LL$ be the straight line passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
or:
:$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} ... | Let $\tuple {x, y}$ be an arbitrary point on the straight line through $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.
The area of the triangle formed by $\tuple {x, y}$, $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ is equal to $0$.
Hence from Area of Triangle in Determinant Form:
:$\AA = \dfrac 1 2 \size {\paren {\begin {... | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be [[Definition:Point|points]] in a [[Definition:Cartesian Plane|cartesian plane]].
Let $\LL$ be the [[Definition:Straight Line|straight line]] passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \d... | Let $\tuple {x, y}$ be an arbitrary [[Definition:Point|point]] on the [[Definition:Straight Line|straight line]] through $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$.
The [[Definition:Area|area]] of the [[Definition:Triangle (Geometry)|triangle]] formed by $\tuple {x, y}$, $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ is... | Equation of Straight Line in Plane/Two-Point Form/Proof 2 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Proof_2 | [
"Two-Point Form of Equation of Straight Line in Plane",
"Equations of Straight Lines in Plane"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/Two-Point Form"
] | [
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Area",
"Definition:Triangle (Geometry)",
"Area of Triangle in Determinant Form",
"Area of Triangle in Determinant Form"
] |
proofwiki-15453 | Equation of Straight Line in Plane/Two-Point Form | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.
Let $\LL$ be the straight line passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
or:
:$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} ... | :400px
Let $P = \tuple {x, y}$ be an arbitrary point on the straight line through $P_1 = \tuple {x_1, y_1}$ and $P_2 = \tuple {x_2, y_2}$.
Construct the straight line $P_1 H K$ perpendicular to the $x$-axis.
We have that $\triangle P_1 H P_2$ and $\triangle P_1 K P$ are similar.
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac ... | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be [[Definition:Point|points]] in a [[Definition:Cartesian Plane|cartesian plane]].
Let $\LL$ be the [[Definition:Straight Line|straight line]] passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \d... | :[[File:Straight-line-2-points-form-Proof-3.png|400px]]
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] on the [[Definition:Straight Line|straight line]] through $P_1 = \tuple {x_1, y_1}$ and $P_2 = \tuple {x_2, y_2}$.
Construct the [[Definition:Straight Line|straight line]] $P_1 H K$ [[Definition... | Equation of Straight Line in Plane/Two-Point Form/Proof 3 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Proof_3 | [
"Two-Point Form of Equation of Straight Line in Plane",
"Equations of Straight Lines in Plane"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/Two-Point Form"
] | [
"File:Straight-line-2-points-form-Proof-3.png",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Right Angle/Perpendicular",
"Definition:Axis/X-Axis",
"Definition:Similar Triangles"
] |
proofwiki-15454 | Equation of Straight Line in Plane/Two-Point Form | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be points in a cartesian plane.
Let $\LL$ be the straight line passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \dfrac {y_2 - y_1} {x_2 - x_1}$
or:
:$\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} ... | :500px
Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$ be the position vectors of the points $A$ and $B$ embedded in the complex plane.
Let $z = x + i y$ be the position vector of an arbitrary point $P$ on the straight line $AB$.
From the diagram:
{{begin-eqn}}
{{eqn | l = OA + AP
| r = OP
| c =
}}
{{eqn |... | Let $P_1 := \tuple {x_1, y_1}$ and $P_2 := \tuple {x_2, y_2}$ be [[Definition:Point|points]] in a [[Definition:Cartesian Plane|cartesian plane]].
Let $\LL$ be the [[Definition:Straight Line|straight line]] passing through $P_1$ and $P_2$.
Then $\LL$ can be described by the equation:
:$\dfrac {y - y_1} {x - x_1} = \d... | :[[File:Straight-line-2-points-form-Proof-4.png|500px]]
Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$ be the [[Definition:Position Vector|position vectors]] of the [[Definition:Point|points]] $A$ and $B$ embedded in the [[Definition:Complex Plane|complex plane]].
Let $z = x + i y$ be the [[Definition:Position Vect... | Equation of Straight Line in Plane/Two-Point Form/Proof 4 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Point_Form/Proof_4 | [
"Two-Point Form of Equation of Straight Line in Plane",
"Equations of Straight Lines in Plane"
] | [
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/Two-Point Form"
] | [
"File:Straight-line-2-points-form-Proof-4.png",
"Definition:Position Vector",
"Definition:Point",
"Definition:Complex Number/Complex Plane",
"Definition:Position Vector",
"Definition:Point",
"Definition:Line/Straight Line",
"Definition:Collinear/Points"
] |
proofwiki-15455 | Equation of Straight Line in Plane/Slope-Intercept Form | Let $\LL$ be the straight line in the Cartesian plane such that:
:the slope of $\LL$ is $m$
:the $y$-intercept of $\LL$ is $c$
Then $\LL$ can be described by the equation:
:$y = m x + c$
such that $m$ is the slope of $\LL$ and $c$ is the $y$-intercept. | Let $\LL$ be the straight line defined by the general equation:
:$\alpha_1 x + \alpha_2 y = \beta$
We have:
{{begin-eqn}}
{{eqn | l = \alpha_1 x + \alpha_2 y
| r = \beta
| c =
}}
{{eqn | ll= \leadsto
| l = \alpha_2 y
| r = y_1 - \alpha_1 x + \beta
| c =
}}
{{eqn | n = 1
| ll= \lead... | Let $\LL$ be the [[Definition:Straight Line|straight line]] in the [[Definition:Cartesian Plane|Cartesian plane]] such that:
:the [[Definition:Slope of Straight Line|slope]] of $\LL$ is $m$
:the [[Definition:Y-Intercept|$y$-intercept]] of $\LL$ is $c$
Then $\LL$ can be described by the equation:
:$y = m x + c$
such... | Let $\LL$ be the [[Definition:Straight Line|straight line]] defined by the [[Equation of Straight Line in Plane/General Equation|general equation]]:
:$\alpha_1 x + \alpha_2 y = \beta$
We have:
{{begin-eqn}}
{{eqn | l = \alpha_1 x + \alpha_2 y
| r = \beta
| c =
}}
{{eqn | ll= \leadsto
| l = \alpha... | Equation of Straight Line in Plane/Slope-Intercept Form/Proof 1 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Slope-Intercept_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Slope-Intercept_Form/Proof_1 | [
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Slope/Straight Line",
"Definition:Intercept/Y-Intercept",
"Definition:Slope/Straight Line",
"Definition:Intercept/Y-Intercept"
] | [
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/General Equation",
"Definition:Intercept/Y-Intercept",
"Definition:Differentiation",
"Definition:Slope/Straight Line",
"Definition:Constant"
] |
proofwiki-15456 | Equation of Straight Line in Plane/Slope-Intercept Form | Let $\LL$ be the straight line in the Cartesian plane such that:
:the slope of $\LL$ is $m$
:the $y$-intercept of $\LL$ is $c$
Then $\LL$ can be described by the equation:
:$y = m x + c$
such that $m$ is the slope of $\LL$ and $c$ is the $y$-intercept. | :500px
Let the straight line $\LL$ make an angle $\psi$ with the $x$-axis.
The slope $m$ of $\LL$ is then given by:
:$m = \tan \psi$
Let $B := \tuple {0, c}$ be the $y$-intercept of $\LL$.
Let $P := \tuple {x, y}$ be an arbitrary point on $\LL$.
Let the perpendicular $MP$ to the $x$-axis be constructed from $P$ to int... | Let $\LL$ be the [[Definition:Straight Line|straight line]] in the [[Definition:Cartesian Plane|Cartesian plane]] such that:
:the [[Definition:Slope of Straight Line|slope]] of $\LL$ is $m$
:the [[Definition:Y-Intercept|$y$-intercept]] of $\LL$ is $c$
Then $\LL$ can be described by the equation:
:$y = m x + c$
such... | :[[File:Straight-line-slope-intercept-form.png|500px]]
Let the [[Definition:Straight Line|straight line]] $\LL$ make an [[Definition:Angle|angle]] $\psi$ with the [[Definition:X-Axis|$x$-axis]].
The [[Definition:Slope of Straight Line|slope]] $m$ of $\LL$ is then given by:
:$m = \tan \psi$
Let $B := \tuple {0, c}$ ... | Equation of Straight Line in Plane/Slope-Intercept Form/Proof 3 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Slope-Intercept_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Slope-Intercept_Form/Proof_3 | [
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Definition:Slope/Straight Line",
"Definition:Intercept/Y-Intercept",
"Definition:Slope/Straight Line",
"Definition:Intercept/Y-Intercept"
] | [
"File:Straight-line-slope-intercept-form.png",
"Definition:Line/Straight Line",
"Definition:Angle",
"Definition:Axis/X-Axis",
"Definition:Slope/Straight Line",
"Definition:Intercept/Y-Intercept",
"Definition:Point",
"Definition:Right Angle/Perpendicular",
"Definition:Axis/X-Axis",
"Definition:Inte... |
proofwiki-15457 | Equation of Straight Line in Plane/Two-Intercept Form | Let $\LL$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.
Then $\LL$ can be described by the equation:
:$\dfrac x a + \dfrac y b = 1$ | :400px
From the General Equation of Straight Line in Plane, $\LL$ can be expressed in the form:
:$(1): \quad \alpha_1 x + \alpha_2 y = \beta$
where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.
Substituting for the two points whose coordinates we know about:
{{begin-eqn}}
{{e... | Let $\LL$ be a [[Definition:Straight Line|straight line]] which [[Definition:Intercept|intercepts]] the [[Definition:X-Axis|$x$-axis]] and [[Definition:Y-Axis|$y$-axis]] respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.
Then $\LL$ can be described by the equation:
:$\dfrac x a + \dfrac y b = 1$ | :[[File:Straight-line-double-intercept-form.png|400px]]
From the [[Equation of Straight Line in Plane/General Equation|General Equation of Straight Line in Plane]], $\LL$ can be expressed in the form:
:$(1): \quad \alpha_1 x + \alpha_2 y = \beta$
where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alph... | Equation of Straight Line in Plane/Two-Intercept Form/Proof 1 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Intercept_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Intercept_Form/Proof_1 | [
"Equation of Straight Line in Plane/Two-Intercept Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Intercept",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis"
] | [
"File:Straight-line-double-intercept-form.png",
"Equation of Straight Line in Plane/General Equation",
"Definition:Point",
"Definition:Cartesian Coordinate System"
] |
proofwiki-15458 | Equation of Straight Line in Plane/Two-Intercept Form | Let $\LL$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.
Then $\LL$ can be described by the equation:
:$\dfrac x a + \dfrac y b = 1$ | By definition, $\LL$ passes through $\tuple {a, 0}$ and $\tuple {0, b}$.
From the Two-Point Form of Equation of Straight Line in Plane, $\LL$ can be expressed in the form:
{{begin-eqn}}
{{eqn | l = \dfrac {y - 0} {x - a}
| r = \dfrac {b - 0} {0 - a}
| c =
}}
{{eqn | ll= \leadsto
| l = -a y
| r ... | Let $\LL$ be a [[Definition:Straight Line|straight line]] which [[Definition:Intercept|intercepts]] the [[Definition:X-Axis|$x$-axis]] and [[Definition:Y-Axis|$y$-axis]] respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.
Then $\LL$ can be described by the equation:
:$\dfrac x a + \dfrac y b = 1$ | By definition, $\LL$ passes through $\tuple {a, 0}$ and $\tuple {0, b}$.
