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-17600 | Number of Parameters of ARMA Model | Let $S$ be a stochastic process based on an equispaced time series.
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t -... | By definition of the parameters of $M$:
{{:Definition:Parameter of ARMA Model}}
Thus:
:there are $p$ parameters of the form $\phi_i$
:there are $q$ parameters of the form $\theta_j$
:$1$ parameter $\mu$
:$1$ parameter $\sigma_a^2$.
That is: $p + q + 1 + 1 = p + q + 2$ parameters.
{{qed}} | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] based on an [[Definition:Equispaced Time Series|equispaced time series]].
Let the values of $S$ at [[Definition:Timestamp of Time Series Observation|timestamps]] $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z... | By definition of the [[Definition:Parameter of ARMA Model|parameters]] of $M$:
{{:Definition:Parameter of ARMA Model}}
Thus:
:there are $p$ [[Definition:Parameter of ARMA Model|parameters]] of the form $\phi_i$
:there are $q$ [[Definition:Parameter of ARMA Model|parameters]] of the form $\theta_j$
:$1$ [[Definition:P... | Number of Parameters of ARMA Model | https://proofwiki.org/wiki/Number_of_Parameters_of_ARMA_Model | https://proofwiki.org/wiki/Number_of_Parameters_of_ARMA_Model | [
"ARMA Models"
] | [
"Definition:Stochastic Process",
"Definition:Time Series/Equispaced",
"Definition:Time Series/Timestamp",
"Definition:Deviation from Mean",
"Definition:Constant Mean Level",
"Definition:Sequence",
"Definition:Independent Shocks",
"Definition:Time Series/Timestamp",
"Definition:Box-Jenkins Model/ARMA... | [
"Definition:ARMA Model/Parameter",
"Definition:ARMA Model/Parameter",
"Definition:ARMA Model/Parameter",
"Definition:ARMA Model/Parameter",
"Definition:ARMA Model/Parameter",
"Definition:ARMA Model/Parameter"
] |
proofwiki-17601 | Characteristic of Field by Annihilator/Characteristic Zero | Suppose that:
:$\map {\mathrm {Ann} } F = \set 0$
That is, the annihilator of $F$ consists of the zero only.
Then:
:$\Char F = 0$
That is, the characteristic of $F$ is zero. | Let the zero of $F$ be $0_F$ and the unity of $F$ be $1_F$.
By definition of characteristic, $\Char F = 0$ {{iff}}:
:$\not \exists n \in \Z, n > 0: \forall r \in F: n \cdot r = 0_F$
That is, there exists no $n \in \Z, n > 0$ such that $n \cdot r = 0_F$ for all $r \in F$.
But note that $\forall r \in F: 0 \cdot r = 0_F$... | Suppose that:
:$\map {\mathrm {Ann} } F = \set 0$
That is, the [[Definition:Annihilator of Ring|annihilator]] of $F$ consists of the [[Definition:Field Zero|zero]] only.
Then:
:$\Char F = 0$
That is, the [[Definition:Characteristic of Field|characteristic]] of $F$ is zero. | Let the [[Definition:Field Zero|zero]] of $F$ be $0_F$ and the [[Definition:Unity of Field|unity]] of $F$ be $1_F$.
By definition of [[Definition:Characteristic of Field|characteristic]], $\Char F = 0$ {{iff}}:
:$\not \exists n \in \Z, n > 0: \forall r \in F: n \cdot r = 0_F$
That is, there exists no $n \in \Z, n > 0... | Characteristic of Field by Annihilator/Characteristic Zero | https://proofwiki.org/wiki/Characteristic_of_Field_by_Annihilator/Characteristic_Zero | https://proofwiki.org/wiki/Characteristic_of_Field_by_Annihilator/Characteristic_Zero | [
"Characteristics of Fields"
] | [
"Definition:Annihilator of Ring",
"Definition:Field Zero",
"Definition:Characteristic of Field"
] | [
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Characteristic of Field",
"Definition:Integral Multiple",
"Definition:Field Zero",
"Definition:Element",
"Non-Trivial Annihilator Contains Positive Integer",
"Definition:Strictly Positive/Integer",
"Definition:Contradiction"
... |
proofwiki-17602 | Characteristic of Field by Annihilator/Prime Characteristic | Suppose that:
:$\exists n \in \map {\mathrm {Ann} } F: n \ne 0$
That is, there exists (at least one) non-zero integer in the annihilator of $F$.
If this is the case, then the characteristic of $F$ is non-zero:
:$\Char F = p \ne 0$
and the annihilator of $F$ consists of the set of integer multiples of $p$:
:$\map {\math... | Let $A := \map {\mathrm {Ann} } F$.
We are told that:
:$\exists n \in A: n \ne 0$
Consider the set $A^+ \set {n \in A: n > 0}$.
From Non-Trivial Annihilator Contains Positive Integer we have that $A^+ \ne \O$.
As $A^+ \subseteq \N$ it follows from the well-ordering principle that $A^+$ has a least value $p$, say.
{{Aim... | Suppose that:
:$\exists n \in \map {\mathrm {Ann} } F: n \ne 0$
That is, there exists (at least one) non-zero [[Definition:Integer|integer]] in the [[Definition:Annihilator of Ring|annihilator]] of $F$.
If this is the case, then the [[Definition:Characteristic of Field|characteristic]] of $F$ is non-zero:
:$\Char F ... | Let $A := \map {\mathrm {Ann} } F$.
We are told that:
:$\exists n \in A: n \ne 0$
Consider the set $A^+ \set {n \in A: n > 0}$.
From [[Non-Trivial Annihilator Contains Positive Integer]] we have that $A^+ \ne \O$.
As $A^+ \subseteq \N$ it follows from the [[Well-Ordering Principle|well-ordering principle]] that $A^... | Characteristic of Field by Annihilator/Prime Characteristic | https://proofwiki.org/wiki/Characteristic_of_Field_by_Annihilator/Prime_Characteristic | https://proofwiki.org/wiki/Characteristic_of_Field_by_Annihilator/Prime_Characteristic | [
"Characteristics of Fields"
] | [
"Definition:Integer",
"Definition:Annihilator of Ring",
"Definition:Characteristic of Field",
"Definition:Annihilator of Ring",
"Definition:Set of Integer Multiples",
"Definition:Prime Number"
] | [
"Non-Trivial Annihilator Contains Positive Integer",
"Well-Ordering Principle",
"Definition:Prime Number",
"Product of Integral Multiples",
"Field has no Proper Zero Divisors",
"Definition:Strictly Positive/Integer",
"Definition:Element",
"Definition:Contradiction",
"Definition:Prime Number",
"Int... |
proofwiki-17603 | Cauchy Sequence is Bounded/Metric Space | Let $M = \struct {A, d}$ be a metric space.
Then every Cauchy sequence in $M$ is bounded. | Let $\sequence {x_n}$ be a Cauchy sequence in $M$.
By definition:
:$\forall \epsilon > 0: \exists N \in \N: \forall m, n > N: \map d {x_n, x_m} < \epsilon$
Particularly, setting $\epsilon = 1$:
:$\exists N_1: \forall m, n > N_1: \map d {x_n, x_m} < 1$
Note that since $N_1 \ge N_1$, this means that:
:$\forall n \ge N_1:... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]].
Then every [[Definition:Cauchy Sequence (Metric Space)|Cauchy sequence]] in $M$ is [[Definition:Bounded Sequence in Metric Space|bounded]]. | Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence (Metric Space)|Cauchy sequence]] in $M$.
By definition:
:$\forall \epsilon > 0: \exists N \in \N: \forall m, n > N: \map d {x_n, x_m} < \epsilon$
Particularly, setting $\epsilon = 1$:
:$\exists N_1: \forall m, n > N_1: \map d {x_n, x_m} < 1$
Note that since $... | Cauchy Sequence is Bounded/Metric Space | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Metric_Space | https://proofwiki.org/wiki/Cauchy_Sequence_is_Bounded/Metric_Space | [
"Cauchy Sequences"
] | [
"Definition:Metric Space",
"Definition:Cauchy Sequence/Metric Space",
"Definition:Bounded Sequence/Metric Space"
] | [
"Definition:Cauchy Sequence/Metric Space",
"Definition:Bounded Metric Space",
"Definition:Bounded Sequence/Metric Space"
] |
proofwiki-17604 | ARIMA Model subsumes ARMA Model | Let $S$ be a stochastic process based on an equispaced time series.
Let $M$ be an ARMA model for $S$.
Then $M$ is also an implementation of an ARIMA model. | Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at times... | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] based on an [[Definition:Equispaced Time Series|equispaced time series]].
Let $M$ be an [[Definition:ARMA Model|ARMA model]] for $S$.
Then $M$ is also an implementation of an [[Definition:ARIMA Model|ARIMA model]]. | Let the values of $S$ at [[Definition:Timestamp of Time Series Observation|timestamps]] $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be [[Definition:Deviation from Mean|deviations]] from a [[Definition:Constant Mean Level|constant mean le... | ARIMA Model subsumes ARMA Model | https://proofwiki.org/wiki/ARIMA_Model_subsumes_ARMA_Model | https://proofwiki.org/wiki/ARIMA_Model_subsumes_ARMA_Model | [
"ARIMA Models",
"ARMA Models"
] | [
"Definition:Stochastic Process",
"Definition:Time Series/Equispaced",
"Definition:Box-Jenkins Model/ARMA",
"Definition:Box-Jenkins Model/ARIMA"
] | [
"Definition:Time Series/Timestamp",
"Definition:Deviation from Mean",
"Definition:Constant Mean Level",
"Definition:Sequence",
"Definition:Independent Shocks",
"Definition:Time Series/Timestamp",
"Definition:Box-Jenkins Model/ARMA",
"Definition:Box-Jenkins Model/ARIMA"
] |
proofwiki-17605 | ARIMA Model subsumes Autoregressive Model | Let $S$ be a stochastic process based on an equispaced time series.
Let $M$ be an autoregressive model for $S$.
Then $M$ is also an implementation of an ARIMA model. | Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at times... | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] based on an [[Definition:Equispaced Time Series|equispaced time series]].
Let $M$ be an [[Definition:Autoregressive Model|autoregressive model]] for $S$.
Then $M$ is also an implementation of an [[Definition:ARIMA Model|ARIMA model]]. | Let the values of $S$ at [[Definition:Timestamp of Time Series Observation|timestamps]] $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be [[Definition:Deviation from Mean|deviations]] from a [[Definition:Constant Mean Level|constant mean le... | ARIMA Model subsumes Autoregressive Model | https://proofwiki.org/wiki/ARIMA_Model_subsumes_Autoregressive_Model | https://proofwiki.org/wiki/ARIMA_Model_subsumes_Autoregressive_Model | [
"ARIMA Models",
"Autoregressive Models"
] | [
"Definition:Stochastic Process",
"Definition:Time Series/Equispaced",
"Definition:Autoregressive Model",
"Definition:Box-Jenkins Model/ARIMA"
] | [
"Definition:Time Series/Timestamp",
"Definition:Deviation from Mean",
"Definition:Constant Mean Level",
"Definition:Sequence",
"Definition:Independent Shocks",
"Definition:Time Series/Timestamp",
"Definition:Autoregressive Model",
"Definition:Box-Jenkins Model/ARIMA"
] |
proofwiki-17606 | ARIMA Model subsumes Moving Average Model | Let $S$ be a stochastic process based on an equispaced time series.
