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-14900 | Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$.
Then:
:$\sequence {x_{n_r} }$ is a Cauchy sequence in $R$. | Let $\epsilon > 0$.
Since $\sequence {x_n}$ is a Cauchy sequence then:
:$\exists N: \forall n,m > N: \norm {x_n - x_m } < \epsilon$
Now let $R = N$.
Then from Strictly Increasing Sequence of Natural Numbers:
:$\forall r, s > R: n_r \ge r$ and $n_s \ge s$
Thus $n_r, n_s > N$ and so:
:$\norm {x_{n_r} - x_{n_s} } < \epsil... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Ring Zero|zero]]: $0$.
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence in $R$]].
Let $\sequence {x_{n_r} }$ be a [[Definition:Subsequence|subsequence]... | Let $\epsilon > 0$.
Since $\sequence {x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] then:
:$\exists N: \forall n,m > N: \norm {x_n - x_m } < \epsilon$
Now let $R = N$.
Then from [[Strictly Increasing Sequence of Natural Numbers]]:
:$\forall r, s > R: n_r \ge r$ and $n_s \ge s$
T... | Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence | https://proofwiki.org/wiki/Subsequence_of_Cauchy_Sequence_in_Normed_Division_Ring_is_Cauchy_Sequence | https://proofwiki.org/wiki/Subsequence_of_Cauchy_Sequence_in_Normed_Division_Ring_is_Cauchy_Sequence | [
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Ring Zero",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Subsequence",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Strictly Increasing Sequence of Natural Numbers"
] |
proofwiki-14901 | Coprime Divisors of Square Number are Square | Let $r$ be a square number.
Let $r = s t$ where $s$ and $t$ are coprime.
Then both $s$ and $t$ are square. | Let $p$ be a prime factor of $s$.
Then for some $n \in \Z_{>0}$:
:$p^{2 n} \divides s t$
where $\divides$ denotes the divisibility relation.
But we have that $s \perp t$ and so $p \nmid t$.
Thus $p^{2 n} \divides s$.
This holds for all prime factors of $s$.
Thus $s$ is the product of squares of primes.
Thus $s$ is squa... | Let $r$ be a [[Definition:Square Number|square number]].
Let $r = s t$ where $s$ and $t$ are [[Definition:Coprime Integers|coprime]].
Then both $s$ and $t$ are [[Definition:Square Number|square]]. | Let $p$ be a [[Definition:Prime Factor|prime factor]] of $s$.
Then for some $n \in \Z_{>0}$:
:$p^{2 n} \divides s t$
where $\divides$ denotes the [[Definition:Divisor of Integer|divisibility relation]].
But we have that $s \perp t$ and so $p \nmid t$.
Thus $p^{2 n} \divides s$.
This holds for all [[Definition:Prime... | Coprime Divisors of Square Number are Square | https://proofwiki.org/wiki/Coprime_Divisors_of_Square_Number_are_Square | https://proofwiki.org/wiki/Coprime_Divisors_of_Square_Number_are_Square | [
"Square Numbers"
] | [
"Definition:Square Number",
"Definition:Coprime/Integers",
"Definition:Square Number"
] | [
"Definition:Prime Factor",
"Definition:Divisor (Algebra)/Integer",
"Definition:Prime Factor",
"Definition:Multiplication/Integers",
"Definition:Square/Function",
"Definition:Prime Number",
"Definition:Square Number"
] |
proofwiki-14902 | Combination Theorem for Cauchy Sequences/Constant Rule | The constant sequence $\tuple {a, a, a, \dots}$ is a Cauchy sequence. | Let $\sequence {x_n}$ be the constant sequence:
:$\forall n, x_n = a$
Given $\epsilon > 0$:
:$\forall n, m \ge 1: \norm {x_n - x_m} = \norm {a - a} = \norm {0} = 0 < \epsilon$
The result follows.
{{qed}} | The constant [[Definition:Sequence|sequence]] $\tuple {a, a, a, \dots}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | Let $\sequence {x_n}$ be the constant [[Definition:Sequence|sequence]]:
:$\forall n, x_n = a$
Given $\epsilon > 0$:
:$\forall n, m \ge 1: \norm {x_n - x_m} = \norm {a - a} = \norm {0} = 0 < \epsilon$
The result follows.
{{qed}} | Combination Theorem for Cauchy Sequences/Constant Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Constant_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Constant_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Sequence"
] |
proofwiki-14903 | Combination Theorem for Cauchy Sequences/Multiple Rule | :$\sequence {a x_n}$ is a Cauchy sequence. | Follows directly from Product Rule for Normed Division Ring Sequences, setting
:$\sequence {y_n} := \sequence {x_n}$
and:
:$\sequence {x_n} := \tuple {a, a, a, \ldots}$
{{qed}} | :$\sequence {a x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | Follows directly from [[Combination Theorem for Cauchy Sequences/Product Rule|Product Rule for Normed Division Ring Sequences]], setting
:$\sequence {y_n} := \sequence {x_n}$
and:
:$\sequence {x_n} := \tuple {a, a, a, \ldots}$
{{qed}} | Combination Theorem for Cauchy Sequences/Multiple Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Multiple_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Multiple_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Combination Theorem for Cauchy Sequences/Product Rule"
] |
proofwiki-14904 | Combination Theorem for Cauchy Sequences/Difference Rule | :$\sequence {x_n - y_n}$ is a Cauchy sequence. | From Multiple Rule for Normed Division Ring Sequences:
:$\sequence {-y_n} = \sequence {\paren {-1} y_n}$ is a Cauchy sequence.
From Sum Rule for Normed Division Ring Sequences:
:$\sequence {x_n - y_n} = \sequence {x_n + \paren {-y_n} }$ is a Cauchy sequence.
{{qed}} | :$\sequence {x_n - y_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | From [[Combination Theorem for Cauchy Sequences/Multiple Rule|Multiple Rule for Normed Division Ring Sequences]]:
:$\sequence {-y_n} = \sequence {\paren {-1} y_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]].
From [[Combination Theorem for Cauchy Sequences/Sum Rule|Sum Rule for Normed ... | Combination Theorem for Cauchy Sequences/Difference Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Difference_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Difference_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Combination Theorem for Cauchy Sequences/Multiple Rule",
"Definition:Cauchy Sequence/Normed Division Ring",
"Combination Theorem for Cauchy Sequences/Sum Rule",
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-14905 | Combination Theorem for Cauchy Sequences/Combined Sum Rule | :$\sequence {a x_n + b y_n }$ is a Cauchy sequence. | From the Multiple Rule for Normed Division Ring Sequences:
:$\sequence {a x_n}$ is a Cauchy sequence
:$\sequence {b y_n}$ is a Cauchy sequence.
The result now follows directly from the Sum Rule for Normed Division Ring Sequences:
:$\sequence {a x_n + b y_n}$ is a Cauchy sequence.
{{qed}} | :$\sequence {a x_n + b y_n }$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]. | From the [[Combination Theorem for Cauchy Sequences/Multiple Rule|Multiple Rule for Normed Division Ring Sequences]]:
:$\sequence {a x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]
:$\sequence {b y_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]].
The re... | Combination Theorem for Cauchy Sequences/Combined Sum Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Combined_Sum_Rule | https://proofwiki.org/wiki/Combination_Theorem_for_Cauchy_Sequences/Combined_Sum_Rule | [
"Combination Theorem for Cauchy Sequences"
] | [
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Combination Theorem for Cauchy Sequences/Multiple Rule",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Combination Theorem for Cauchy Sequences/Sum Rule",
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-14906 | Congruent Integers in Same Residue Class | Let $m \in \Z_{>0}$ be a (strictly) positive integer.
Let $\Z_m$ be the set of residue classes modulo $m$:
:$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$
Let $a, b \in \set {0, 1, \ldots, m -1 }$.
Then:
:$\eqclass a m = \eqclass b m \iff a \equiv b \pmod m$ | By definition of the set of residue classes modulo $m$, $\Z_m$ is the quotient set of congruence modulo $m$:
:$\Z_m = \dfrac \Z {\RR_m}$
where $\RR_m$ is the congruence relation modulo $m$ on the set of all $a, b \in \Z$:
:$\RR_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$
By the Fundamental... | Let $m \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\Z_m$ be the [[Definition:Set of Residue Classes|set of residue classes modulo $m$]]:
:$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$
Let $a, b \in \set {0, 1, \ldots, m -1 }$.
Then:
:$\eqcla... | By definition of the [[Definition:Set of Residue Classes|set of residue classes modulo $m$]], $\Z_m$ is the [[Definition:Quotient Set|quotient set]] of [[Definition:Congruence Modulo Integer|congruence modulo $m$]]:
:$\Z_m = \dfrac \Z {\RR_m}$
where $\RR_m$ is the [[Definition:Congruence Modulo Integer|congruence rela... | Congruent Integers in Same Residue Class | https://proofwiki.org/wiki/Congruent_Integers_in_Same_Residue_Class | https://proofwiki.org/wiki/Congruent_Integers_in_Same_Residue_Class | [
"Residue Classes"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set of Residue Classes"
] | [
"Definition:Set of Residue Classes",
"Definition:Quotient Set",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Set",
"Fundamental Theorem on Equivalence Relations",
"Definition:Set Partition"
] |
proofwiki-14907 | Residue Classes form Partition of Integers | Let $m \in \Z_{>0}$ be a (strictly) positive integer.
Let $\Z_m$ be the set of residue classes modulo $m$:
:$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$
Then $\Z_m$ forms a partition of $\Z$. | By definition of the set of residue classes modulo $m$, $\Z_m$ is the quotient set of congruence modulo $m$:
:$\Z_m = \dfrac \Z {\RR_m}$
where $\RR_m$ is the congruence relation modulo $m$ on the set of all $a, b \in \Z$:
:$\RR_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$
By the Fundamental... | Let $m \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\Z_m$ be the [[Definition:Set of Residue Classes|set of residue classes modulo $m$]]:
:$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$
Then $\Z_m$ forms a [[Definition:Set Partition|partition]]... | By definition of the [[Definition:Set of Residue Classes|set of residue classes modulo $m$]], $\Z_m$ is the [[Definition:Quotient Set|quotient set]] of [[Definition:Congruence Modulo Integer|congruence modulo $m$]]:
:$\Z_m = \dfrac \Z {\RR_m}$
where $\RR_m$ is the [[Definition:Congruence Modulo Integer|congruence rela... | Residue Classes form Partition of Integers | https://proofwiki.org/wiki/Residue_Classes_form_Partition_of_Integers | https://proofwiki.org/wiki/Residue_Classes_form_Partition_of_Integers | [
"Residue Classes"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set of Residue Classes",
"Definition:Set Partition"
] | [
"Definition:Set of Residue Classes",
"Definition:Quotient Set",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Set",
"Fundamental Theorem on Equivalence Relations",
"Definition:Set Partition"
] |
proofwiki-14908 | Cardinality of Set of Residue Classes | Let $m \in \Z_{>0}$ be a (strictly) positive integer.
Let $\Z_m$ be the set of residue classes modulo $m$.
Then:
:$\card {Z_m} = m$
where $\card {\, \cdot \,}$ denotes cardinality. | By definition of the set of residue classes modulo $m$, $Z_m$ is the quotient set of congruence modulo $m$:
:$\Z_m = \dfrac \Z {\RR_m}$
where $\RR_m$ is the congruence relation modulo $m$ on the set of all $a, b \in \Z$:
:$\RR_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$
Thus by definition ... | Let $m \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\Z_m$ be the [[Definition:Set of Residue Classes|set of residue classes modulo $m$]].
Then:
:$\card {Z_m} = m$
where $\card {\, \cdot \,}$ denotes [[Definition:Cardinality|cardinality]]. | By definition of the [[Definition:Set of Residue Classes|set of residue classes modulo $m$]], $Z_m$ is the [[Definition:Quotient Set|quotient set]] of [[Definition:Congruence Modulo Integer|congruence modulo $m$]]:
:$\Z_m = \dfrac \Z {\RR_m}$
where $\RR_m$ is the [[Definition:Congruence Modulo Integer|congruence relat... | Cardinality of Set of Residue Classes | https://proofwiki.org/wiki/Cardinality_of_Set_of_Residue_Classes | https://proofwiki.org/wiki/Cardinality_of_Set_of_Residue_Classes | [
"Residue Classes"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set of Residue Classes",
"Definition:Cardinality"
] | [
"Definition:Set of Residue Classes",
"Definition:Quotient Set",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Set",
"Definition:Set of Residue Classes"
] |
proofwiki-14909 | Structure Induced by Ring with Unity Operations is Ring with Unity | Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$.
Then $\struct {R^S, +', \circ'}$ is a ring with unity whose unity is $f_{1_R}: S \to R$, defined by:
:$\forall s \in S: \map {f_{1_R} } s = 1_R... | By Structure Induced by Ring Operations is Ring then $\struct {R^S, +', \circ'}$ is a ring.
We have from Induced Structure Identity that the constant mapping $f_{1_R}: S \to R$ defined as:
:$\forall x \in S: \map {f_{1_R} } x = 1_R$
is the identity for $\struct {R^S, \circ'}$.
The result follows by definition of ring w... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring with Unity|ring with unity]] whose [[Definition:Unity of Ring|unity]] is $1_R$.
Let $S$ be a [[Definition:Set|set]].
Let $\struct {R^S, +', \circ'}$ be the [[Definition:Induced Structure|structure on $R^S$ induced]] by $+'$ and $\circ'$.
Then $\struct {R^S, +', \cir... | By [[Structure Induced by Ring Operations is Ring]] then $\struct {R^S, +', \circ'}$ is a [[Definition:Ring (Abstract Algebra)|ring]].
We have from [[Induced Structure Identity]] that the [[Definition:Constant Mapping|constant mapping]] $f_{1_R}: S \to R$ defined as:
:$\forall x \in S: \map {f_{1_R} } x = 1_R$
is th... | Structure Induced by Ring with Unity Operations is Ring with Unity | https://proofwiki.org/wiki/Structure_Induced_by_Ring_with_Unity_Operations_is_Ring_with_Unity | https://proofwiki.org/wiki/Structure_Induced_by_Ring_with_Unity_Operations_is_Ring_with_Unity | [
"Rings of Mappings",
"Rings with Unity"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring"
] | [
"Structure Induced by Ring Operations is Ring",
"Definition:Ring (Abstract Algebra)",
"Induced Structure Identity",
"Definition:Constant Mapping",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Category:Rings of M... |
proofwiki-14910 | Cauchy Sequences form Ring with Unity | Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\struct {R^\N, +, \circ}$ be the ring of sequences over $R$ with unity $\tuple {1, 1, 1, \dotsc}$.
Let $\CC \subset R^\N$ be the set of Cauchy sequences on $R$.
Then:
:$\struct {\CC, +, \circ}$ is a subring of $R^\N$ with unity $\tuple {1... | The Subring Test used to prove the result.
By Constant Rule for Cauchy sequences:
:the constant sequence $\tuple {1, 1, 1, \dotsc}$ is a Cauchy sequences.
Hence:
:$\CC \neq \O$
Let $\sequence {x_n}, \sequence {y_n} \in \CC$.
By definition of pointwise addition:
:$\sequence {x_n} + \paren {-\sequence {y_n}} = \sequence ... | Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\struct {R^\N, +, \circ}$ be the [[Definition:Ring of Sequences|ring of sequences over $R$]] with [[Definition:Ring with Unity|unity]] $\tuple {1, 1, 1, \dotsc}$.
Let $\CC \subset R^\N$ be the [[Defin... | The [[Subring Test]] used to prove the result.
By [[Combination Theorem for Cauchy Sequences/Constant Rule|Constant Rule for Cauchy sequences]]:
:the constant [[Definition:Sequence|sequence]] $\tuple {1, 1, 1, \dotsc}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequences]].
Hence:
:$\CC \neq \O... | Cauchy Sequences form Ring with Unity | https://proofwiki.org/wiki/Cauchy_Sequences_form_Ring_with_Unity | https://proofwiki.org/wiki/Cauchy_Sequences_form_Ring_with_Unity | [
"Cauchy Sequences",
"Normed Division Rings",
"Cauchy Sequences in Normed Division Rings",
"Rings of Sequences"
] | [
"Definition:Normed Division Ring",
"Definition:Ring of Sequences",
"Definition:Ring with Unity",
"Definition:Set",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Subring",
"Definition:Unity (Abstract Algebra)/Ring"
] | [
"Subring Test",
"Combination Theorem for Cauchy Sequences/Constant Rule",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Pointwise Addition",
"Combination Theorem for Cauchy Sequences/Difference Rule",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definitio... |
proofwiki-14911 | Equivalence Relation on Power Set induced by Intersection with Subset | Let $A, T$ be sets such that $A \subseteq T$.
Let $S = \powerset T$ denote the power set of $T$.
Let $\alpha$ denote the relation defined on $S$ by:
:$\forall X, Y \in S: X \mathrel \alpha Y \iff X \cap A = Y \cap A$
Then $\alpha$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | Let $A, T$ be [[Definition:Set|sets]] such that $A \subseteq T$.
Let $S = \powerset T$ denote the [[Definition:Power Set|power set]] of $T$.
Let $\alpha$ denote the [[Definition:Relation|relation]] defined on $S$ by:
:$\forall X, Y \in S: X \mathrel \alpha Y \iff X \cap A = Y \cap A$
Then $\alpha$ is an [[Definiti... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Equivalence Relation on Power Set induced by Intersection with Subset | https://proofwiki.org/wiki/Equivalence_Relation_on_Power_Set_induced_by_Intersection_with_Subset | https://proofwiki.org/wiki/Equivalence_Relation_on_Power_Set_induced_by_Intersection_with_Subset | [
"Examples of Equivalence Relations",
"Equivalence Relation on Power Set induced by Intersection with Subset"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-14912 | Structure Induced by Commutative Ring Operations is Commutative Ring | Let $\struct {R, +, \circ}$ be a commutative ring.
Let $S$ be a set.
Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$.
Then $\struct {R^S, +', \circ'}$ is a commutative ring. | By Structure Induced by Ring Operations is Ring then $\struct {R^S, +', \circ'}$ is a ring.
From Structure Induced by Commutative Operation is Commutative, so is the pointwise operation $\circ$ induces on $R^S$.
The result follows by definition of commutative ring.
{{qed}}
Category:Rings of Mappings
d9gcno26rkxm8oz0gfx... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]].
Let $S$ be a [[Definition:Set|set]].
Let $\struct {R^S, +', \circ'}$ be the [[Definition:Induced Structure|structure on $R^S$ induced]] by $+'$ and $\circ'$.
Then $\struct {R^S, +', \circ'}$ is a [[Definition:Commutative Ring|commutat... | By [[Structure Induced by Ring Operations is Ring]] then $\struct {R^S, +', \circ'}$ is a [[Definition:Ring (Abstract Algebra)|ring]].
From [[Structure Induced by Commutative Operation is Commutative]], so is the [[Definition:Pointwise Operation|pointwise operation $\circ$ induces]] on $R^S$.
The result follows by de... | Structure Induced by Commutative Ring Operations is Commutative Ring | https://proofwiki.org/wiki/Structure_Induced_by_Commutative_Ring_Operations_is_Commutative_Ring | https://proofwiki.org/wiki/Structure_Induced_by_Commutative_Ring_Operations_is_Commutative_Ring | [
"Rings of Mappings"
] | [
"Definition:Commutative Ring",
"Definition:Set",
"Definition:Pointwise Operation/Induced Structure",
"Definition:Commutative Ring"
] | [
"Structure Induced by Ring Operations is Ring",
"Definition:Ring (Abstract Algebra)",
"Structure Induced by Commutative Operation is Commutative",
"Definition:Pointwise Operation",
"Definition:Commutative Ring",
"Category:Rings of Mappings"
] |
proofwiki-14913 | Equivalence Relation on Power Set induced by Intersection with Subset/Equivalence Class of Empty Set | The equivalence class of $\O$ in $S$ with respect to $\alpha$ is given by:
:$\eqclass \O \alpha = \powerset {T \setminus A}$ | That $\alpha$ is an equivalence relation is proved in Equivalence Relation on Power Set induced by Intersection with Subset.