From the [[Two-Point Form of Equation of Straight Line in Plane]], $\LL$ can be expressed in the form:
{{begin-eqn}}
{{eqn | l = \dfrac {y - 0} {x - a}
| r = \dfrac {b - 0} {0 - a}
| c =
}}
{{eqn | ll= \leadsto
| l = -a y
... | Equation of Straight Line in Plane/Two-Intercept Form/Proof 2 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Intercept_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Intercept_Form/Proof_2 | [
"Equation of Straight Line in Plane/Two-Intercept Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Intercept",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis"
] | [
"Equation of Straight Line in Plane/Two-Point Form"
] |
proofwiki-15459 | Equation of Straight Line in Plane/Two-Intercept Form | Let $\LL$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.
Then $\LL$ can be described by the equation:
:$\dfrac x a + \dfrac y b = 1$ | :400px
We have that $\LL$ is passes through the two points $A = \tuple {a, 0}$ and $B = \tuple {0, b}$.
Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.
We have:
{{begin-eqn}}
{{eqn | l = \triangle OBP + \triangle OAP
| r = \triangle OAB
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {b x} 2 + \df... | Let $\LL$ be a [[Definition:Straight Line|straight line]] which [[Definition:Intercept|intercepts]] the [[Definition:X-Axis|$x$-axis]] and [[Definition:Y-Axis|$y$-axis]] respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.
Then $\LL$ can be described by the equation:
:$\dfrac x a + \dfrac y b = 1$ | :[[File:Straight-line-double-intercept-form-Proof-3.png|400px]]
We have that $\LL$ is passes through the two [[Definition:Point|points]] $A = \tuple {a, 0}$ and $B = \tuple {0, b}$.
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] on $\LL$.
We have:
{{begin-eqn}}
{{eqn | l = \triangle OBP + \trian... | Equation of Straight Line in Plane/Two-Intercept Form/Proof 3 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Intercept_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Two-Intercept_Form/Proof_3 | [
"Equation of Straight Line in Plane/Two-Intercept Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Intercept",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis"
] | [
"File:Straight-line-double-intercept-form-Proof-3.png",
"Definition:Point",
"Definition:Point",
"Area of Triangle in Terms of Side and Altitude",
"Definition:Multiplication/Real Numbers",
"Definition:Division/Field/Real Numbers"
] |
proofwiki-15460 | Equation of Straight Line in Plane/Normal Form | Let $\LL$ be a straight line such that:
:the perpendicular distance from $\LL$ to the origin is $p$
:the angle made between that perpendicular and the $x$-axis is $\alpha$.
Then $\LL$ can be defined by the equation:
:$x \cos \alpha + y \sin \alpha = p$ | :400px
Let $A$ be the $x$-intercept of $\LL$.
Let $B$ be the $y$-intercept of $\LL$.
Let $A = \tuple {a, 0}$ and $B = \tuple {0, b}$.
From the Equation of Straight Line in Plane: Two-Intercept Form, $\LL$ can be expressed in the form:
:$(1): \quad \dfrac x a + \dfrac y a = 1$
Then:
{{begin-eqn}}
{{eqn | l = p
| r... | Let $\LL$ be a [[Definition:Straight Line|straight line]] such that:
:the [[Definition:Perpendicular Distance between Point and Straight Line|perpendicular distance]] from $\LL$ to the [[Definition:Origin|origin]] is $p$
:the [[Definition:Angle|angle]] made between that [[Definition:Perpendicular|perpendicular]] and th... | :[[File:Straight-line-normal-form.png|400px]]
Let $A$ be the [[Definition:X-Intercept|$x$-intercept]] of $\LL$.
Let $B$ be the [[Definition:Y-Intercept|$y$-intercept]] of $\LL$.
Let $A = \tuple {a, 0}$ and $B = \tuple {0, b}$.
From the [[Equation of Straight Line in Plane/Two-Intercept Form|Equation of Straight L... | Equation of Straight Line in Plane/Normal Form/Proof 1 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Normal_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Normal_Form/Proof_1 | [
"Equation of Straight Line in Plane/Normal Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Right Angle/Perpendicular",
"Definition:Axis/X-Axis",
"Definition:Equation of Geometric Figure"
] | [
"File:Straight-line-normal-form.png",
"Definition:Intercept/X-Intercept",
"Definition:Intercept/Y-Intercept",
"Equation of Straight Line in Plane/Two-Intercept Form"
] |
proofwiki-15461 | Equation of Straight Line in Plane/Normal Form | Let $\LL$ be a straight line such that:
:the perpendicular distance from $\LL$ to the origin is $p$
:the angle made between that perpendicular and the $x$-axis is $\alpha$.
Then $\LL$ can be defined by the equation:
:$x \cos \alpha + y \sin \alpha = p$ | :400px
Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.
Let $O$ be the origin of the Cartesian plane in which $\LL$ is embedded.
Let $PQ$ be the perpendicular dropped from $P$ to the $x$-axis.
Let $QS$ be the perpendicular dropped from $Q$ to the line $ON$.
Let $PR$ be the perpendicular dropped from $P$ to the l... | Let $\LL$ be a [[Definition:Straight Line|straight line]] such that:
:the [[Definition:Perpendicular Distance between Point and Straight Line|perpendicular distance]] from $\LL$ to the [[Definition:Origin|origin]] is $p$
:the [[Definition:Angle|angle]] made between that [[Definition:Perpendicular|perpendicular]] and th... | :[[File:Straight-line-normal-form-Proof-2.png|400px]]
Let $P = \tuple {x, y}$ be an arbitrary [[Definition:Point|point]] on $\LL$.
Let $O$ be the [[Definition:Origin|origin]] of the [[Definition:Cartesian Plane|Cartesian plane]] in which $\LL$ is embedded.
Let $PQ$ be the [[Definition:Perpendicular|perpendicular]] d... | Equation of Straight Line in Plane/Normal Form/Proof 2 | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Normal_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Normal_Form/Proof_2 | [
"Equation of Straight Line in Plane/Normal Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Coordinate System/Origin",
"Definition:Angle",
"Definition:Right Angle/Perpendicular",
"Definition:Axis/X-Axis",
"Definition:Equation of Geometric Figure"
] | [
"File:Straight-line-normal-form-Proof-2.png",
"Definition:Point",
"Definition:Coordinate System/Origin",
"Definition:Cartesian Plane",
"Definition:Right Angle/Perpendicular",
"Definition:Axis/X-Axis",
"Definition:Right Angle/Perpendicular",
"Definition:Line/Straight Line",
"Definition:Right Angle/Pe... |
proofwiki-15462 | Perpendicular Distance from Straight Line in Plane to Point/General Form | Let $\LL$ be a straight line embedded in a cartesian plane, given by the equation:
:$a x + b y + c = 0$
Let $P$ be a point in the cartesian plane whose coordinates are given by:
:$P = \tuple {x_0, y_0}$
Then the perpendicular distance $d$ from $P$ to $\LL$ is given by:
:$d = \dfrac {\size {a x_0 + b y_0 + c} } {\sqrt {... | We have that $\LL$ has the equation:
:$(1): \quad a x + b y + c = 0$
500px
Let a perpendicular be dropped from $P$ to $\LL$ at $Q$.
The perpendicular distance $d$ that we are to find is then $PQ$.
In order to simplify the algebra that will inevitably follow, we are to make a transformation as follows.
Let $\MM$ be cons... | Let $\LL$ be a [[Definition:Straight Line|straight line]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], given by the [[Equation of Straight Line in Plane/General Equation|equation]]:
:$a x + b y + c = 0$
Let $P$ be a [[Definition:Point|point]] in the [[Definition:Cartesian Plane|cartesian plane]] whose... | We have that $\LL$ has the [[Equation of Straight Line in Plane/General Equation|equation]]:
:$(1): \quad a x + b y + c = 0$
[[File:Distance-from-Straight-Line-to-Point.png|500px]]
Let a [[Definition:Perpendicular|perpendicular]] be dropped from $P$ to $\LL$ at $Q$.
The [[Definition:Perpendicular Distance between... | Perpendicular Distance from Straight Line in Plane to Point/General Form/Proof 1 | https://proofwiki.org/wiki/Perpendicular_Distance_from_Straight_Line_in_Plane_to_Point/General_Form | https://proofwiki.org/wiki/Perpendicular_Distance_from_Straight_Line_in_Plane_to_Point/General_Form/Proof_1 | [
"Perpendicular Distance from Straight Line in Plane to Point"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/General Equation",
"Definition:Point",
"Definition:Cartesian Plane",
"Definition:Cartesian Coordinate System",
"Definition:Perpendicular Distance between Point and Straight Line"
] | [
"Equation of Straight Line in Plane/General Equation",
"File:Distance-from-Straight-Line-to-Point.png",
"Definition:Right Angle/Perpendicular",
"Definition:Perpendicular Distance between Point and Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Right Angle/Perpendicular",
"Definition... |
proofwiki-15463 | Equation of Straight Line in Plane/Point-Slope Form | Let $\LL$ be a straight line embedded in a cartesian plane, given in slope-intercept form as:
:$y = m x + c$
where $m$ is the slope of $\LL$.
Let $\LL$ pass through the point $\tuple {x_0, y_0}$.
Then $\LL$ can be expressed by the equation:
:$y - y_0 = m \paren {x - x_0}$ | As $\tuple {x_0, y_0}$ is on $\LL$, it follows that:
{{begin-eqn}}
{{eqn | l = y_0
| r = m x_0 + c
| c =
}}
{{eqn | ll= \leadsto
| l = c
| r = y_0 - m x_0
| c =
}}
{{end-eqn}}
Substituting back into the equation for $\LL$:
{{begin-eqn}}
{{eqn | l = y
| r = m x + \paren {y_0 - m x_... | Let $\LL$ be a [[Definition:Straight Line|straight line]] embedded in a [[Definition:Cartesian Plane|cartesian plane]], given in [[Equation of Straight Line in Plane/Slope-Intercept Form|slope-intercept form]] as:
:$y = m x + c$
where $m$ is the [[Definition:Slope of Straight Line|slope]] of $\LL$.
Let $\LL$ pass thro... | As $\tuple {x_0, y_0}$ is on $\LL$, it follows that:
{{begin-eqn}}
{{eqn | l = y_0
| r = m x_0 + c
| c =
}}
{{eqn | ll= \leadsto
| l = c
| r = y_0 - m x_0
| c =
}}
{{end-eqn}}
Substituting back into the equation for $\LL$:
{{begin-eqn}}
{{eqn | l = y
| r = m x + \paren {y_0 - m... | Equation of Straight Line in Plane/Point-Slope Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Point-Slope_Form | https://proofwiki.org/wiki/Equation_of_Straight_Line_in_Plane/Point-Slope_Form | [
"Equation of Straight Line in Plane/Point-Slope Form",
"Equations of Straight Lines in Plane"
] | [
"Definition:Line/Straight Line",
"Definition:Cartesian Plane",
"Equation of Straight Line in Plane/Slope-Intercept Form",
"Definition:Slope/Straight Line",
"Definition:Point"
] | [] |
proofwiki-15464 | Shortest Possible Distance between Lattice Points on Straight Line in Cartesian Plane | Let $\LL$ be the straight line defined by the equation:
:$a x - b y = c$
Let $p_1$ and $p_2$ be lattice points on $\LL$.
Then the shortest possible distance $d$ between $p_1$ and $p_2$ is:
:$d = \dfrac {\sqrt {a^2 + b^2} } {\gcd \set {a, b} }$
where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.... | Let $p_1 = \tuple {x_1, y_1}$ and $p_2 = \tuple {x_2, y_2}$ be on $\LL$.