Let $M$ be a moving average model for $S$.
Then $M$ is also an implementation of an ARIMA model. | Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
:$\tilde z_t = z_t - \mu$
Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at times... | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] based on an [[Definition:Equispaced Time Series|equispaced time series]].
Let $M$ be a [[Definition:Moving Average Model|moving average model]] for $S$.
Then $M$ is also an implementation of an [[Definition:ARIMA Model|ARIMA model]]. | Let the values of $S$ at [[Definition:Timestamp of Time Series Observation|timestamps]] $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$
Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be [[Definition:Deviation from Mean|deviations]] from a [[Definition:Constant Mean Level|constant mean le... | ARIMA Model subsumes Moving Average Model | https://proofwiki.org/wiki/ARIMA_Model_subsumes_Moving_Average_Model | https://proofwiki.org/wiki/ARIMA_Model_subsumes_Moving_Average_Model | [
"ARIMA Models",
"Moving Average Models"
] | [
"Definition:Stochastic Process",
"Definition:Time Series/Equispaced",
"Definition:Moving Average Model",
"Definition:Box-Jenkins Model/ARIMA"
] | [
"Definition:Time Series/Timestamp",
"Definition:Deviation from Mean",
"Definition:Constant Mean Level",
"Definition:Sequence",
"Definition:Independent Shocks",
"Definition:Time Series/Timestamp",
"Definition:Moving Average Model",
"Definition:Box-Jenkins Model/ARIMA"
] |
proofwiki-17607 | Real Division is not Closed | The operation of division on the set of real numbers $\R$ is not closed. | From Division by Zero, we have that for all $a \in \R$, the operation $\dfrac a 0$ is not defined.
{{qed}} | The [[Definition:Binary Operation|operation]] of [[Definition:Real Division|division]] on the [[Definition:Set|set]] of [[Definition:Real Number|real numbers]] $\R$ is not [[Definition:Closed Operation|closed]]. | From [[Division by Zero]], we have that for all $a \in \R$, the operation $\dfrac a 0$ is not defined.
{{qed}} | Real Division is not Closed | https://proofwiki.org/wiki/Real_Division_is_not_Closed | https://proofwiki.org/wiki/Real_Division_is_not_Closed | [
"Real Division",
"Algebraic Closure"
] | [
"Definition:Operation/Binary Operation",
"Definition:Division/Field/Real Numbers",
"Definition:Set",
"Definition:Real Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure"
] | [
"Division by Zero"
] |
proofwiki-17608 | Strictly Positive Real Numbers are not Closed under Subtraction | The set $\R_{>0}$ of strictly positive real numbers is not closed under subtraction. | ;Proof by Counterexample
Let $a = 1$ and $b = 2$.
Then:
:$a - b = -1$
but $-1$ is not a (strictly) positive real number.
{{qed}} | The [[Definition:Set|set]] $\R_{>0}$ of [[Definition:Strictly Positive Real Number|strictly positive real numbers]] is not [[Definition:Closed Algebraic Structure|closed]] under [[Definition:Real Subtraction|subtraction]]. | ;[[Proof by Counterexample]]
Let $a = 1$ and $b = 2$.
Then:
:$a - b = -1$
but $-1$ is not a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
{{qed}} | Strictly Positive Real Numbers are not Closed under Subtraction | https://proofwiki.org/wiki/Strictly_Positive_Real_Numbers_are_not_Closed_under_Subtraction | https://proofwiki.org/wiki/Strictly_Positive_Real_Numbers_are_not_Closed_under_Subtraction | [
"Real Subtraction",
"Algebraic Closure"
] | [
"Definition:Set",
"Definition:Strictly Positive/Real Number",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Subtraction/Real Numbers"
] | [
"Proof by Counterexample",
"Definition:Strictly Positive/Real Number"
] |
proofwiki-17609 | Zero of Field is Unique | Let $\struct {F, +, \times}$ be a field.
The zero of $F$ is unique. | By definition, a field is a ring whose ring product less zero is an abelian group.
The result follows from Ring Zero is Unique.
{{qed}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
The [[Definition:Field Zero|zero]] of $F$ is unique. | By definition, a [[Definition:Field (Abstract Algebra)|field]] is a [[Definition:Ring (Abstract Algebra)|ring]] whose [[Definition:Ring Product|ring product]] less [[Definition:Ring Zero|zero]] is an [[Definition:Abelian Group|abelian group]].
The result follows from [[Ring Zero is Unique]].
{{qed}} | Zero of Field is Unique/Proof 1 | https://proofwiki.org/wiki/Zero_of_Field_is_Unique | https://proofwiki.org/wiki/Zero_of_Field_is_Unique/Proof_1 | [
"Field Theory",
"Zero of Field is Unique"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Ring Zero",
"Definition:Abelian Group",
"Ring Zero is Unique"
] |
proofwiki-17610 | Zero of Field is Unique | Let $\struct {F, +, \times}$ be a field.
The zero of $F$ is unique. | Let $0_1$ and $0_2$ both be elements of $F$ such that:
:$\forall a \in F: a + 0_1 = a$
:$\forall a \in F: a + 0_2 = a$
Then:
:$0_1 + 0_2 = 0_2$
because $0_1$ is a zero element
:$0_1 + 0_2 = 0_1$
because $0_2$ is a zero element
Hence:
:$0_1 = 0_2$
and the two zero elements are the same.
{{qed}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
The [[Definition:Field Zero|zero]] of $F$ is unique. | Let $0_1$ and $0_2$ both be [[Definition:Element|elements]] of $F$ such that:
:$\forall a \in F: a + 0_1 = a$
:$\forall a \in F: a + 0_2 = a$
Then:
:$0_1 + 0_2 = 0_2$
because $0_1$ is a [[Definition:Zero Element|zero element]]
:$0_1 + 0_2 = 0_1$
because $0_2$ is a [[Definition:Zero Element|zero element]]
Hence:
:$0... | Zero of Field is Unique/Proof 2 | https://proofwiki.org/wiki/Zero_of_Field_is_Unique | https://proofwiki.org/wiki/Zero_of_Field_is_Unique/Proof_2 | [
"Field Theory",
"Zero of Field is Unique"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero"
] | [
"Definition:Element",
"Definition:Zero Element",
"Definition:Zero Element",
"Definition:Field Zero"
] |
proofwiki-17611 | Negative of Element in Field is Unique | Let $\struct {F, +, \times}$ be a field.
Let $a \in F$.
Then the negative $-a$ of $a$ is unique. | By definition, a field is a ring whose ring product less zero is an abelian group.
The result follows from Ring Negative is Unique.
{{qed}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $a \in F$.
Then the [[Definition:Field Negative|negative]] $-a$ of $a$ is [[Definition:Unique|unique]]. | By definition, a [[Definition:Field (Abstract Algebra)|field]] is a [[Definition:Ring (Abstract Algebra)|ring]] whose [[Definition:Ring Product|ring product]] less [[Definition:Ring Zero|zero]] is an [[Definition:Abelian Group|abelian group]].
The result follows from [[Ring Negative is Unique]].
{{qed}} | Negative of Element in Field is Unique/Proof 1 | https://proofwiki.org/wiki/Negative_of_Element_in_Field_is_Unique | https://proofwiki.org/wiki/Negative_of_Element_in_Field_is_Unique/Proof_1 | [
"Field Theory",
"Negative of Element in Field is Unique"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Negative",
"Definition:Unique"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Ring Zero",
"Definition:Abelian Group",
"Ring Negative is Unique"
] |
proofwiki-17612 | Negative of Element in Field is Unique | Let $\struct {F, +, \times}$ be a field.
Let $a \in F$.
Then the negative $-a$ of $a$ is unique. | Let $b, c \in F$ such that both $a + b = 0$ and $a + c = 0$.
Thus:
{{begin-eqn}}
{{eqn | l = b + \paren {a + c}
| r = b + 0
| c = as $c$ is a negative of $a$
}}
{{eqn | r = b
| c = {{Defof|Field Zero}}
}}
{{end-eqn}}
But also:
{{begin-eqn}}
{{eqn | l = \paren {b + a} + c
| r = \paren {a + b} + c... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $a \in F$.
Then the [[Definition:Field Negative|negative]] $-a$ of $a$ is [[Definition:Unique|unique]]. | Let $b, c \in F$ such that both $a + b = 0$ and $a + c = 0$.
Thus:
{{begin-eqn}}
{{eqn | l = b + \paren {a + c}
| r = b + 0
| c = as $c$ is a [[Definition:Field Negative|negative]] of $a$
}}
{{eqn | r = b
| c = {{Defof|Field Zero}}
}}
{{end-eqn}}
But also:
{{begin-eqn}}
{{eqn | l = \paren {b + a} +... | Negative of Element in Field is Unique/Proof 2 | https://proofwiki.org/wiki/Negative_of_Element_in_Field_is_Unique | https://proofwiki.org/wiki/Negative_of_Element_in_Field_is_Unique/Proof_2 | [
"Field Theory",
"Negative of Element in Field is Unique"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Negative",
"Definition:Unique"
] | [
"Definition:Field Negative",
"Definition:Field Negative"
] |
proofwiki-17613 | Space of Bounded Sequences with Supremum Norm forms Banach Space | Let $\struct {\map {\ell^\infty} \R, \norm {\, \cdot \,}_\infty}$ be the normed vector space of bounded sequences on $\R$.
Then $\struct {\map {\ell^\infty} \R, \norm {\, \cdot \,}_\infty}$ is a Banach space. | A Banach space is a normed vector space, where a Cauchy sequence converges {{WRT}} the supplied norm.
To prove the theorem, we need to show that a Cauchy sequence in $\struct {\map {\ell^\infty} \R, \norm {\,\cdot\,}_\infty}$ converges.
We take a Cauchy sequence $\sequence {x_n}_{n \mathop \in \N}$ in $\struct {\map {\... | Let $\struct {\map {\ell^\infty} \R, \norm {\, \cdot \,}_\infty}$ be the [[Definition:Normed Vector Space of Bounded Sequences|normed vector space of bounded sequences on $\R$]].
Then $\struct {\map {\ell^\infty} \R, \norm {\, \cdot \,}_\infty}$ is a [[Definition:Banach Space|Banach space]]. | A [[Definition:Banach Space|Banach space]] is a [[Definition:Normed Vector Space|normed vector space]], where a [[Definition:Cauchy Sequence|Cauchy sequence]] [[Definition:Convergent Sequence in Normed Vector Space|converges]] {{WRT}} the supplied [[Definition:Norm on Vector Space|norm]].