We have that:
:$\eqclass \O \alpha = \set {X \in S: X \cap A = \O \cap A = \O}$
Thus:
{{begin-eqn}}
{{eqn | l = X
| o = \in
| r = \eqclass \O \alpha
| c =
}}
{{eqn | ll= \leads... | The [[Definition:Equivalence Class|equivalence class]] of $\O$ in $S$ with respect to $\alpha$ is given by:
:$\eqclass \O \alpha = \powerset {T \setminus A}$ | That $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Power Set induced by Intersection with Subset]].
We have that:
:$\eqclass \O \alpha = \set {X \in S: X \cap A = \O \cap A = \O}$
Thus:
{{begin-eqn}}
{{eqn | l = X
| o = \in
| r = \eqclass... | Equivalence Relation on Power Set induced by Intersection with Subset/Equivalence Class of Empty Set | https://proofwiki.org/wiki/Equivalence_Relation_on_Power_Set_induced_by_Intersection_with_Subset/Equivalence_Class_of_Empty_Set | https://proofwiki.org/wiki/Equivalence_Relation_on_Power_Set_induced_by_Intersection_with_Subset/Equivalence_Class_of_Empty_Set | [
"Equivalence Relation on Power Set induced by Intersection with Subset"
] | [
"Definition:Equivalence Class"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Power Set induced by Intersection with Subset",
"Empty Intersection iff Subset of Complement"
] |
proofwiki-14914 | Equivalence Relation on Power Set induced by Intersection with Subset/Cardinality of Set of Equivalence Classes | Let $A$ be finite with $\card A = n$, where $\card {\, \cdot \,}$ denotes cardinality.
The cardinality of the set of $\alpha$-equivalence classes is given by:
:$\card {\set {\eqclass X \alpha: X \in S} } = 2^n$ | That $\alpha$ is an equivalence relation is proved in Equivalence Relation on Power Set induced by Intersection with Subset.
Let $U \subseteq A$.
Then $U$ is in its own $\alpha$-equivalence class.
Now suppose $U' \subseteq A$ such that $U \ne U'$.
From Intersection with Subset is Subset:
:$U \cap A = U$
and:
:$U' \cap ... | Let $A$ be [[Definition:Finite Set|finite]] with $\card A = n$, where $\card {\, \cdot \,}$ denotes [[Definition:Cardinality|cardinality]].
The [[Definition:Cardinality|cardinality]] of the [[Definition:Set|set]] of [[Definition:Equivalence Class|$\alpha$-equivalence classes]] is given by:
:$\card {\set {\eqclass X \... | That $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Power Set induced by Intersection with Subset]].
Let $U \subseteq A$.
Then $U$ is in its own [[Definition:Equivalence Class|$\alpha$-equivalence class]].
Now suppose $U' \subseteq A$ such that $U \ne ... | Equivalence Relation on Power Set induced by Intersection with Subset/Cardinality of Set of Equivalence Classes | https://proofwiki.org/wiki/Equivalence_Relation_on_Power_Set_induced_by_Intersection_with_Subset/Cardinality_of_Set_of_Equivalence_Classes | https://proofwiki.org/wiki/Equivalence_Relation_on_Power_Set_induced_by_Intersection_with_Subset/Cardinality_of_Set_of_Equivalence_Classes | [
"Equivalence Relation on Power Set induced by Intersection with Subset"
] | [
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Cardinality",
"Definition:Set",
"Definition:Equivalence Class"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Power Set induced by Intersection with Subset",
"Definition:Equivalence Class",
"Intersection with Subset is Subset",
"Definition:Equivalence Class",
"Definition:Subset",
"Cardinality of Power Set of Finite Set"
] |
proofwiki-14915 | Reflexive and Symmetric Relation is not necessarily Transitive | Let $S$ be a set.
Let $\alpha \subseteq S \times S$ be a relation on $S$.
Let $\alpha$ be both reflexive and symmetric.
Then it is not necessarily the case that $\alpha$ is also transitive. | Proof by Counterexample:
Let $S = \set {a, b, c}$.
Let:
: $\alpha = \set {\tuple {a, a}, \tuple {b, b}, \tuple {c, c}, \tuple {a, b}, \tuple {b, a}, \tuple {b, c}, \tuple {c, b} }$
By inspection it is seen that $\alpha$ is both reflexive and symmetric.
However, we have:
:$a \mathrel \alpha b$ and $b \mathrel \alpha c$
... | Let $S$ be a [[Definition:Set|set]].
Let $\alpha \subseteq S \times S$ be a [[Definition:Endorelation|relation]] on $S$.
Let $\alpha$ be both [[Definition:Reflexive Relation|reflexive]] and [[Definition:Symmetric Relation|symmetric]].
Then it is not necessarily the case that $\alpha$ is also [[Definition:Transitive... | [[Proof by Counterexample]]:
Let $S = \set {a, b, c}$.
Let:
: $\alpha = \set {\tuple {a, a}, \tuple {b, b}, \tuple {c, c}, \tuple {a, b}, \tuple {b, a}, \tuple {b, c}, \tuple {c, b} }$
By inspection it is seen that $\alpha$ is both [[Definition:Reflexive Relation|reflexive]] and [[Definition:Symmetric Relation|symme... | Reflexive and Symmetric Relation is not necessarily Transitive/Proof 1 | https://proofwiki.org/wiki/Reflexive_and_Symmetric_Relation_is_not_necessarily_Transitive | https://proofwiki.org/wiki/Reflexive_and_Symmetric_Relation_is_not_necessarily_Transitive/Proof_1 | [
"Reflexive Relations",
"Symmetric Relations",
"Transitive Relations",
"Reflexive and Symmetric Relation is not necessarily Transitive"
] | [
"Definition:Set",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation"
] | [
"Proof by Counterexample",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation"
] |
proofwiki-14916 | Reflexive and Symmetric Relation is not necessarily Transitive | Let $S$ be a set.
Let $\alpha \subseteq S \times S$ be a relation on $S$.
Let $\alpha$ be both reflexive and symmetric.
Then it is not necessarily the case that $\alpha$ is also transitive. | Proof by Counterexample:
Let $S = \Z$ be the set of integers.
Let $n \in \Z$ such that $n > 1$.
Let $\alpha$ be the relation on $S$ defined as:
:$\forall x, y \in S: x \mathrel \alpha y \iff \size {x - y} < n$
where $\size x$ denotes the absolute value of $x$.
It is seen that:
:$\forall x \in \Z: \size {x - x} = 0 < n$... | Let $S$ be a [[Definition:Set|set]].
Let $\alpha \subseteq S \times S$ be a [[Definition:Endorelation|relation]] on $S$.
Let $\alpha$ be both [[Definition:Reflexive Relation|reflexive]] and [[Definition:Symmetric Relation|symmetric]].
Then it is not necessarily the case that $\alpha$ is also [[Definition:Transitive... | [[Proof by Counterexample]]:
Let $S = \Z$ be the set of [[Definition:Integer|integers]].
Let $n \in \Z$ such that $n > 1$.
Let $\alpha$ be the [[Definition:Endorelation|relation]] on $S$ defined as:
:$\forall x, y \in S: x \mathrel \alpha y \iff \size {x - y} < n$
where $\size x$ denotes the [[Definition:Absolute ... | Reflexive and Symmetric Relation is not necessarily Transitive/Proof 2 | https://proofwiki.org/wiki/Reflexive_and_Symmetric_Relation_is_not_necessarily_Transitive | https://proofwiki.org/wiki/Reflexive_and_Symmetric_Relation_is_not_necessarily_Transitive/Proof_2 | [
"Reflexive Relations",
"Symmetric Relations",
"Transitive Relations",
"Reflexive and Symmetric Relation is not necessarily Transitive"
] | [
"Definition:Set",
"Definition:Endorelation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation"
] | [
"Proof by Counterexample",
"Definition:Integer",
"Definition:Endorelation",
"Definition:Absolute Value",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Rela... |
proofwiki-14917 | Symmetric and Transitive Relation is not necessarily Reflexive | Let $S$ be a set.
Let $\alpha \subseteq S \times S$ be a relation on $S$.
Let $\alpha$ be both symmetric and transitive.
Then it is not necessarily the case that $\alpha$ is also reflexive. | Proof by Counterexample:
Let $S = \Z$ be the set of integers.
Let $\alpha$ be the relation on $S$ defined as:
:$\forall x, y \in S: x \mathrel \alpha y \iff x = y = 0$
{{begin-eqn}}
{{eqn | l = x
| o = \alpha
| r = y
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y = 0
| c =
}}
{{eqn ... | Let $S$ be a [[Definition:Set|set]].
Let $\alpha \subseteq S \times S$ be a [[Definition:Endorelation|relation]] on $S$.
Let $\alpha$ be both [[Definition:Symmetric Relation|symmetric]] and [[Definition:Transitive Relation|transitive]].
Then it is not necessarily the case that $\alpha$ is also [[Definition:Reflexiv... | [[Proof by Counterexample]]:
Let $S = \Z$ be the set of [[Definition:Integer|integers]].
Let $\alpha$ be the [[Definition:Endorelation|relation]] on $S$ defined as:
:$\forall x, y \in S: x \mathrel \alpha y \iff x = y = 0$
{{begin-eqn}}
{{eqn | l = x
| o = \alpha
| r = y
| c =
}}
{{eqn | ll= \le... | Symmetric and Transitive Relation is not necessarily Reflexive/Proof 2 | https://proofwiki.org/wiki/Symmetric_and_Transitive_Relation_is_not_necessarily_Reflexive | https://proofwiki.org/wiki/Symmetric_and_Transitive_Relation_is_not_necessarily_Reflexive/Proof_2 | [
"Reflexive Relations",
"Symmetric Relations",
"Transitive Relations",
"Symmetric and Transitive Relation is not necessarily Reflexive"
] | [
"Definition:Set",
"Definition:Endorelation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Reflexive Relation"
] | [
"Proof by Counterexample",
"Definition:Integer",
"Definition:Endorelation",
"Definition:Symmetric Relation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Reflexive Relation"
] |
proofwiki-14918 | Symmetric and Transitive Relation is not necessarily Reflexive | Let $S$ be a set.
Let $\alpha \subseteq S \times S$ be a relation on $S$.
Let $\alpha$ be both symmetric and transitive.
Then it is not necessarily the case that $\alpha$ is also reflexive. | Proof by Counterexample:
Let $S = \set {1, 2}$ be a set.
Let $\RR$ be the relation on $S$ defined as:
:$\forall x, y \in S: x \mathrel \RR y \iff x = y = 2$
{{begin-eqn}}
{{eqn | l = x
| o = \RR
| r = y
| c =
}}
{{eqn | ll= \leadsto
| l = x
| r = y = 2
| c =
}}
{{eqn | ll= \leadsto... | Let $S$ be a [[Definition:Set|set]].
Let $\alpha \subseteq S \times S$ be a [[Definition:Endorelation|relation]] on $S$.
Let $\alpha$ be both [[Definition:Symmetric Relation|symmetric]] and [[Definition:Transitive Relation|transitive]].
Then it is not necessarily the case that $\alpha$ is also [[Definition:Reflexiv... | [[Proof by Counterexample]]:
Let $S = \set {1, 2}$ be a [[Definition:Set|set]].
Let $\RR$ be the [[Definition:Endorelation|relation]] on $S$ defined as:
:$\forall x, y \in S: x \mathrel \RR y \iff x = y = 2$
{{begin-eqn}}
{{eqn | l = x
| o = \RR
| r = y
| c =
}}
{{eqn | ll= \leadsto
| l = ... | Symmetric and Transitive Relation is not necessarily Reflexive/Proof 3 | https://proofwiki.org/wiki/Symmetric_and_Transitive_Relation_is_not_necessarily_Reflexive | https://proofwiki.org/wiki/Symmetric_and_Transitive_Relation_is_not_necessarily_Reflexive/Proof_3 | [
"Reflexive Relations",
"Symmetric Relations",
"Transitive Relations",
"Symmetric and Transitive Relation is not necessarily Reflexive"
] | [
"Definition:Set",
"Definition:Endorelation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Reflexive Relation"
] | [
"Proof by Counterexample",
"Definition:Set",
"Definition:Endorelation",
"Definition:Symmetric Relation",
"Definition:Reflexive Relation",
"Definition:Symmetric Relation",
"Definition:Transitive Relation",
"Definition:Reflexive Relation"
] |
proofwiki-14919 | Equivalence Relation on Natural Numbers such that Quotient is Power of Two | Let $\alpha$ denote the relation defined on the natural numbers $\N$ by:
:$\forall x, y \in \N: x \mathrel \alpha y \iff \exists n \in \Z: x = 2^n y$
Then $\alpha$ is an equivalence relation. | Checking in turn each of the criteria for equivalence: | Let $\alpha$ denote the [[Definition:Relation|relation]] defined on the [[Definition:Natural Numbers|natural numbers]] $\N$ by:
:$\forall x, y \in \N: x \mathrel \alpha y \iff \exists n \in \Z: x = 2^n y$
Then $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]]. | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Equivalence Relation on Natural Numbers such that Quotient is Power of Two | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two | [
"Examples of Equivalence Relations",
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two"
] | [
"Definition:Relation",
"Definition:Natural Numbers",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-14920 | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class of Prime | Let $\eqclass p \alpha$ be the $\alpha$-equivalence class of a prime number $p$.
Then $\eqclass p \alpha$ contains no other prime number other than $p$. | That $\alpha$ is an equivalence relation is proved in Equivalence Relation on Natural Numbers such that Quotient is Power of Two.
Let $p$ be a prime number whose $\alpha$-equivalence class is $\eqclass p \alpha$.
{{AimForCont}} $\eqclass p \alpha$ contains a prime number $q$ such that $q \ne p$.
Then:
:$p = 2^n q$
for ... | Let $\eqclass p \alpha$ be the [[Definition:Equivalence Class|$\alpha$-equivalence class]] of a [[Definition:Prime Number|prime number]] $p$.
Then $\eqclass p \alpha$ contains no other [[Definition:Prime Number|prime number]] other than $p$. | That $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Natural Numbers such that Quotient is Power of Two]].
Let $p$ be a [[Definition:Prime Number|prime number]] whose [[Definition:Equivalence Class|$\alpha$-equivalence class]] is $\eqclass p \alpha$.
{{A... | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class of Prime | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/Equivalence_Class_of_Prime | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/Equivalence_Class_of_Prime | [
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two"
] | [
"Definition:Equivalence Class",
"Definition:Prime Number",
"Definition:Prime Number"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two",
"Definition:Prime Number",
"Definition:Equivalence Class",
"Definition:Prime Number",
"Definition:Contradiction",
"Definition:Composite Number",
"Definition:Divisor (Algebra)/Integer",
"D... |
proofwiki-14921 | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Smallest Equivalence Class with no Prime | Let $\eqclass x \alpha$ denote the $\alpha$-equivalence class of a natural number $x$.
Let $r$ be the smallest natural number such that $\eqclass r \alpha$ contains no prime number.
Then $r = 9$. | That $\alpha$ is an equivalence relation is proved in Equivalence Relation on Natural Numbers such that Quotient is Power of Two.
{{AimForCont}} $r$ is even.
Then $r = 2 s$ for some $s \in \N$.
Thus $s \in \eqclass r \alpha$ such that $s < r$.
This contradicts the supposition that $r$ is the smallest such natural numbe... | Let $\eqclass x \alpha$ denote the [[Definition:Equivalence Class|$\alpha$-equivalence class]] of a [[Definition:Natural Number|natural number]] $x$.
Let $r$ be the [[Definition:Smallest Element|smallest]] [[Definition:Natural Number|natural number]] such that $\eqclass r \alpha$ contains no [[Definition:Prime Number|... | That $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Natural Numbers such that Quotient is Power of Two]].
{{AimForCont}} $r$ is [[Definition:Even Integer|even]].
Then $r = 2 s$ for some $s \in \N$.
Thus $s \in \eqclass r \alpha$ such that $s < r$.
Thi... | Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Smallest Equivalence Class with no Prime | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/Smallest_Equivalence_Class_with_no_Prime | https://proofwiki.org/wiki/Equivalence_Relation_on_Natural_Numbers_such_that_Quotient_is_Power_of_Two/Smallest_Equivalence_Class_with_no_Prime | [
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two"
] | [
"Definition:Equivalence Class",
"Definition:Natural Numbers",
"Definition:Smallest Element",
"Definition:Natural Numbers",
"Definition:Prime Number"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Natural Numbers such that Quotient is Power of Two",
"Definition:Even Integer",
"Definition:Contradiction",
"Definition:Natural Numbers",
"Definition:Odd Integer",
"Definition:Composite Number"
] |
proofwiki-14922 | Equivalence Relation on Square Matrices induced by Positive Integer Powers | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $S$ be the set of all square matrices of order $n$.
Let $\alpha$ denote the relation defined on $S$ by:
:$\forall \mathbf A, \mathbf B \in S: \mathbf A \mathrel \alpha \mathbf B \iff \exists r, s \in \N: \mathbf A^r = \mathbf B^s$
Then $\alpha$ is an equivalence... | Checking in turn each of the criteria for equivalence: | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $S$ be the [[Definition:Set|set]] of all [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $n$]].
Let $\alpha$ denote the [[Definition:Relation|relation]] defined on $S$ by:
:$\fo... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Equivalence Relation on Square Matrices induced by Positive Integer Powers | https://proofwiki.org/wiki/Equivalence_Relation_on_Square_Matrices_induced_by_Positive_Integer_Powers | https://proofwiki.org/wiki/Equivalence_Relation_on_Square_Matrices_induced_by_Positive_Integer_Powers | [
"Examples of Equivalence Relations",
"Matrix Algebra"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Relation",
"Definition:Equivalence Relation"
] | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-14923 | Equivalence Relation on Integers Modulo 5 induced by Squaring | Let $\beta$ denote the relation defined on the integers $\Z$ by:
:$\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$
Then $\beta$ is an equivalence relation.
=== Number of $\beta$-Equivalence Classes ===
{{:Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes}}
... | Checking in turn each of the criteria for equivalence: | Let $\beta$ denote the [[Definition:Relation|relation]] defined on the [[Definition:Integer|integers]] $\Z$ by:
:$\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$
Then $\beta$ is an [[Definition:Equivalence Relation|equivalence relation]].
=== [[Equivalence Relation on Integers Modulo 5 induced ... | Checking in turn each of the criteria for [[Definition:Equivalence Relation|equivalence]]: | Equivalence Relation on Integers Modulo 5 induced by Squaring | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring | [
"Examples of Equivalence Relations",
"Examples of Modulo Arithmetic",
"Equivalence Relation on Integers Modulo 5 induced by Squaring"
] | [
"Definition:Relation",
"Definition:Integer",
"Definition:Equivalence Relation",
"Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes",
"Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not Well-Defined",
"Equivalence Relation on ... | [
"Definition:Equivalence Relation",
"Definition:Equivalence Relation"
] |
proofwiki-14924 | Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes | The number of distinct $\beta$-equivalence classes is $3$:
{{begin-eqn}}
{{eqn | l = \eqclass 0 \beta
| o =
}}
{{eqn | l = \eqclass 1 \beta
| r = \eqclass 4 \beta
| c =
}}
{{eqn | l = \eqclass 2 \beta
| r = \eqclass 3 \beta
| c =
}}
{{end-eqn}} | That $\beta$ is an equivalence relation is proved in Equivalence Relation on Integers Modulo 5 induced by Squaring.
The set of residue classes modulo $5$ is:
:$\set {\eqclass 0 5, \eqclass 1 5, \eqclass 2 5, \eqclass 3 5, \eqclass 4 5}$
Then:
{{begin-eqn}}
{{eqn | l = 0 \times 0
| r = 0
| c =
}}
{{eqn | ll... | The number of [[Definition:Distinct|distinct]] [[Definition:Equivalence Class|$\beta$-equivalence classes]] is $3$:
{{begin-eqn}}
{{eqn | l = \eqclass 0 \beta
| o =
}}
{{eqn | l = \eqclass 1 \beta
| r = \eqclass 4 \beta
| c =
}}
{{eqn | l = \eqclass 2 \beta
| r = \eqclass 3 \beta
| c = ... | That $\beta$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Integers Modulo 5 induced by Squaring]].