Thus:
{{begin-eqn}}
{{eqn | l = a x_1 - b y_1
| r = c
}}
{{eqn | l = a x_2 - b y_2
| r = c
| c =
}}
{{end-eqn}}
From Solution of Linear Diophantine Equation, it is necessary and sufficient that:
:$\gcd \set {a, b} \divides c$
fo... | Let $\LL$ be the [[Definition:Straight Line|straight line]] defined by the [[Equation of Straight Line in Plane/General Equation|equation]]:
:$a x - b y = c$
Let $p_1$ and $p_2$ be [[Definition:Lattice Point of Cartesian Coordinate System|lattice points]] on $\LL$.
Then the shortest possible [[Definition:Distance be... | Let $p_1 = \tuple {x_1, y_1}$ and $p_2 = \tuple {x_2, y_2}$ be on $\LL$.
Thus:
{{begin-eqn}}
{{eqn | l = a x_1 - b y_1
| r = c
}}
{{eqn | l = a x_2 - b y_2
| r = c
| c =
}}
{{end-eqn}}
From [[Solution of Linear Diophantine Equation]], it is [[Definition:Necessary and Sufficient|necessary and suffi... | Shortest Possible Distance between Lattice Points on Straight Line in Cartesian Plane | https://proofwiki.org/wiki/Shortest_Possible_Distance_between_Lattice_Points_on_Straight_Line_in_Cartesian_Plane | https://proofwiki.org/wiki/Shortest_Possible_Distance_between_Lattice_Points_on_Straight_Line_in_Cartesian_Plane | [
"Straight Lines",
"Linear Diophantine Equations"
] | [
"Definition:Line/Straight Line",
"Equation of Straight Line in Plane/General Equation",
"Definition:Lattice Point/Cartesian Coordinate System",
"Definition:Distance between Points",
"Definition:Greatest Common Divisor"
] | [
"Solution of Linear Diophantine Equation",
"Definition:Biconditional/Semantics of Biconditional/Necessary and Sufficient",
"Definition:Lattice Point/Cartesian Coordinate System",
"Solution of Linear Diophantine Equation",
"Definition:Lattice Point/Cartesian Coordinate System",
"Definition:Distance between... |
proofwiki-15465 | Decomposition into Even-Odd Integers is not always Unique | For every even integer $n$ such that $n > 1$, if $n$ can be expressed as the product of one or more even-times odd integers, it is not necessarily the case that this product is unique. | Let $n \in \Z$ be of the form $2^2 p q$ where $p$ and $q$ are odd primes.
Then:
:$n = \paren {2 p} \times \paren {2 q} = 2 \times \paren {2 p q}$
A specific example that can be cited is $n = 60$:
:$60 = 6 \times 10$
and:
:$60 = 2 \times 30$.
Each of $2, 6, 10, 30$ are even-times odd integers:
{{begin-eqn}}
{{eqn | l = ... | For every [[Definition:Even Integer|even integer]] $n$ such that $n > 1$, if $n$ can be expressed as the [[Definition:Integer Multiplication|product]] of one or more [[Definition:Even-Times Odd Integer|even-times odd integers]], it is not necessarily the case that this [[Definition:Integer Multiplication|product]] is [... | Let $n \in \Z$ be of the form $2^2 p q$ where $p$ and $q$ are [[Definition:Odd Prime|odd primes]].
Then:
:$n = \paren {2 p} \times \paren {2 q} = 2 \times \paren {2 p q}$
A specific example that can be cited is $n = 60$:
:$60 = 6 \times 10$
and:
:$60 = 2 \times 30$.
Each of $2, 6, 10, 30$ are [[Definition:Even-Time... | Decomposition into Even-Odd Integers is not always Unique | https://proofwiki.org/wiki/Decomposition_into_Even-Odd_Integers_is_not_always_Unique | https://proofwiki.org/wiki/Decomposition_into_Even-Odd_Integers_is_not_always_Unique | [
"Even Integers"
] | [
"Definition:Even Integer",
"Definition:Multiplication/Integers",
"Definition:Even Integer/Even-Times Odd",
"Definition:Multiplication/Integers",
"Definition:Unique"
] | [
"Definition:Odd Prime",
"Definition:Even Integer/Even-Times Odd",
"Definition:Divisor (Algebra)/Integer",
"Definition:Even Integer/Even-Times Odd"
] |
proofwiki-15466 | Decomposition into Product of Power of 2 and Odd Integer is Unique | Let $n \in \Z$ be an integer.
Then $n$ can be decomposed into the product of a power of $2$ and an odd integer. | {{AimForCont}} there exists $n \in \Z$ which can be decomposed into a power of $2$ and an odd integer in more than one way.
That is:
:$n = 2^a r$
and:
:$n = 2^b s$
where:
:$a, b \in \Z_{\ge 0}$
:$r$ and $s$ are odd integers.
:either $a \ne b$ or $r \ne s$.
Suppose $r = s$.
Then:
:$\dfrac n r = \dfrac n s = 2^a = 2^b$
w... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then $n$ can be decomposed into the [[Definition:Integer Multiplication|product]] of a [[Definition:Integer Power|power of $2$]] and an [[Definition:Odd Integer|odd integer]]. | {{AimForCont}} there exists $n \in \Z$ which can be decomposed into a [[Definition:Integer Power|power of $2$]] and an [[Definition:Odd Integer|odd integer]] in more than one way.
That is:
:$n = 2^a r$
and:
:$n = 2^b s$
where:
:$a, b \in \Z_{\ge 0}$
:$r$ and $s$ are [[Definition:Odd Integer|odd integers]].
:either $a ... | Decomposition into Product of Power of 2 and Odd Integer is Unique | https://proofwiki.org/wiki/Decomposition_into_Product_of_Power_of_2_and_Odd_Integer_is_Unique | https://proofwiki.org/wiki/Decomposition_into_Product_of_Power_of_2_and_Odd_Integer_is_Unique | [
"Integers"
] | [
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:Odd Integer"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Odd Integer",
"Definition:Odd Integer",
"Definition:Contradiction",
"Definition:Contradiction",
"Definition:Even Integer",
"Definition:Contradiction",
"Definition:Odd Integer",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Mul... |
proofwiki-15467 | Prime Decomposition of Integer is Unique | Let $n$ be an integer such that $n > 1$.
Then the prime decomposition of $n$ is unique. | From Integer is Expressible as Product of Primes, $n$ can be expressed as the product of one or more primes.
Let $n = q_1 q_2 \dotsm q_s$ where $q_1, q_2, \ldots, q_s$ are all primes such that:
:$(1): \quad n = q_1 \le q_2 \le \dotsb \le q_s$
From Expression for Integer as Product of Primes is Unique, the expression fo... | Let $n$ be an [[Definition:Integer|integer]] such that $n > 1$.
Then the [[Definition:Prime Decomposition|prime decomposition]] of $n$ is [[Definition:Unique|unique]]. | From [[Integer is Expressible as Product of Primes]], $n$ can be expressed as the [[Definition:Integer Multiplication|product]] of one or more [[Definition:Prime Number|primes]].
Let $n = q_1 q_2 \dotsm q_s$ where $q_1, q_2, \ldots, q_s$ are all [[Definition:Prime Number|primes]] such that:
:$(1): \quad n = q_1 \le q_... | Prime Decomposition of Integer is Unique | https://proofwiki.org/wiki/Prime_Decomposition_of_Integer_is_Unique | https://proofwiki.org/wiki/Prime_Decomposition_of_Integer_is_Unique | [
"Prime Decompositions"
] | [
"Definition:Integer",
"Definition:Prime Decomposition",
"Definition:Unique"
] | [
"Integer is Expressible as Product of Primes",
"Definition:Multiplication/Integers",
"Definition:Prime Number",
"Definition:Prime Number",
"Expression for Integer as Product of Primes is Unique",
"Definition:Unique",
"Fundamental Theorem on Equivalence Relations",
"Definition:Set Partition",
"Defini... |
proofwiki-15468 | Expression for Integers as Powers of Same Primes | Let $a, b \in \Z$ be integers.
Let their prime decompositions be given by:
{{begin-eqn}}
{{eqn | l = a
| r = {q_1}^{e_1} {q_2}^{e_2} \cdots {q_r}^{e_r}
}}
{{eqn | r = \prod_{\substack {q_i \mathop \divides a \\ \text {$q_i$ is prime} } } {q_i}^{e_i}
}}
{{eqn | l = b
| r = {s_1}^{f_1} {s_2}^{f_2} \cdots {s_u... | In the prime decompositions $(1)$ and $(2)$, we have that:
:$q_1 < q_2 < \dotsb < q_r$
and:
:$s_1 < s_2 < \dotsb < s_u$
Hence we can define:
{{begin-eqn}}
{{eqn | l = E
| r = \set {q_1, q_2, \ldots, q_r}
}}
{{eqn | l = F
| r = \set {s_1, s_2, \ldots, s_u}
}}
{{end-eqn}}
as all the $q_1, q_2, \dotsc, q_r$ ar... | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Let their [[Definition:Prime Decomposition|prime decompositions]] be given by:
{{begin-eqn}}
{{eqn | l = a
| r = {q_1}^{e_1} {q_2}^{e_2} \cdots {q_r}^{e_r}
}}
{{eqn | r = \prod_{\substack {q_i \mathop \divides a \\ \text {$q_i$ is prime} } } {q_i}^{e_i}
}}
{... | In the [[Definition:Prime Decomposition|prime decompositions]] $(1)$ and $(2)$, we have that:
:$q_1 < q_2 < \dotsb < q_r$
and:
:$s_1 < s_2 < \dotsb < s_u$
Hence we can define:
{{begin-eqn}}
{{eqn | l = E
| r = \set {q_1, q_2, \ldots, q_r}
}}
{{eqn | l = F
| r = \set {s_1, s_2, \ldots, s_u}
}}
{{end-eqn}... | Expression for Integers as Powers of Same Primes | https://proofwiki.org/wiki/Expression_for_Integers_as_Powers_of_Same_Primes | https://proofwiki.org/wiki/Expression_for_Integers_as_Powers_of_Same_Primes | [
"Prime Decompositions"
] | [
"Definition:Integer",
"Definition:Prime Decomposition",
"Definition:Prime Number"
] | [
"Definition:Prime Decomposition",
"Definition:Distinct/Plural",
"Definition:Distinct/Plural",
"Definition:Element",
"Definition:Distinct/Plural",
"Definition:Inclusion Mapping",
"Definition:Inclusion Mapping",
"Definition:Multiplication/Integers",
"Definition:Power (Algebra)/Integer",
"Definition:... |
proofwiki-15469 | Expression for Integers as Powers of Same Primes/General Result | Let $a_1, a_2, \dotsc, a_n \in \Z$ be integers.
Let their prime decompositions be given by:
:$\ds a_i = \prod_{\substack {p_{i j} \mathop \divides a_i \\ \text {$p_{i j}$ is prime} } } {p_{i j} }^{e_{i j} }$
Then there exists a set $T$ of prime numbers:
:$T = \set {t_1, t_2, \dotsc, t_v}$
such that:
:$t_1 < t_2 < \dots... | The proof proceeds by induction.
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
:for all $a_i \in \set {a_1, a_2, \ldots, a_n}$: there exists a set $T = \set {t_1, t_2, \dotsc, t_v}$ of prime numbers such that $t_1 < t_2 < \dotsb < t_v$ such that:
::$\ds a_i = \prod_{j \mathop = 1}^v {t_j}^{g_{i j} }$ | Let $a_1, a_2, \dotsc, a_n \in \Z$ be [[Definition:Integer|integers]].