To prove the theorem, we need... | Space of Bounded Sequences with Supremum Norm forms Banach Space | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_with_Supremum_Norm_forms_Banach_Space | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_with_Supremum_Norm_forms_Banach_Space | [
"Functional Analysis",
"Banach Spaces",
"Space of Bounded Sequences"
] | [
"Definition:Space of Bounded Sequences/Normed Vector Space",
"Definition:Banach Space"
] | [
"Definition:Banach Space",
"Definition:Normed Vector Space",
"Definition:Cauchy Sequence",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Norm/Vector Space",
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Cauchy Se... |
proofwiki-17614 | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 2 | :$\forall B_1, B_2 \in \mathscr B : \card{B_1} = \card{B_2}$
where $\card{B_1}$ and $\card{B_2}$ denote the cardinality of the sets $B_1$ and $B_2$ respectively. | {{AimForCont}}:
:$\exists B_1, B_2 \in \mathscr B : \card{B_1} \ne \card{B_2}$
{{WLOG}} and from Max Operation Equals an Operand let:
:$\card{B_1 \cap B_2} = \max \set{\card{C_1 \cap C_2} : C_1, C_2 \in \mathscr B : \card{C_1} \ne \card{C_2}}$
{{WLOG}} let:
:$\card{B_1} > \card{B_2}$
From Set Difference of Larger Set w... | :$\forall B_1, B_2 \in \mathscr B : \card{B_1} = \card{B_2}$
where $\card{B_1}$ and $\card{B_2}$ denote the [[Definition:Cardinality|cardinality]] of the [[Definition:Set|sets]] $B_1$ and $B_2$ respectively. | {{AimForCont}}:
:$\exists B_1, B_2 \in \mathscr B : \card{B_1} \ne \card{B_2}$
{{WLOG}} and from [[Max Operation Equals an Operand]] let:
:$\card{B_1 \cap B_2} = \max \set{\card{C_1 \cap C_2} : C_1, C_2 \in \mathscr B : \card{C_1} \ne \card{C_2}}$
{{WLOG}} let:
:$\card{B_1} > \card{B_2}$
From [[Set Difference of ... | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 2 | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Lemma_2 | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Lemma_2 | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom"
] | [
"Definition:Cardinality",
"Definition:Set"
] | [
"Max Operation Equals an Operand",
"Set Difference of Larger Set with Smaller is Not Empty",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Finite Set Formed by Substitution has Larger Intersection",
"Finite Set Formed by Substitution has Same Cardinality",
"Definition:Contradiction",
"Category:Matroid Ba... |
proofwiki-17615 | Negative of Field Negative | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.
Let $a \in F$ and let $-a$ be the field negative of $a$.
Then:
:$-\paren {-a} = a$ | {{begin-eqn}}
{{eqn | l = \paren {-a} + a
| r = a + \paren {-a}
| c = {{Field-axiom|A2}}
}}
{{eqn | r = 0_F
| c = {{Field-axiom|A4}}
}}
{{eqn | ll= \leadsto
| l = a
| r = -\paren {-a}
| c = {{Defof|Field Negative}}
}}
{{end-eqn}}
{{qed}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$.
Let $a \in F$ and let $-a$ be the [[Definition:Field Negative|field negative]] of $a$.
Then:
:$-\paren {-a} = a$ | {{begin-eqn}}
{{eqn | l = \paren {-a} + a
| r = a + \paren {-a}
| c = {{Field-axiom|A2}}
}}
{{eqn | r = 0_F
| c = {{Field-axiom|A4}}
}}
{{eqn | ll= \leadsto
| l = a
| r = -\paren {-a}
| c = {{Defof|Field Negative}}
}}
{{end-eqn}}
{{qed}} | Negative of Field Negative | https://proofwiki.org/wiki/Negative_of_Field_Negative | https://proofwiki.org/wiki/Negative_of_Field_Negative | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Field Negative"
] | [] |
proofwiki-17616 | Field Product with Zero | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.
Let $a \in F$.
Then:
:$a \times 0_F = 0_F$ | {{begin-eqn}}
{{eqn | l = a
| r = a \times 1_F
| c = {{Field-axiom|M3}}: $1_F$ is the unity of $F$
}}
{{eqn | r = a \times \paren {0_F + 1_F}
| c = {{Field-axiom|A3}}
}}
{{eqn | r = a \times 0_F + a \times 1_F
| c = {{Field-axiom|D}}
}}
{{eqn | r = a \times 0_F + a
| c = {{Field-axiom|M3}}... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$.
Let $a \in F$.
Then:
:$a \times 0_F = 0_F$ | {{begin-eqn}}
{{eqn | l = a
| r = a \times 1_F
| c = {{Field-axiom|M3}}: $1_F$ is the [[Definition:Unity of Field|unity]] of $F$
}}
{{eqn | r = a \times \paren {0_F + 1_F}
| c = {{Field-axiom|A3}}
}}
{{eqn | r = a \times 0_F + a \times 1_F
| c = {{Field-axiom|D}}
}}
{{eqn | r = a \times 0_F + a
... | Field Product with Zero | https://proofwiki.org/wiki/Field_Product_with_Zero | https://proofwiki.org/wiki/Field_Product_with_Zero | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero"
] | [
"Definition:Multiplicative Identity"
] |
proofwiki-17617 | Product with Field Negative | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $a, b \in F$.
Then:
:$-\paren {a \times b} = a \times \paren {-b} = \paren {-a} \times b$ | {{begin-eqn}}
{{eqn | l = a \times b + a \times \paren {-b}
| r = a \times \paren {b + \paren {-b} }
| c = {{Field-axiom|A1}}
}}
{{eqn | r = a \times 0_F
| c = {{Field-axiom|A4}}
}}
{{eqn | r = 0_F
| c = Field Product with Zero
}}
{{eqn | ll= \leadsto
| l = -\paren {a \times b}
| r =... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$ and whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $a, b \in F$.
Then:
:$-\paren {a \times b} = a \times \paren {-b} = \paren {-a} \times b$ | {{begin-eqn}}
{{eqn | l = a \times b + a \times \paren {-b}
| r = a \times \paren {b + \paren {-b} }
| c = {{Field-axiom|A1}}
}}
{{eqn | r = a \times 0_F
| c = {{Field-axiom|A4}}
}}
{{eqn | r = 0_F
| c = [[Field Product with Zero]]
}}
{{eqn | ll= \leadsto
| l = -\paren {a \times b}
|... | Product with Field Negative | https://proofwiki.org/wiki/Product_with_Field_Negative | https://proofwiki.org/wiki/Product_with_Field_Negative | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity"
] | [
"Field Product with Zero",
"Field Product with Zero"
] |
proofwiki-17618 | Product with Field Negative/Corollary | :$\paren {-1_F} \times a = \paren {-a}$ | {{begin-eqn}}
{{eqn | l = a + \paren {-1_F} \times a
| r = 1_F \times a + \paren {-1_F} \times a
| c = {{Field-axiom|M3}}
}}
{{eqn | r = \paren {1_F + \paren {-1_F} } \times a
| c = {{Field-axiom|D}}
}}
{{eqn | r = 0_F \times a
| c = {{Field-axiom|A4}}
}}
{{eqn | r = 0_F
| c = Field Produc... | :$\paren {-1_F} \times a = \paren {-a}$ | {{begin-eqn}}
{{eqn | l = a + \paren {-1_F} \times a
| r = 1_F \times a + \paren {-1_F} \times a
| c = {{Field-axiom|M3}}
}}
{{eqn | r = \paren {1_F + \paren {-1_F} } \times a
| c = {{Field-axiom|D}}
}}
{{eqn | r = 0_F \times a
| c = {{Field-axiom|A4}}
}}
{{eqn | r = 0_F
| c = [[Field Prod... | Product with Field Negative/Corollary | https://proofwiki.org/wiki/Product_with_Field_Negative/Corollary | https://proofwiki.org/wiki/Product_with_Field_Negative/Corollary | [
"Field Theory"
] | [] | [
"Field Product with Zero"
] |
proofwiki-17619 | Condition for Division by Field Elements to be Unity | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $a, b \in F$.
Then:
:$\dfrac a b = 1_F$
{{iff}}:
:$a = b$
where $\dfrac a b$ denotes division. | === Necessary Condition ===
Let $a = b$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac a b
| r = a \times b^{-1}
| c = {{Defof|Division over Field}}
}}
{{eqn | r = b \times b^{-1}
| c = as $a = b$
}}
{{eqn | r = 1_F
| c = {{Field-axiom|M4}}
}}
{{end-eqn}}
{{qed|lemma}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$ and whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $a, b \in F$.
Then:
:$\dfrac a b = 1_F$
{{iff}}:
:$a = b$
where $\dfrac a b$ denotes [[Definition:Division over Field|division]]. | === Necessary Condition ===
Let $a = b$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac a b
| r = a \times b^{-1}
| c = {{Defof|Division over Field}}
}}
{{eqn | r = b \times b^{-1}
| c = as $a = b$
}}
{{eqn | r = 1_F
| c = {{Field-axiom|M4}}
}}
{{end-eqn}}
{{qed|lemma}} | Condition for Division by Field Elements to be Unity | https://proofwiki.org/wiki/Condition_for_Division_by_Field_Elements_to_be_Unity | https://proofwiki.org/wiki/Condition_for_Division_by_Field_Elements_to_be_Unity | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Division/Field"
] | [] |
proofwiki-17620 | Field Product with Non-Zero Element yields Unique Solution | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $a, b, x \in F$ such that $b \ne 0_F$.
Let:
:$b \times x = a$
Then:
:$x = a b^{-1}$
That is:
:$x = \dfrac a b$
where $\dfrac a b$ denotes division. | {{begin-eqn}}
{{eqn | l = b \times x
| r = a
| c = with $b \ne 0_F$
}}
{{eqn | ll= \leadsto
| l = b^{-1} \times \paren {b \times x}
| r = b^{-1} \times a
| c = multiplying both sides by $b^{-1}$, which exists because $b \ne 0$
}}
{{eqn | ll= \leadsto
| l = \paren {b^{-1} \times b} \t... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$ and whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $a, b, x \in F$ such that $b \ne 0_F$.
Let:
:$b \times x = a$
Then:
:$x = a b^{-1}$
That is:
:$x = \dfrac a b$
where $\dfrac a b$... | {{begin-eqn}}
{{eqn | l = b \times x
| r = a
| c = with $b \ne 0_F$
}}
{{eqn | ll= \leadsto
| l = b^{-1} \times \paren {b \times x}
| r = b^{-1} \times a
| c = multiplying both sides by $b^{-1}$, which exists because $b \ne 0$
}}
{{eqn | ll= \leadsto
| l = \paren {b^{-1} \times b} \t... | Field Product with Non-Zero Element yields Unique Solution | https://proofwiki.org/wiki/Field_Product_with_Non-Zero_Element_yields_Unique_Solution | https://proofwiki.org/wiki/Field_Product_with_Non-Zero_Element_yields_Unique_Solution | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Division/Field"
] | [] |
proofwiki-17621 | Field Unity Divided by Element equals Multiplicative Inverse | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $a \in F$.