The [[Definition:Set of Residue Classes|set of residue classes modulo $5$]] is:
:$\set {\eqclass 0 5, \eqclass 1 5, \eqclass 2 5, \eqclass 3 5, \eqclass 4 5}$
T... | Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring/Number_of_Equivalence_Classes | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring/Number_of_Equivalence_Classes | [
"Equivalence Relation on Integers Modulo 5 induced by Squaring"
] | [
"Definition:Distinct",
"Definition:Equivalence Class"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Integers Modulo 5 induced by Squaring",
"Definition:Set of Residue Classes"
] |
proofwiki-14925 | Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not Well-Defined | Let the $+_\beta$ operator ("addition") on the $\beta$-equivalence classes be defined as:
:$\eqclass a \beta +_\beta \eqclass b \beta := \eqclass {a + b} \beta$
Then such an operation is not well-defined. | That $\beta$ is an equivalence relation is proved in Equivalence Relation on Integers Modulo 5 induced by Squaring.
From Number of Equivalence Classes we have:
We have:
{{begin-eqn}}
{{eqn | l = \eqclass 1 \beta
| r = \eqclass 4 \beta
| c =
}}
{{eqn | l = \eqclass 2 \beta
| r = \eqclass 3 \beta
... | Let the $+_\beta$ [[Definition:Operator|operator]] ("addition") on the [[Definition:Equivalence Class|$\beta$-equivalence classes]] be defined as:
:$\eqclass a \beta +_\beta \eqclass b \beta := \eqclass {a + b} \beta$
Then such an [[Definition:Operation|operation]] is not [[Definition:Well-Defined Operation|well-defi... | That $\beta$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Integers Modulo 5 induced by Squaring]].
From [[Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes|Number of Equivalence Classes]] we have:
We have:
{{begin-eqn... | Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not Well-Defined | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring/Addition_Modulo_Beta_is_not_Well-Defined | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring/Addition_Modulo_Beta_is_not_Well-Defined | [
"Equivalence Relation on Integers Modulo 5 induced by Squaring"
] | [
"Definition:Operation/Operator",
"Definition:Equivalence Class",
"Definition:Operation",
"Definition:Well-Defined/Operation"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Integers Modulo 5 induced by Squaring",
"Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes"
] |
proofwiki-14926 | Equivalence Relation on Integers Modulo 5 induced by Squaring/Multiplication Modulo Beta is Well-Defined | Let the $\times_\beta$ operator ("multiplication") on the $\beta$-equivalence classes be defined as:
:$\eqclass a \beta \times_\beta \eqclass b \beta := \eqclass {a \times b} \beta$
Then such an operation is well-defined. | That $\beta$ is an equivalence relation is proved in Equivalence Relation on Integers Modulo 5 induced by Squaring.
Let:
{{begin-eqn}}
{{eqn | l = x, x'
| o = \in
| r = \eqclass x \beta
}}
{{eqn | l = y, y'
| o = \in
| r = \eqclass y \beta
}}
{{end-eqn}}
We have:
{{begin-eqn}}
{{eqn | l = x^2
... | Let the $\times_\beta$ [[Definition:Operator|operator]] ("multiplication") on the [[Definition:Equivalence Class|$\beta$-equivalence classes]] be defined as:
:$\eqclass a \beta \times_\beta \eqclass b \beta := \eqclass {a \times b} \beta$
Then such an [[Definition:Operation|operation]] is [[Definition:Well-Defined Op... | That $\beta$ is an [[Definition:Equivalence Relation|equivalence relation]] is proved in [[Equivalence Relation on Integers Modulo 5 induced by Squaring]].
Let:
{{begin-eqn}}
{{eqn | l = x, x'
| o = \in
| r = \eqclass x \beta
}}
{{eqn | l = y, y'
| o = \in
| r = \eqclass y \beta
}}
{{end-eqn}}... | Equivalence Relation on Integers Modulo 5 induced by Squaring/Multiplication Modulo Beta is Well-Defined | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring/Multiplication_Modulo_Beta_is_Well-Defined | https://proofwiki.org/wiki/Equivalence_Relation_on_Integers_Modulo_5_induced_by_Squaring/Multiplication_Modulo_Beta_is_Well-Defined | [
"Equivalence Relation on Integers Modulo 5 induced by Squaring"
] | [
"Definition:Operation/Operator",
"Definition:Equivalence Class",
"Definition:Operation",
"Definition:Well-Defined/Operation"
] | [
"Definition:Equivalence Relation",
"Equivalence Relation on Integers Modulo 5 induced by Squaring"
] |
proofwiki-14927 | Constant Sequence Converges to Constant in Normed Division Ring | :the constant sequence $\tuple {\lambda, \lambda, \lambda, \dots}$ is convergent and $\ds \lim_{n \mathop \to \infty} \lambda = \lambda$ | Let $\sequence {x_n}$ be the constant sequence:
:$\forall n \in \N: x_n = \lambda$
Given $\epsilon \in \R_{>0}$:
:$\forall n \ge 1: \norm {x_n - \lambda} = \norm {\lambda - \lambda} = \norm 0 = 0 < \epsilon$
The result follows.
{{qed}}
Category:Sequences
Category:Normed Division Rings
h73emd7nlxa5xjkduobvr6lacofekfi | :the constant [[Definition:Sequence|sequence]] $\tuple {\lambda, \lambda, \lambda, \dots}$ is [[Definition:Convergent Sequence in Normed Division Ring|convergent]] and $\ds \lim_{n \mathop \to \infty} \lambda = \lambda$ | Let $\sequence {x_n}$ be the constant [[Definition:Sequence|sequence]]:
:$\forall n \in \N: x_n = \lambda$
Given $\epsilon \in \R_{>0}$:
:$\forall n \ge 1: \norm {x_n - \lambda} = \norm {\lambda - \lambda} = \norm 0 = 0 < \epsilon$
The result follows.
{{qed}}
[[Category:Sequences]]
[[Category:Normed Division Rings]... | Constant Sequence Converges to Constant in Normed Division Ring | https://proofwiki.org/wiki/Constant_Sequence_Converges_to_Constant_in_Normed_Division_Ring | https://proofwiki.org/wiki/Constant_Sequence_Converges_to_Constant_in_Normed_Division_Ring | [
"Sequences",
"Normed Division Rings"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Sequence",
"Category:Sequences",
"Category:Normed Division Rings"
] |
proofwiki-14928 | Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0$.
Let $\sequence {x_n}$ and $\sequence {y_n} $ be sequences in $R$.
Let $\sequence {x_n}$ converge to $0$.
Let $\sequence {y_n}$ be a Cauchy sequence.
Then:
:$\sequence {x_n y_n}$ and $\sequence {y_n x_n}$ converge to $0$. | By Cauchy Sequence in Normed Division Ring is Bounded:
:$\exists M \in \R_{>0}: \forall n, \norm {x_n} \le M$
Given $\epsilon > 0$.
Since $\sequence {x_n}$ converges to $0$ then:
:$\exists N \in \N: \forall n > N, \norm {x_n} < \dfrac \epsilon M$
Hence:
{{begin-eqn}}
{{eqn | l = \norm {x_n y_n - 0}
| r = \norm {... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]] with [[Definition:Ring Zero|zero]] $0$.
Let $\sequence {x_n}$ and $\sequence {y_n} $ be [[Definition:Sequence|sequences in $R$]].
Let $\sequence {x_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converge]... | By [[Cauchy Sequence in Normed Division Ring is Bounded]]:
:$\exists M \in \R_{>0}: \forall n, \norm {x_n} \le M$
Given $\epsilon > 0$.
Since $\sequence {x_n}$ [[Definition:Convergent Sequence in Normed Division Ring|converges]] to $0$ then:
:$\exists N \in \N: \forall n > N, \norm {x_n} < \dfrac \epsilon M$
Hence:... | Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero | https://proofwiki.org/wiki/Product_of_Sequence_Converges_to_Zero_with_Cauchy_Sequence_Converges_to_Zero | https://proofwiki.org/wiki/Product_of_Sequence_Converges_to_Zero_with_Cauchy_Sequence_Converges_to_Zero | [
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Ring Zero",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Cauchy Sequence is Bounded/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Real",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] |
proofwiki-14929 | Cauchy Sequence is Bounded/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Every Cauchy sequence in $R$ is bounded. | Let $\sequence {x_n} $ be a Cauchy sequence in $R$.
Then by definition:
:$\forall \epsilon \in \R_{\gt 0}: \exists N \in \N : \forall n, m \ge N: \norm {x_n - x_m} < \epsilon$
Let $n_1$ satisfy:
:$\forall n, m \ge n_1: \norm {x_n - x_m} < 1$
Then $\forall n \ge n_1$:
{{begin-eqn}}
{{eqn | l = \norm {x_n}
| r = \n... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Every [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence in $R$]] is [[Definition:Bounded Sequence in Normed Division Ring|bounded]]. | Let $\sequence {x_n} $ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence in $R$]].
Then by definition:
:$\forall \epsilon \in \R_{\gt 0}: \exists N \in \N : \forall n, m \ge N: \norm {x_n - x_m} < \epsilon$
Let $n_1$ satisfy:
:$\forall n, m \ge n_1: \norm {x_n - x_m} < 1$
Then $\forall n \g... | Cauchy Sequence is Bounded/Normed Division Ring/Proof 1 | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Division_Ring | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Division_Ring/Proof_1 | [
"Cauchy Sequence is Bounded",
"Cauchy Sequences in Normed Division Rings",
"Cauchy Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] |
proofwiki-14930 | Cauchy Sequence is Bounded/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Every Cauchy sequence in $R$ is bounded. | Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$.
Let $\sequence {x_n} $ be a Cauchy sequence in $\struct {R, \norm {\,\cdot\,}}$.
By the definition of a Cauchy sequence in a normed division ring, $\sequence {x_n} $ is a Cauchy sequence in $\struct {R, d}$.
By Cauchy Sequence in Metric Space is Boun... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Every [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence in $R$]] is [[Definition:Bounded Sequence in Normed Division Ring|bounded]]. | Let $d$ be the [[Definition:Metric Induced by Norm on Division Ring|metric induced]] on $R$ be the [[Definition:Norm on Division Ring|norm]] $\norm {\,\cdot\,}$.
Let $\sequence {x_n} $ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in $\struct {R, \norm {\,\cdot\,}}$.
By the definition of... | Cauchy Sequence is Bounded/Normed Division Ring/Proof 2 | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Division_Ring | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Division_Ring/Proof_2 | [
"Cauchy Sequence is Bounded",
"Cauchy Sequences in Normed Division Rings",
"Cauchy Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] | [
"Definition:Metric Induced by Norm on Division Ring",
"Definition:Norm/Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Metric Space",
"Cauchy Sequence is Bounded/Metric Space",
"Definition:Bounded Sequence/... |
proofwiki-14931 | Cauchy Sequence is Bounded/Normed Division Ring | Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring.
Every Cauchy sequence in $R$ is bounded. | Let $\sequence {x_n} $ be a Cauchy sequence in $R$.
By Norm Sequence of Cauchy Sequence has Limit, $\sequence {\norm {x_n} }$ is a convergent sequence in $\R$.
By Convergent Real Sequence is Bounded, $\sequence {\norm {x_n} }$ is bounded.
That is:
:$\exists M \in \R_{\gt 0}: \forall n \in \N: \norm {x_n} = \size {\norm... | Let $\struct {R, \norm {\,\cdot\,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Every [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence in $R$]] is [[Definition:Bounded Sequence in Normed Division Ring|bounded]]. | Let $\sequence {x_n} $ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence in $R$]].
By [[Norm Sequence of Cauchy Sequence has Limit]], $\sequence {\norm {x_n} }$ is a [[Definition:Convergent Real Sequence|convergent sequence in $\R$]].
By [[Convergent Real Sequence is Bounded]], $\sequence {\no... | Cauchy Sequence is Bounded/Normed Division Ring/Proof 3 | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Division_Ring | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Normed_Division_Ring/Proof_3 | [
"Cauchy Sequence is Bounded",
"Cauchy Sequences in Normed Division Rings",
"Cauchy Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Bounded Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring",
"Norm Sequence of Cauchy Sequence has Limit",
"Definition:Convergent Sequence/Real Numbers",
"Convergent Real Sequence is Bounded",
"Definition:Bounded Sequence/Real",
"Definition:Bounded Sequence/Normed Division Ring"
] |
proofwiki-14932 | Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $N \in \N$
Let $\sequence {y_n}$ be the sequence defined by:
:$\forall n, y_n = x_{N + n}$
Let $\sequence {y_n}$ be a Cauchy sequence in $R$.
Then:
:$\sequence {x_n}$ is a Cauchy sequence in $R$. | Given $\epsilon > 0$:
By the definition of a Cauchy sequence then:
:$\exists N': \forall n, m > N', \norm {y_n - y_m} < \epsilon$
Hence $\forall n, m > \paren {N' + N}$:
{{begin-eqn}}
{{eqn | l = \norm {x_n - x_m }
| r = \norm {y_{n - N} - y_{m - N} }
| c = $n, m > N$
}}
{{eqn | o = <
| r = \epsilon
... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $R$.
Let $N \in \N$
Let $\sequence {y_n}$ be the [[Definition:Sequence|sequence]] defined by:
:$\forall n, y_n = x_{N + n}$
Let $\sequence {y_n}$ be a... | Given $\epsilon > 0$:
By the definition of a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] then:
:$\exists N': \forall n, m > N', \norm {y_n - y_m} < \epsilon$
Hence $\forall n, m > \paren {N' + N}$:
{{begin-eqn}}
{{eqn | l = \norm {x_n - x_m }
| r = \norm {y_{n - N} - y_{m - N} }
... | Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence | https://proofwiki.org/wiki/Cauchy_Sequence_with_Finite_Elements_Prepended_is_Cauchy_Sequence | https://proofwiki.org/wiki/Cauchy_Sequence_with_Finite_Elements_Prepended_is_Cauchy_Sequence | [
"Cauchy Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-14933 | Convergent Sequence with Finite Elements Prepended is Convergent Sequence | Let $\struct {R, \norm { \, \cdot \, } }$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Let $N \in \N$.
Let $\sequence {y_n}$ be the sequence defined by:
:$\forall n \in \N: y_n = x_{N + n}$
Let $\sequence {y_n}$ be a convergent sequence in $R$ with limit $l$.
Then:
:$\sequence {x_n}$ is a conv... | Let $\epsilon \in \R_{>0}$ be given.
By the definition of a convergent sequence in $R$ with limit $l$:
:$\exists N' \in \R_{>0}: \forall n \in \N: n > N' \implies \norm {y_n - l} < \epsilon$
Hence:
{{begin-eqn}}
{{eqn | q = \forall n > \paren {N' + N}
| l = \norm {x_n - l}
| r = \norm {y_{n - N} - l}
... | Let $\struct {R, \norm { \, \cdot \, } }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $R$.
Let $N \in \N$.
Let $\sequence {y_n}$ be the [[Definition:Sequence|sequence]] defined by:
:$\forall n \in \N: y_n = x_{N + n}$
Let $\sequence {... | Let $\epsilon \in \R_{>0}$ be given.
By the definition of a [[Definition:Convergent Sequence in Normed Division Ring|convergent sequence]] in $R$ with limit $l$:
:$\exists N' \in \R_{>0}: \forall n \in \N: n > N' \implies \norm {y_n - l} < \epsilon$
Hence:
{{begin-eqn}}
{{eqn | q = \forall n > \paren {N' + N}
... | Convergent Sequence with Finite Elements Prepended is Convergent Sequence | https://proofwiki.org/wiki/Convergent_Sequence_with_Finite_Elements_Prepended_is_Convergent_Sequence | https://proofwiki.org/wiki/Convergent_Sequence_with_Finite_Elements_Prepended_is_Convergent_Sequence | [
"Convergent Sequences in Normed Division Rings",
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Sequence",
"Definition:Sequence",
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring"
] |
proofwiki-14934 | Maximal Left and Right Ideal iff Quotient Ring is Division Ring | Let $R$ be a ring with unity.
Let $J$ be an ideal of $R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$J$ is a maximal left ideal}}
{{item|(2):|$J$ is a maximal right ideal}}
{{item|(3):|the quotient ring $R / J$ is a division ring}}
{{end-itemize}} | === Maximal Left Ideal implies Quotient Ring is Division Ring ===
{{:Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring}}{{qed|lemma}} | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
{{TFAE}}
{{begin-itemize}}
{{item|(1):|$J$ is a [[Definition:Maximal Left Ideal of Ring|maximal left ideal]]}}
{{item|(2):|$J$ is a [[Definition:Maximal Right Ideal of Ring|maximal right ideal]]}}
{{i... | === [[Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring|Maximal Left Ideal implies Quotient Ring is Division Ring]] ===
{{:Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring}}{{... | Maximal Left and Right Ideal iff Quotient Ring is Division Ring | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring | [
"Quotient Rings",
"Maximal Ideals of Rings"
] | [
"Definition:Ring with Unity",
"Definition:Ideal of Ring",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Maximal Ideal of Ring/Right",
"Definition:Quotient Ring",
"Definition:Division Ring"
] | [
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring",
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring"
] |
proofwiki-14935 | Definition:Constructed Semantics/Instance 2/Rule of Addition | The Rule of Addition:
:$q \implies (q \lor p)$
is a tautology in Instance 2 of constructed semantics. | {{handwaving}}
By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Addition can be written as:
: $\neg q \lor \left({p \lor q}\right)$
This evaluates as follows:
:$\begin{array}{|cc|c|ccc|} \hline
\neg & q & \lor & (p & \lor & q)... | The [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]]:
:$q \implies (q \lor p)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 2|Instance 2]] of [[Definition:Constructed Semantics|constructed semantics]]. | {{handwaving}}
By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]] can be written as:
: $\neg ... | Definition:Constructed Semantics/Instance 2/Rule of Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_2/Rule_of_Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_2/Rule_of_Addition | [
"Formal Semantics"
] | [
"Rule of Addition/Sequent Form/Formulation 2/Form 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 2",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Addition/Sequent Form/Formulation 2/Form 2"
] |
proofwiki-14936 | Definition:Constructed Semantics/Instance 2/Rule of Commutation | The Rule of Commutation:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a tautology in Instance 2 of constructed semantics. | {{handwaving}}
By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Commutation can be written as:
:$\neg \left({p \lor q}\right) \lor \left({q \lor p}\right)$
This evaluates as follows:
:$\begin{array}{|cccc|c|ccc|} \hline
\neg &... | The [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]]:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 2|Instance 2]] of [[Definition:Constructed Semantics|cons... | {{handwaving}}
By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]] can be wri... | Definition:Constructed Semantics/Instance 2/Rule of Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_2/Rule_of_Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_2/Rule_of_Commutation | [
"Formal Semantics"
] | [
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 2",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication"
] |
proofwiki-14937 | Definition:Constructed Semantics/Instance 2/Factor Principle | The Factor Principle:
:$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$
is a tautology in Instance 2 of constructed semantics. | {{handwaving}}
By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Factor Principle can be written as:
:$\neg \left({\neg p \lor q}\right) \lor \left({\neg \left({r \lor p}\right) \lor \left ({r \lor q}\right)}\right)$
This evaluates as ... | The [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]]:
:$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 2|Instance 2]] of [[D... | {{handwaving}}
By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]] can be written as:
:$\neg ... | Definition:Constructed Semantics/Instance 2/Factor Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_2/Factor_Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_2/Factor_Principle | [
"Formal Semantics"
] | [
"Factor Principles/Disjunction on Left/Formulation 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 2",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Factor Principles/Disjunction on Left/Formulation 2"
] |
proofwiki-14938 | Test for Left Ideal | Let $J$ be a subset of a ring $\struct {R, +, \circ}$.
Then $J$ is an left ideal of $\struct {R, +, \circ}$ {{iff}} these all hold:
:$(1): \quad J \ne \O$
:$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$
:$(3): \quad \forall j \in J, r \in R: r \circ j \in J$ | === Necessary Condition ===
Let $J$ be a left ideal of $\struct {R, +, \circ}$.