Let their [[Definition:Prime Decomposition|prime decompositions]] be given by:
:$\ds a_i = \prod_{\substack {p_{i j} \mathop \divides a_i \\ \text {$p_{i j}$ is prime} } } {p_{i j} }^{e_{i j} }$
Then there exists a [[Definition:Set|set]] $T$ of ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:for all $a_i \in \set {a_1, a_2, \ldots, a_n}$: there exists a [[Definition:Set|set]] $T = \set {t_1, t_2, \dotsc, t_v}$ of [[Definition:Prime Number|prime... | Expression for Integers as Powers of Same Primes/General Result | https://proofwiki.org/wiki/Expression_for_Integers_as_Powers_of_Same_Primes/General_Result | https://proofwiki.org/wiki/Expression_for_Integers_as_Powers_of_Same_Primes/General_Result | [
"Prime Decompositions"
] | [
"Definition:Integer",
"Definition:Prime Decomposition",
"Definition:Set",
"Definition:Prime Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Set",
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Set",
"Definition:Prime Number",
"Definition:Set",
"Definition:Prime Number",
"Principle of Mathematical Induction",
"Definition:Set",
"Definitio... |
proofwiki-15470 | GCD from Prime Decomposition/General Result | Let $n \in \N$ be a natural number such that $n \ge 2$.
Let $\N_n$ be defined as:
:$\N_n := \set {1, 2, \dotsc, n}$
Let $A_n = \set {a_1, a_2, \dotsc, a_n} \subseteq \Z$ be a set of $n$ integers.
From Expression for Integers as Powers of Same Primes, let:
:$\ds \forall i \in \N_n: a_i = \prod_{p_j \mathop \in T} {p_j}^... | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \map \gcd {A_n} = \prod_{j \mathop \in \N_r} {p_j}^{\min \set {e_{i j}: \, i \in \N_n} }$
=== Basis for the Induction ===
$\map P 2$ is the case:
:$\ds \gcd \set {a_1, a_2} = \prod_{j \mathop \in \N_r} {p_j}^{\min \set... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n \ge 2$.
Let $\N_n$ be defined as:
:$\N_n := \set {1, 2, \dotsc, n}$
Let $A_n = \set {a_1, a_2, \dotsc, a_n} \subseteq \Z$ be a [[Definition:Set|set]] of $n$ [[Definition:Integer|integers]].
From [[Expression for Integers as Powers of Same... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map \gcd {A_n} = \prod_{j \mathop \in \N_r} {p_j}^{\min \set {e_{i j}: \, i \in \N_n} }$
=== Basis for the Induction ===
$\map P 2$ is the case:
... | GCD from Prime Decomposition/General Result | https://proofwiki.org/wiki/GCD_from_Prime_Decomposition/General_Result | https://proofwiki.org/wiki/GCD_from_Prime_Decomposition/General_Result | [
"Greatest Common Divisor",
"Prime Decompositions"
] | [
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Integer",
"Expression for Integers as Powers of Same Primes",
"Definition:Divisor (Algebra)/Integer",
"Definition:Greatest Common Divisor/Integers/General Definition"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"GCD from Prime Decomposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Principle of Mathematical Induction"
] |
proofwiki-15471 | LCM from Prime Decomposition/General Result | Let $n \in \N$ be a natural number such that $n \ge 2$.
Let $\N_n$ be defined as:
:$\N_n := \set {1, 2, \dotsc, n}$
Let $A_n = \set {a_1, a_2, \dotsc, a_n} \subseteq \Z$ be a set of $n$ integers.
From Expression for Integers as Powers of Same Primes, let:
:$\ds \forall i \in \N_n: a_i = \prod_{p_j \mathop \in T} {p_j}^... | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \map \lcm {A_n} = \prod_{j \mathop \in \N_r} {p_j}^{\max \set {e_{i j}: \, i \in \N_n} }$ | Let $n \in \N$ be a [[Definition:Natural Number|natural number]] such that $n \ge 2$.
Let $\N_n$ be defined as:
:$\N_n := \set {1, 2, \dotsc, n}$
Let $A_n = \set {a_1, a_2, \dotsc, a_n} \subseteq \Z$ be a [[Definition:Set|set]] of $n$ [[Definition:Integer|integers]].
From [[Expression for Integers as Powers of Same... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \map \lcm {A_n} = \prod_{j \mathop \in \N_r} {p_j}^{\max \set {e_{i j}: \, i \in \N_n} }$ | LCM from Prime Decomposition/General Result | https://proofwiki.org/wiki/LCM_from_Prime_Decomposition/General_Result | https://proofwiki.org/wiki/LCM_from_Prime_Decomposition/General_Result | [
"Lowest Common Multiple",
"Prime Decompositions"
] | [
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Integer",
"Expression for Integers as Powers of Same Primes",
"Definition:Divisor (Algebra)/Integer",
"Definition:Greatest Common Divisor/Integers/General Definition"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-15472 | LCM of 3 Integers in terms of GCDs of Pairs of those Integers | Let $a, b, c \in \Z_{>0}$ be strictly positive integers.
Then:
:$\lcm \set {a, b, c} = \dfrac {a b c \gcd \set {a, b, c} } {d_1 d_2 d_3}$
where:
:$\gcd$ denotes greatest common divisor
:$\lcm$ denotes lowest common multiple
:$d_1 = \gcd \set {a, b}$
:$d_2 = \gcd \set {b, c}$
:$d_3 = \gcd \set {a, c}$ | {{begin-eqn}}
{{eqn | l = \lcm \set {a, b, c}
| r = \lcm \set {a, \lcm \set {b, c} }
}}
{{eqn | r = \frac {a \lcm \set {b, c} } {\gcd \set {a, \lcm \set {b, c} } }
| c = Product of GCD and LCM
}}
{{eqn | r = \frac {a b c} {\gcd \set {b, c} } \paren {\frac 1 {\gcd \set {a, \lcm \set {b, c} } } }
| c = ... | Let $a, b, c \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|strictly positive integers]].
Then:
:$\lcm \set {a, b, c} = \dfrac {a b c \gcd \set {a, b, c} } {d_1 d_2 d_3}$
where:
:$\gcd$ denotes [[Definition:Greatest Common Divisor of Integers|greatest common divisor]]
:$\lcm$ denotes [[Definition:Lowest Comm... | {{begin-eqn}}
{{eqn | l = \lcm \set {a, b, c}
| r = \lcm \set {a, \lcm \set {b, c} }
}}
{{eqn | r = \frac {a \lcm \set {b, c} } {\gcd \set {a, \lcm \set {b, c} } }
| c = [[Product of GCD and LCM]]
}}
{{eqn | r = \frac {a b c} {\gcd \set {b, c} } \paren {\frac 1 {\gcd \set {a, \lcm \set {b, c} } } }
| ... | LCM of 3 Integers in terms of GCDs of Pairs of those Integers | https://proofwiki.org/wiki/LCM_of_3_Integers_in_terms_of_GCDs_of_Pairs_of_those_Integers | https://proofwiki.org/wiki/LCM_of_3_Integers_in_terms_of_GCDs_of_Pairs_of_those_Integers | [
"Lowest Common Multiple",
"Greatest Common Divisor",
"LCM of 3 Integers in terms of GCDs of Pairs of those Integers"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Greatest Common Divisor/Integers",
"Definition:Lowest Common Multiple/Integers"
] | [
"Product of GCD and LCM",
"Product of GCD and LCM",
"GCD and LCM Distribute Over Each Other",
"Product of GCD and LCM",
"LCM of 3 Integers in terms of GCDs of Pairs of those Integers/Lemma"
] |
proofwiki-15473 | Alternating Summation of Binomial Coefficient of Summation of Binomial Coefficient of Sequence | Let $\sequence a, \sequence b$ be real sequences which satisfy the condition:
:$a_n = \ds \sum_{r \mathop = 0}^n \binom n r b_r$
Then:
:$\ds \paren {-1}^n b_n = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s a_s$ | {{begin-eqn}}
{{eqn | l = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s a_s
| r = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s \paren {\sum_{r \mathop = 0}^s \binom s r b_r}
| c =
}}
{{eqn | r = \sum_{s \mathop = 0}^n \sum_{\substack {r \mathop \le 0 \mathop \le n \\ s \mathop \ge r} } \binom n s \bin... | Let $\sequence a, \sequence b$ be [[Definition:Real Sequence|real sequences]] which satisfy the condition:
:$a_n = \ds \sum_{r \mathop = 0}^n \binom n r b_r$
Then:
:$\ds \paren {-1}^n b_n = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s a_s$ | {{begin-eqn}}
{{eqn | l = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s a_s
| r = \sum_{s \mathop = 0}^n \binom n s \paren {-1}^s \paren {\sum_{r \mathop = 0}^s \binom s r b_r}
| c =
}}
{{eqn | r = \sum_{s \mathop = 0}^n \sum_{\substack {r \mathop \le 0 \mathop \le n \\ s \mathop \ge r} } \binom n s \bin... | Alternating Summation of Binomial Coefficient of Summation of Binomial Coefficient of Sequence | https://proofwiki.org/wiki/Alternating_Summation_of_Binomial_Coefficient_of_Summation_of_Binomial_Coefficient_of_Sequence | https://proofwiki.org/wiki/Alternating_Summation_of_Binomial_Coefficient_of_Summation_of_Binomial_Coefficient_of_Sequence | [
"Binomial Coefficients"
] | [
"Definition:Real Sequence"
] | [
"Binomial Theorem/Integral Index"
] |
proofwiki-15474 | Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient | Let $F_n$ denote the $n$th Fibonacci number.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_{2 n}
| r = \sum_{k \mathop = 1}^n \dbinom n k F_k
| c =
}}
{{eqn | r = \dbinom n 1 F_1 + \dbinom n 2 F_2 + \dbinom n 3 F_3 + \dotsb + \dbinom n {n - 1} F_{n - 1} + \dbinom n n F_n
| c =
... | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$F_{2 n} = \ds \sum_{k \mathop = 1}^n \dbinom n k F_k$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = F_0
| r = 0
| c =
}}
{{eqn | r = \sum_{k \mathop = 1}^0 \dbinom 0 k F_k
| c = vacuously
}}
{{end-eq... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]].
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_{2 n}
| r = \sum_{k \mathop = 1}^n \dbinom n k F_k
| c =
}}
{{eqn | r = \dbinom n 1 F_1 + \dbinom n 2 F_2 + \dbinom n 3 F_3 + \dotsb + \dbinom n {n - 1} F_{n -... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$F_{2 n} = \ds \sum_{k \mathop = 1}^n \dbinom n k F_k$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = F_0
| r = 0
| c =
}}
{{eqn | r = \... | Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient/Proof 1 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Product_of_Fibonacci_Number_with_Binomial_Coefficient | https://proofwiki.org/wiki/Sum_of_Sequence_of_Product_of_Fibonacci_Number_with_Binomial_Coefficient/Proof_1 | [
"Fibonacci Numbers",
"Binomial Coefficients",
"Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient"
] | [
"Definition:Fibonacci Number",
"Definition:Binomial Coefficient"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Summation/Vacuous Summation",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Pascal's Rule",
"Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient",
... |
proofwiki-15475 | Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient | Let $F_n$ denote the $n$th Fibonacci number.
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_{2 n}
| r = \sum_{k \mathop = 1}^n \dbinom n k F_k
| c =
}}
{{eqn | r = \dbinom n 1 F_1 + \dbinom n 2 F_2 + \dbinom n 3 F_3 + \dotsb + \dbinom n {n - 1} F_{n - 1} + \dbinom n n F_n
| c =
... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \dbinom n k F_k
| r = \sum_{k \mathop = 1}^n \dbinom n k \paren {\frac {\phi^k - \hat \phi^k} {\sqrt 5} }
| c = Euler-Binet Formula
}}
{{eqn | r = \frac 1 {\sqrt 5} \paren {\sum_{k \mathop = 1}^n \dbinom n k \phi^k - \sum_{k \mathop = 1}^n \dbinom n k \hat \p... | Let $F_n$ denote the $n$th [[Definition:Fibonacci Numbers|Fibonacci number]].