Then:
:$\dfrac {1_F} a = a^{-1}$
where $\dfrac {1_F} a$ denotes division. | {{begin-eqn}}
{{eqn | l = \dfrac {1_F} a
| r = 1_F \times a^{-1}
| c = {{Defof|Division over Field}}
}}
{{eqn | r = a^{-1} \times 1_F
| c = {{Field-axiom|M2}}
}}
{{eqn | r = a^{-1}
| c = {{Field-axiom|M3}}
}}
{{end-eqn}}
{{qed}} | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$ and whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $a \in F$.
Then:
:$\dfrac {1_F} a = a^{-1}$
where $\dfrac {1_F} a$ denotes [[Definition:Division over Field|division]]. | {{begin-eqn}}
{{eqn | l = \dfrac {1_F} a
| r = 1_F \times a^{-1}
| c = {{Defof|Division over Field}}
}}
{{eqn | r = a^{-1} \times 1_F
| c = {{Field-axiom|M2}}
}}
{{eqn | r = a^{-1}
| c = {{Field-axiom|M3}}
}}
{{end-eqn}}
{{qed}} | Field Unity Divided by Element equals Multiplicative Inverse | https://proofwiki.org/wiki/Field_Unity_Divided_by_Element_equals_Multiplicative_Inverse | https://proofwiki.org/wiki/Field_Unity_Divided_by_Element_equals_Multiplicative_Inverse | [
"Field Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity",
"Definition:Division/Field"
] | [] |
proofwiki-17622 | Cancellation Law for Field Product | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $a, b, c \in F$.
Then:
:$a \times b = a \times c \implies a = 0_F \text { or } b = c$ | Let $a \times b = a \times c$.
Then:
{{begin-eqn}}
{{eqn | l = a
| o = \ne
| r = 0_F
| c =
}}
{{eqn | ll= \leadsto
| q = \exists a^{-1} \in F
| l = a^{-1} \times a
| r = 1_F
| c =
}}
{{eqn | ll= \leadsto
| l = a^{-1} \times \paren {a \times b}
| r = a^{-1} \times ... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$ and whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $a, b, c \in F$.
Then:
:$a \times b = a \times c \implies a = 0_F \text { or } b = c$ | Let $a \times b = a \times c$.
Then:
{{begin-eqn}}
{{eqn | l = a
| o = \ne
| r = 0_F
| c =
}}
{{eqn | ll= \leadsto
| q = \exists a^{-1} \in F
| l = a^{-1} \times a
| r = 1_F
| c =
}}
{{eqn | ll= \leadsto
| l = a^{-1} \times \paren {a \times b}
| r = a^{-1} \time... | Cancellation Law for Field Product | https://proofwiki.org/wiki/Cancellation_Law_for_Field_Product | https://proofwiki.org/wiki/Cancellation_Law_for_Field_Product | [
"Field Theory",
"Cancellation Laws"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity"
] | [
"Rule of Transposition"
] |
proofwiki-17623 | Inverse of Field Product | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $a, b \in F$ such that $a \ne 0$ and $b \ne 0$.
Then:
:$\paren {a \times b}^{-1} = b^{-1} \times a^{-1}$ | We are given that $a \ne 0$ and $b \ne 0$.
From Field has no Proper Zero Divisors and Rule of Transposition, we have:
:$a \times b \ne 0$
By {{Field-axiom|M4}} we have that $\paren {a \times b}^{-1}$ exists.
Then we have:
{{begin-eqn}}
{{eqn | l = \paren {b^{-1} \times a^{-1} } \times \paren {a \times b}
| r = b^... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$ and whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $a, b \in F$ such that $a \ne 0$ and $b \ne 0$.
Then:
:$\paren {a \times b}^{-1} = b^{-1} \times a^{-1}$ | We are given that $a \ne 0$ and $b \ne 0$.
From [[Field has no Proper Zero Divisors]] and [[Rule of Transposition]], we have:
:$a \times b \ne 0$
By {{Field-axiom|M4}} we have that $\paren {a \times b}^{-1}$ exists.
Then we have:
{{begin-eqn}}
{{eqn | l = \paren {b^{-1} \times a^{-1} } \times \paren {a \times b}
... | Inverse of Field Product | https://proofwiki.org/wiki/Inverse_of_Field_Product | https://proofwiki.org/wiki/Inverse_of_Field_Product | [
"Field Theory",
"Inverse Elements"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity"
] | [
"Field has no Proper Zero Divisors",
"Rule of Transposition",
"Definition:Multiplicative Inverse/Field"
] |
proofwiki-17624 | Inverse of Field Product | Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $a, b \in F$ such that $a \ne 0$ and $b \ne 0$.
Then:
:$\paren {a \times b}^{-1} = b^{-1} \times a^{-1}$ | By definition, a field is a non-trivial division ring whose ring product is commutative.
By definition, a division ring is a ring with unity such that every non-zero element is a unit.
Hence we can use Inverse of Division Product:
:$\paren {\dfrac a b}^{-1} = \dfrac {1_R} {\paren {a / b}} = \dfrac b a$
which applies t... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]] whose [[Definition:Field Zero|zero]] is $0_F$ and whose [[Definition:Unity of Field|unity]] is $1_F$.
Let $a, b \in F$ such that $a \ne 0$ and $b \ne 0$.
Then:
:$\paren {a \times b}^{-1} = b^{-1} \times a^{-1}$ | By definition, a [[Definition:Field (Abstract Algebra)|field]] is a [[Definition:Non-Trivial Ring|non-trivial]] [[Definition:Division Ring|division ring]] whose [[Definition:Ring Product|ring product]] is [[Definition:Commutative Operation|commutative]].
By definition, a [[Definition:Division Ring|division ring]] is a... | Inverse of Field Product with Inverse/Proof 1 | https://proofwiki.org/wiki/Inverse_of_Field_Product | https://proofwiki.org/wiki/Inverse_of_Field_Product_with_Inverse/Proof_1 | [
"Field Theory",
"Inverse Elements"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Field Zero",
"Definition:Multiplicative Identity"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Non-Trivial Ring",
"Definition:Division Ring",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Commutative/Operation",
"Definition:Division Ring",
"Definition:Ring with Unity",
"Definition:Ring Zero",
"Definition:Element",
"Definition:Un... |
proofwiki-17625 | Set Difference of Larger Set with Smaller is Not Empty | Let $S$ and $T$ be finite sets.
Let $\card S > \card T$.
Then:
:$S \setminus T \ne \O$ | From Cardinality of Subset of Finite Set:
:$S \nsubseteq T$
From the contrapositive statement of Set Difference with Superset is Empty Set:
:$S \setminus T \ne \O$.
{{qed}}
Category:Set Difference
Category:Cardinality
n3frdoief51ck6ehmgy4zv0wz97nrbu | Let $S$ and $T$ be [[Definition:Finite Set|finite]] [[Definition:Set|sets]].
Let $\card S > \card T$.
Then:
:$S \setminus T \ne \O$ | From [[Cardinality of Subset of Finite Set]]:
:$S \nsubseteq T$
From the [[Definition:Contrapositive|contrapositive statement]] of [[Set Difference with Superset is Empty Set]]:
:$S \setminus T \ne \O$.
{{qed}}
[[Category:Set Difference]]
[[Category:Cardinality]]
n3frdoief51ck6ehmgy4zv0wz97nrbu | Set Difference of Larger Set with Smaller is Not Empty | https://proofwiki.org/wiki/Set_Difference_of_Larger_Set_with_Smaller_is_Not_Empty | https://proofwiki.org/wiki/Set_Difference_of_Larger_Set_with_Smaller_is_Not_Empty | [
"Set Difference",
"Cardinality"
] | [
"Definition:Finite Set",
"Definition:Set"
] | [
"Cardinality of Subset of Finite Set",
"Definition:Contrapositive Statement",
"Set Difference with Superset is Empty Set",
"Category:Set Difference",
"Category:Cardinality"
] |
proofwiki-17626 | Autocovariance Matrix for Stationary Process is Variance by Autocorrelation Matrix | Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.
Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:
:$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$
Let $\boldsymbol \Gamma_n$ denote the '''autocovariance matrix''' associated with $S$ for $\sequence {s_n}$.
Let ... | From Autocorrelation of Strictly Stationary Stochastic Process:
:$\rho_k = \dfrac {\gamma_k} {\gamma_0}$
Then from Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance:
:$\gamma_0 = \sigma_z^2$
Hence:
:$\gamma_k = \sigma_z^2 \rho_k$
and the result follows.
{{qed}} | Let $S$ be a [[Definition:Strictly Stationary Stochastic Process|strictly stationary stochastic process]] giving rise to a [[Definition:Time Series|time series]] $T$.
Let $\sequence {s_n}$ be a [[Definition:Sequence|sequence]] of $n$ [[Definition:Successive Values of Equispaced Time Series|successive values]] of $T$:
... | From [[Autocorrelation of Strictly Stationary Stochastic Process]]:
:$\rho_k = \dfrac {\gamma_k} {\gamma_0}$
Then from [[Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance]]:
:$\gamma_0 = \sigma_z^2$
Hence:
:$\gamma_k = \sigma_z^2 \rho_k$
and the result follows.
{{qed}} | Autocovariance Matrix for Stationary Process is Variance by Autocorrelation Matrix | https://proofwiki.org/wiki/Autocovariance_Matrix_for_Stationary_Process_is_Variance_by_Autocorrelation_Matrix | https://proofwiki.org/wiki/Autocovariance_Matrix_for_Stationary_Process_is_Variance_by_Autocorrelation_Matrix | [
"Autocovariance Matrices",
"Autocorrelation Matrices"
] | [
"Definition:Strictly Stationary Stochastic Process",
"Definition:Time Series",
"Definition:Sequence",
"Definition:Successive Values of Time Series/Equispaced",
"Definition:Autocovariance Matrix",
"Definition:Autocorrelation Matrix",
"Definition:Variance"
] | [
"Autocorrelation of Strictly Stationary Stochastic Process",
"Autocovariance at Zero Lag for Strictly Stationary Stochastic Process is Variance"
] |
proofwiki-17627 | Determinant of Autocorrelation Matrix is Strictly Positive | Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.
Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:
:$\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$
Let $\mathbf P_n$ denote the '''autocorrelation matrix''' associated with $S$ for $\sequence {s_n}$.
The determin... | We have that the Autocorrelation Matrix is Positive Definite.
The result follows from Determinant of Positive Definite Matrix is Strictly Positive.