Then conditions $(1)$ to $(3)$ hold by virtue of the ring axioms and $J$ being a left ideal.
{{qed|lemma}} | Let $J$ be a [[Definition:Subset|subset]] of a [[Definition:Ring (Abstract Algebra)|ring]] $\struct {R, +, \circ}$.
Then $J$ is an [[Definition:Left Ideal of Ring|left ideal]] of $\struct {R, +, \circ}$ {{iff}} these all hold:
:$(1): \quad J \ne \O$
:$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$
:$(3): \qu... | === Necessary Condition ===
Let $J$ be a [[Definition:Left Ideal of Ring|left ideal]] of $\struct {R, +, \circ}$.
Then conditions $(1)$ to $(3)$ hold by virtue of the [[Axiom:Ring Axioms|ring axioms]] and $J$ being a [[Definition:Left Ideal of Ring|left ideal]].
{{qed|lemma}} | Test for Left Ideal | https://proofwiki.org/wiki/Test_for_Left_Ideal | https://proofwiki.org/wiki/Test_for_Left_Ideal | [
"Ideal Theory"
] | [
"Definition:Subset",
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring/Left Ideal"
] | [
"Definition:Ideal of Ring/Left Ideal",
"Axiom:Ring Axioms",
"Definition:Ideal of Ring/Left Ideal",
"Definition:Ideal of Ring/Left Ideal"
] |
proofwiki-14939 | Test for Right Ideal | Let $J$ be a subset of a ring $\struct {R, +, \circ}$.
Then $J$ is a right ideal of $\struct {R, +, \circ}$ {{iff}} these all hold:
:$(1): \quad J \ne \O$
:$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$
:$(3): \quad \forall j \in J, r \in R: j \circ r \in J$ | === Necessary Condition ===
Let $J$ be a right ideal of $\struct {R, +, \circ}$.
Then conditions $(1)$ to $(3)$ hold by virtue of the ring axioms and $J$ being a right ideal.
{{qed|lemma}} | Let $J$ be a [[Definition:Subset|subset]] of a [[Definition:Ring (Abstract Algebra)|ring]] $\struct {R, +, \circ}$.
Then $J$ is a [[Definition:Right Ideal of Ring|right ideal]] of $\struct {R, +, \circ}$ {{iff}} these all hold:
:$(1): \quad J \ne \O$
:$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$
:$(3): \q... | === Necessary Condition ===
Let $J$ be a [[Definition:Right Ideal of Ring|right ideal]] of $\struct {R, +, \circ}$.
Then conditions $(1)$ to $(3)$ hold by virtue of the [[Axiom:Ring Axioms|ring axioms]] and $J$ being a [[Definition:Right Ideal of Ring|right ideal]].
{{qed|lemma}} | Test for Right Ideal | https://proofwiki.org/wiki/Test_for_Right_Ideal | https://proofwiki.org/wiki/Test_for_Right_Ideal | [
"Ideal Theory"
] | [
"Definition:Subset",
"Definition:Ring (Abstract Algebra)",
"Definition:Ideal of Ring/Right Ideal"
] | [
"Definition:Ideal of Ring/Right Ideal",
"Axiom:Ring Axioms",
"Definition:Ideal of Ring/Right Ideal",
"Definition:Ideal of Ring/Right Ideal"
] |
proofwiki-14940 | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring | Let $R$ be a ring with unity.
Let $J$ be an ideal of $R$.
If $J$ is a maximal left ideal then the quotient ring $R / J$ is a division ring. | Since $J \subset R$, it follows from Quotient Ring of Ring with Unity is Ring with Unity that $R / J$ is a ring with unity.
We now need to prove that every non-zero element of $\struct {R / J, +, \circ}$ has an inverse for $\circ$ in $R / J$.
By Left Inverse for All is Right Inverse it is sufficient to show that $\str... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
If $J$ is a [[Definition:Maximal Left Ideal of Ring|maximal left ideal]] then the [[Definition:Quotient Ring|quotient ring]] $R / J$ is a [[Definition:Division Ring|division ring]]. | Since $J \subset R$, it follows from [[Quotient Ring of Ring with Unity is Ring with Unity]] that $R / J$ is a [[Definition:Ring with Unity|ring with unity]].
We now need to prove that every non-[[Definition:Ring Zero|zero element]] of $\struct {R / J, +, \circ}$ has an [[Definition:Inverse Element|inverse]] for $\c... | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Maximal_Left_Ideal_implies_Quotient_Ring_is_Division_Ring | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Maximal_Left_Ideal_implies_Quotient_Ring_is_Division_Ring | [
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring"
] | [
"Definition:Ring with Unity",
"Definition:Ideal of Ring",
"Definition:Maximal Ideal of Ring/Left",
"Definition:Quotient Ring",
"Definition:Division Ring"
] | [
"Quotient Ring of Ring with Unity is Ring with Unity",
"Definition:Ring with Unity",
"Definition:Ring Zero",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Left Inverse for All is Right Inverse",
"Definition:Inverse (Abstract Algebra)/Left Inverse",
"Definition:Ring Zero",
"Definition:Ring Zero",
... |
proofwiki-14941 | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Right Ideal implies Quotient Ring is Division Ring | Let $R$ be a ring with unity.
Let $J$ be an ideal of $R$.
If $J$ is a maximal right ideal then the quotient ring $R / J$ is a division ring. | Since $J \subset R$, it follows from Quotient Ring of Ring with Unity is Ring with Unity that $R / J$ is a ring with unity.
We now need to prove that every non-zero element of $\struct {R / J, +, \circ}$ has an inverse for $\circ$ in $R / J$.
By Right Inverse for All is Left Inverse it is sufficient to show that $\str... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
If $J$ is a [[Definition:Maximal Right Ideal of Ring|maximal right ideal]] then the [[Definition:Quotient Ring|quotient ring]] $R / J$ is a [[Definition:Division Ring|division ring]]. | Since $J \subset R$, it follows from [[Quotient Ring of Ring with Unity is Ring with Unity]] that $R / J$ is a [[Definition:Ring with Unity|ring with unity]].
We now need to prove that every non-zero element of $\struct {R / J, +, \circ}$ has an [[Definition:Inverse Element|inverse]] for $\circ$ in $R / J$.
By [[Ri... | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Right Ideal implies Quotient Ring is Division Ring | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Maximal_Right_Ideal_implies_Quotient_Ring_is_Division_Ring | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Maximal_Right_Ideal_implies_Quotient_Ring_is_Division_Ring | [
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring"
] | [
"Definition:Ring with Unity",
"Definition:Ideal of Ring",
"Definition:Maximal Ideal of Ring/Right",
"Definition:Quotient Ring",
"Definition:Division Ring"
] | [
"Quotient Ring of Ring with Unity is Ring with Unity",
"Definition:Ring with Unity",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Right Inverse for All is Left Inverse",
"Definition:Right Inverse",
"Definition:Ring Zero",
"Definition:Subset",
"Test for Ideal",
"Definition:Ideal of Ring",
"Tes... |
proofwiki-14942 | Inverse of Injective and Surjective Mapping is Mapping | Let $f: S \to T$ be a mapping such that:
:$(1): \quad f$ is an injection
:$(2): \quad f$ is a surjection.
Then the inverse $f^{-1}$ of $f$ is itself a mapping. | Recall the definition of the inverse of $f$:
$f^{-1} \subseteq T \times S$ is the relation defined as:
:$f^{-1} = \set {\tuple {t, s}: t = \map f s}$
Let $f: S \to T$ be a mapping such that:
::$(1): \quad f$ is an injection
::$(2): \quad f$ is a surjection.
By Inverse of Injection is Many-to-One Relation, $f^{-1}$ is m... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that:
:$(1): \quad f$ is an [[Definition:Injection|injection]]
:$(2): \quad f$ is a [[Definition:Surjection|surjection]].
Then the [[Definition:Inverse of Mapping|inverse]] $f^{-1}$ of $f$ is itself a [[Definition:Mapping|mapping]]. | Recall the definition of the [[Definition:Inverse of Mapping|inverse of $f$]]:
$f^{-1} \subseteq T \times S$ is the [[Definition:Relation|relation]] defined as:
:$f^{-1} = \set {\tuple {t, s}: t = \map f s}$
Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that:
::$(1): \quad f$ is an [[Definition:Injection... | Inverse of Injective and Surjective Mapping is Mapping/Proof 1 | https://proofwiki.org/wiki/Inverse_of_Injective_and_Surjective_Mapping_is_Mapping | https://proofwiki.org/wiki/Inverse_of_Injective_and_Surjective_Mapping_is_Mapping/Proof_1 | [
"Mapping is Injection and Surjection iff Inverse is Mapping"
] | [
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Inverse of Mapping",
"Definition:Mapping"
] | [
"Definition:Inverse of Mapping",
"Definition:Relation",
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Inverse of Injection is Many-to-One Relation",
"Definition:Many-to-One Relation",
"Inverse of Surjection is Relation both Left-Total and Right-Total",
"Definition:Left-Tota... |
proofwiki-14943 | Inverse of Injective and Surjective Mapping is Mapping | Let $f: S \to T$ be a mapping such that:
:$(1): \quad f$ is an injection
:$(2): \quad f$ is a surjection.
Then the inverse $f^{-1}$ of $f$ is itself a mapping. | Let $f: S \to T$ be a mapping such that:
::$(1): \quad f$ is an injection
::$(2): \quad f$ is a surjection.
Let $t \in T$.
Then as $f$ is a surjection:
:$\exists s \in S: t = \map f s$
As $f$ is an injection, there is only one $s \in S$ such that $t = \map f s$.
Define $\map g t = s$.
As $t \in T$ is arbitrary, it foll... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that:
:$(1): \quad f$ is an [[Definition:Injection|injection]]
:$(2): \quad f$ is a [[Definition:Surjection|surjection]].
Then the [[Definition:Inverse of Mapping|inverse]] $f^{-1}$ of $f$ is itself a [[Definition:Mapping|mapping]]. | Let $f: S \to T$ be a [[Definition:Mapping|mapping]] such that:
::$(1): \quad f$ is an [[Definition:Injection|injection]]
::$(2): \quad f$ is a [[Definition:Surjection|surjection]].
Let $t \in T$.
Then as $f$ is a [[Definition:Surjection|surjection]]:
:$\exists s \in S: t = \map f s$
As $f$ is an [[Definition:Inje... | Inverse of Injective and Surjective Mapping is Mapping/Proof 2 | https://proofwiki.org/wiki/Inverse_of_Injective_and_Surjective_Mapping_is_Mapping | https://proofwiki.org/wiki/Inverse_of_Injective_and_Surjective_Mapping_is_Mapping/Proof_2 | [
"Mapping is Injection and Surjection iff Inverse is Mapping"
] | [
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Inverse of Mapping",
"Definition:Mapping"
] | [
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection",
"Definition:Surjection",
"Definition:Injection",
"Definition:Unique",
"Definition:Mapping",
"Definition:Injection",
"Definition:Inverse Mapping"
] |
proofwiki-14944 | Inverse is Mapping implies Mapping is Injection and Surjection | Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping.
Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping.
Then:
:$(1): \quad f$ is an injection
:$(2): \quad f$ is a surjection. | This is divided into two parts: | Let $S$ and $T$ be [[Definition:Set|sets]].
Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let the [[Definition:Inverse of Mapping|inverse]] $f^{-1} \subseteq T \times S$ itself be a [[Definition:Mapping|mapping]].
Then:
:$(1): \quad f$ is an [[Definition:Injection|injection]]
:$(2): \quad f$ is a [[Definiti... | This is divided into two parts: | Inverse is Mapping implies Mapping is Injection and Surjection | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Injection_and_Surjection | https://proofwiki.org/wiki/Inverse_is_Mapping_implies_Mapping_is_Injection_and_Surjection | [
"Mapping is Injection and Surjection iff Inverse is Mapping"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Mapping",
"Definition:Injection",
"Definition:Surjection"
] | [] |
proofwiki-14945 | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Right Ideal | Let $R$ be a ring with unity.
Let $J$ be an ideal of $R$.
If the quotient ring $R / J$ is a division ring then $J$ is a maximal right ideal. | Let $K$ be a right ideal of $R$ such that $J \subsetneq K \subset R$.
Let $x \in K \setminus J$.
As $x \notin J$ then $x + J \ne J$, the zero in $R / J$.
As $R / J$ is a division ring then $x + J \in R / J$ has an inverse, say $s + J$.
That is:
:$1_R + J = \paren {x + J} \circ \paren {s + J} = \paren {x \circ s} + J$
... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
If the [[Definition:Quotient Ring|quotient ring]] $R / J$ is a [[Definition:Division Ring|division ring]] then $J$ is a [[Definition:Maximal Right Ideal of Ring|maximal right ideal]]. | Let $K$ be a [[Definition:Right Ideal of Ring|right ideal]] of $R$ such that $J \subsetneq K \subset R$.
Let $x \in K \setminus J$.
As $x \notin J$ then $x + J \ne J$, the [[Definition:Ring Zero|zero]] in $R / J$.
As $R / J$ is a [[Definition:Division Ring|division ring]] then $x + J \in R / J$ has an [[Definition:... | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Right Ideal | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Quotient_Ring_is_Division_Ring_implies_Maximal_Right_Ideal | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Quotient_Ring_is_Division_Ring_implies_Maximal_Right_Ideal | [
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring"
] | [
"Definition:Ring with Unity",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Division Ring",
"Definition:Maximal Ideal of Ring/Right"
] | [
"Definition:Ideal of Ring/Right Ideal",
"Definition:Ring Zero",
"Definition:Division Ring",
"Definition:Product Inverse",
"Left Cosets are Equal iff Product with Inverse in Subgroup",
"Definition:Ideal of Ring/Right Ideal"
] |
proofwiki-14946 | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Left Ideal | Let $R$ be a ring with unity.
Let $J$ be an ideal of $R$.
If the quotient ring $R / J$ is a division ring then $J$ is a maximal left ideal. | Let $K$ be a left ideal of $R$ such that $J \subsetneq K \subset R$.
Let $x \in K \setminus J$.
As $x \notin J$, then $x + J \ne J$, the zero in $R / J$.
As $R / J$ is a division ring, $x + J \in R / J$ has an inverse, say $s + J$.
That is:
:$1_R + J = \paren {s + J} \circ \paren {x + J} = \paren {s \circ x} + J$
By L... | Let $R$ be a [[Definition:Ring with Unity|ring with unity]].
Let $J$ be an [[Definition:Ideal of Ring|ideal]] of $R$.
If the [[Definition:Quotient Ring|quotient ring]] $R / J$ is a [[Definition:Division Ring|division ring]] then $J$ is a [[Definition:Maximal Left Ideal of Ring|maximal left ideal]]. | Let $K$ be a [[Definition:Left Ideal of Ring|left ideal]] of $R$ such that $J \subsetneq K \subset R$.
Let $x \in K \setminus J$.
As $x \notin J$, then $x + J \ne J$, the [[Definition:Ring Zero|zero]] in $R / J$.
As $R / J$ is a [[Definition:Division Ring|division ring]], $x + J \in R / J$ has an [[Definition:Invers... | Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Left Ideal | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Quotient_Ring_is_Division_Ring_implies_Maximal_Left_Ideal | https://proofwiki.org/wiki/Maximal_Left_and_Right_Ideal_iff_Quotient_Ring_is_Division_Ring/Quotient_Ring_is_Division_Ring_implies_Maximal_Left_Ideal | [
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring"
] | [
"Definition:Ring with Unity",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Division Ring",
"Definition:Maximal Ideal of Ring/Left"
] | [
"Definition:Ideal of Ring/Left Ideal",
"Definition:Ring Zero",
"Definition:Division Ring",
"Definition:Inverse (Abstract Algebra)/Inverse",
"Left Cosets are Equal iff Product with Inverse in Subgroup",
"Definition:Ideal of Ring/Left Ideal"
] |
proofwiki-14947 | Quotient Ring of Cauchy Sequences is Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\CC$ be the ring of Cauchy sequences over $R$.
Let $\NN$ be the set of null sequences.
Then the quotient ring $\CC / \NN$ is a division ring. | By Null Sequences form Maximal Left and Right Ideal then $\NN$ is an ideal of the ring $\CC$ that is also a maximal left ideal.
By Maximal Left and Right Ideal iff Quotient Ring is Division Ring then the quotient ring $\CC / \NN$ is a division ring
{{qed}} | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|ring of Cauchy sequences over $R$]].
Let $\NN$ be the [[Definition:Set|set]] of [[Definition:Null Sequence in Normed Division Ring|null sequences]].
Then the [... | By [[Null Sequences form Maximal Left and Right Ideal]] then $\NN$ is an [[Definition:Ideal of Ring|ideal]] of the [[Definition:Ring of Sequences|ring $\CC$]] that is also a [[Definition:Maximal Left Ideal of Ring|maximal left ideal]].
By [[Maximal Left and Right Ideal iff Quotient Ring is Division Ring]] then the [[D... | Quotient Ring of Cauchy Sequences is Division Ring | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Division_Ring | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Division_Ring | [
"Cauchy Sequences",
"Normed Division Rings",
"Completion of Normed Division Ring",
"Rings of Sequences"
] | [
"Definition:Normed Division Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Quotient Ring",
"Definition:Division Ring"
] | [
"Null Sequences form Maximal Left and Right Ideal",
"Definition:Ideal of Ring",
"Definition:Ring of Sequences",
"Definition:Maximal Ideal of Ring/Left",
"Maximal Left and Right Ideal iff Quotient Ring is Division Ring",
"Definition:Quotient Ring",
"Definition:Division Ring"
] |
proofwiki-14948 | Hilbert Proof System Instance 2 Independence Results/Independence of A1 | Axiom $(A1)$ is independent from $(A2)$, $(A3)$, $(A4)$. | Denote with $\mathscr H_2 - (A1)$ the proof system resulting from $\mathscr H_2$ by removing axiom $(A1)$.
Consider $\mathscr C_2$, Instance 2 of constructed semantics.
We will prove that:
* $\mathscr H_2 - (A1)$ is sound for $\mathscr C_2$;
* Axiom $(A1)$ is not a tautology in $\mathscr C_2$
which leads to the conclus... | [[Definition:Axiom (Formal Systems)|Axiom]] $(A1)$ is [[Definition:Independent Axiom|independent]] from $(A2)$, $(A3)$, $(A4)$. | Denote with $\mathscr H_2 - (A1)$ the [[Definition:Proof System|proof system]] resulting from $\mathscr H_2$ by removing [[Definition:Axiom (Formal Systems)|axiom]] $(A1)$.
Consider $\mathscr C_2$, [[Definition:Constructed Semantics/Instance 2|Instance 2]] of [[Definition:Constructed Semantics|constructed semantics]].... | Hilbert Proof System Instance 2 Independence Results/Independence of A1 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A1 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A1 | [
"Hilbert Proof System Instance 2"
] | [
"Definition:Axiom/Formal Systems",
"Definition:Independent Axiom"
] | [
"Definition:Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Constructed Semantics/Instance 2",
"Definition:Constructed Semantics",
"Definition:Sound Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Tautology/Formal Semantics",
"Definition:Theorem/Formal System",
"Definitio... |
proofwiki-14949 | Embedding Ring into Ring Structure Induced by Ring Operations | Let $\struct {R, +, \circ}$ be a ring.
Let $S$ be a non-empty set.
Let $\struct {R^S, +', \circ'}$ be the ring of mappings, where $+'$ and $\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\circ$.