Then:
{{begin-eqn}}
{{eqn | q = \forall n \in \Z_{>0}
| l = F_{2 n}
| r = \sum_{k \mathop = 1}^n \dbinom n k F_k
| c =
}}
{{eqn | r = \dbinom n 1 F_1 + \dbinom n 2 F_2 + \dbinom n 3 F_3 + \dotsb + \dbinom n {n - 1} F_{n -... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n \dbinom n k F_k
| r = \sum_{k \mathop = 1}^n \dbinom n k \paren {\frac {\phi^k - \hat \phi^k} {\sqrt 5} }
| c = [[Euler-Binet Formula]]
}}
{{eqn | r = \frac 1 {\sqrt 5} \paren {\sum_{k \mathop = 1}^n \dbinom n k \phi^k - \sum_{k \mathop = 1}^n \dbinom n k \ha... | Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient/Proof 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Product_of_Fibonacci_Number_with_Binomial_Coefficient | https://proofwiki.org/wiki/Sum_of_Sequence_of_Product_of_Fibonacci_Number_with_Binomial_Coefficient/Proof_2 | [
"Fibonacci Numbers",
"Binomial Coefficients",
"Sum of Sequence of Product of Fibonacci Number with Binomial Coefficient"
] | [
"Definition:Fibonacci Number",
"Definition:Binomial Coefficient"
] | [
"Euler-Binet Formula",
"Summation is Linear",
"Binomial Coefficient with Zero",
"Binomial Theorem",
"Golden Mean as Root of Quadratic",
"Euler-Binet Formula"
] |
proofwiki-15476 | Integer and Fifth Power have same Last Digit | Let $n \in \Z$ be an integer.
Then $n^5$ has the same last digit as $n$ when both are expressed in conventional decimal notation. | It suffices to check $10$ digit cases:
{{explain|what does the above line mean?}}
{{begin-eqn}}
{{eqn | l = 0^5 = 0 \times 0 \times 0 \times 0 \times 0
| r = 0
| rr= \equiv 0 \pmod {10}
| c =
}}
{{eqn | l = 1^5 = 1 \times 1 \times 1 \times 1 \times 1
| r = 1
| rr= \equiv 1 \pmod {10}
... | Let $n \in \Z$ be an [[Definition:Integer|integer]].
Then $n^5$ has the same last [[Definition:Digit|digit]] as $n$ when both are expressed in conventional [[Definition:Decimal Notation|decimal notation]]. | It suffices to check $10$ digit cases:
{{explain|what does the above line mean?}}
{{begin-eqn}}
{{eqn | l = 0^5 = 0 \times 0 \times 0 \times 0 \times 0
| r = 0
| rr= \equiv 0 \pmod {10}
| c =
}}
{{eqn | l = 1^5 = 1 \times 1 \times 1 \times 1 \times 1
| r = 1
| rr= \equiv 1 \pmod {10}
... | Integer and Fifth Power have same Last Digit/Proof 2 | https://proofwiki.org/wiki/Integer_and_Fifth_Power_have_same_Last_Digit | https://proofwiki.org/wiki/Integer_and_Fifth_Power_have_same_Last_Digit/Proof_2 | [
"Integer and Fifth Power have same Last Digit",
"Fifth Powers"
] | [
"Definition:Integer",
"Definition:Digit",
"Definition:Decimal Notation"
] | [] |
proofwiki-15477 | Sufficient Condition for 5 to divide n^2+1 | Let:
{{begin-eqn}}
{{eqn | l = 5
| o = \nmid
| r = n - 1
}}
{{eqn | l = 5
| o = \nmid
| r = n
}}
{{eqn | l = 5
| o = \nmid
| r = n + 1
}}
{{end-eqn}}
where $\nmid$ denotes non-divisibility.
Then:
:$5 \divides n^2 + 1$
where $\divides$ denotes divisibility. | We have that:
{{begin-eqn}}
{{eqn | l = 5
| o = \nmid
| r = n - 1
}}
{{eqn | ll= \leadsto
| l = n - 1
| o = \not \equiv
| r = 0
| rr= \pmod 5
}}
{{eqn | ll= \leadsto
| l = n
| o = \not \equiv
| r = 1
| rr= \pmod 5
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 5
... | Let:
{{begin-eqn}}
{{eqn | l = 5
| o = \nmid
| r = n - 1
}}
{{eqn | l = 5
| o = \nmid
| r = n
}}
{{eqn | l = 5
| o = \nmid
| r = n + 1
}}
{{end-eqn}}
where $\nmid$ denotes non-[[Definition:Divisor of Integer|divisibility]].
Then:
:$5 \divides n^2 + 1$
where $\divides$ denotes [[De... | We have that:
{{begin-eqn}}
{{eqn | l = 5
| o = \nmid
| r = n - 1
}}
{{eqn | ll= \leadsto
| l = n - 1
| o = \not \equiv
| r = 0
| rr= \pmod 5
}}
{{eqn | ll= \leadsto
| l = n
| o = \not \equiv
| r = 1
| rr= \pmod 5
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 5
... | Sufficient Condition for 5 to divide n^2+1 | https://proofwiki.org/wiki/Sufficient_Condition_for_5_to_divide_n^2+1 | https://proofwiki.org/wiki/Sufficient_Condition_for_5_to_divide_n^2+1 | [
"Modulo Arithmetic"
] | [
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [] |
proofwiki-15478 | Wilson's Theorem/Necessary Condition | Let $p$ be a prime number.
Then:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an odd prime.
Let $p$ be prime.
Consider $n \in \Z, 1 \le n < p$.
As $p$ is prime, $n \perp p$.
From Law of Inverses (Modulo Arithmetic), we have:
:$\exists n' \in \Z, 1 \le n' < p: n n' \equiv 1 \pmod p$
By Solution of Linear Congruence, for each $n$ th... | Let $p$ be a [[Definition:Prime Number|prime number]].
Then:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an [[Definition:Odd Prime|odd prime]].
Let $p$ be [[Definition:Prime Number|prime]].
Consider $n \in \Z, 1 \le n < p$.
As $p$ is [[Definition:Prime Number|prime]], $n \perp p$.
From [[Law of Inverses (Modulo Arithmetic)]], we have:
:$\exists n' \i... | Wilson's Theorem/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition/Proof_1 | [
"Wilson's Theorem"
] | [
"Definition:Prime Number"
] | [
"Definition:Odd Prime",
"Definition:Prime Number",
"Definition:Prime Number",
"Law of Inverses (Modulo Arithmetic)",
"Solution of Linear Congruence",
"Difference of Two Squares",
"Definition:Odd Prime",
"Negative Number is Congruent to Modulus minus Number",
"Definition:Congruence (Number Theory)/In... |
proofwiki-15479 | Wilson's Theorem/Necessary Condition | Let $p$ be a prime number.
Then:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an odd prime.
Consider $p$ points on the circumference of a circle $C$ dividing it into $p$ equal arcs.
By joining these points in a cycle, we can create polygons, some of them stellated.
From Number of Different n-gons that can be Inscribed in Circle, t... | Let $p$ be a [[Definition:Prime Number|prime number]].
Then:
:$\paren {p - 1}! \equiv -1 \pmod p$ | If $p = 2$ the result is obvious.
Therefore we assume that $p$ is an [[Definition:Odd Prime|odd prime]].
Consider $p$ [[Definition:Point|points]] on the [[Definition:Circumference of Circle|circumference]] of a [[Definition:Circle|circle]] $C$ dividing it into $p$ equal [[Definition:Arc of Circle|arcs]].
By joining... | Wilson's Theorem/Necessary Condition/Proof 2 | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition/Proof_2 | [
"Wilson's Theorem"
] | [
"Definition:Prime Number"
] | [
"Definition:Odd Prime",
"Definition:Point",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Circle/Arc",
"Definition:Point",
"Definition:Cyclic Permutation",
"Definition:Polygon",
"Definition:Stellation/Polygon",
"Number of Different n-gons that can be Inscribed in Circle",
"... |
proofwiki-15480 | Wilson's Theorem/Necessary Condition | Let $p$ be a prime number.
Then:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Let $p$ be prime.
Consider $\struct {\Z_p, +, \times}$, the ring of integers modulo $m$.
From Ring of Integers Modulo Prime is Field, $\struct {\Z_p, +, \times}$ is a field.
Hence, apart from $\eqclass 0 p$, all elements of $\struct {\Z_p, +, \times}$ are units
As $\struct {\Z_p, +, \times}$ is a field, it is also by d... | Let $p$ be a [[Definition:Prime Number|prime number]].
Then:
:$\paren {p - 1}! \equiv -1 \pmod p$ | Let $p$ be [[Definition:Prime Number|prime]].
Consider $\struct {\Z_p, +, \times}$, the [[Definition:Ring of Integers Modulo m|ring of integers modulo $m$]].
From [[Ring of Integers Modulo Prime is Field]], $\struct {\Z_p, +, \times}$ is a [[Definition:Field (Abstract Algebra)|field]].
Hence, apart from $\eqclass 0 ... | Wilson's Theorem/Necessary Condition/Proof 3 | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition | https://proofwiki.org/wiki/Wilson's_Theorem/Necessary_Condition/Proof_3 | [
"Wilson's Theorem"
] | [
"Definition:Prime Number"
] | [
"Definition:Prime Number",
"Definition:Ring of Integers Modulo m",
"Ring of Integers Modulo Prime is Field",
"Definition:Field (Abstract Algebra)",
"Definition:Element",
"Definition:Unit of Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Integral Domain",
"Product of Units of Integral Doma... |
proofwiki-15481 | Wilson's Theorem/Sufficient Condition | Let $p$ be a (strictly) positive integer such that:
:$\paren {p - 1}! \equiv -1 \pmod p$
Then $p$ is a prime number. | Assume $p$ is composite, and $q$ is a prime such that $q \divides p$.
Then both $p$ and $\paren {p - 1}!$ are divisible by $q$.
{{AimForCont}} the congruence $\paren {p - 1}! \equiv -1 \pmod p$ were satisfied.
From Congruence by Divisor of Modulus:
:$\paren {p - 1}! \equiv -1 \pmod q$
However, this amounts to:
:$0 \equ... | Let $p$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]] such that:
:$\paren {p - 1}! \equiv -1 \pmod p$
Then $p$ is a [[Definition:Prime Number|prime number]]. | Assume $p$ is [[Definition:Composite Number|composite]], and $q$ is a [[Definition:Prime Number|prime]] such that $q \divides p$.
Then both $p$ and $\paren {p - 1}!$ are [[Definition:Divisor of Integer|divisible]] by $q$.
{{AimForCont}} the [[Definition:Congruence Modulo Integer|congruence]] $\paren {p - 1}! \equiv -... | Wilson's Theorem/Sufficient Condition | https://proofwiki.org/wiki/Wilson's_Theorem/Sufficient_Condition | https://proofwiki.org/wiki/Wilson's_Theorem/Sufficient_Condition | [
"Wilson's Theorem"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Prime Number"
] | [
"Definition:Composite Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Congruence (Number Theory)/Integers",
"Congruence by Divisor of Modulus/Integer Modulus",
"Definition:Contradiction",
"Definition:Composite Number",
"Definition:Congruence (Number Theory)/Inte... |
proofwiki-15482 | Necessary Condition for Integral Functional to have Extremum/Two Variables/Lemma | Let $D \subset \R^2$.
Let $\Gamma$ be the boundary of $D$.
Let $\alpha : D \to \R$ be a continuous mapping.
Let $h : D \to \R$ be a twice differentiable mapping such that $\map h \Gamma = 0$.