{{qed}} | Let $S$ be a [[Definition:Strictly Stationary Stochastic Process|strictly stationary stochastic process]] giving rise to a [[Definition:Time Series|time series]] $T$.
Let $\sequence {s_n}$ be a [[Definition:Sequence|sequence]] of $n$ [[Definition:Successive Values of Equispaced Time Series|successive values]] of $T$:
... | We have that the [[Autocorrelation Matrix is Positive Definite]].
The result follows from [[Determinant of Positive Definite Matrix is Strictly Positive]].
{{qed}} | Determinant of Autocorrelation Matrix is Strictly Positive | https://proofwiki.org/wiki/Determinant_of_Autocorrelation_Matrix_is_Strictly_Positive | https://proofwiki.org/wiki/Determinant_of_Autocorrelation_Matrix_is_Strictly_Positive | [
"Autocorrelation Matrices",
"Determinant of Autocorrelation Matrix is Strictly Positive"
] | [
"Definition:Strictly Stationary Stochastic Process",
"Definition:Time Series",
"Definition:Sequence",
"Definition:Successive Values of Time Series/Equispaced",
"Definition:Autocorrelation Matrix",
"Definition:Determinant",
"Definition:Strictly Positive/Real Number"
] | [
"Autocorrelation Matrix is Positive Definite",
"Determinant of Positive Definite Matrix is Strictly Positive"
] |
proofwiki-17628 | Sum of Wholly Real Numbers is Wholly Real | Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be wholly real complex numbers.
Then $x + y$ is also wholly real. | We have:
{{begin-eqn}}
{{eqn | l = x + y
| r = \tuple {a, 0} + \tuple {b, 0}
| c =
}}
{{eqn | r = \tuple {a + b, 0 + 0}
| c = {{Defof|Complex Addition}}
}}
{{eqn | r = \tuple {a + b, 0}
| c =
}}
{{end-eqn}}
{{qed}} | Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be [[Definition:Wholly Real|wholly real]] [[Definition:Complex Number|complex numbers]].
Then $x + y$ is also [[Definition:Wholly Real|wholly real]]. | We have:
{{begin-eqn}}
{{eqn | l = x + y
| r = \tuple {a, 0} + \tuple {b, 0}
| c =
}}
{{eqn | r = \tuple {a + b, 0 + 0}
| c = {{Defof|Complex Addition}}
}}
{{eqn | r = \tuple {a + b, 0}
| c =
}}
{{end-eqn}}
{{qed}} | Sum of Wholly Real Numbers is Wholly Real | https://proofwiki.org/wiki/Sum_of_Wholly_Real_Numbers_is_Wholly_Real | https://proofwiki.org/wiki/Sum_of_Wholly_Real_Numbers_is_Wholly_Real | [
"Complex Addition"
] | [
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number",
"Definition:Complex Number/Wholly Real"
] | [] |
proofwiki-17629 | Product of Wholly Real Numbers is Wholly Real | Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be wholly real complex numbers.
Then $x y$ is also wholly real. | We have:
{{begin-eqn}}
{{eqn | l = x y
| r = \tuple {a, 0} \tuple {b, 0}
| c =
}}
{{eqn | r = \tuple {a \times b - 0 \times 0, a \times 0 + 0 \times b}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = \tuple {a \times b, 0}
| c =
}}
{{end-eqn}}
{{qed}} | Let $x = \tuple {a, 0}$ and $y = \tuple {b, 0}$ be [[Definition:Wholly Real|wholly real]] [[Definition:Complex Number|complex numbers]].
Then $x y$ is also [[Definition:Wholly Real|wholly real]]. | We have:
{{begin-eqn}}
{{eqn | l = x y
| r = \tuple {a, 0} \tuple {b, 0}
| c =
}}
{{eqn | r = \tuple {a \times b - 0 \times 0, a \times 0 + 0 \times b}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = \tuple {a \times b, 0}
| c =
}}
{{end-eqn}}
{{qed}} | Product of Wholly Real Numbers is Wholly Real | https://proofwiki.org/wiki/Product_of_Wholly_Real_Numbers_is_Wholly_Real | https://proofwiki.org/wiki/Product_of_Wholly_Real_Numbers_is_Wholly_Real | [
"Complex Multiplication"
] | [
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number",
"Definition:Complex Number/Wholly Real"
] | [] |
proofwiki-17630 | Product of Imaginary Unit with Itself | Let $\tuple {0, 1}$ denote the imaginary unit.
Then:
:$\tuple {0, 1} \times \tuple {0, 1} = \tuple {-1, 0}$
where $\times$ denotes complex multiplication. | {{begin-eqn}}
{{eqn | l = \tuple {0, 1} \times \tuple {0, 1}
| r = \tuple {0 \times 0 - 1 \times 1, 0 \times 1 + 0 \times 1}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = \tuple {-1, 0}
| c =
}}
{{end-eqn}}
{{qed}} | Let $\tuple {0, 1}$ denote the [[Definition:Imaginary Unit|imaginary unit]].
Then:
:$\tuple {0, 1} \times \tuple {0, 1} = \tuple {-1, 0}$
where $\times$ denotes [[Definition:Complex Multiplication|complex multiplication]]. | {{begin-eqn}}
{{eqn | l = \tuple {0, 1} \times \tuple {0, 1}
| r = \tuple {0 \times 0 - 1 \times 1, 0 \times 1 + 0 \times 1}
| c = {{Defof|Complex Multiplication}}
}}
{{eqn | r = \tuple {-1, 0}
| c =
}}
{{end-eqn}}
{{qed}} | Product of Imaginary Unit with Itself | https://proofwiki.org/wiki/Product_of_Imaginary_Unit_with_Itself | https://proofwiki.org/wiki/Product_of_Imaginary_Unit_with_Itself | [
"Imaginary Parts",
"Complex Multiplication"
] | [
"Definition:Complex Number/Imaginary Unit",
"Definition:Multiplication/Complex Numbers"
] | [] |
proofwiki-17631 | Element of Matroid Base and Circuit has Substitute | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $B \subseteq S$ be a base of $M$.
Let $C \subseteq S$ be a circuit of $M$.
Let $x \in B \cap C$.
Then:
:$\exists y \in C \setminus B : \paren{B \setminus \set x} \cup \set y$ is a base of $M$
That is, there exists $y \in C \setminus B$ such that substituting $y$ for $... | By definition of a circuit we have: | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $B \subseteq S$ be a [[Definition:Base of Matroid|base]] of $M$.
Let $C \subseteq S$ be a [[Definition:Circuit (Matroid)|circuit]] of $M$.
Let $x \in B \cap C$.
Then:
:$\exists y \in C \setminus B : \paren{B \setminus \set x} \cup \set y$ i... | By definition of a [[Definition:Circuit (Matroid)|circuit]] we have: | Element of Matroid Base and Circuit has Substitute | https://proofwiki.org/wiki/Element_of_Matroid_Base_and_Circuit_has_Substitute | https://proofwiki.org/wiki/Element_of_Matroid_Base_and_Circuit_has_Substitute | [
"Matroid Bases",
"Matroid Circuits",
"Element of Matroid Base and Circuit has Substitute"
] | [
"Definition:Matroid",
"Definition:Base of Matroid",
"Definition:Circuit (Matroid)",
"Definition:Base of Matroid",
"Definition:Substitution (Set Theory)",
"Definition:Base of Matroid"
] | [
"Definition:Circuit (Matroid)"
] |
proofwiki-17632 | Set Difference with Non-Empty Proper Subset is Non-Empty Proper Subset | Let $S$ be a set.
Let $T \subsetneq S$ be a non-empty proper subset of $S$.
Let $S \setminus T$ denote the set difference between $S$ and $T$.
Then:
:$S \setminus T$ is a non-empty proper subset of $S$ | From Set Difference is Subset:
:$S \setminus T \subseteq S$
From Set Difference with Proper Subset:
:$S \setminus T \ne \O$
Then {{hypothesis}}:
:$T \ne \O$
From Intersection with Subset is Subset:
:$S \cap T = T$
Hence:
:$S \cap T \ne \O$
From the contrapositive statement of Set Difference with Disjoint Set:
:$S \setm... | Let $S$ be a [[Definition:Set|set]].
Let $T \subsetneq S$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Proper Subset|proper subset]] of $S$.
Let $S \setminus T$ denote the [[Definition:Set Difference|set difference]] between $S$ and $T$.
Then:
:$S \setminus T$ is a [[Definition:Non-Empty Set|non-empty... | From [[Set Difference is Subset]]:
:$S \setminus T \subseteq S$
From [[Set Difference with Proper Subset]]:
:$S \setminus T \ne \O$
Then {{hypothesis}}:
:$T \ne \O$
From [[Intersection with Subset is Subset]]:
:$S \cap T = T$
Hence:
:$S \cap T \ne \O$
From the [[Definition:Contrapositive Statement|contrapositive s... | Set Difference with Non-Empty Proper Subset is Non-Empty Proper Subset | https://proofwiki.org/wiki/Set_Difference_with_Non-Empty_Proper_Subset_is_Non-Empty_Proper_Subset | https://proofwiki.org/wiki/Set_Difference_with_Non-Empty_Proper_Subset_is_Non-Empty_Proper_Subset | [
"Set Difference",
"Proper Subsets"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Proper Subset",
"Definition:Set Difference",
"Definition:Non-Empty Set",
"Definition:Proper Subset"
] | [
"Set Difference is Subset",
"Set Difference with Proper Subset",
"Intersection with Subset is Subset",
"Definition:Contrapositive Statement",
"Set Difference with Disjoint Set",
"Definition:Non-Empty Set",
"Definition:Proper Subset",
"Category:Set Difference",
"Category:Proper Subsets"
] |
proofwiki-17633 | Subspace of Normed Vector Space with Induced Norm forms Normed Vector Space | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a normed vector space.
Let $Y \subseteq X$ be a vector subspace.
Let $\norm {\, \cdot \,}_Y$ be the induced norm on $Y$.
Then $\struct {Y, \norm {\, \cdot \,}_Y}$ is a normed vector space. | === Positive definiteness ===
By definition of induced norm:
:$\forall y \in Y : \norm {y}_Y = \norm {y}_X > 0$
Suppose $y \in Y: \norm y_Y = 0$.
Since $\norm {\, \cdot \,}_Y$ is an induced norm in $\struct {X, \norm {\, \cdot \,}_X}$:
:$\norm y_X = 0$
Therefore:
:$y = \mathbf 0 \in X$
By definition of a vector subspac... | Let $\struct {X, \norm {\, \cdot \,}_X}$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $Y \subseteq X$ be a [[Definition:Vector Subspace|vector subspace]].
Let $\norm {\, \cdot \,}_Y$ be the [[Definition:Induced Norm|induced norm]] on $Y$.
Then $\struct {Y, \norm {\, \cdot \,}_Y}$ is a [[Definiti... | === Positive definiteness ===
By definition of [[Definition:Induced Norm|induced norm]]:
:$\forall y \in Y : \norm {y}_Y = \norm {y}_X > 0$
Suppose $y \in Y: \norm y_Y = 0$.