For each $r \in R$, let $f_r: S \to R$ be the mapping defined by:
:$\forall s \in S, \map {f_r} s = r$
Tha... | By the definition of a ring monomorphism it is sufficient to prove for all $r, r' \in R$ that:
:$\quad \map \phi {r + r'} = \map \phi r +' \map \phi {r'}$
:$\quad \map \phi {r \circ r'} = \map \phi r \circ' \map \phi {r'}$
:$\quad r \ne r' \implies \map \phi r \ne \map \phi {r'}$
That is, for all $r, r' \in R$ it needs... | Let $\struct {R, +, \circ}$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $S$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\struct {R^S, +', \circ'}$ be the [[Definition:Ring of Mappings|ring of mappings]], where $+'$ and $\circ'$ are the [[Definition:Pointwise Operation|pointwise operations induced]... | By the definition of a [[Definition:Ring Monomorphism|ring monomorphism]] it is sufficient to prove for all $r, r' \in R$ that:
:$\quad \map \phi {r + r'} = \map \phi r +' \map \phi {r'}$
:$\quad \map \phi {r \circ r'} = \map \phi r \circ' \map \phi {r'}$
:$\quad r \ne r' \implies \map \phi r \ne \map \phi {r'}$
That ... | Embedding Ring into Ring Structure Induced by Ring Operations | https://proofwiki.org/wiki/Embedding_Ring_into_Ring_Structure_Induced_by_Ring_Operations | https://proofwiki.org/wiki/Embedding_Ring_into_Ring_Structure_Induced_by_Ring_Operations | [
"Rings of Mappings",
"Ring Monomorphisms"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Non-Empty Set",
"Definition:Ring of Mappings",
"Definition:Pointwise Operation",
"Definition:Mapping",
"Definition:Constant Mapping",
"Definition:Mapping",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Monomorphism"
] | [
"Definition:Ring Monomorphism"
] |
proofwiki-14950 | Embedding Normed Division Ring into Ring of Cauchy Sequences | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\CC$ be the ring of Cauchy sequences over $R$
Let $\phi: R \to \CC$ be the mapping from $R$ to $\CC$ defined as:
:$\forall a \in R: \map \phi a = \tuple {a, a, a, \dots}$
where $\tuple {a, a, a, \dots}$ is the constant sequence.
Then $\phi$ is a ri... | By Cauchy Sequences form Ring with Unity, $\CC$ is a subring of the ring of sequences over $R$.
Let $i: \CC \to R^\N$ be the inclusion mapping of $\CC$ into the ring of sequences.
By Embedding Ring into Ring Structure Induced by Ring Operations the composition $i \circ \phi: R \to R^\N$ is a ring monomorphism.
Since f... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|ring of Cauchy sequences over $R$]]
Let $\phi: R \to \CC$ be the [[Definition:Mapping|mapping]] from $R$ to $\CC$ defined as:
:$\forall a \in R: \map \phi a = \... | By [[Cauchy Sequences form Ring with Unity]], $\CC$ is a [[Definition:Subring|subring]] of the [[Definition:Ring of Sequences|ring of sequences]] over $R$.
Let $i: \CC \to R^\N$ be the [[Definition:Inclusion Mapping|inclusion mapping]] of $\CC$ into the [[Definition:Ring of Sequences|ring of sequences]].
By [[Embedd... | Embedding Normed Division Ring into Ring of Cauchy Sequences | https://proofwiki.org/wiki/Embedding_Normed_Division_Ring_into_Ring_of_Cauchy_Sequences | https://proofwiki.org/wiki/Embedding_Normed_Division_Ring_into_Ring_of_Cauchy_Sequences | [
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Mapping",
"Definition:Constant Sequence",
"Definition:Ring Monomorphism"
] | [
"Cauchy Sequences form Ring with Unity",
"Definition:Subring",
"Definition:Ring of Sequences",
"Definition:Inclusion Mapping",
"Definition:Ring of Sequences",
"Embedding Ring into Ring Structure Induced by Ring Operations",
"Definition:Composition",
"Definition:Ring Monomorphism"
] |
proofwiki-14951 | Hilbert Proof System Instance 2 Independence Results/Independence of A2 | Axiom $(A2)$ is independent from $(A1)$, $(A3)$, $(A4)$. | Denote with $\mathscr H_2 - (A2)$ the proof system resulting from $\mathscr H_2$ by removing axiom $(A2)$.
Consider $\mathscr C_3$, Instance 3 of constructed semantics.
We will prove that:
* $\mathscr H_2 - (A2)$ is sound for $\mathscr C_3$;
* Axiom $(A2)$ is not a tautology in $\mathscr C_3$
which leads to the conclus... | [[Definition:Axiom (Formal Systems)|Axiom]] $(A2)$ is [[Definition:Independent Axiom|independent]] from $(A1)$, $(A3)$, $(A4)$. | Denote with $\mathscr H_2 - (A2)$ the [[Definition:Proof System|proof system]] resulting from $\mathscr H_2$ by removing [[Definition:Axiom (Formal Systems)|axiom]] $(A2)$.
Consider $\mathscr C_3$, [[Definition:Constructed Semantics/Instance 3|Instance 3]] of [[Definition:Constructed Semantics|constructed semantics]].... | Hilbert Proof System Instance 2 Independence Results/Independence of A2 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A2 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A2 | [
"Hilbert Proof System Instance 2"
] | [
"Definition:Axiom/Formal Systems",
"Definition:Independent Axiom"
] | [
"Definition:Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Constructed Semantics/Instance 3",
"Definition:Constructed Semantics",
"Definition:Sound Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Tautology/Formal Semantics",
"Definition:Theorem/Formal System",
"Definitio... |
proofwiki-14952 | Definition:Constructed Semantics/Instance 3/Rule of Idempotence | The Rule of Idempotence:
:$(p \lor p) \implies p$
is a tautology in Instance 3 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Idempotence can be written as:
: $\neg \left({p \lor p}\right) \lor p$
This evaluates as follows:
:$\begin{array}{|cccc|c|c|} \hline
\neg & (p & \lor & p) & \lor & p \\
\hline
2... | The [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]]:
:$(p \lor p) \implies p$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 3|Instance 3]] of [[Definition:Constructed Semantics|constructed semantics]]. | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]] can be written as:
: $\n... | Definition:Constructed Semantics/Instance 3/Rule of Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_3/Rule_of_Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_3/Rule_of_Idempotence | [
"Formal Semantics"
] | [
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 3",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Category:Formal Semantics"
] |
proofwiki-14953 | Definition:Constructed Semantics/Instance 3/Rule of Commutation | The Rule of Commutation:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a tautology in Instance 3 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Commutation can be written as:
:$\neg \left({p \lor q}\right) \lor \left({q \lor p}\right)$
This evaluates as follows:
:$\begin{array}{|cccc|c|ccc|} \hline
\neg & (p & \lor & q)... | The [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]]:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 3|Instance 3]] of [[Definition:Constructed Semantics|cons... | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]] can be written as:
:$\ne... | Definition:Constructed Semantics/Instance 3/Rule of Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_3/Rule_of_Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_3/Rule_of_Commutation | [
"Formal Semantics"
] | [
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 3",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication",
"Category:Formal Semantics"
] |
proofwiki-14954 | Definition:Constructed Semantics/Instance 3/Factor Principle | The Factor Principle:
:$\paren {p \implies q} \implies \paren {\paren {r \lor p} \implies \paren {r \lor q} }$
is a tautology in Instance 3 of constructed semantics. | {{handwaving}}
By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Factor Principle can be written as:
:$\neg \paren {\neg p \lor q} \lor \paren {\neg \paren {r \lor p} \lor \paren {r \lor q} }$
This evaluates as follows:
:$\begin{array}... | The [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]]:
:$\paren {p \implies q} \implies \paren {\paren {r \lor p} \implies \paren {r \lor q} }$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 3|Instance 3]] of [[Definition:Constructed Se... | {{handwaving}}
By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]] can be written as:
:$\neg ... | Definition:Constructed Semantics/Instance 3/Factor Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_3/Factor_Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_3/Factor_Principle | [
"Formal Semantics"
] | [
"Factor Principles/Disjunction on Left/Formulation 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 3",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Factor Principles/Disjunction on Left/Formulation 2",
"Category:Formal Semantics"
] |
proofwiki-14955 | Hilbert Proof System Instance 2 Independence Results/Independence of A3 | Axiom $(A3)$ is independent from $(A1)$, $(A2)$, $(A4)$. | {{tidy}}
Denote with $\mathscr H_2 - (A3)$ the proof system resulting from $\mathscr H_2$ by removing axiom $(A3)$.
Consider $\mathscr C_4$, Instance 4 of constructed semantics.
We will prove that:
* $\mathscr H_2 - (A3)$ is sound for $\mathscr C_4$;
* Axiom $(A3)$ is not a tautology in $\mathscr C_4$
which leads to th... | [[Definition:Axiom (Formal Systems)|Axiom]] $(A3)$ is [[Definition:Independent Axiom|independent]] from $(A1)$, $(A2)$, $(A4)$. | {{tidy}}
Denote with $\mathscr H_2 - (A3)$ the [[Definition:Proof System|proof system]] resulting from $\mathscr H_2$ by removing [[Definition:Axiom (Formal Systems)|axiom]] $(A3)$.
Consider $\mathscr C_4$, [[Definition:Constructed Semantics/Instance 4|Instance 4]] of [[Definition:Constructed Semantics|constructed se... | Hilbert Proof System Instance 2 Independence Results/Independence of A3 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A3 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A3 | [
"Hilbert Proof System Instance 2"
] | [
"Definition:Axiom/Formal Systems",
"Definition:Independent Axiom"
] | [
"Definition:Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Constructed Semantics/Instance 4",
"Definition:Constructed Semantics",
"Definition:Sound Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Tautology/Formal Semantics",
"Definition:Theorem/Formal System",
"Definitio... |
proofwiki-14956 | Definition:Constructed Semantics/Instance 4/Rule of Idempotence | The Rule of Idempotence:
:$(p \lor p) \implies p$
is a tautology in Instance 4 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Idempotence can be written as:
: $\neg \left({p \lor p}\right) \lor p$
This evaluates as follows:
:$\begin{array}{|cccc|c|c|} \hline
\neg & (p & \lor & p) & \lor & p \\
\hline
1... | The [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]]:
:$(p \lor p) \implies p$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 4|Instance 4]] of [[Definition:Constructed Semantics|constructed semantics]]. | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]] can be written as:
: $\n... | Definition:Constructed Semantics/Instance 4/Rule of Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_4/Rule_of_Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_4/Rule_of_Idempotence | [
"Formal Semantics"
] | [
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 4",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Category:Formal Semantics"
] |
proofwiki-14957 | Definition:Constructed Semantics/Instance 4/Rule of Addition | The Rule of Addition:
:$q \implies (q \lor p)$
is a tautology in Instance 4 of constructed semantics. | {{handwaving}}
By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Addition can be written as:
: $\neg q \lor \left({p \lor q}\right)$
This evaluates as follows:
:$\begin{array}{|cc|c|ccc|} \hline
\neg & q & \lor & (p & \lor & q)... | The [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]]:
:$q \implies (q \lor p)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 4|Instance 4]] of [[Definition:Constructed Semantics|constructed semantics]]. | {{handwaving}}
By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]] can be written as:
: $\neg ... | Definition:Constructed Semantics/Instance 4/Rule of Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_4/Rule_of_Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_4/Rule_of_Addition | [
"Formal Semantics"
] | [
"Rule of Addition/Sequent Form/Formulation 2/Form 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 4",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Addition/Sequent Form/Formulation 2/Form 2",
"Category:Formal Semantics"
] |
proofwiki-14958 | Definition:Constructed Semantics/Instance 4/Factor Principle | The Factor Principle:
{{:Factor Principles/Disjunction on Left/Formulation 2}}
:$\paren {p \implies q} \implies \paren {\paren {r \lor p} \implies \paren {r \lor q} }$
is a tautology in Instance 4 of constructed semantics. | {{handwaving}}
By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Factor Principle can be written as:
:$\neg \paren {\neg p \lor q} \lor \paren {\neg \paren {r \lor p} \lor \paren {r \lor q} }$
This evaluates as follows:
:$\begin{array}... | The [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]]:
{{:Factor Principles/Disjunction on Left/Formulation 2}}
:$\paren {p \implies q} \implies \paren {\paren {r \lor p} \implies \paren {r \lor q} }$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semanti... | {{handwaving}}
By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Factor Principles/Disjunction on Left/Formulation 2|Factor Principle]] can be written as:
:$\neg ... | Definition:Constructed Semantics/Instance 4/Factor Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_4/Factor_Principle | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_4/Factor_Principle | [
"Formal Semantics"
] | [
"Factor Principles/Disjunction on Left/Formulation 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 4",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Factor Principles/Disjunction on Left/Formulation 2",
"Category:Formal Semantics"
] |
proofwiki-14959 | Cardinality of Set of Surjections | Let $S$ and $T$ be finite sets.
Let $\card S = m, \card T = n$.
Let $C$ be the number of surjections from $S$ to $T$.
Then:
:$C = n! \ds {m \brace n}$
where $\ds {m \brace n}$ denotes a Stirling number of the second kind. | Let $T$ be the codomain of a surjection $f$ from $S$ to $T$.
By the Quotient Theorem for Surjections, $f$ induces an equivalence $\RR_f$ on $T$:
:$f = r \circ q_{\RR_f}$
where:
:$\RR_f$ is the equivalence induced by $f$ on $T$
:$r: S / \RR_f \to T$ is a bijection from the quotient set $S / \RR_f$ to $T$
:$q_{\RR_f}: S ... | Let $S$ and $T$ be [[Definition:Finite Set|finite sets]].
Let $\card S = m, \card T = n$.
Let $C$ be the number of [[Definition:Surjection|surjections]] from $S$ to $T$.
Then:
:$C = n! \ds {m \brace n}$
where $\ds {m \brace n}$ denotes a [[Definition:Stirling Numbers of the Second Kind|Stirling number of the second... | Let $T$ be the [[Definition:Codomain of Mapping|codomain]] of a [[Definition:Surjection|surjection]] $f$ from $S$ to $T$.
By the [[Quotient Theorem for Surjections]], $f$ [[Definition:Equivalence Relation Induced by Mapping|induces an equivalence]] $\RR_f$ on $T$:
:$f = r \circ q_{\RR_f}$
where:
:$\RR_f$ is the [[Def... | Cardinality of Set of Surjections | https://proofwiki.org/wiki/Cardinality_of_Set_of_Surjections | https://proofwiki.org/wiki/Cardinality_of_Set_of_Surjections | [
"Surjections",
"Combinatorics",
"Counting Arguments",
"Cardinality of Set of Surjections"
] | [
"Definition:Finite Set",
"Definition:Surjection",
"Definition:Stirling Numbers of the Second Kind"
] | [
"Definition:Codomain (Set Theory)/Mapping",
"Definition:Surjection",
"Quotient Theorem for Surjections",
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Equivalence Relation Induced by Mapping",
"Definition:Bijection",
"Definition:Quotient Set",
"Definition:Quotient Mapping",
"Fund... |
proofwiki-14960 | Hilbert Proof System Instance 2 Independence Results/Independence of A4 | Axiom $(\text A 4)$ is independent from $(\text A 1)$, $(\text A 2)$, $(\text A 3)$. | Denote with $\mathscr H_2 - (\text A 4)$ the proof system resulting from $\mathscr H_2$ by removing axiom $(\text A 4)$.
Consider $\mathscr C_5$, Instance 5 of constructed semantics.
We will prove that:
* $\mathscr H_2 - (\text A 4)$ is sound for $\mathscr C_5$;
* Axiom $(\text A 4)$ is not a tautology in $\mathscr C_5... | [[Definition:Axiom (Formal Systems)|Axiom]] $(\text A 4)$ is [[Definition:Independent Axiom|independent]] from $(\text A 1)$, $(\text A 2)$, $(\text A 3)$. | Denote with $\mathscr H_2 - (\text A 4)$ the [[Definition:Proof System|proof system]] resulting from $\mathscr H_2$ by removing [[Definition:Axiom (Formal Systems)|axiom]] $(\text A 4)$.
Consider $\mathscr C_5$, [[Definition:Constructed Semantics/Instance 5|Instance 5]] of [[Definition:Constructed Semantics|constructe... | Hilbert Proof System Instance 2 Independence Results/Independence of A4 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A4 | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/Independence_of_A4 | [
"Hilbert Proof System Instance 2"
] | [
"Definition:Axiom/Formal Systems",
"Definition:Independent Axiom"
] | [
"Definition:Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Constructed Semantics/Instance 5",
"Definition:Constructed Semantics",
"Definition:Sound Proof System",
"Definition:Axiom/Formal Systems",
"Definition:Tautology/Formal Semantics",
"Definition:Theorem/Formal System",
"Definitio... |
proofwiki-14961 | Image of Set Difference under Mapping/Corollary 3 | Let $f: S \to T$ be a surjection.
Let $A \subseteq S$ be a subset of $S$.
Then:
:$T \setminus f \sqbrk A \subseteq f \sqbrk {S \setminus A}$
where $\setminus$ denotes set difference. | As $T$ is a surjection, $T = f \sqbrk S$.
Thus Image of Set Difference under Mapping can be applied:
:$f \sqbrk S \setminus f \sqbrk A \subseteq f \sqbrk {S \setminus A}$
{{qed}} | Let $f: S \to T$ be a [[Definition:Surjection|surjection]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Then:
:$T \setminus f \sqbrk A \subseteq f \sqbrk {S \setminus A}$
where $\setminus$ denotes [[Definition:Set Difference|set difference]]. | As $T$ is a [[Definition:Surjection|surjection]], $T = f \sqbrk S$.
Thus [[Image of Set Difference under Mapping]] can be applied:
:$f \sqbrk S \setminus f \sqbrk A \subseteq f \sqbrk {S \setminus A}$
{{qed}} | Image of Set Difference under Mapping/Corollary 3 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Corollary_3 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Corollary_3 | [
"Image of Set Difference under Mapping"
] | [
"Definition:Surjection",
"Definition:Subset",
"Definition:Set Difference"
] | [
"Definition:Surjection",
"Image of Set Difference under Mapping"
] |
proofwiki-14962 | Definition:Constructed Semantics/Instance 5/Rule of Idempotence | The Rule of Idempotence:
:$(p \lor p) \implies p$
is a tautology in Instance 5 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Idempotence can be written as:
: $\neg \left({p \lor p}\right) \lor p$
This evaluates as follows:
:$\begin{array}{|cccc|c|c|} \hline
\neg & (p & \lor & p) & \lor & p \\
\hline
1... | The [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]]:
:$(p \lor p) \implies p$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 5|Instance 5]] of [[Definition:Constructed Semantics|constructed semantics]]. | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication|Rule of Idempotence]] can be written as:
: $\n... | Definition:Constructed Semantics/Instance 5/Rule of Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_5/Rule_of_Idempotence | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_5/Rule_of_Idempotence | [
"Formal Semantics"
] | [
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 5",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Idempotence/Disjunction/Formulation 2/Reverse Implication",
"Category:Formal Semantics"
] |
proofwiki-14963 | Additive Function is Linear for Rational Factors | Let $f: \R \to \R$ be an additive function.
Then:
:$\forall r \in \Q, x \in \R: \map f {x r} = r \map f x$ | Trivially, we have:
:$\forall x \in \R: \map f {1 \cdot x} = 1 \map f x$
Next, suppose that:
:$\map f {n x} = n \map f x$
By additivity of $f$, we have:
{{begin-eqn}}
{{eqn | l = \map f {\paren {n + 1} x}
| r = \map f {n x + x}
| c =
}}
{{eqn | r = \map f {n x} + \map f x = n \map f x + \map f x
| c ... | Let $f: \R \to \R$ be an [[Definition:Additive Function (Algebra)|additive function]].