Suppose for every $h$ we have that:
:$\ds \iint_D \map \alpha {x, y} \map h {x,y} \rd x \rd y = 0$.
Then:
:$\ds \forall x, y \in... | {{AimForCont}} that:
:$\ds \exists x_0,y_0 \in D : \map \alpha {x_0,y_0} > 0$
$\alpha$ is continuous in $D$.
Therefore, there exists a closed ball $B^-_{\epsilon}$ defined by:
:$\map {B^-_{\epsilon}} {x_0, y_0} := \set {\tuple{x,y} \in D : \paren {x - x_0}^2 + \paren {y - y_0}^2 \le \epsilon^2}$
such that:
:$\forall x... | Let $D \subset \R^2$.
Let $\Gamma$ be the [[Definition:Boundary (Geometry)|boundary]] of $D$.
Let $\alpha : D \to \R$ be a [[Definition:Continuous Real Function on Subset|continuous mapping]].
Let $h : D \to \R$ be a twice [[Definition:Differentiability Class|differentiable mapping]] such that $\map h \Gamma = 0$.
... | {{AimForCont}} that:
:$\ds \exists x_0,y_0 \in D : \map \alpha {x_0,y_0} > 0$
$\alpha$ is [[Definition:Continuous Real Function on Interval|continuous]] in $D$.
Therefore, there exists a [[Definition:Closed Ball|closed ball]] $B^-_{\epsilon}$ defined by:
:$\map {B^-_{\epsilon}} {x_0, y_0} := \set {\tuple{x,y} \in ... | Necessary Condition for Integral Functional to have Extremum/Two Variables/Lemma | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum/Two_Variables/Lemma | https://proofwiki.org/wiki/Necessary_Condition_for_Integral_Functional_to_have_Extremum/Two_Variables/Lemma | [
"Calculus of Variations"
] | [
"Definition:Boundary (Geometry)",
"Definition:Continuous Real Function/Subset",
"Definition:Differentiability Class"
] | [
"Definition:Continuous Real Function/Interval",
"Definition:Closed Ball",
"Definition:Lemma",
"Definition:Positive",
"Definition:Closed Ball",
"Definition:Definite Integral",
"Definition:Positive",
"Definition:Contradiction",
"Definition:Lemma"
] |
proofwiki-15483 | Number of Different n-gons that can be Inscribed in Circle | Let $C$ be a circle on whose circumference $n$ points are placed which divide $C$ into $n$ equal arcs.
The number of different $n$-gons (either stellated or otherwise) that can be described on $C$ whose vertices are those $n$ points is:
:$S_n = \dfrac {\paren {n - 1}!} 2$ | An $n$-gon (either stellated or otherwise) is constructed by connecting all $n$ points in some order.
From Number of Permutations, there are ${}^n P_n = n!$ ways of ordering $n$ points.
However, for each $n$-gon formed in this way, we have:
:$n$ ways of choosing the first vertex
:$2$ different sides from that vertex to... | Let $C$ be a [[Definition:Circle|circle]] on whose [[Definition:Circumference of Circle|circumference]] $n$ [[Definition:Point|points]] are placed which divide $C$ into $n$ equal [[Definition:Arc of Circle|arcs]].
The number of different [[Definition:Polygon|$n$-gons]] (either [[Definition:Stellated Polygon|stellated]... | An [[Definition:Polygon|$n$-gon]] (either [[Definition:Stellated Polygon|stellated]] or otherwise) is constructed by connecting all $n$ [[Definition:Point|points]] in some order.
From [[Number of Permutations]], there are ${}^n P_n = n!$ ways of ordering $n$ [[Definition:Point|points]].
However, for each [[Definition... | Number of Different n-gons that can be Inscribed in Circle | https://proofwiki.org/wiki/Number_of_Different_n-gons_that_can_be_Inscribed_in_Circle | https://proofwiki.org/wiki/Number_of_Different_n-gons_that_can_be_Inscribed_in_Circle | [
"Number of Different n-gons that can be Inscribed in Circle",
"Stellated Polygons",
"Polygons",
"Circles",
"Combinatorics"
] | [
"Definition:Circle",
"Definition:Circle/Circumference",
"Definition:Point",
"Definition:Circle/Arc",
"Definition:Polygon",
"Definition:Stellation/Polygon",
"Definition:Polygon/Vertex"
] | [
"Definition:Polygon",
"Definition:Stellation/Polygon",
"Definition:Point",
"Number of Permutations",
"Definition:Point",
"Definition:Polygon",
"Definition:Polygon/Vertex",
"Definition:Polygon/Side",
"Definition:Polygon/Vertex",
"Definition:Clockwise",
"Definition:Anticlockwise",
"Definition:Po... |
proofwiki-15484 | Number of Regular Stellated Odd n-gons | Let $n \in \Z_{>0}$ be a strictly positive odd integer.
Then there are $\dfrac {n - 1} 2$ distinct regular stellated $n$-gons. | Let $P$ be a regular stellated $n$-gons.
Let the $n$ vertices of $P$ be $p_1, p_2, \dotsc, p_n$.
These will be arranged on the circumference of a circle $C$, dividing $C$ into $n$ arcs of equal length.
Once we have chosen the first side of $P$, the others are all the same length and are completely determined by that fi... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive]] [[Definition:Odd Integer|odd integer]].
Then there are $\dfrac {n - 1} 2$ [[Definition:Distinct Elements|distinct]] [[Definition:Regular Stellated Polygon|regular stellated $n$-gons]]. | Let $P$ be a [[Definition:Regular Stellated Polygon|regular stellated $n$-gons]].
Let the $n$ [[Definition:Vertex of Polygon|vertices]] of $P$ be $p_1, p_2, \dotsc, p_n$.
These will be arranged on the [[Definition:Circumference of Circle|circumference]] of a [[Definition:Circle|circle]] $C$, dividing $C$ into $n$ [[D... | Number of Regular Stellated Odd n-gons | https://proofwiki.org/wiki/Number_of_Regular_Stellated_Odd_n-gons | https://proofwiki.org/wiki/Number_of_Regular_Stellated_Odd_n-gons | [
"Regular Stellated Polygons"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Odd Integer",
"Definition:Distinct/Plural",
"Definition:Stellation/Polygon/Regular"
] | [
"Definition:Stellation/Polygon/Regular",
"Definition:Polygon/Vertex",
"Definition:Circle/Circumference",
"Definition:Circle",
"Definition:Circle/Arc",
"Definition:Arc Length",
"Definition:Polygon/Side",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Polygon/Vertex",
"... |
proofwiki-15485 | Wilson's Theorem/Corollary 1 | Let $p$ be a prime number.
Then $p$ is the smallest prime number which divides $\paren {p - 1}! + 1$. | From Wilson's Theorem, $p$ divides $\paren {p - 1}! + 1$.
Let $q$ be a prime number less than $p$.
Then $q$ is a divisor of $\paren {p - 1}!$ and so does not divide $\paren {p - 1}! + 1$.
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Then $p$ is the smallest [[Definition:Prime Number|prime number]] which [[Definition:Divisor of Integer|divides]] $\paren {p - 1}! + 1$. | From [[Wilson's Theorem]], $p$ [[Definition:Divisor of Integer|divides]] $\paren {p - 1}! + 1$.
Let $q$ be a [[Definition:Prime Number|prime number]] less than $p$.
Then $q$ is a [[Definition:Divisor of Integer|divisor]] of $\paren {p - 1}!$ and so does not [[Definition:Divisor of Integer|divide]] $\paren {p - 1}! + ... | Wilson's Theorem/Corollary 1 | https://proofwiki.org/wiki/Wilson's_Theorem/Corollary_1 | https://proofwiki.org/wiki/Wilson's_Theorem/Corollary_1 | [
"Wilson's Theorem"
] | [
"Definition:Prime Number",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer"
] | [
"Wilson's Theorem",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-15486 | Generating Function for Natural Numbers/Corollary | Let $\sequence {a_n}$ be the sequence defined as:
:$\forall n \in \N_{> 0}: a_n = n - 1$
That is:
:$\sequence {a_n} = 1, 2, 3, 4, \ldots$
Then the generating function for $\sequence {a_n}$ is given as:
:$H \paren z = \dfrac z {\paren {1 - z}^2}$ | From Generating Function for Natural Numbers:
:$\sequence {a_n} = 0, 1, 2, 3, 4, \ldots$
has the generating function:
:$G \paren z = \dfrac 1 {\paren {1 - z}^2}$
Then by Generating Function by Power of Parameter:
:$z G \paren z = \dfrac z {\paren {1 - z}^2}$
is the generating function for the sequence defined as:
:$\fo... | Let $\sequence {a_n}$ be the [[Definition:Sequence|sequence]] defined as:
:$\forall n \in \N_{> 0}: a_n = n - 1$
That is:
:$\sequence {a_n} = 1, 2, 3, 4, \ldots$
Then the [[Definition:Generating Function|generating function]] for $\sequence {a_n}$ is given as:
:$H \paren z = \dfrac z {\paren {1 - z}^2}$ | From [[Generating Function for Natural Numbers]]:
:$\sequence {a_n} = 0, 1, 2, 3, 4, \ldots$
has the [[Definition:Generating Function|generating function]]:
:$G \paren z = \dfrac 1 {\paren {1 - z}^2}$
Then by [[Generating Function by Power of Parameter]]:
:$z G \paren z = \dfrac z {\paren {1 - z}^2}$
is the [[Definit... | Generating Function for Natural Numbers/Corollary | https://proofwiki.org/wiki/Generating_Function_for_Natural_Numbers/Corollary | https://proofwiki.org/wiki/Generating_Function_for_Natural_Numbers/Corollary | [
"Examples of Generating Functions",
"Natural Numbers"
] | [
"Definition:Sequence",
"Definition:Generating Function"
] | [
"Generating Function for Natural Numbers",
"Definition:Generating Function",
"Generating Function by Power of Parameter",
"Definition:Generating Function",
"Definition:Sequence"
] |
proofwiki-15487 | Generating Function for Triangular Numbers/Corollary | Let $\sequence {b_n}$ be the sequence defined as:
:$\forall n \in \N_{> 0}: b_n = \dfrac {\paren {n + 1} \paren {n + 2} } 2$
That is:
:$\sequence {b_n}_{n \mathop \ge 0} = 1, 3, 6, 10, \ldots, \dbinom {n + 2} 2, \ldots$
Then the generating function for $\sequence {b_n}$ is given as:
:$H \paren z = \dfrac 1 {\paren {1 -... | From Generating Function for Triangular Numbers:
:$H \paren z = \dfrac z {\paren {1 - z}^3}$
is the generating function for $\sequence {a_n}$ as given by:
:$\sequence {a_n} = 1, 3, 6, 10, \ldots, \dbinom {n + 1} 2, \ldots$
that is, such that:
:$a_1 = 1, a_2 = 3, a_3 = 6, \ldots$
The result follows from Generating Funct... | Let $\sequence {b_n}$ be the [[Definition:Sequence|sequence]] defined as:
:$\forall n \in \N_{> 0}: b_n = \dfrac {\paren {n + 1} \paren {n + 2} } 2$
That is:
:$\sequence {b_n}_{n \mathop \ge 0} = 1, 3, 6, 10, \ldots, \dbinom {n + 2} 2, \ldots$
Then the [[Definition:Generating Function|generating function]] for $\sequ... | From [[Generating Function for Triangular Numbers]]:
:$H \paren z = \dfrac z {\paren {1 - z}^3}$
is the [[Definition:Generating Function|generating function]] for $\sequence {a_n}$ as given by:
:$\sequence {a_n} = 1, 3, 6, 10, \ldots, \dbinom {n + 1} 2, \ldots$
that is, such that:
:$a_1 = 1, a_2 = 3, a_3 = 6, \ldots$... | Generating Function for Triangular Numbers/Corollary | https://proofwiki.org/wiki/Generating_Function_for_Triangular_Numbers/Corollary | https://proofwiki.org/wiki/Generating_Function_for_Triangular_Numbers/Corollary | [
"Examples of Generating Functions"
] | [
"Definition:Sequence",
"Definition:Generating Function"
] | [
"Generating Function for Triangular Numbers",
"Definition:Generating Function",
"Generating Function Divided by Power of Parameter"
] |
proofwiki-15488 | Square Modulo n Congruent to Square of Inverse Modulo n | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
:$a^2 \equiv \paren {n - a}^2 \pmod n$
where the notation denotes congruence modulo $n$. | {{begin-eqn}}
{{eqn | l = \paren {n - a}^2
| r = n^2 - 2 n - a^2
| c =
}}
{{eqn | o = \equiv
| r = a^2
| rr= \pmod n
| c =
}}
{{end-eqn}}
{{qed}}
Category:Modulo Arithmetic
oaexkl44pkr2mnoa7jpv1g88idzc81s | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Then:
:$a^2 \equiv \paren {n - a}^2 \pmod n$
where the notation denotes [[Definition:Congruence Modulo Integer|congruence modulo $n$]]. | {{begin-eqn}}
{{eqn | l = \paren {n - a}^2
| r = n^2 - 2 n - a^2
| c =
}}
{{eqn | o = \equiv
| r = a^2
| rr= \pmod n
| c =
}}
{{end-eqn}}
{{qed}}
[[Category:Modulo Arithmetic]]
oaexkl44pkr2mnoa7jpv1g88idzc81s | Square Modulo n Congruent to Square of Inverse Modulo n | https://proofwiki.org/wiki/Square_Modulo_n_Congruent_to_Square_of_Inverse_Modulo_n | https://proofwiki.org/wiki/Square_Modulo_n_Congruent_to_Square_of_Inverse_Modulo_n | [
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Congruence (Number Theory)/Integers"
] | [
"Category:Modulo Arithmetic"
] |
proofwiki-15489 | Partition of Integer into Powers of 2 for Consecutive Integers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map b n$ denote the number of ways $n$ can be partitioned into (integer) powers of $2$.