Since $\norm {\, \cdot \,}_Y$ is an [[Definition:Induced Norm|induced norm]] in $\struct {X, \norm {\, \cdot \,}_X}$:
:$\norm y_X = 0$
Therefo... | Subspace of Normed Vector Space with Induced Norm forms Normed Vector Space | https://proofwiki.org/wiki/Subspace_of_Normed_Vector_Space_with_Induced_Norm_forms_Normed_Vector_Space | https://proofwiki.org/wiki/Subspace_of_Normed_Vector_Space_with_Induced_Norm_forms_Normed_Vector_Space | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Vector Subspace",
"Definition:Induced Norm",
"Definition:Normed Vector Space"
] | [
"Definition:Induced Norm",
"Definition:Induced Norm",
"Definition:Vector Subspace"
] |
proofwiki-17634 | All Bases of Matroid have same Cardinality/Corollary | Let $X \subseteq S$ be any independent subset of $M$.
Then:
:$\card X \le \card B$ | From Independent Subset is Contained in Maximal Independent Subset :
:$\exists B' \subseteq S : X \subseteq B'$ and $B'$ is a maximal independent subset of $S$
By definition of a base:
:$B'$ is a base of $M$
From Cardinality of Subset of Finite Set:
:$\card X \le \card {B'}$
From All Bases of Matroid have same Cardinal... | Let $X \subseteq S$ be any [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Then:
:$\card X \le \card B$ | From [[Independent Subset is Contained in Maximal Independent Subset ]]:
:$\exists B' \subseteq S : X \subseteq B'$ and $B'$ is a [[Definition:Maximal Set|maximal]] [[Definition:Independent Subset (Matroid)|independent subset]] of $S$
By definition of a [[Definition:Base of Matroid|base]]:
:$B'$ is a [[Definition:Base... | All Bases of Matroid have same Cardinality/Corollary | https://proofwiki.org/wiki/All_Bases_of_Matroid_have_same_Cardinality/Corollary | https://proofwiki.org/wiki/All_Bases_of_Matroid_have_same_Cardinality/Corollary | [
"Matroid Bases"
] | [
"Definition:Matroid/Independent Set"
] | [
"Independent Subset is Contained in Maximal Independent Subset ",
"Definition:Maximal/Set",
"Definition:Matroid/Independent Set",
"Definition:Base of Matroid",
"Definition:Base of Matroid",
"Cardinality of Subset of Finite Set",
"All Bases of Matroid have same Cardinality",
"Category:Matroid Bases"
] |
proofwiki-17635 | Independent Set can be Augmented by Larger Independent Set/Corollary | Let $B \subseteq S$ be a base of $M$.
Then:
:$\exists Z \subseteq B \setminus X : \card{X \cup Z} = \card B : X \cup Z$ is a base of $M$ | From Cardinality of Independent Set of Matroid is Smaller or Equal to Base:
:$\card X \le \card B$ | Let $B \subseteq S$ be a [[Definition:Base of Matroid|base]] of $M$.
Then:
:$\exists Z \subseteq B \setminus X : \card{X \cup Z} = \card B : X \cup Z$ is a [[Definition:Base of Matroid|base]] of $M$ | From [[Cardinality of Independent Set of Matroid is Smaller or Equal to Base]]:
:$\card X \le \card B$ | Independent Set can be Augmented by Larger Independent Set/Corollary | https://proofwiki.org/wiki/Independent_Set_can_be_Augmented_by_Larger_Independent_Set/Corollary | https://proofwiki.org/wiki/Independent_Set_can_be_Augmented_by_Larger_Independent_Set/Corollary | [
"Matroid Independent Subsets"
] | [
"Definition:Base of Matroid",
"Definition:Base of Matroid"
] | [
"All Bases of Matroid have same Cardinality/Corollary"
] |
proofwiki-17636 | Independent Subset is Base if Cardinality Equals Rank of Matroid/Corollary | Let $B \subseteq S$ be a base of $M$.
Let $X \subseteq S$ be any independent subset of $M$.
Let $\card X = \card B$.
Then:
:$X$ is a base of $M$. | From All Bases of Matroid have same Cardinality:
:$\card B = \map \rho S$
where $\rho$ denotes the rank function on $M$.
Hence:
:$\card X = \map \rho S$
From Independent Subset is Base if Cardinality Equals Rank of Matroid:
:$X$ is a base of $M$.
{{qed}}
Category:Matroid Independent Subsets
Category:Matroid Bases
8o2nh... | Let $B \subseteq S$ be a [[Definition:Base of Matroid|base]] of $M$.
Let $X \subseteq S$ be any [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Let $\card X = \card B$.
Then:
:$X$ is a [[Definition:Base of Matroid|base]] of $M$. | From [[All Bases of Matroid have same Cardinality]]:
:$\card B = \map \rho S$
where $\rho$ denotes the [[Definition:Rank Function (Matroid)|rank function]] on $M$.
Hence:
:$\card X = \map \rho S$
From [[Independent Subset is Base if Cardinality Equals Rank of Matroid]]:
:$X$ is a [[Definition:Base of Matroid|base]] o... | Independent Subset is Base if Cardinality Equals Rank of Matroid/Corollary | https://proofwiki.org/wiki/Independent_Subset_is_Base_if_Cardinality_Equals_Rank_of_Matroid/Corollary | https://proofwiki.org/wiki/Independent_Subset_is_Base_if_Cardinality_Equals_Rank_of_Matroid/Corollary | [
"Matroid Independent Subsets",
"Matroid Bases"
] | [
"Definition:Base of Matroid",
"Definition:Matroid/Independent Set",
"Definition:Base of Matroid"
] | [
"All Bases of Matroid have same Cardinality",
"Definition:Rank Function (Matroid)",
"Independent Subset is Base if Cardinality Equals Rank of Matroid",
"Definition:Base of Matroid",
"Category:Matroid Independent Subsets",
"Category:Matroid Bases"
] |
proofwiki-17637 | Matroid Unique Circuit Property | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $X \subseteq S$ be an independent subset of $M$.
Let $x \in S$ such that:
:$X \cup \set x$ is a dependent subset of $M$.
Then there exists a unique circuit $C$ such that:
:$x \in C \subseteq X \cup \set x$ | From Dependent Subset Contains a Circuit:
:there exists a circuit $C$ such that $C \subseteq X \cup \set x$
From Dependent Subset of Independent Set Union Singleton Contains Singleton:
:$x \in C$
{{AimForCont}} $C'$ is circuit of $M$ such that:
:$C' \ne C$
:$C' \subseteq X \cup \set x$
From Dependent Subset of Independ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $X \subseteq S$ be an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Let $x \in S$ such that:
:$X \cup \set x$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$.
Then there exists a unique [[Defin... | From [[Dependent Subset Contains a Circuit]]:
:there exists a [[Definition:Circuit (Matroid)|circuit]] $C$ such that $C \subseteq X \cup \set x$
From [[Dependent Subset of Independent Set Union Singleton Contains Singleton]]:
:$x \in C$
{{AimForCont}} $C'$ is [[Definition:Circuit (Matroid)|circuit]] of $M$ such that... | Matroid Unique Circuit Property/Proof 1 | https://proofwiki.org/wiki/Matroid_Unique_Circuit_Property | https://proofwiki.org/wiki/Matroid_Unique_Circuit_Property/Proof_1 | [
"Matroid Circuits",
"Matroid Unique Circuit Property"
] | [
"Definition:Matroid",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set",
"Definition:Circuit (Matroid)"
] | [
"Dependent Subset Contains a Circuit",
"Definition:Circuit (Matroid)",
"Dependent Subset of Independent Set Union Singleton Contains Singleton",
"Definition:Circuit (Matroid)",
"Dependent Subset of Independent Set Union Singleton Contains Singleton",
"Circuits of Matroid iff Matroid Circuit Axioms",
"De... |
proofwiki-17638 | Matroid Unique Circuit Property | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $X \subseteq S$ be an independent subset of $M$.
Let $x \in S$ such that:
:$X \cup \set x$ is a dependent subset of $M$.
Then there exists a unique circuit $C$ such that:
:$x \in C \subseteq X \cup \set x$ | From Dependent Subset Contains a Circuit:
:there exists a circuit $C$ such that $C \subseteq X \cup \set x$
From Dependent Subset of Independent Set Union Singleton Contains Singleton:
:$x \in C$
{{AimForCont}} $C'$ is circuit of $M$ such that:
:$C' \ne C$
:$C' \subseteq X \cup \set x$
From Dependent Subset of Independ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $X \subseteq S$ be an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Let $x \in S$ such that:
:$X \cup \set x$ is a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$.
Then there exists a unique [[Defin... | From [[Dependent Subset Contains a Circuit]]:
:there exists a [[Definition:Circuit (Matroid)|circuit]] $C$ such that $C \subseteq X \cup \set x$
From [[Dependent Subset of Independent Set Union Singleton Contains Singleton]]:
:$x \in C$
{{AimForCont}} $C'$ is [[Definition:Circuit (Matroid)|circuit]] of $M$ such that... | Matroid Unique Circuit Property/Proof 2 | https://proofwiki.org/wiki/Matroid_Unique_Circuit_Property | https://proofwiki.org/wiki/Matroid_Unique_Circuit_Property/Proof_2 | [
"Matroid Circuits",
"Matroid Unique Circuit Property"
] | [
"Definition:Matroid",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set",
"Definition:Circuit (Matroid)"
] | [
"Dependent Subset Contains a Circuit",
"Definition:Circuit (Matroid)",
"Dependent Subset of Independent Set Union Singleton Contains Singleton",
"Definition:Circuit (Matroid)",
"Dependent Subset of Independent Set Union Singleton Contains Singleton",
"Definition:Minimal/Set",
"Definition:Matroid/Depende... |
proofwiki-17639 | Absolute Value of Negative | Let $x \in \R$ be a real number.
Then:
:$\size x = \size {-x}$
where $\size x$ denotes the absolute value of $x$. | Let $x \ge 0$.
By definition of absolute value:
:$\size x = x$
We have that:
:$-x < 0$
and so by definition of absolute value:
:$\size {-x} = -\paren {-x} = x$
{{qed|lemma}}
Now let $x < 0$.
By definition of absolute value:
:$\size x = -x$
We have that:
:$-x > 0$
and so by definition of absolute value:
:$\size {-x} = -... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\size x = \size {-x}$
where $\size x$ denotes the [[Definition:Absolute Value|absolute value]] of $x$. | Let $x \ge 0$.
By definition of [[Definition:Absolute Value|absolute value]]:
:$\size x = x$
We have that:
:$-x < 0$
and so by definition of [[Definition:Absolute Value|absolute value]]:
:$\size {-x} = -\paren {-x} = x$
{{qed|lemma}}
Now let $x < 0$.