Then:
:$\forall r \in \Q, x \in \R: \map f {x r} = r \map f x$ | Trivially, we have:
:$\forall x \in \R: \map f {1 \cdot x} = 1 \map f x$
Next, suppose that:
:$\map f {n x} = n \map f x$
By [[Definition:Additive Function (Algebra)|additivity]] of $f$, we have:
{{begin-eqn}}
{{eqn | l = \map f {\paren {n + 1} x}
| r = \map f {n x + x}
| c =
}}
{{eqn | r = \map f {n ... | Additive Function is Linear for Rational Factors | https://proofwiki.org/wiki/Additive_Function_is_Linear_for_Rational_Factors | https://proofwiki.org/wiki/Additive_Function_is_Linear_for_Rational_Factors | [
"Additive Functions"
] | [
"Definition:Additive Function (Algebra)"
] | [
"Definition:Additive Function (Algebra)",
"Principle of Mathematical Induction",
"Additive Function is Odd Function",
"Odd Function of Zero is Zero",
"Category:Additive Functions"
] |
proofwiki-14964 | Definition:Constructed Semantics/Instance 5/Rule of Addition | The Rule of Addition:
:$q \implies (q \lor p)$
is a tautology in Instance 5 of constructed semantics. | {{handwaving}}
By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Addition can be written as:
: $\neg q \lor \left({p \lor q}\right)$
This evaluates as follows:
:$\begin{array}{|cc|c|ccc|} \hline
\neg & q & \lor & (p & \lor & q)... | The [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]]:
:$q \implies (q \lor p)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 5|Instance 5]] of [[Definition:Constructed Semantics|constructed semantics]]. | {{handwaving}}
By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Addition/Sequent Form/Formulation 2/Form 2|Rule of Addition]] can be written as:
: $\neg ... | Definition:Constructed Semantics/Instance 5/Rule of Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_5/Rule_of_Addition | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_5/Rule_of_Addition | [
"Formal Semantics"
] | [
"Rule of Addition/Sequent Form/Formulation 2/Form 2",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 5",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Addition/Sequent Form/Formulation 2/Form 2",
"Category:Formal Semantics"
] |
proofwiki-14965 | Definition:Constructed Semantics/Instance 5/Rule of Commutation | The Rule of Commutation:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a tautology in Instance 5 of constructed semantics. | By the definitional abbreviation for the conditional:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the Rule of Commutation can be written as:
:$\neg \left({p \lor q}\right) \lor \left({q \lor p}\right)$
This evaluates as follows:
:$\begin{array}{|cccc|c|ccc|} \hline
\neg & (p & \lor & q)... | The [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]]:
:$\left({p \lor q}\right) \implies \left({q \lor p}\right)$
is a [[Definition:Tautology (Formal Semantics)|tautology]] in [[Definition:Constructed Semantics/Instance 5|Instance 5]] of [[Definition:Constructed Semantics|cons... | By the [[Definition:Definitional Abbreviation|definitional abbreviation]] for the [[Definition:Conditional|conditional]]:
:$\mathbf A \implies \mathbf B =_{\text{def}} \neg \mathbf A \lor \mathbf B$
the [[Rule of Commutation/Disjunction/Formulation 2/Forward Implication|Rule of Commutation]] can be written as:
:$\ne... | Definition:Constructed Semantics/Instance 5/Rule of Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_5/Rule_of_Commutation | https://proofwiki.org/wiki/Definition:Constructed_Semantics/Instance_5/Rule_of_Commutation | [
"Formal Semantics"
] | [
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication",
"Definition:Tautology/Formal Semantics",
"Definition:Constructed Semantics/Instance 5",
"Definition:Constructed Semantics"
] | [
"Definition:Definitional Abbreviation",
"Definition:Conditional",
"Rule of Commutation/Disjunction/Formulation 2/Forward Implication",
"Category:Formal Semantics"
] |
proofwiki-14966 | Additive Function of Zero is Zero | Let $f: \R \to \R$ be an additive function.
Then:
:$\map f 0 = 0$ | As $f$ is additive, we have:
{{begin-eqn}}
{{eqn | l = \map f 1
| r = \map f {0 + 1}
| c = Real Addition Identity is Zero
}}
{{eqn | r = \map f 0 + \map f 1
| c = {{Defof|Additive Function (Algebra)|Additive Function}}
}}
{{end-eqn}}
that is:
:$\map f 0 = 0$
{{qed}}
Category:Additive Functions
3ioacdf... | Let $f: \R \to \R$ be an [[Definition:Additive Function (Algebra)|additive function]].
Then:
:$\map f 0 = 0$ | As $f$ is [[Definition:Additive Function (Algebra)|additive]], we have:
{{begin-eqn}}
{{eqn | l = \map f 1
| r = \map f {0 + 1}
| c = [[Real Addition Identity is Zero]]
}}
{{eqn | r = \map f 0 + \map f 1
| c = {{Defof|Additive Function (Algebra)|Additive Function}}
}}
{{end-eqn}}
that is:
:$\map f ... | Additive Function of Zero is Zero | https://proofwiki.org/wiki/Additive_Function_of_Zero_is_Zero | https://proofwiki.org/wiki/Additive_Function_of_Zero_is_Zero | [
"Additive Functions"
] | [
"Definition:Additive Function (Algebra)"
] | [
"Definition:Additive Function (Algebra)",
"Real Addition Identity is Zero",
"Category:Additive Functions"
] |
proofwiki-14967 | Additive Function is Odd Function | Let $f: \R \to \R$ be an additive function.
Then $f$ is an odd function. | From Additive Function of Zero is Zero:
:$\map f 0 = 0$
Thus, for all $x \in \R$, we have:
{{begin-eqn}}
{{eqn | l = 0
| r = \map f 0
| c =
}}
{{eqn | r = \map f {x + \paren {-x} }
| c =
}}
{{eqn | r = \map f x + \map f {-x}
| c =
}}
{{end-eqn}}
It follows that the function $f$ is odd:
:$\for... | Let $f: \R \to \R$ be an [[Definition:Additive Function (Algebra)|additive function]].
Then $f$ is an [[Definition:Odd Function|odd function]]. | From [[Additive Function of Zero is Zero]]:
:$\map f 0 = 0$
Thus, for all $x \in \R$, we have:
{{begin-eqn}}
{{eqn | l = 0
| r = \map f 0
| c =
}}
{{eqn | r = \map f {x + \paren {-x} }
| c =
}}
{{eqn | r = \map f x + \map f {-x}
| c =
}}
{{end-eqn}}
It follows that the function $f$ is [[De... | Additive Function is Odd Function | https://proofwiki.org/wiki/Additive_Function_is_Odd_Function | https://proofwiki.org/wiki/Additive_Function_is_Odd_Function | [
"Additive Functions",
"Odd Functions"
] | [
"Definition:Additive Function (Algebra)",
"Definition:Odd Function"
] | [
"Additive Function of Zero is Zero",
"Definition:Odd Function",
"Category:Additive Functions",
"Category:Odd Functions"
] |
proofwiki-14968 | Difference of Images under Mapping not necessarily equal to Image of Difference | Let $f: S \to T$ be a mapping.
The image of the set difference of two subsets of $S$ is not necessarily equal to the set difference of the images.
That is:
Let $S_1$ and $S_2$ be subsets of $S$.
Then it is not always the case that:
:$f \sqbrk {S_1} \setminus f \sqbrk {S_2} = f \sqbrk {S_1 \setminus S_2}$
where $\setmin... | Note that from Image of Set Difference under Mapping:
:$f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$
By Proof by Counterexample it is demonstrated that the inclusion does not necessarily apply in the other direction.
Let:
:$S_1 = \set {x \in \Z: x \le 0}$
:$S_2 = \set {x \in \Z: x \ge... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
The [[Definition:Image of Subset under Mapping|image]] of the [[Definition:Set Difference|set difference]] of two [[Definition:Subset|subsets]] of $S$ is not necessarily equal to the [[Definition:Set Difference|set difference]] of the [[Definition:Image of Subset u... | Note that from [[Image of Set Difference under Mapping]]:
:$f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$
By [[Proof by Counterexample]] it is demonstrated that the [[Definition:Subset|inclusion]] does not necessarily apply in the other direction.
Let:
:$S_1 = \set {x \in \Z: x \le... | Difference of Images under Mapping not necessarily equal to Image of Difference | https://proofwiki.org/wiki/Difference_of_Images_under_Mapping_not_necessarily_equal_to_Image_of_Difference | https://proofwiki.org/wiki/Difference_of_Images_under_Mapping_not_necessarily_equal_to_Image_of_Difference | [
"Images",
"Set Difference",
"Image of Set Difference under Mapping"
] | [
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Set Difference",
"Definition:Subset",
"Definition:Set Difference",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subset",
"Definition:Set Difference"
] | [
"Image of Set Difference under Mapping",
"Proof by Counterexample",
"Definition:Subset",
"Set Difference with Self is Empty Set"
] |
proofwiki-14969 | Cross-Relation on Real Numbers is Equivalence Relation | Let $\R^2$ denote the cartesian plane.
Let $\alpha$ denote the relation defined on $\R^2$ by:
:$\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
Then $\alpha$ is an equivalence relation on $\R^2$. | $\alpha$ is an instance of a cross-relation.
We also have that Real Addition is Commutative.
The result therefore follows from Cross-Relation is Equivalence Relation.
{{qed}} | Let $\R^2$ denote the [[Definition:Cartesian Plane|cartesian plane]].
Let $\alpha$ denote the [[Definition:Endorelation|relation]] defined on $\R^2$ by:
:$\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$
Then $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]] on $\R... | $\alpha$ is an instance of a [[Definition:Cross-Relation|cross-relation]].
We also have that [[Real Addition is Commutative]].
The result therefore follows from [[Cross-Relation is Equivalence Relation]].
{{qed}} | Cross-Relation on Real Numbers is Equivalence Relation | https://proofwiki.org/wiki/Cross-Relation_on_Real_Numbers_is_Equivalence_Relation | https://proofwiki.org/wiki/Cross-Relation_on_Real_Numbers_is_Equivalence_Relation | [
"Real Numbers",
"Examples of Equivalence Relations",
"Cross-Relations",
"Cross-Relation on Real Numbers is Equivalence Relation"
] | [
"Definition:Cartesian Plane",
"Definition:Endorelation",
"Definition:Equivalence Relation"
] | [
"Definition:Cross-Relation",
"Real Addition is Commutative",
"Cross-Relation is Equivalence Relation"
] |
proofwiki-14970 | Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation | The equivalence classes of $\alpha$, when interpreted as points in the plane, are the straight lines of slope $1$. | We have from Cross-Relation on Real Numbers is Equivalence Relation that $\alpha$ is an equivalence relation.
Thus:
{{begin-eqn}}
{{eqn | l = \tuple {x_1, y_1}
| o = \alpha
| r = \tuple {x_2, y_2}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x_1 + y_2
| r = x_2 + y_1
| c =
}}
{{eqn | l... | The [[Definition:Equivalence Class|equivalence classes]] of $\alpha$, when interpreted as [[Definition:Point|points]] in [[Definition:The Plane|the plane]], are the [[Definition:Straight Line|straight lines]] of [[Definition:Slope of Straight Line|slope]] $1$. | We have from [[Cross-Relation on Real Numbers is Equivalence Relation]] that $\alpha$ is an [[Definition:Equivalence Relation|equivalence relation]].
Thus:
{{begin-eqn}}
{{eqn | l = \tuple {x_1, y_1}
| o = \alpha
| r = \tuple {x_2, y_2}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x_1 + y_2
... | Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation | https://proofwiki.org/wiki/Cross-Relation_on_Real_Numbers_is_Equivalence_Relation/Geometrical_Interpretation | https://proofwiki.org/wiki/Cross-Relation_on_Real_Numbers_is_Equivalence_Relation/Geometrical_Interpretation | [
"Cross-Relation on Real Numbers is Equivalence Relation"
] | [
"Definition:Equivalence Class",
"Definition:Point",
"Definition:Plane Surface/The Plane",
"Definition:Line/Straight Line",
"Definition:Slope/Straight Line"
] | [
"Cross-Relation on Real Numbers is Equivalence Relation",
"Definition:Equivalence Relation",
"Definition:Equivalence Class",
"Definition:Set",
"Definition:Point",
"Equation of Straight Line in Plane",
"Definition:Line/Straight Line",
"Definition:Slope/Straight Line"
] |
proofwiki-14971 | Subtraction on Numbers is Anticommutative/Natural Numbers | The operation of subtraction on the natural numbers $\N$ is anticommutative, and defined only when $a = b$:
That is:
:$a - b = b - a \iff a = b$ | $a - b$ is defined on $\N$ only if $a \ge b$.
If $a > b$, then although $a - b$ is defined, $b - a$ is not.
So for $a - b = b - a$ it is necessary for both to be defined.
This happens only when $a = b$.
Hence the result. | The operation of [[Definition:Natural Number Subtraction|subtraction]] on the [[Definition:Natural Number|natural numbers]] $\N$ is [[Definition:Anticommutative|anticommutative]], and defined only when $a = b$:
That is:
:$a - b = b - a \iff a = b$ | $a - b$ is defined on $\N$ only if $a \ge b$.
If $a > b$, then although $a - b$ is defined, $b - a$ is not.
So for $a - b = b - a$ it is necessary for both to be defined.
This happens only when $a = b$.
Hence the result. | Subtraction on Numbers is Anticommutative/Natural Numbers | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Anticommutative/Natural_Numbers | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Anticommutative/Natural_Numbers | [
"Subtraction on Numbers is Anticommutative"
] | [
"Definition:Subtraction/Natural Numbers",
"Definition:Natural Numbers",
"Definition:Anticommutative"
] | [] |
proofwiki-14972 | Subtraction on Numbers is Anticommutative/Integral Domains | The operation of subtraction on the numbers is anticommutative.
That is:
:$a - b = b - a \iff a = b$ | Let $a, b$ be elements of one of the standard number sets: $\Z, \Q, \R, \C$.
Each of those systems is an integral domain, and so is closed under the operation of subtraction.
=== Necessary Condition ===
Let $a = b$.
Then $a - b = 0 = b - a$.
{{qed|lemma}}
=== Sufficient Condition ===
Let $a - b = b - a$.
Then:
{{begin-... | The [[Definition:Binary Operation|operation]] of [[Definition:Subtraction|subtraction]] on the [[Definition:Number|numbers]] is [[Definition:Anticommutative|anticommutative]].
That is:
:$a - b = b - a \iff a = b$ | Let $a, b$ be [[Definition:Element|elements]] of one of the [[Definition:Standard Number System|standard number sets]]: $\Z, \Q, \R, \C$.
Each of those systems is an [[Definition:Integral Domain|integral domain]], and so is [[Definition:Closed Algebraic Structure|closed]] under the [[Definition:Binary Operation|operat... | Subtraction on Numbers is Anticommutative/Integral Domains | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Anticommutative/Integral_Domains | https://proofwiki.org/wiki/Subtraction_on_Numbers_is_Anticommutative/Integral_Domains | [
"Subtraction on Numbers is Anticommutative"
] | [
"Definition:Operation/Binary Operation",
"Definition:Subtraction",
"Definition:Number",
"Definition:Anticommutative"
] | [
"Definition:Element",
"Definition:Number",
"Definition:Integral Domain",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Operation/Binary Operation",
"Definition:Subtraction",
"Commutative Law of Addition"
] |
proofwiki-14973 | Natural Number Subtraction is not Closed | The operation of subtraction on the natural numbers is not closed. | By definition of natural number subtraction:
:$n - m = p$
where $p \in \N$ such that $n = m + p$.
However, when $m > n$ there exists no $p \in \N$ such that $n = m + p$.
{{qed}} | The [[Definition:Binary Operation|operation]] of [[Definition:Natural Number Subtraction|subtraction]] on the [[Definition:Natural Numbers|natural numbers]] is not [[Definition:Closed Operation|closed]]. | By definition of [[Definition:Natural Number Subtraction|natural number subtraction]]:
:$n - m = p$
where $p \in \N$ such that $n = m + p$.
However, when $m > n$ there exists no $p \in \N$ such that $n = m + p$.
{{qed}} | Natural Number Subtraction is not Closed | https://proofwiki.org/wiki/Natural_Number_Subtraction_is_not_Closed | https://proofwiki.org/wiki/Natural_Number_Subtraction_is_not_Closed | [
"Subtraction",
"Natural Numbers"
] | [
"Definition:Operation/Binary Operation",
"Definition:Subtraction/Natural Numbers",
"Definition:Natural Numbers",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Definition:Subtraction/Natural Numbers"
] |
proofwiki-14974 | Integers under Addition form Semigroup | The set of integers under addition $\struct {\Z, +}$ forms a semigroup. | === Closure ===
Integer Addition is Closed.
{{qed|lemma}} | The [[Definition:Set|set]] of [[Definition:Integer|integers]] under [[Definition:Integer Addition|addition]] $\struct {\Z, +}$ forms a [[Definition:Semigroup|semigroup]]. | === Closure ===
[[Integer Addition is Closed]].
{{qed|lemma}} | Integers under Addition form Semigroup | https://proofwiki.org/wiki/Integers_under_Addition_form_Semigroup | https://proofwiki.org/wiki/Integers_under_Addition_form_Semigroup | [
"Integer Addition",
"Examples of Semigroups"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Semigroup"
] | [
"Integer Addition is Closed"
] |
proofwiki-14975 | Natural Numbers under Multiplication form Subsemigroup of Integers | Let $\struct {\N, \times}$ denote the set of natural numbers under multiplication.
Let $\struct {\Z, \times}$ denote the set of integers under multiplication.
Then $\struct {\N, \times}$ is a subsemigroup of $\struct {\Z, \times}$. | We have from Natural Numbers under Multiplication form Semigroup that $\struct {\N, \times}$ forms a semigroup.
We have from Integers under Multiplication form Semigroup that $\struct {\Z, \times}$ forms a semigroup.
From Natural Numbers are Non-Negative Integers, we have that $\N \subseteq \Z$.
From the definition of ... | Let $\struct {\N, \times}$ denote the [[Definition:Natural Numbers|set of natural numbers]] under [[Definition:Natural Number Multiplication|multiplication]].
Let $\struct {\Z, \times}$ denote the [[Definition:Integer|set of integers]] under [[Definition:Integer Multiplication|multiplication]].
Then $\struct {\N, \t... | We have from [[Natural Numbers under Multiplication form Semigroup]] that $\struct {\N, \times}$ forms a [[Definition:Semigroup|semigroup]].
We have from [[Integers under Multiplication form Semigroup]] that $\struct {\Z, \times}$ forms a [[Definition:Semigroup|semigroup]].
From [[Natural Numbers are Non-Negative Int... | Natural Numbers under Multiplication form Subsemigroup of Integers | https://proofwiki.org/wiki/Natural_Numbers_under_Multiplication_form_Subsemigroup_of_Integers | https://proofwiki.org/wiki/Natural_Numbers_under_Multiplication_form_Subsemigroup_of_Integers | [
"Examples of Subsemigroups",
"Natural Number Multiplication",
"Integer Multiplication"
] | [
"Definition:Natural Numbers",
"Definition:Multiplication/Natural Numbers",
"Definition:Integer",
"Definition:Multiplication/Integers",
"Definition:Subsemigroup"
] | [
"Natural Numbers under Multiplication form Semigroup",
"Definition:Semigroup",
"Integers under Multiplication form Semigroup",
"Definition:Semigroup",
"Natural Numbers are Non-Negative Integers",
"Definition:Multiplication/Integers",
"Definition:Extension of Operation"
] |
proofwiki-14976 | Cauchy Sequence Is Eventually Bounded Away From Non-Limit | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Suppose $\sequence {x_n}$ does not converge to $l \in R$.
Then:
:$\exists K \in \N$ and $C \in \R_{>0}: \forall n > K: C < \norm {x_n - l}$ | Because $\sequence {x_n}$ does not converge to $l$:
:$\exists \epsilon \in \R_{>0}: \forall n \in \N: \exists m \ge n: \norm {x_m - l} \ge \epsilon$
Because $\sequence {x_n}$ is a Cauchy sequence:
:$\exists K \in \N: \forall n, m \ge K: \norm {x_n - x_m} < \dfrac \epsilon 2$
Let $M \ge K: \norm {x_M - l} \ge \epsilon$.... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring |normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence in $R$]].