Then:
:$\map b {2 n} = \map b {2 n + 1}$ | We prove the theorem by establishing a bijection between the set of partitions of $2 n$ with that of $2 n + 1$, under the constraint where each partition is an integer power of $2$.
We have that $2^k$ is even for all $k > 0$.
Also, we have that $2 n + 1$ is odd for all $n$.
So, for each partition of $2 n + 1$ into inte... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\map b n$ denote the number of ways $n$ can be [[Definition:Integer Partition|partitioned]] into [[Definition:Integer Power|(integer) powers of $2$]].
Then:
:$\map b {2 n} = \map b {2 n + 1}$ | We prove the theorem by establishing a [[Definition:Bijection|bijection]] between the set of [[Definition:Integer Partition|partitions]] of $2 n$ with that of $2 n + 1$, under the constraint where each [[Definition:Integer Partition|partition]] is an [[Definition:Integer Power|integer power of $2$]].
We have that $2^... | Partition of Integer into Powers of 2 for Consecutive Integers | https://proofwiki.org/wiki/Partition_of_Integer_into_Powers_of_2_for_Consecutive_Integers | https://proofwiki.org/wiki/Partition_of_Integer_into_Powers_of_2_for_Consecutive_Integers | [
"Partition Theory"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Integer Partition",
"Definition:Power (Algebra)/Integer"
] | [
"Definition:Bijection",
"Definition:Integer Partition",
"Definition:Integer Partition",
"Definition:Power (Algebra)/Integer",
"Definition:Even Integer",
"Definition:Odd Integer",
"Definition:Integer Partition",
"Definition:Power (Algebra)/Integer",
"Definition:Integer Partition/Part",
"Definition:... |
proofwiki-15490 | Number of Partitions with no Multiple of 3 equals Number of Partitions where Parts appear No More than Twice | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\map t n$ denote the number of ways $n$ can be partitioned into parts which are specifically not multiples of $3$.
Let $\map v n$ denote the number of ways $n$ can be partitioned such that no part appears twice.
Then:
:$\forall n \in \Z_{>0}: \map t n = \map v ... | {{ProofWanted|Chapter $12$ of {{BookReference|Number Theory|1971|George E. Andrews}} }} | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\map t n$ denote the number of ways $n$ can be [[Definition:Integer Partition|partitioned]] into [[Definition:Part of Integer Partition|parts]] which are specifically not [[Definition:Integer Multiplication|multiples]] ... | {{ProofWanted|Chapter $12$ of {{BookReference|Number Theory|1971|George E. Andrews}} }} | Number of Partitions with no Multiple of 3 equals Number of Partitions where Parts appear No More than Twice | https://proofwiki.org/wiki/Number_of_Partitions_with_no_Multiple_of_3_equals_Number_of_Partitions_where_Parts_appear_No_More_than_Twice | https://proofwiki.org/wiki/Number_of_Partitions_with_no_Multiple_of_3_equals_Number_of_Partitions_where_Parts_appear_No_More_than_Twice | [
"Examples of Integer Partitions"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Integer Partition",
"Definition:Integer Partition/Part",
"Definition:Multiplication/Integers",
"Definition:Integer Partition",
"Definition:Integer Partition/Part"
] | [] |
proofwiki-15491 | Congruent Numbers are not necessarily Equal | Let $x, y, z \in \R$ be real numbers such that:
:$x \equiv y \pmod z$
where $x \equiv y \pmod z$ denotes congruence modulo $z$.
Then it is not necessarily the case that $x = y$. | Proof by Counterexample:
We have that:
:$11 - 5 = 6 = 3 \times 2$
and so by definition of congruence modulo $2$:
:$10 \equiv 4 \pmod 2$
But $11 \ne 5$.
{{qed}} | Let $x, y, z \in \R$ be [[Definition:Real Number|real numbers]] such that:
:$x \equiv y \pmod z$
where $x \equiv y \pmod z$ denotes [[Definition:Congruence (Number Theory)|congruence modulo $z$]].
Then it is not necessarily the case that $x = y$. | [[Proof by Counterexample]]:
We have that:
:$11 - 5 = 6 = 3 \times 2$
and so by definition of [[Definition:Congruence (Number Theory)|congruence modulo $2$]]:
:$10 \equiv 4 \pmod 2$
But $11 \ne 5$.
{{qed}} | Congruent Numbers are not necessarily Equal | https://proofwiki.org/wiki/Congruent_Numbers_are_not_necessarily_Equal | https://proofwiki.org/wiki/Congruent_Numbers_are_not_necessarily_Equal | [
"Modulo Arithmetic"
] | [
"Definition:Real Number",
"Definition:Congruence (Number Theory)"
] | [
"Proof by Counterexample",
"Definition:Congruence (Number Theory)"
] |
proofwiki-15492 | Congruence Modulo Negative Number | Let $a, b, c \in \R$ be real numbers.
Then:
:$a \equiv b \pmod c \iff a \equiv b \pmod {-c}$ | {{begin-eqn}}
{{eqn | l = a
| o = \equiv
| r = b
| rr= \pmod c
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {a - b}
| r = k c
| c = {{Defof|Congruence (Number Theory)|Congruence Modulo $c$}}: for some $k \in \Z$
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {a - b}
... | Let $a, b, c \in \R$ be [[Definition:Real Number|real numbers]].
Then:
:$a \equiv b \pmod c \iff a \equiv b \pmod {-c}$ | {{begin-eqn}}
{{eqn | l = a
| o = \equiv
| r = b
| rr= \pmod c
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {a - b}
| r = k c
| c = {{Defof|Congruence (Number Theory)|Congruence Modulo $c$}}: for some $k \in \Z$
}}
{{eqn | ll= \leadstoandfrom
| l = \paren {a - b}
... | Congruence Modulo Negative Number | https://proofwiki.org/wiki/Congruence_Modulo_Negative_Number | https://proofwiki.org/wiki/Congruence_Modulo_Negative_Number | [
"Modulo Arithmetic"
] | [
"Definition:Real Number"
] | [] |
proofwiki-15493 | P-adic Norm and Absolute Value are Not Equivalent | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.
Let $\size{\,\cdot\,}$ be the absolute value on the rationals $\Q$.
Then $\norm {\,\cdot\,}_p$ and $\size{\,\cdot\,}$ are not equivalent norms.
That is, the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topolo... | By definition of the $p$-adic norm:
:$\norm p_p = \dfrac 1 p < 1$
By definition of the absolute value:
:$\size p = p > 1$
By definition of open unit ball equivalence, $\norm {\,\cdot\,}_p$ and $\size {\,\cdot\,}$ are not equivalent norms.
By Equivalence of Definitions of Equivalent Division Ring Norms and the definitio... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime number]] $p$.
Let $\size{\,\cdot\,}$ be the [[Definition:Absolute Value|absolute value]] on the [[Definition:Rational Numbers|rationals $\Q$]].
Then... | By definition of the [[Definition:P-adic Norm|$p$-adic norm]]:
:$\norm p_p = \dfrac 1 p < 1$
By definition of the [[Definition:Absolute Value|absolute value]]:
:$\size p = p > 1$
By definition of [[Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent|open unit ball equivalence]], $\norm {\,\cdot\,}_p$ ... | P-adic Norm and Absolute Value are Not Equivalent/Proof 1 | https://proofwiki.org/wiki/P-adic_Norm_and_Absolute_Value_are_Not_Equivalent | https://proofwiki.org/wiki/P-adic_Norm_and_Absolute_Value_are_Not_Equivalent/Proof_1 | [
"Norm Theory",
"P-adic Number Theory",
"P-adic Norm and Absolute Value are Not Equivalent"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Absolute Value",
"Definition:Rational Number",
"Definition:Equivalent Division Ring Norms",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Topology Induced by Division Ring Norm"
] | [
"Definition:P-adic Norm",
"Definition:Absolute Value",
"Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent",
"Definition:Equivalent Division Ring Norms",
"Equivalence of Definitions of Equivalent Division Ring Norms",
"Definition:Equivalent Division Ring Norms/Topologically Equivalent",
... |
proofwiki-15494 | P-adic Norm and Absolute Value are Not Equivalent | Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime number $p$.
Let $\size{\,\cdot\,}$ be the absolute value on the rationals $\Q$.
Then $\norm {\,\cdot\,}_p$ and $\size{\,\cdot\,}$ are not equivalent norms.
That is, the topology induced by $\norm {\,\cdot\,}_p$ does not equal the topolo... | It is noted that:
:$\sup \set {\size n: n \in \Z} = +\infty$
By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|3}}, $\size {\,\cdot\,}$ is Archimedean.
From P-adic Norm on Rational Numbers is Non-Archimedean Norm, $\norm {\,\cdot\,}_p$ is non-Archimedean.
By Equivalent Norms are both Non-Archimedea... | Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals $\Q$]] for some [[Definition:Prime Number|prime number]] $p$.
Let $\size{\,\cdot\,}$ be the [[Definition:Absolute Value|absolute value]] on the [[Definition:Rational Numbers|rationals $\Q$]].
Then... | It is noted that:
:$\sup \set {\size n: n \in \Z} = +\infty$
By {{Corollary|Characterisation of Non-Archimedean Division Ring Norms|3}}, $\size {\,\cdot\,}$ is [[Definition:Archimedean Division Ring Norm|Archimedean]].