By definition of [[Definition:Absolute Value|absolute value]]:
... | Absolute Value of Negative | https://proofwiki.org/wiki/Absolute_Value_of_Negative | https://proofwiki.org/wiki/Absolute_Value_of_Negative | [
"Absolute Value Function"
] | [
"Definition:Real Number",
"Definition:Absolute Value"
] | [
"Definition:Absolute Value",
"Definition:Absolute Value",
"Definition:Absolute Value",
"Definition:Absolute Value",
"Category:Absolute Value Function"
] |
proofwiki-17640 | Dependent Subset Contains a Circuit | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $\mathscr C$ denote the set of all circuits of $M$.
Let $A$ be a dependent subset.
Then:
:$\exists C \in \mathscr C : C \subseteq A$ | Consider the ordered set $\struct {\powerset S \setminus \mathscr I, \subseteq}$.
From Element of Finite Ordered Set is Between Maximal and Minimal Elements:
:$\exists C \in \mathscr I : C \subseteq A$ and $A$ is minimal in $\struct {\powerset S \setminus \mathscr I, \subseteq}$.
By definition of a circuit:
:$C \in \ma... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $\mathscr C$ denote the set of all [[Definition:Circuit (Matroid)|circuits]] of $M$.
Let $A$ be a [[Definition:Dependent Subset (Matroid)|dependent subset]].
Then:
:$\exists C \in \mathscr C : C \subseteq A$ | Consider the [[Definition:Ordered Set|ordered set]] $\struct {\powerset S \setminus \mathscr I, \subseteq}$.
From [[Element of Finite Ordered Set is Between Maximal and Minimal Elements]]:
:$\exists C \in \mathscr I : C \subseteq A$ and $A$ is [[Definition:Minimal Element|minimal]] in $\struct {\powerset S \setminus \... | Dependent Subset Contains a Circuit | https://proofwiki.org/wiki/Dependent_Subset_Contains_a_Circuit | https://proofwiki.org/wiki/Dependent_Subset_Contains_a_Circuit | [
"Matroid Dependent Subsets",
"Matroid Circuits"
] | [
"Definition:Matroid",
"Definition:Circuit (Matroid)",
"Definition:Matroid/Dependent Set"
] | [
"Definition:Ordered Set",
"Finite Non-Empty Subset of Ordered Set has Maximal and Minimal Elements/Corollary",
"Definition:Minimal/Element",
"Definition:Circuit (Matroid)",
"Category:Matroid Dependent Subsets",
"Category:Matroid Circuits"
] |
proofwiki-17641 | Condition for Linear Operation on Complex Numbers to be of Finite Order | Let $A$ be the operation on the complex numbers $\C$ defined as:
:$\map A x = \alpha x + \beta$
Then $A$ is of finite order greater than $1$ {{iff}} $\alpha$ is a root of unity other than $1$. | {{ProofWanted|Not completely sure I understand the context}} | Let $A$ be the [[Definition:Operation|operation]] on the [[Definition:Complex Number|complex numbers]] $\C$ defined as:
:$\map A x = \alpha x + \beta$
Then $A$ is of [[Definition:Finite Order|finite order]] greater than $1$ {{iff}} $\alpha$ is a [[Definition:Complex Root of Unity|root of unity]] other than $1$. | {{ProofWanted|Not completely sure I understand the context}} | Condition for Linear Operation on Complex Numbers to be of Finite Order | https://proofwiki.org/wiki/Condition_for_Linear_Operation_on_Complex_Numbers_to_be_of_Finite_Order | https://proofwiki.org/wiki/Condition_for_Linear_Operation_on_Complex_Numbers_to_be_of_Finite_Order | [
"Roots of Unity"
] | [
"Definition:Operation",
"Definition:Complex Number",
"Definition:Finite Order",
"Definition:Complex Root of Unity"
] | [] |
proofwiki-17642 | Positive Rational Numbers under Division do not form Group | Let $\struct {\Q_{>0}, /}$ denote the algebraic structure consisting of the set of (strictly) positive rational numbers $\Q_{>0}$ under the operation $/$ of division.
We have that $\struct {\Q_{>0}, /}$ is not a group. | In order to be a group, it is necessary that $\struct {\Q_{>0}, /}$ be an associative structure.
But consider the elements $2, 6, 12$ of $\Q_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {12 / 6} / 2
| r = 2 / 1
| c =
}}
{{eqn | r = 1
| c =
}}
{{end-eqn}}
whereas:
{{begin-eqn}}
{{eqn | l = 12 / \... | Let $\struct {\Q_{>0}, /}$ denote the [[Definition:Algebraic Structure with One Operation|algebraic structure]] consisting of the [[Definition:Set|set]] of [[Definition:Strictly Positive Rational Number|(strictly) positive rational numbers]] $\Q_{>0}$ under the [[Definition:Binary Operation|operation]] $/$ of [[Definit... | In order to be a [[Definition:Group|group]], it is necessary that $\struct {\Q_{>0}, /}$ be an [[Definition:Associative Algebraic Structure|associative structure]].
But consider the [[Definition:Element|elements]] $2, 6, 12$ of $\Q_{>0}$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {12 / 6} / 2
| r = 2 / 1
... | Positive Rational Numbers under Division do not form Group | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Division_do_not_form_Group | https://proofwiki.org/wiki/Positive_Rational_Numbers_under_Division_do_not_form_Group | [
"Rational Numbers",
"Rational Division"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Set",
"Definition:Strictly Positive/Rational Number",
"Definition:Operation/Binary Operation",
"Definition:Division/Field/Rational Numbers",
"Definition:Group"
] | [
"Definition:Group",
"Definition:Semigroup",
"Definition:Element",
"Definition:Associative Operation",
"Definition:Group"
] |
proofwiki-17643 | Derivative of Real Area Hyperbolic Sine of x over a/Corollary 1 | :$\map {\dfrac \d {\d x} } {\ln \size {x + \sqrt {x^2 + a^2} } } = \dfrac 1 {\sqrt {x^2 + a^2} }$ | {{begin-eqn}}
{{eqn | l = \map \arsinh {\frac x a}
| r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 + a^2} }
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | r = \map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 + a^2} }
| c =
}}
{{eqn | r = \map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 + a^2} } }... | :$\map {\dfrac \d {\d x} } {\ln \size {x + \sqrt {x^2 + a^2} } } = \dfrac 1 {\sqrt {x^2 + a^2} }$ | {{begin-eqn}}
{{eqn | l = \map \arsinh {\frac x a}
| r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 + a^2} }
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | r = \map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 + a^2} }
| c =
}}
{{eqn | r = \map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 + a^2} } }... | Derivative of Real Area Hyperbolic Sine of x over a/Corollary 1 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Sine_of_x_over_a/Corollary_1 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Sine_of_x_over_a/Corollary_1 | [
"Derivative of Inverse Hyperbolic Sine"
] | [] | [
"Difference of Logarithms",
"Derivative of Real Area Hyperbolic Sine of x over a",
"Derivative of Constant"
] |
proofwiki-17644 | Negative of Logarithm of x plus Root x squared plus a squared | Let $x \in \R$ be a real number.
Then:
:$-\map \ln {x + \sqrt {x^2 + a^2} } = \map \ln {-x + \sqrt {x^2 + a^2} } - \map \ln {a^2}$ | We have that $\sqrt {x^2 + a^2} > x$ for all $x$.
Thus:
:$x + \sqrt {x^2 + a^2} > 0$
and so $\map \ln {x + \sqrt {x^2 + a^2} }$ is defined for all $x$.
Then we have:
{{begin-eqn}}
{{eqn | l = -\map \ln {x + \sqrt {x^2 + a^2} }
| r = \map \ln {\dfrac 1 {x + \sqrt {x^2 + a^2} } }
| c = Logarithm of Reciprocal... | Let $x \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$-\map \ln {x + \sqrt {x^2 + a^2} } = \map \ln {-x + \sqrt {x^2 + a^2} } - \map \ln {a^2}$ | We have that $\sqrt {x^2 + a^2} > x$ for all $x$.
Thus:
:$x + \sqrt {x^2 + a^2} > 0$
and so $\map \ln {x + \sqrt {x^2 + a^2} }$ is defined for all $x$.