Suppose $\sequence {x_n}$ does not [[Definition:Convergent Sequence in Normed Division Ring|converge]] to $l... | Because $\sequence {x_n}$ does not [[Definition:Convergent Sequence in Normed Division Ring|converge]] to $l$:
:$\exists \epsilon \in \R_{>0}: \forall n \in \N: \exists m \ge n: \norm {x_m - l} \ge \epsilon$
Because $\sequence {x_n}$ is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]]:
:$\exist... | Cauchy Sequence Is Eventually Bounded Away From Non-Limit | https://proofwiki.org/wiki/Cauchy_Sequence_Is_Eventually_Bounded_Away_From_Non-Limit | https://proofwiki.org/wiki/Cauchy_Sequence_Is_Eventually_Bounded_Away_From_Non-Limit | [
"Cauchy Sequence in Normed Division Ring is Bounded"
] | [
"Definition:Normed Division Ring ",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"Definition:Convergent Sequence/Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Category:Cauchy Sequence in Normed Division Ring is Bounded"
] |
proofwiki-14977 | Embedding Division Ring into Quotient Ring of Cauchy Sequences | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\CC$ be the ring of Cauchy sequences over $R$
Let $\NN = \set {\sequence {x_n}: \ds \lim_{n \mathop \to \infty} x_n = 0}$
Let $\norm {\, \cdot \,}: \CC \, \big / \NN \to \R_{\ge 0}$ be the norm on the quotient ring $\CC \, \big / \NN$ defined by:
:... | By the definition of a distance-preserving mapping and a ring monomorphism it has to be shown that:
:$(1): \quad \phi$ is a homomorphism.
:$(2): \quad \phi$ is an injection.
:$(3): \quad \phi$ is distance-preserving. | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring |normed division ring]].
Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|ring of Cauchy sequences over $R$]]
Let $\NN = \set {\sequence {x_n}: \ds \lim_{n \mathop \to \infty} x_n = 0}$
Let $\norm {\, \cdot \,}: \CC \, \big / \NN \t... | By the definition of a [[Definition:Distance-Preserving Mapping|distance-preserving mapping]] and a [[Definition:Ring Monomorphism|ring monomorphism]] it has to be shown that:
:$(1): \quad \phi$ is a [[Definition:Ring Homomorphism|homomorphism]].
:$(2): \quad \phi$ is an [[Definition:Injection|injection]].
:$(3): \quad... | Embedding Division Ring into Quotient Ring of Cauchy Sequences | https://proofwiki.org/wiki/Embedding_Division_Ring_into_Quotient_Ring_of_Cauchy_Sequences | https://proofwiki.org/wiki/Embedding_Division_Ring_into_Quotient_Ring_of_Cauchy_Sequences | [
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring ",
"Definition:Ring of Cauchy Sequences",
"Quotient Ring of Cauchy Sequences is Normed Division Ring",
"Definition:Quotient Ring",
"Definition:Coset/Left Coset",
"Definition:Sequence",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism"
] | [
"Definition:Distance-Preserving Mapping",
"Definition:Ring Monomorphism",
"Definition:Ring Homomorphism",
"Definition:Injection",
"Definition:Distance-Preserving Mapping",
"Definition:Ring Homomorphism",
"Definition:Ring Monomorphism",
"Definition:Ring Homomorphism",
"Definition:Injection",
"Defin... |
proofwiki-14978 | Identity of Submonoid is not necessarily Identity of Monoid | Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.
Let $\struct {T, \circ}$ be a submonoid of $\struct {S, \circ}$ whose identity is $e_T$.
Then it is not necessarily the case that $e_T = e_S$. | Let $\struct {S, \times}$ be the semigroup formed by the set of order $2$ square matrices over the real numbers $R$ under (conventional) matrix multiplication.
Let $T$ be the subset of $S$ consisting of the matrices of the form $\begin{bmatrix} x & 0 \\ 0 & 0 \end{bmatrix}$ for $x \in \R$.
From Matrices of the Form $\b... | Let $\struct {S, \circ}$ be a [[Definition:Monoid|monoid]] whose [[Definition:Identity Element|identity]] is $e_S$.
Let $\struct {T, \circ}$ be a [[Definition:Submonoid|submonoid]] of $\struct {S, \circ}$ whose [[Definition:Identity Element|identity]] is $e_T$.
Then it is not necessarily the case that $e_T = e_S$. | Let $\struct {S, \times}$ be the [[Definition:Semigroup|semigroup]] formed by the [[Definition:Set|set]] of [[Definition:Order of Square Matrix|order $2$]] [[Definition:Square Matrix|square matrices]] over the [[Definition:Real Number|real numbers]] $R$ under [[Definition:Matrix Product (Conventional)|(conventional) ma... | Identity of Submonoid is not necessarily Identity of Monoid | https://proofwiki.org/wiki/Identity_of_Submonoid_is_not_necessarily_Identity_of_Monoid | https://proofwiki.org/wiki/Identity_of_Submonoid_is_not_necessarily_Identity_of_Monoid | [
"Submonoids"
] | [
"Definition:Monoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Submonoid",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Semigroup",
"Definition:Set",
"Definition:Matrix/Square Matrix/Order",
"Definition:Matrix/Square Matrix",
"Definition:Real Number",
"Definition:Matrix Product (Conventional)",
"Definition:Subset",
"Definition:Matrix/Square Matrix",
"Subsemigroup/Examples/2x2 Matrices with One Non-Zero En... |
proofwiki-14979 | Non-Zero Real Numbers under Multiplication form Group | Let $\R_{\ne 0}$ be the set of real numbers without zero:
:$\R_{\ne 0} = \R \setminus \set 0$
The structure $\struct {\R_{\ne 0}, \times}$ forms a group. | <onlyinclude>
Taking the group axioms in turn: | Let $\R_{\ne 0}$ be the set of [[Definition:Real Number|real numbers]] without [[Definition:Zero (Number)|zero]]:
:$\R_{\ne 0} = \R \setminus \set 0$
The [[Definition:Algebraic Structure with One Operation|structure]] $\struct {\R_{\ne 0}, \times}$ forms a [[Definition:Group|group]]. | <onlyinclude>
Taking the [[Axiom:Group Axioms|group axioms]] in turn: | Non-Zero Real Numbers under Multiplication form Group | https://proofwiki.org/wiki/Non-Zero_Real_Numbers_under_Multiplication_form_Group | https://proofwiki.org/wiki/Non-Zero_Real_Numbers_under_Multiplication_form_Group | [
"Real Multiplication",
"Examples of Groups"
] | [
"Definition:Real Number",
"Definition:Zero (Number)",
"Definition:Algebraic Structure/One Operation",
"Definition:Group"
] | [
"Axiom:Group Axioms"
] |
proofwiki-14980 | Symmetric Group on n Letters is Isomorphic to Symmetric Group | The symmetric group on $n$ letters $\struct {S_n, \circ}$ is isomorphic to the symmetric group on the $n$ elements of any set $T$ whose cardinality is $n$.
That is:
:$\forall T \subseteq \mathbb U, \card T = n: \struct {S_n, \circ} \cong \struct {\Gamma \paren T, \circ}$
where:
:$\map \Gamma T$ denotes the set of permu... | The fact that $\struct {S_n, \circ}$ is a group is a direct implementation of the result Symmetric Group is Group.
By definition of cardinality, as $\card T = n$ we can find a bijection between $T$ and $\N_n$.
From Number of Permutations, it is immediate that $\order {\paren {\Gamma \paren T, \circ} } = n! = \order {\s... | The [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]] $\struct {S_n, \circ}$ is [[Definition:Group Isomorphism|isomorphic]] to the [[Definition:Symmetric Group|symmetric group]] on the $n$ [[Definition:Element|elements]] of any set $T$ whose [[Definition:Cardinality|cardinality]] is $n$.
That... | The fact that $\struct {S_n, \circ}$ is a [[Definition:Group|group]] is a direct implementation of the result [[Symmetric Group is Group]].
By definition of [[Definition:Cardinality|cardinality]], as $\card T = n$ we can find a [[Definition:Bijection|bijection]] between $T$ and $\N_n$.
From [[Number of Permutations]... | Symmetric Group on n Letters is Isomorphic to Symmetric Group | https://proofwiki.org/wiki/Symmetric_Group_on_n_Letters_is_Isomorphic_to_Symmetric_Group | https://proofwiki.org/wiki/Symmetric_Group_on_n_Letters_is_Isomorphic_to_Symmetric_Group | [
"Symmetric Groups"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Symmetric Group",
"Definition:Element",
"Definition:Cardinality",
"Definition:Set",
"Definition:Permutation",
"Definition:Algebraic Structure/One Operation",
"Definition:Composition of ... | [
"Definition:Group",
"Symmetric Group is Group",
"Definition:Cardinality",
"Definition:Bijection",
"Number of Permutations",
"Definition:Bijection",
"Transplanting Theorem"
] |
proofwiki-14981 | Symmetric Groups of Same Order are Isomorphic | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $T_1$ and $T_2$ be sets whose cardinality $\card {T_1}$ and $\card {T_2}$ are both $n$.
Let $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ be the symmetric group on $S$ and $T$ respectively.
Then $\struct {\map \Gamma {T_1}, \circ}... | Consider the symmetric group on $n$ letters $S_n$.
From Symmetric Group on n Letters is Isomorphic to Symmetric Group we have that:
:$\struct {\map \Gamma {T_1}, \circ}$ is isomorphic to $S_n$
:$\struct {\map \Gamma {T_2}, \circ}$ is isomorphic to $S_n$
and hence from Isomorphism is Equivalence Relation:
:$\struct {\ma... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $T_1$ and $T_2$ be [[Definition:Set|sets]] whose [[Definition:Cardinality|cardinality]] $\card {T_1}$ and $\card {T_2}$ are both $n$.
Let $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ be ... | Consider the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]] $S_n$.
From [[Symmetric Group on n Letters is Isomorphic to Symmetric Group]] we have that:
:$\struct {\map \Gamma {T_1}, \circ}$ is [[Definition:Group Isomorphism|isomorphic]] to $S_n$
:$\struct {\map \Gamma {T_2}, \circ}$ is [[... | Symmetric Groups of Same Order are Isomorphic/Proof 1 | https://proofwiki.org/wiki/Symmetric_Groups_of_Same_Order_are_Isomorphic | https://proofwiki.org/wiki/Symmetric_Groups_of_Same_Order_are_Isomorphic/Proof_1 | [
"Symmetric Groups of Same Order are Isomorphic",
"Symmetric Groups",
"Group Isomorphisms"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Cardinality",
"Definition:Symmetric Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Symmetric Group/n Letters",
"Symmetric Group on n Letters is Isomorphic to Symmetric Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Isomorphism is Equivalence Relation",
"Definition:Isomorphism (Abstract Al... |
proofwiki-14982 | Symmetric Groups of Same Order are Isomorphic | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $T_1$ and $T_2$ be sets whose cardinality $\card {T_1}$ and $\card {T_2}$ are both $n$.
Let $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ be the symmetric group on $S$ and $T$ respectively.
Then $\struct {\map \Gamma {T_1}, \circ}... | Let us define a bijection:
:$\alpha: T_1 \to T_2$
Let $\theta: \struct {\map \Gamma {T_1}, \circ} \to \struct {\map \Gamma {T_2}, \circ}$ be defined as:
:$\forall f \in \struct {\map \Gamma {T_1}, \circ}: \map \theta f = \alpha \circ f \circ \alpha^{-1}$
Let $f, g \in \map \Gamma {T_1}$.
We have:
{{begin-eqn}}
{{eqn | ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $T_1$ and $T_2$ be [[Definition:Set|sets]] whose [[Definition:Cardinality|cardinality]] $\card {T_1}$ and $\card {T_2}$ are both $n$.
Let $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ be ... | Let us define a [[Definition:Bijection|bijection]]:
:$\alpha: T_1 \to T_2$
Let $\theta: \struct {\map \Gamma {T_1}, \circ} \to \struct {\map \Gamma {T_2}, \circ}$ be defined as:
:$\forall f \in \struct {\map \Gamma {T_1}, \circ}: \map \theta f = \alpha \circ f \circ \alpha^{-1}$
Let $f, g \in \map \Gamma {T_1}$.
W... | Symmetric Groups of Same Order are Isomorphic/Proof 2 | https://proofwiki.org/wiki/Symmetric_Groups_of_Same_Order_are_Isomorphic | https://proofwiki.org/wiki/Symmetric_Groups_of_Same_Order_are_Isomorphic/Proof_2 | [
"Symmetric Groups of Same Order are Isomorphic",
"Symmetric Groups",
"Group Isomorphisms"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Set",
"Definition:Cardinality",
"Definition:Symmetric Group",
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism"
] | [
"Definition:Bijection",
"Definition:Group Homomorphism",
"Definition:Injection",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Surjection",
"Definition:Injection",
"Definition:Surjection",
"Definition:Bijection",
"Definition:Bijection",
"Definition:Group Homomorphism",
"Definition... |
proofwiki-14983 | Quotient Ring of Cauchy Sequences is Normed Division Ring | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\CC$ be the ring of Cauchy sequences over $R$
Let $\NN$ be the set of null sequences.
For all $\sequence {x_n} \in \CC$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \NN$
Let $\norm {\, \cdot \,}_1: \CC \,\big / \NN \to \R_{\ge ... | By Quotient Ring of Cauchy Sequences is Division Ring then $\CC \,\big / \NN$ is a division ring.
It remains to be proved that:
:$\norm {\, \cdot \,}_1$ is well-defined
:$\norm {\, \cdot \,}_1$ satisfies the norm axioms. | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\CC$ be the [[Definition:Ring of Cauchy Sequences|ring of Cauchy sequences over $R$]]
Let $\NN$ be the [[Definition:Set|set]] of [[Definition:Null Sequence in Normed Division Ring|null sequences]].
For all $\... | By [[Quotient Ring of Cauchy Sequences is Division Ring]] then $\CC \,\big / \NN$ is a [[Definition:Division Ring| division ring]].
It remains to be proved that:
:$\norm {\, \cdot \,}_1$ is [[Definition:Well-Defined Mapping|well-defined]]
:$\norm {\, \cdot \,}_1$ satisfies the [[Axiom:Multiplicative Norm Axioms|norm a... | Quotient Ring of Cauchy Sequences is Normed Division Ring | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring | https://proofwiki.org/wiki/Quotient_Ring_of_Cauchy_Sequences_is_Normed_Division_Ring | [
"Cauchy Sequences",
"Normed Division Rings",
"Completion of Normed Division Ring"
] | [
"Definition:Normed Division Ring",
"Definition:Ring of Cauchy Sequences",
"Definition:Set",
"Definition:Null Sequence/Normed Division Ring",
"Definition:Coset/Left Coset",
"Definition:Normed Division Ring"
] | [
"Quotient Ring of Cauchy Sequences is Division Ring",
"Definition:Division Ring",
"Definition:Well-Defined/Mapping",
"Axiom:Multiplicative Norm Axioms"
] |
proofwiki-14984 | Inequality Rule for Real Sequences | Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} x_n
| r = l
}}
{{eqn | l = \lim_{n \mathop \to \infty} y_n
| r = m
}}
{{end-eqn}}
Let there exist $N \in ... | Suppose $l > m$.
Then:
:$m = \dfrac m 2 + \dfrac m 2 < \dfrac {l + m} 2 < \dfrac l 2 + \dfrac l 2 = l$
Let $\epsilon = \dfrac {l - m} 2$.
Then:
:$\epsilon > 0$
We are given that:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
By definition of the limit of a real sequence, we can find $N_1$ such that:
:$\forall n \ge N_1: \... | Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Real Sequence|sequences in $\R$]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Convergent Real Sequence|convergent]] to the following [[Definition:Limit of Real Sequence|limits]]:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} x_n
... | Suppose $l > m$.
Then:
:$m = \dfrac m 2 + \dfrac m 2 < \dfrac {l + m} 2 < \dfrac l 2 + \dfrac l 2 = l$
Let $\epsilon = \dfrac {l - m} 2$.
Then:
:$\epsilon > 0$
We are given that:
:$\ds \lim_{n \mathop \to \infty} x_n = l$
By definition of the [[Definition:Limit of Real Sequence|limit of a real sequence]], we can ... | Inequality Rule for Real Sequences/Proof 1 | https://proofwiki.org/wiki/Inequality_Rule_for_Real_Sequences | https://proofwiki.org/wiki/Inequality_Rule_for_Real_Sequences/Proof_1 | [
"Inequality Rule for Real Sequences",
"Limits of Sequences",
"Real Sequences",
"Named Theorems"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Definition:Limit of Sequence/Real Numbers",
"Definition:Absolute Value",
"Definition:Contrapositive Statement"
] |
proofwiki-14985 | Inequality Rule for Real Sequences | Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} x_n
| r = l
}}
{{eqn | l = \lim_{n \mathop \to \infty} y_n
| r = m
}}
{{end-eqn}}
Let there exist $N \in ... | Consider the sequence $\sequence {z_n}$ defined by:
:$z_n := y_n - x_n$
From Sum Rule for Real Sequences:
:$z_n \to m - l$ as $n \to \infty$
Furthermore, the assumption that $x_n \le y_n$ for all $n \in \N$ means that:
:$\forall n \in \N: z_n \ge 0$
Applying the Lower and Upper Bounds for Sequences to the sequence $\se... | Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Real Sequence|sequences in $\R$]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Convergent Real Sequence|convergent]] to the following [[Definition:Limit of Real Sequence|limits]]:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty} x_n
... | Consider the [[Definition:Real Sequence|sequence]] $\sequence {z_n}$ defined by:
:$z_n := y_n - x_n$
From [[Sum Rule for Real Sequences]]:
:$z_n \to m - l$ as $n \to \infty$
Furthermore, the assumption that $x_n \le y_n$ for all $n \in \N$ means that:
:$\forall n \in \N: z_n \ge 0$
Applying the [[Lower and Upper Bo... | Inequality Rule for Real Sequences/Proof 2 | https://proofwiki.org/wiki/Inequality_Rule_for_Real_Sequences | https://proofwiki.org/wiki/Inequality_Rule_for_Real_Sequences/Proof_2 | [
"Inequality Rule for Real Sequences",
"Limits of Sequences",
"Real Sequences",
"Named Theorems"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Definition:Real Sequence",
"Combination Theorem for Sequences/Real/Sum Rule",
"Lower and Upper Bounds for Sequences",
"Definition:Real Sequence"
] |
proofwiki-14986 | Rule of Association/Disjunction/Formulation 2/Forward Implication | :$\vdash \paren {p \lor \paren {q \lor r} } \implies \paren {\paren {p \lor q} \lor r}$ | {{BeginTableau|\vdash \paren {p \lor \paren {q \lor r} } \implies \paren {\paren {p \lor q} \lor r}|Instance 2 of the Hilbert-style systems}}
{{TableauLine
| n = 1
| f = \paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }
| rlnk = Rule of Association/Disjunction/Formulation 2/Reverse Implic... | :$\vdash \paren {p \lor \paren {q \lor r} } \implies \paren {\paren {p \lor q} \lor r}$ | {{BeginTableau|\vdash \paren {p \lor \paren {q \lor r} } \implies \paren {\paren {p \lor q} \lor r}|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
{{TableauLine
| n = 1
| f = \paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }
| rlnk = Rule of Assoc... | Rule of Association/Disjunction/Formulation 2/Forward Implication | https://proofwiki.org/wiki/Rule_of_Association/Disjunction/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Rule_of_Association/Disjunction/Formulation_2/Forward_Implication | [
"Rule of Association",
"Hilbert Proof System Instance 2"
] | [] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-14987 | Rule of Association/Disjunction/Formulation 2/Reverse Implication | :$\vdash \paren {p \lor \paren {q \lor r} } \impliedby \paren {\paren {p \lor q} \lor r}$ | By definition of $\impliedby$, we prove:
:$\vdash \paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }$
{{BeginTableau|\vdash \paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }|Instance 2 of the Hilbert-style systems}}
{{TableauLine
| n = 1
| f = r \implies \paren {... | :$\vdash \paren {p \lor \paren {q \lor r} } \impliedby \paren {\paren {p \lor q} \lor r}$ | By definition of $\impliedby$, we prove:
:$\vdash \paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }$
{{BeginTableau|\vdash \paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
... | Rule of Association/Disjunction/Formulation 2/Reverse Implication | https://proofwiki.org/wiki/Rule_of_Association/Disjunction/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/Rule_of_Association/Disjunction/Formulation_2/Reverse_Implication | [
"Rule of Association",
"Hilbert Proof System Instance 2"
] | [] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-14988 | Left Regular Representation of Subset Product | Let $\struct {S, \circ}$ be a magma.
Let $T \subseteq S$ be a subset of $S$.
Let $\lambda_a: S \to S$ be the left regular representation of $S$ with respect to $a$.