From [[P-adic Norm on Rational Numbers is Non-Archimedean Norm]], $\norm {\,\cdot\,}_p$ is [[Defini... | P-adic Norm and Absolute Value are Not Equivalent/Proof 2 | https://proofwiki.org/wiki/P-adic_Norm_and_Absolute_Value_are_Not_Equivalent | https://proofwiki.org/wiki/P-adic_Norm_and_Absolute_Value_are_Not_Equivalent/Proof_2 | [
"Norm Theory",
"P-adic Number Theory",
"P-adic Norm and Absolute Value are Not Equivalent"
] | [
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Prime Number",
"Definition:Absolute Value",
"Definition:Rational Number",
"Definition:Equivalent Division Ring Norms",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Topology Induced by Division Ring Norm"
] | [
"Definition:Non-Archimedean/Norm (Division Ring)/Archimedean",
"P-adic Norm forms Non-Archimedean Valued Field/Rational Numbers",
"Definition:Non-Archimedean/Norm (Division Ring)",
"Equivalent Norms are both Non-Archimedean or both Archimedean",
"Definition:Equivalent Division Ring Norms"
] |
proofwiki-15495 | P-adic Norms are Not Equivalent | Let $p_1$ and $p_2$ be prime numbers such that $p_1 \ne p_2$.
Let $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ be the $p$-adic norms on the rationals $\Q$.
Then $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ are not equivalent norms.
That is, the topology induced by $\norm {\,\cdot\,}_{p_1}$ does no... | Consider $p_1/p_2 \in \Q$.
With $\norm {\,\cdot\,}_{p_1}$:
{{begin-eqn}}
{{eqn | l = \norm {p_1/p_2}_{p_1}
| r = \norm {p_1}_{p_1} \norm {1/p_2}_{p_1}
| c = {{NormAxiomNonArch|2}}
}}
{{eqn | r = \norm {p_1}_{p_1} \times 1
| c = $p_1$ does not divide $p_2$
}}
{{eqn | r = 1 / {p_1}
}}
{{eqn | o = \lt
... | Let $p_1$ and $p_2$ be [[Definition:Prime Number|prime numbers]] such that $p_1 \ne p_2$.
Let $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ be the [[Definition:P-adic Norm|$p$-adic norms]] on the [[Definition:Rational Numbers|rationals $\Q$]].
Then $\norm {\,\cdot\,}_{p_1}$ and $\norm {\,\cdot\,}_{p_2}$ ar... | Consider $p_1/p_2 \in \Q$.
With $\norm {\,\cdot\,}_{p_1}$:
{{begin-eqn}}
{{eqn | l = \norm {p_1/p_2}_{p_1}
| r = \norm {p_1}_{p_1} \norm {1/p_2}_{p_1}
| c = {{NormAxiomNonArch|2}}
}}
{{eqn | r = \norm {p_1}_{p_1} \times 1
| c = $p_1$ does not divide $p_2$
}}
{{eqn | r = 1 / {p_1}
}}
{{eqn | o = \lt... | P-adic Norms are Not Equivalent | https://proofwiki.org/wiki/P-adic_Norms_are_Not_Equivalent | https://proofwiki.org/wiki/P-adic_Norms_are_Not_Equivalent | [
"Norm Theory",
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:Equivalent Division Ring Norms",
"Definition:Topology Induced by Division Ring Norm",
"Definition:Topology Induced by Division Ring Norm"
] | [
"Definition:Equivalent Division Ring Norms/Open Unit Ball Equivalent",
"Definition:Equivalent Division Ring Norms",
"Equivalence of Definitions of Equivalent Division Ring Norms",
"Definition:Equivalent Division Ring Norms/Topologically Equivalent",
"Definition:Topology Induced by Division Ring Norm",
"De... |
proofwiki-15496 | Polynomials of Congruent Integers are Congruent | Let $x, y, m \in \Z$ be integers where $m \ne 0$.
Let:
:$x \equiv y \pmod m$
where the notation indicates congruence modulo $m$.
Let $a_0, a_1, \ldots, a_r$ be integers.
Then:
:$\ds \sum_{k \mathop = 0}^r a_k x^k \equiv \sum_{k \mathop = 0}^r a_k y^k \pmod m$ | We have that:
:$x \equiv y \pmod m$
From Congruence of Powers:
:$x^k \equiv y^k \pmod m$
From Modulo Multiplication is Well-Defined:
:$\forall k \in \set {0, 2, \ldots, r}: a_k x^k \equiv a_k y^k \pmod m$
The result follows from Modulo Addition is Well-Defined.
{{qed}} | Let $x, y, m \in \Z$ be [[Definition:Integer|integers]] where $m \ne 0$.
Let:
:$x \equiv y \pmod m$
where the notation indicates [[Definition:Congruence Modulo Integer|congruence modulo $m$]].
Let $a_0, a_1, \ldots, a_r$ be [[Definition:Integer|integers]].
Then:
:$\ds \sum_{k \mathop = 0}^r a_k x^k \equiv \sum_{k \... | We have that:
:$x \equiv y \pmod m$
From [[Congruence of Powers]]:
:$x^k \equiv y^k \pmod m$
From [[Modulo Multiplication is Well-Defined]]:
:$\forall k \in \set {0, 2, \ldots, r}: a_k x^k \equiv a_k y^k \pmod m$
The result follows from [[Modulo Addition is Well-Defined]].
{{qed}} | Polynomials of Congruent Integers are Congruent | https://proofwiki.org/wiki/Polynomials_of_Congruent_Integers_are_Congruent | https://proofwiki.org/wiki/Polynomials_of_Congruent_Integers_are_Congruent | [
"Polynomial Theory",
"Modulo Arithmetic"
] | [
"Definition:Integer",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Integer"
] | [
"Congruence of Powers",
"Modulo Multiplication is Well-Defined",
"Modulo Addition is Well-Defined"
] |
proofwiki-15497 | Congruent Integers less than Half Modulus are Equal | Let $k \in \Z_{>0}$ be a strictly positive integer.
Let $a, b \in \Z$ such that $\size a < \dfrac k 2$ and $\size b < \dfrac k 2$.
Then:
:$a \equiv b \pmod k \implies a = b$
where $\equiv$ denotes congruence modulo $k$. | We have that:
:$-\dfrac k 2 < a < \dfrac k 2$
and:
:$-\dfrac k 2 < -b < \dfrac k 2$
Thus:
:$-k < a - b < k$
Let $a \equiv b \pmod k$
Then:
:$a - b = n k$
for some $n \in \Z$.
But as $-k < n k < k$ it must be the case that $n = 0$.
Thus $a - b = 0$ and the result follows.
{{qed}} | Let $k \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $a, b \in \Z$ such that $\size a < \dfrac k 2$ and $\size b < \dfrac k 2$.
Then:
:$a \equiv b \pmod k \implies a = b$
where $\equiv$ denotes [[Definition:Congruence Modulo Integer|congruence modulo $k$]]. | We have that:
:$-\dfrac k 2 < a < \dfrac k 2$
and:
:$-\dfrac k 2 < -b < \dfrac k 2$
Thus:
:$-k < a - b < k$
Let $a \equiv b \pmod k$
Then:
:$a - b = n k$
for some $n \in \Z$.
But as $-k < n k < k$ it must be the case that $n = 0$.
Thus $a - b = 0$ and the result follows.
{{qed}} | Congruent Integers less than Half Modulus are Equal | https://proofwiki.org/wiki/Congruent_Integers_less_than_Half_Modulus_are_Equal | https://proofwiki.org/wiki/Congruent_Integers_less_than_Half_Modulus_are_Equal | [
"Modulo Arithmetic"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Congruence (Number Theory)/Integers"
] | [] |
proofwiki-15498 | Complete Residue System Modulo m has m Elements | Let $m \in \Z_{\ne 0}$ be a non-zero integer.
Let $S := \set {r_1, r_2, \dotsb, r_s}$ be a complete residue system modulo $m$.
Then $s = m$. | Let:
:$t_0 = 0, t_1 = 1, \dots, t_{m - 1} = m - 1$
Let $n \in \Z$.
Then from the Division Theorem there exist unique integers $q$ and $u$ such that:
:$n = m q + u$
such that $0 \le u < m$.
That is:
:$n \equiv u \pmod m$
and $u$ is one of $t_0, t_1, \ldots, t_{m - 1}$.
Also, since $\size {t_i - t_j} < m$, no two element... | Let $m \in \Z_{\ne 0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Integer|integer]].
Let $S := \set {r_1, r_2, \dotsb, r_s}$ be a [[Definition:Complete Residue System|complete residue system modulo $m$]].
Then $s = m$. | Let:
:$t_0 = 0, t_1 = 1, \dots, t_{m - 1} = m - 1$
Let $n \in \Z$.
Then from the [[Division Theorem]] there exist [[Definition:Unique|unique]] [[Definition:Integer|integers]] $q$ and $u$ such that:
:$n = m q + u$
such that $0 \le u < m$.
That is:
:$n \equiv u \pmod m$
and $u$ is one of $t_0, t_1, \ldots, t_{m - 1}$... | Complete Residue System Modulo m has m Elements | https://proofwiki.org/wiki/Complete_Residue_System_Modulo_m_has_m_Elements | https://proofwiki.org/wiki/Complete_Residue_System_Modulo_m_has_m_Elements | [
"Residue Classes"
] | [
"Definition:Zero (Number)",
"Definition:Integer",
"Definition:Complete Residue System"
] | [
"Division Theorem",
"Definition:Unique",
"Definition:Integer",
"Definition:Element",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Complete Residue System",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Unique",
"Definition:Element",
"Definition:Complete Residue Sys... |
proofwiki-15499 | Initial Segment of Natural Numbers forms Complete Residue System | Let $m \in \Z_{\ne 0}$ be a non-zero integer.
Let $\N_m = \set {0, 1, 2, \ldots, m - 1}$ denote the initial segment of $\N$
Then $\N_m$ is a complete residue system modulo $m$. | Let $n \in \Z$.
From the Division Theorem there exist unique integers $q$ and $u$ such that:
:$n = m q + u$
such that $0 \le u < m$.
That is:
:$n \equiv u \pmod m$
and $u$ is one of $0, 1, \ldots, m - 1$.
Also, since $\forall i, j \in \N_m: \size {i - j} < m$, no two elements of $\N_m$ are congruent.
Thus $\N_m = \set ... | Let $m \in \Z_{\ne 0}$ be a non-[[Definition:Zero (Number)|zero]] [[Definition:Integer|integer]].
Let $\N_m = \set {0, 1, 2, \ldots, m - 1}$ denote the [[Definition:Initial Segment of Zero-Based Natural Numbers|initial segment of $\N$]]
Then $\N_m$ is a [[Definition:Complete Residue System|complete residue system]] ... | Let $n \in \Z$.
From the [[Division Theorem]] there exist [[Definition:Unique|unique]] [[Definition:Integer|integers]] $q$ and $u$ such that:
:$n = m q + u$
such that $0 \le u < m$.
That is:
:$n \equiv u \pmod m$
and $u$ is one of $0, 1, \ldots, m - 1$.
Also, since $\forall i, j \in \N_m: \size {i - j} < m$, no two ... | Initial Segment of Natural Numbers forms Complete Residue System | https://proofwiki.org/wiki/Initial_Segment_of_Natural_Numbers_forms_Complete_Residue_System | https://proofwiki.org/wiki/Initial_Segment_of_Natural_Numbers_forms_Complete_Residue_System | [
"Residue Classes",
"Complete Residue Systems"
] | [
"Definition:Zero (Number)",
"Definition:Integer",
"Definition:Initial Segment of Natural Numbers/Zero-Based",
"Definition:Complete Residue System"
] | [
"Division Theorem",
"Definition:Unique",
"Definition:Integer",
"Definition:Element",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Complete Residue System"
] |
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