Then:
:$\lambda_a \sqbrk T = \set a \circ T = a \circ T$
where $a \circ T$ denotes subset product with a singleton. | {{begin-eqn}}
{{eqn | l = \lambda_a \sqbrk T
| r = \set {s \in S: \exists t \in T: s = \map {\lambda_a} t}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {s \in S: \exists t \in T: s = a \circ t}
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | r = \set {a \circ t: t \in T}
... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\lambda_a: S \to S$ be the [[Definition:Left Regular Representation|left regular representation]] of $S$ with respect to $a$.
Then:
:$\lambda_a \sqbrk T = \set a \circ T = a \circ T$
where $a... | {{begin-eqn}}
{{eqn | l = \lambda_a \sqbrk T
| r = \set {s \in S: \exists t \in T: s = \map {\lambda_a} t}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {s \in S: \exists t \in T: s = a \circ t}
| c = {{Defof|Left Regular Representation}}
}}
{{eqn | r = \set {a \circ t: t \in T}
... | Left Regular Representation of Subset Product | https://proofwiki.org/wiki/Left_Regular_Representation_of_Subset_Product | https://proofwiki.org/wiki/Left_Regular_Representation_of_Subset_Product | [
"Regular Representations"
] | [
"Definition:Magma",
"Definition:Subset",
"Definition:Regular Representations/Left Regular Representation",
"Definition:Subset Product/Singleton"
] | [] |
proofwiki-14989 | Right Regular Representation of Subset Product | Let $\struct {S, \circ}$ be a magma.
Let $T \subseteq S$ be a subset of $S$.
Let $\rho_a: S \to S$ be the right regular representation of $S$ with respect to $a$.
Then:
:$\rho_a \sqbrk T = T \circ \set a = T \circ a$
where $T \circ a$ denotes subset product with a singleton. | {{begin-eqn}}
{{eqn | l = \rho_a \sqbrk T
| r = \set {s \in S: \exists t \in T: s = \map {\rho_a} t}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {s \in S: \exists t \in T: s = t \circ a}
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | r = \set {t \circ a: t \in T}
|... | Let $\struct {S, \circ}$ be a [[Definition:Magma|magma]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $\rho_a: S \to S$ be the [[Definition:Right Regular Representation|right regular representation]] of $S$ with respect to $a$.
Then:
:$\rho_a \sqbrk T = T \circ \set a = T \circ a$
where $T \ci... | {{begin-eqn}}
{{eqn | l = \rho_a \sqbrk T
| r = \set {s \in S: \exists t \in T: s = \map {\rho_a} t}
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | r = \set {s \in S: \exists t \in T: s = t \circ a}
| c = {{Defof|Right Regular Representation}}
}}
{{eqn | r = \set {t \circ a: t \in T}
|... | Right Regular Representation of Subset Product | https://proofwiki.org/wiki/Right_Regular_Representation_of_Subset_Product | https://proofwiki.org/wiki/Right_Regular_Representation_of_Subset_Product | [
"Regular Representations"
] | [
"Definition:Magma",
"Definition:Subset",
"Definition:Regular Representations/Right Regular Representation",
"Definition:Subset Product/Singleton"
] | [] |
proofwiki-14990 | Order of Cycle is Length of Cycle | Let $S_n$ denote the symmetric group on $n$ letters.
Let $\pi \in S_n$ be a cyclic permutation of length $k$.
Then:
:$\order \pi = k$
where:
:$\order \pi$ denotes the order of $\pi$ in $S_n$. | Let $\pi = \tuple {a_0, a_1, \ldots, a_{k - 1} }$.
Observe that:
{{begin-eqn}}
{{eqn | l = \paren {\paren {j + n} \pmod k} + 1
| r = \paren {j + n + 1} \pmod k
| c =
}}
{{eqn | ll=\leadsto
| l = \map \pi {a_{\paren {j + n} \pmod k} }
| r = a_{\paren {j + n + 1} \pmod k}
| c =
}}
{{eqn | ... | Let $S_n$ denote the [[Definition:Symmetric Group on n Letters|symmetric group on $n$ letters]].
Let $\pi \in S_n$ be a [[Definition:Cyclic_Permutation|cyclic permutation of length $k$]].
Then:
:$\order \pi = k$
where:
:$\order \pi$ denotes the [[Definition:Order of Group Element|order]] of $\pi$ in $S_n$. | Let $\pi = \tuple {a_0, a_1, \ldots, a_{k - 1} }$.
Observe that:
{{begin-eqn}}
{{eqn | l = \paren {\paren {j + n} \pmod k} + 1
| r = \paren {j + n + 1} \pmod k
| c =
}}
{{eqn | ll=\leadsto
| l = \map \pi {a_{\paren {j + n} \pmod k} }
| r = a_{\paren {j + n + 1} \pmod k}
| c =
}}
{{eqn ... | Order of Cycle is Length of Cycle | https://proofwiki.org/wiki/Order_of_Cycle_is_Length_of_Cycle | https://proofwiki.org/wiki/Order_of_Cycle_is_Length_of_Cycle | [
"Symmetric Groups"
] | [
"Definition:Symmetric Group/n Letters",
"Definition:Cyclic_Permutation",
"Definition:Order of Group Element"
] | [
"Definition:Identity Mapping",
"Definition:Order of Group Element"
] |
proofwiki-14991 | Norm Sequence of Cauchy Sequence has Limit | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Then $\sequence {\norm {x_n} }$ has a limit in $\R$.
That is,
:$\exists l \in \R: \ds \lim_{n \mathop \to \infty} \norm {x_n} = l$ | It is first shown that $\sequence {\norm {x_n} }$ is a real Cauchy sequence in $\R$.
Let $\epsilon \in \R_{>0}$ be given.
By the definition of Cauchy sequence then:
:$\exists N \in \N: \forall n, m > N, \norm {x_n - x_m} < \epsilon$
By Reverse Triangle Inequality on Normed Division Ring, then:
:$\forall n, m > N: \cmod... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence (Normed Division Ring)|Cauchy sequence]] in $R$.
Then $\sequence {\norm {x_n} }$ has a [[Definition:Limit of Real Sequence|limit]] in $\R$.
That is,
:$\exists... | It is first shown that $\sequence {\norm {x_n} }$ is a [[Definition:Real Cauchy Sequence|real Cauchy sequence]] in $\R$.
Let $\epsilon \in \R_{>0}$ be given.
By the definition of [[Definition:Cauchy Sequence (Normed Division Ring)|Cauchy sequence]] then:
:$\exists N \in \N: \forall n, m > N, \norm {x_n - x_m} < \epsi... | Norm Sequence of Cauchy Sequence has Limit | https://proofwiki.org/wiki/Norm_Sequence_of_Cauchy_Sequence_has_Limit | https://proofwiki.org/wiki/Norm_Sequence_of_Cauchy_Sequence_has_Limit | [
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Cauchy Sequence/Normed Division Ring",
"Reverse Triangle Inequality/Normed Division Ring",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy's Convergence Criterion/Real Numbers",
"Definition:Limit of Sequ... |
proofwiki-14992 | Equivalent Cauchy Sequences have Equal Limits of Norm Sequences | Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ and $\sequence {y_n}$ be Cauchy sequences in $R$.
Let $\ds \lim_{n \mathop \to \infty} {x_n - y_n} = 0$.
Then:
:$\ds \lim_{n \mathop \to \infty} \norm {x_n} = \lim_{n \mathop \to \infty} \norm {y_n}$ | Let:
:$l = \ds \lim_{n \mathop \to \infty} \norm {x_n}$
and:
:$m = \ds \lim_{n \mathop \to \infty} \norm {y_n}$
By Norm Sequence of Cauchy Sequence has Limit, both of these limits exist.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty } \paren {\norm {x_n} - \norm {y_n} }
| r = l - m
| c = Diffe... | Let $\struct {R, \norm {\, \cdot \,} }$ be a [[Definition:Normed Division Ring|normed division ring]].
Let $\sequence {x_n}$ and $\sequence {y_n}$ be [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequences]] in $R$.
Let $\ds \lim_{n \mathop \to \infty} {x_n - y_n} = 0$.
Then:
:$\ds \lim_{n \mathop \to ... | Let:
:$l = \ds \lim_{n \mathop \to \infty} \norm {x_n}$
and:
:$m = \ds \lim_{n \mathop \to \infty} \norm {y_n}$
By [[Norm Sequence of Cauchy Sequence has Limit]], both of these [[Definition:Limit of Sequence (Normed Division Ring)|limits]] exist.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{n \mathop \to \infty } \paren {\... | Equivalent Cauchy Sequences have Equal Limits of Norm Sequences | https://proofwiki.org/wiki/Equivalent_Cauchy_Sequences_have_Equal_Limits_of_Norm_Sequences | https://proofwiki.org/wiki/Equivalent_Cauchy_Sequences_have_Equal_Limits_of_Norm_Sequences | [
"Cauchy Sequences",
"Normed Division Rings"
] | [
"Definition:Normed Division Ring",
"Definition:Cauchy Sequence/Normed Division Ring"
] | [
"Norm Sequence of Cauchy Sequence has Limit",
"Definition:Limit of Sequence/Normed Division Ring",
"Combination Theorem for Sequences/Real/Difference Rule",
"Modulus of Limit",
"Reverse Triangle Inequality/Normed Division Ring",
"Squeeze Theorem",
"Category:Cauchy Sequences",
"Category:Normed Division... |
proofwiki-14993 | Rule of Conjunction/Sequent Form/Formulation 1 | {{begin-eqn}}
{{eqn | l = p
| o =
}}
{{eqn | l = q
| o =
}}
{{eqn | ll= \vdash
| l = p \land q
| o =
}}
{{end-eqn}} | {{BeginTableau|p, q \vdash p \land q}}
{{Premise|1|p}}
{{Premise|2|q}}
{{Conjunction|3|1, 2|p \land q|1|2}}
{{EndTableau|qed}} | {{begin-eqn}}
{{eqn | l = p
| o =
}}
{{eqn | l = q
| o =
}}
{{eqn | ll= \vdash
| l = p \land q
| o =
}}
{{end-eqn}} | {{BeginTableau|p, q \vdash p \land q}}
{{Premise|1|p}}
{{Premise|2|q}}
{{Conjunction|3|1, 2|p \land q|1|2}}
{{EndTableau|qed}} | Rule of Conjunction/Sequent Form/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Rule_of_Conjunction/Sequent_Form/Formulation_1 | https://proofwiki.org/wiki/Rule_of_Conjunction/Sequent_Form/Formulation_1/Proof_1 | [
"Rule of Conjunction"
] | [] | [] |
proofwiki-14994 | Rule of Conjunction/Sequent Form/Formulation 1 | {{begin-eqn}}
{{eqn | l = p
| o =
}}
{{eqn | l = q
| o =
}}
{{eqn | ll= \vdash
| l = p \land q
| o =
}}
{{end-eqn}} | We apply the Method of Truth Tables.
$\begin{array}{|c|c||ccc|} \hline
p & q & p & \land & q\\
\hline
\F & \F & \F & \F & \F \\
\F & \T & \F & \F & \T \\
\T & \F & \T & \F & \F \\
\T & \T & \T & \T & \T \\
\hline
\end{array}$
As can be seen, only when both $p$ and $q$ are true, then so is $p \land q$.
{{qed}} | {{begin-eqn}}
{{eqn | l = p
| o =
}}
{{eqn | l = q
| o =
}}
{{eqn | ll= \vdash
| l = p \land q
| o =
}}
{{end-eqn}} | We apply the [[Method of Truth Tables]].
$\begin{array}{|c|c||ccc|} \hline
p & q & p & \land & q\\
\hline
\F & \F & \F & \F & \F \\
\F & \T & \F & \F & \T \\
\T & \F & \T & \F & \F \\
\T & \T & \T & \T & \T \\
\hline
\end{array}$
As can be seen, only when both $p$ and $q$ are [[Definition:True|true]], then so is $p \... | Rule of Conjunction/Sequent Form/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Rule_of_Conjunction/Sequent_Form/Formulation_1 | https://proofwiki.org/wiki/Rule_of_Conjunction/Sequent_Form/Formulation_1/Proof_by_Truth_Table | [
"Rule of Conjunction"
] | [] | [
"Method of Truth Tables",
"Definition:True"
] |
proofwiki-14995 | Rule of Conjunction/Sequent Form/Formulation 2 | :$\vdash p \implies \paren {q \implies \paren {p \land q} }$ | {{BeginTableau|\vdash p \implies \paren {q \implies \paren {p \land q} }|Instance 2 of the Hilbert-style systems}}
{{TableauLine
| n = 1
| f = \neg p \lor p
| rlnk = Law of Excluded Middle/Sequent Form/Proof 2
| rtxt = Law of Excluded Middle
}}
{{TableauLine
| n = 2
| f = \paren {\neg p \lor p} \implies \paren {p... | :$\vdash p \implies \paren {q \implies \paren {p \land q} }$ | {{BeginTableau|\vdash p \implies \paren {q \implies \paren {p \land q} }|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
{{TableauLine
| n = 1
| f = \neg p \lor p
| rlnk = Law of Excluded Middle/Sequent Form/Proof 2
| rtxt = Law of Excluded Middle
}}
{{TableauLine
| n = 2
... | Rule of Conjunction/Sequent Form/Formulation 2 | https://proofwiki.org/wiki/Rule_of_Conjunction/Sequent_Form/Formulation_2 | https://proofwiki.org/wiki/Rule_of_Conjunction/Sequent_Form/Formulation_2 | [
"Rule of Conjunction",
"Hilbert Proof System Instance 2"
] | [] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-14996 | Hilbert Proof System Instance 2 Independence Results/RST4 is Derivable | Rule of inference $RST \, 4$ is derivable from $RST \, 1, RST \, 2, RST \, 3$ and the axioms $(A1)$ through $(A4)$. | Recall the statement of $RST \, 4$:
:If $\mathbf A$ and $\mathbf B$ are theorems of $\mathscr H_2$, then so is $\mathbf A \land \mathbf B$.
Suppose that $\mathbf A$ and $\mathbf B$ are theorems of $\mathscr H_2$.
From Rule of Conjunction/Sequent Form/Formulation 2, we have as a theorem:
:$p \implies \paren{ q \implies ... | [[Definition:Rule of Inference|Rule of inference]] $RST \, 4$ is [[Definition:Derivable Rule of Inference|derivable]] from $RST \, 1, RST \, 2, RST \, 3$ and the [[Definition:Axiom (Formal Systems)|axioms]] $(A1)$ through $(A4)$. | Recall the statement of $RST \, 4$:
:If $\mathbf A$ and $\mathbf B$ are [[Definition:Theorem (Formal Systems)|theorems]] of $\mathscr H_2$, then so is $\mathbf A \land \mathbf B$.
Suppose that $\mathbf A$ and $\mathbf B$ are [[Definition:Theorem (Formal Systems)|theorems]] of $\mathscr H_2$.
From [[Rule of Conjunct... | Hilbert Proof System Instance 2 Independence Results/RST4 is Derivable | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/RST4_is_Derivable | https://proofwiki.org/wiki/Hilbert_Proof_System_Instance_2_Independence_Results/RST4_is_Derivable | [
"Hilbert Proof System Instance 2"
] | [
"Definition:Rule of Inference",
"Definition:Derivable Rule of Inference",
"Definition:Axiom/Formal Systems"
] | [
"Definition:Theorem/Formal System",
"Definition:Theorem/Formal System",
"Rule of Conjunction/Sequent Form/Formulation 2",
"Definition:Theorem/Formal System"
] |
proofwiki-14997 | Subgroup Generated by One Element is Cyclic | Let $G$ be a group.
Let $a \in G$.
Then $\gen a$, the subgroup generated by $a$, is cyclic: | By Subgroup Generated by One Element is Set of Powers:
:$\gen a = \set {a^n : n \in \Z}$
The result follows by definition of cyclic group.
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $a \in G$.
Then $\gen a$, the [[Definition:Generated Subgroup|subgroup generated]] by $a$, is [[Definition:Cyclic Group|cyclic]]: | By [[Subgroup Generated by One Element is Set of Powers]]:
:$\gen a = \set {a^n : n \in \Z}$
The result follows by definition of [[Definition:Cyclic Group|cyclic group]].
{{qed}} | Subgroup Generated by One Element is Cyclic | https://proofwiki.org/wiki/Subgroup_Generated_by_One_Element_is_Cyclic | https://proofwiki.org/wiki/Subgroup_Generated_by_One_Element_is_Cyclic | [
"Generated Subgroups"
] | [
"Definition:Group",
"Definition:Generated Subgroup",
"Definition:Cyclic Group"
] | [
"Subgroup Generated by One Element is Set of Powers",
"Definition:Cyclic Group"
] |
proofwiki-14998 | Subgroup Generated by Infinite Order Element is Infinite | Let $G$ be a group.
Let $a \in G$ be of infinite order.
Let $\gen a$ be the subgroup generated by $a$.
Then $\gen a$ is of infinite order. | {{AimForCont}} $\gen a$ is of finite order.
We have that $a \in \gen a$ by definition.
From Element of Finite Group is of Finite Order it follows that $a$ is of finite order.
From this contradiction it follows that $\gen a$ must be of infinite order after all.
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $a \in G$ be of [[Definition:Infinite Order Element|infinite order]].
Let $\gen a$ be the [[Definition:Generated Subgroup|subgroup generated]] by $a$.
Then $\gen a$ is of [[Definition:Infinite Group|infinite order]]. | {{AimForCont}} $\gen a$ is of [[Definition:Finite Group|finite order]].
We have that $a \in \gen a$ by definition.
From [[Element of Finite Group is of Finite Order]] it follows that $a$ is of [[Definition:Finite Order Element|finite order]].
From this [[Definition:Contradiction|contradiction]] it follows that $\gen... | Subgroup Generated by Infinite Order Element is Infinite | https://proofwiki.org/wiki/Subgroup_Generated_by_Infinite_Order_Element_is_Infinite | https://proofwiki.org/wiki/Subgroup_Generated_by_Infinite_Order_Element_is_Infinite | [
"Infinite Groups",
"Order of Group Elements",
"Generated Subgroups"
] | [
"Definition:Group",
"Definition:Order of Group Element/Infinite",
"Definition:Generated Subgroup",
"Definition:Infinite Group"
] | [
"Definition:Finite Group",
"Element of Finite Group is of Finite Order",
"Definition:Order of Group Element/Finite",
"Definition:Contradiction",
"Definition:Infinite Group"
] |
proofwiki-14999 | Element of Cyclic Group is not necessarily Generator | Let $\gen g = G$ be a cyclic group.
Let $a \in G$
Then it is not necessarily the case that $a$ is also a generator of $G$. | Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.
Consider the subgroup $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ generated by $2$.
By definition, $\gen 2$ is a cyclic group.
Consider the element $4 \in \struct {\R_{\ne 0}, \times}$.
We have that $4 = 2^2$.
Thus $4 \in \gen 2$.
There ex... | Let $\gen g = G$ be a [[Definition:Cyclic Group|cyclic group]].
Let $a \in G$
Then it is not necessarily the case that $a$ is also a [[Definition:Generator of Cyclic Group|generator]] of $G$. | Consider the [[Definition:Multiplicative Group of Real Numbers|multiplicative group of real numbers]] $\struct {\R_{\ne 0}, \times}$.
Consider the [[Definition:Subgroup|subgroup]] $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ [[Definition:Generated Subgroup|generated by]] $2$.
By definition, $\gen 2$ is a [[Definition:C... | Element of Cyclic Group is not necessarily Generator | https://proofwiki.org/wiki/Element_of_Cyclic_Group_is_not_necessarily_Generator | https://proofwiki.org/wiki/Element_of_Cyclic_Group_is_not_necessarily_Generator | [
"Cyclic Groups"
] | [
"Definition:Cyclic Group",
"Definition:Cyclic Group/Generator"
] | [
"Definition:Multiplicative Group of Real Numbers",
"Definition:Subgroup",
"Definition:Generated Subgroup",
"Definition:Cyclic Group",
"Definition:Element",
"Definition:Generated Subgroup",
"Definition:Cyclic Group/Generator",
"Proof by Counterexample"
] |
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