Then we have:
{{begin-eqn}}
{{eqn | l = -\map \ln {x + \sqrt {x^2 + a^2} }
| r = \map \ln {\dfrac 1 {x + \sqrt {x^2 + a^2} } }
| c = [[Logarithm of Recip... | Negative of Logarithm of x plus Root x squared plus a squared | https://proofwiki.org/wiki/Negative_of_Logarithm_of_x_plus_Root_x_squared_plus_a_squared | https://proofwiki.org/wiki/Negative_of_Logarithm_of_x_plus_Root_x_squared_plus_a_squared | [
"Negative of Logarithm of x plus Root x squared plus a squared",
"Logarithms"
] | [
"Definition:Real Number"
] | [
"Logarithm of Reciprocal",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Difference of Two Squares",
"Difference of Logarithms"
] |
proofwiki-17645 | Derivative of Real Area Hyperbolic Sine of x over a/Corollary 2 | :$\map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } } = -\dfrac 1 {\sqrt {x^2 + a^2} }$ | {{begin-eqn}}
{{eqn | l = -\map \arsinh {\frac x a}
| r = \map \arsinh {-\frac x a}
| c = Inverse Hyperbolic Sine is Odd Function
}}
{{eqn | r = \map \ln {-\paren {\frac x a} + \sqrt {\paren {-\frac x a}^2 + a^2} }
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | r = \map \ln {-\frac x a + \dfrac ... | :$\map {\dfrac \d {\d x} } {\ln \size {x - \sqrt {x^2 + a^2} } } = -\dfrac 1 {\sqrt {x^2 + a^2} }$ | {{begin-eqn}}
{{eqn | l = -\map \arsinh {\frac x a}
| r = \map \arsinh {-\frac x a}
| c = [[Inverse Hyperbolic Sine is Odd Function]]
}}
{{eqn | r = \map \ln {-\paren {\frac x a} + \sqrt {\paren {-\frac x a}^2 + a^2} }
| c = {{Defof|Real Area Hyperbolic Sine}}
}}
{{eqn | r = \map \ln {-\frac x a + \df... | Derivative of Real Area Hyperbolic Sine of x over a/Corollary 2 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Sine_of_x_over_a/Corollary_2 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Sine_of_x_over_a/Corollary_2 | [
"Derivative of Inverse Hyperbolic Sine"
] | [] | [
"Inverse Hyperbolic Sine is Odd Function",
"Difference of Logarithms",
"Sum of Logarithms",
"Derivative of Real Area Hyperbolic Sine of x over a",
"Derivative of Constant"
] |
proofwiki-17646 | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Corollary | :$\ds \int \frac {\d x} {-\sqrt {x^2 + a^2} } = \ln \size {x - \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 + a^2} }
| r = \map \ln {x + \sqrt {x^2 + a^2} } + C
| c = Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ in Logarithm Form
}}
{{eqn | l = \int \frac {\d x} {-\sqrt {x^2 + a^2} }
| r = -\map \ln {x + \sqrt {x^2 + a^2} } + C
| c = Primitive of... | :$\ds \int \frac {\d x} {-\sqrt {x^2 + a^2} } = \ln \size {x - \sqrt {x^2 + a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 + a^2} }
| r = \map \ln {x + \sqrt {x^2 + a^2} } + C
| c = [[Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form|Primitive of $\dfrac 1 {\sqrt {x^2 + a^2} }$ in Logarithm Form]]
}}
{{eqn | l = \int \frac {\d x} {-\sqrt {x^2 + a^2... | Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_plus_a_squared/Logarithm_Form/Corollary | [
"Primitive of Reciprocal of Root of x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of Root of x squared plus a squared/Logarithm Form",
"Primitive of Constant Multiple of Function"
] |
proofwiki-17647 | Derivative of Real Area Hyperbolic Cosine of x over a/Corollary 1 | :$\map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 - a^2} } } = \dfrac 1 {\sqrt {x^2 - a^2} }$
for $x > a$. | {{begin-eqn}}
{{eqn | l = \map \arcosh {\frac x a}
| r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - a^2} }
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | r = \map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 - a^2} }
| c =
}}
{{eqn | r = \map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 - a^2} }... | :$\map {\dfrac \d {\d x} } {\map \ln {x + \sqrt {x^2 - a^2} } } = \dfrac 1 {\sqrt {x^2 - a^2} }$
for $x > a$. | {{begin-eqn}}
{{eqn | l = \map \arcosh {\frac x a}
| r = \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - a^2} }
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | r = \map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 - a^2} }
| c =
}}
{{eqn | r = \map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 - a^2} }... | Derivative of Real Area Hyperbolic Cosine of x over a/Corollary 1 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_x_over_a/Corollary_1 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_x_over_a/Corollary_1 | [
"Derivative of Real Area Hyperbolic Cosine"
] | [] | [
"Difference of Logarithms",
"Derivative of Real Area Hyperbolic Cosine of x over a",
"Derivative of Constant",
"Definition:Real Function/Domain"
] |
proofwiki-17648 | Derivative of Real Area Hyperbolic Cosine of x over a/Corollary 2 | :$\map {\dfrac \d {\d x} } {\map \ln {x - \sqrt {x^2 - a^2} } } = -\dfrac 1 {\sqrt {x^2 - a^2} }$
for $x > a$. | {{begin-eqn}}
{{eqn | l = -\map {\arcosh} {\frac x a}
| r = -\map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - a^2} }
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | r = -\map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 - a^2} }
| c =
}}
{{eqn | r = -\map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 - ... | :$\map {\dfrac \d {\d x} } {\map \ln {x - \sqrt {x^2 - a^2} } } = -\dfrac 1 {\sqrt {x^2 - a^2} }$
for $x > a$. | {{begin-eqn}}
{{eqn | l = -\map {\arcosh} {\frac x a}
| r = -\map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - a^2} }
| c = {{Defof|Real Area Hyperbolic Cosine}}
}}
{{eqn | r = -\map \ln {\frac x a + \dfrac 1 a \sqrt {x^2 - a^2} }
| c =
}}
{{eqn | r = -\map \ln {\dfrac 1 a \paren {x + \sqrt {x^2 - ... | Derivative of Real Area Hyperbolic Cosine of x over a/Corollary 2 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_x_over_a/Corollary_2 | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cosine_of_x_over_a/Corollary_2 | [
"Derivative of Real Area Hyperbolic Cosine"
] | [] | [
"Difference of Logarithms",
"Negative of Logarithm of x plus Root x squared minus a squared",
"Sum of Logarithms",
"Derivative of Real Area Hyperbolic Cosine of x over a",
"Derivative of Constant",
"Definition:Real Function/Domain"
] |
proofwiki-17649 | Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Corollary | :$\ds \int \frac {\d x} {-\sqrt {x^2 - a^2} } = \ln \size {x - \sqrt {x^2 - a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 - a^2} }
| r = \ln \size {x + \sqrt {x^2 - a^2} } + C
| c = Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ in Logarithm Form
}}
{{eqn | l = \int \frac {\d x} {-\sqrt {x^2 + a^2} }
| r = -\ln \size {x + \sqrt {x^2 - a^2} } + C
| c = Primitive ... | :$\ds \int \frac {\d x} {-\sqrt {x^2 - a^2} } = \ln \size {x - \sqrt {x^2 - a^2} } + C$ | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {\sqrt {x^2 - a^2} }
| r = \ln \size {x + \sqrt {x^2 - a^2} } + C
| c = [[Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form|Primitive of $\dfrac 1 {\sqrt {x^2 - a^2} }$ in Logarithm Form]]
}}
{{eqn | l = \int \frac {\d x} {-\sqrt {x^2 + a... | Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Logarithm_Form/Corollary | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_Root_of_x_squared_minus_a_squared/Logarithm_Form/Corollary | [
"Primitive of Reciprocal of Root of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form",
"Primitive of Constant Multiple of Function",
"Negative of Logarithm of x plus Root x squared minus a squared"
] |
proofwiki-17650 | Derivative of Real Area Hyperbolic Tangent of x over a/Corollary | :$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } } = \dfrac 1 {a^2 - x^2}$
where $\size x < a$. | {{begin-eqn}}
{{eqn | l = \dfrac 1 a \map {\tanh^{-1} } {\frac x a}
| r = \dfrac 1 a \cdot \dfrac 1 2 \map \ln {\dfrac {1 + \frac x a} {1 - \frac x a} }
| c = {{Defof|Real Area Hyperbolic Tangent}}
}}
{{eqn | r = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} }
| c = multiplying top and bottom of arg... | :$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } } = \dfrac 1 {a^2 - x^2}$
where $\size x < a$. | {{begin-eqn}}
{{eqn | l = \dfrac 1 a \map {\tanh^{-1} } {\frac x a}
| r = \dfrac 1 a \cdot \dfrac 1 2 \map \ln {\dfrac {1 + \frac x a} {1 - \frac x a} }
| c = {{Defof|Real Area Hyperbolic Tangent}}
}}
{{eqn | r = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} }
| c = multiplying [[Definition:Numerato... | Derivative of Real Area Hyperbolic Tangent of x over a/Corollary | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Tangent_of_x_over_a/Corollary | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Tangent_of_x_over_a/Corollary | [
"Derivative of Inverse Hyperbolic Tangent"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Derivative of Real Area Hyperbolic Tangent of x over a"
] |
proofwiki-17651 | Derivative of Real Area Hyperbolic Cotangent of x over a/Corollary | :$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } } = \dfrac 1 {a^2 - x^2}$
where $\size x > a$. | {{begin-eqn}}
{{eqn | l = \dfrac 1 a \map \arcoth {\frac x a}
| r = \dfrac 1 a \cdot \dfrac 1 2 \map \ln {\dfrac {\frac x a + 1} {\frac x a - 1} }
| c = {{Defof|Real Area Hyperbolic Cotangent}}
}}
{{eqn | r = \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} }
| c = multiplying top and bottom of argumen... | :$\map {\dfrac \d {\d x} } {\dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } } = \dfrac 1 {a^2 - x^2}$
where $\size x > a$. | {{begin-eqn}}
{{eqn | l = \dfrac 1 a \map \arcoth {\frac x a}
| r = \dfrac 1 a \cdot \dfrac 1 2 \map \ln {\dfrac {\frac x a + 1} {\frac x a - 1} }
| c = {{Defof|Real Area Hyperbolic Cotangent}}
}}
{{eqn | r = \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} }
| c = multiplying [[Definition:Numerator|to... | Derivative of Real Area Hyperbolic Cotangent of x over a/Corollary | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cotangent_of_x_over_a/Corollary | https://proofwiki.org/wiki/Derivative_of_Real_Area_Hyperbolic_Cotangent_of_x_over_a/Corollary | [
"Derivative of Inverse Hyperbolic Cotangent"
] | [] | [
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Derivative of Real Area Hyperbolic Cotangent of x over a"
] |
proofwiki-17652 | Rank of Matroid Circuit is One Less Than Cardinality/Lemma | :$C \setminus \set x$ is a maximal independent subset of $C$ | From Set Difference is Subset:
:$C \setminus \set x \subseteq C$
Because $x \in C$ and $x \notin C \setminus \set x$:
:$C \setminus \set x \ne C$
From Proper Subset of Matroid Circuit is Independent and matroid axiom $(\text I 1)$:
:$C \setminus \set x \in \mathscr I$
Let $X$ be an independent subset such that:
:$C \se... | :$C \setminus \set x$ is a [[Definition:Maximal Set|maximal]] [[Definition:Independent Subset (Matroid)|independent subset]] of $C$ | From [[Set Difference is Subset]]:
:$C \setminus \set x \subseteq C$
Because $x \in C$ and $x \notin C \setminus \set x$:
:$C \setminus \set x \ne C$
From [[Proper Subset of Matroid Circuit is Independent]] and [[Axiom:Matroid Axioms|matroid axiom $(\text I 1)$]]:
:$C \setminus \set x \in \mathscr I$
Let $X$ be an ... | Rank of Matroid Circuit is One Less Than Cardinality/Lemma | https://proofwiki.org/wiki/Rank_of_Matroid_Circuit_is_One_Less_Than_Cardinality/Lemma | https://proofwiki.org/wiki/Rank_of_Matroid_Circuit_is_One_Less_Than_Cardinality/Lemma | [
"Rank of Matroid Circuit is One Less Than Cardinality"
] | [
"Definition:Maximal/Set",
"Definition:Matroid/Independent Set"
] | [
"Set Difference is Subset",
"Proper Subset of Matroid Circuit is Independent",
"Axiom:Matroid Axioms",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set",
"Singleton of Element is Subset",
"Union with Superset is Superset",
"Set Difference over Subset",
"Set Difference with Uni... |
proofwiki-17653 | Union of Matroid Base with Element of Complement is Dependent | :$B \cup \set x$ is a dependent superset of $B$ | From Set is Subset of Union:
:$B \subseteq B \cup \set x$
Because $x \in B \cup \set x$ and $x \notin B$:
:$B \ne B \cup \set x$
Hence:
:$B \subsetneq B \cup \set x$
By definition of base:
:$B$ is a maximal independent subset
Hence:
:$B \cup \set x \notin \mathscr I$
{{qed}}
Category:Matroid Bases
Category:Matroid Depe... | :$B \cup \set x$ is a [[Definition:Dependent Subset (Matroid)|dependent]] [[Definition:Superset|superset]] of $B$ | From [[Set is Subset of Union]]:
:$B \subseteq B \cup \set x$
Because $x \in B \cup \set x$ and $x \notin B$:
:$B \ne B \cup \set x$
Hence:
:$B \subsetneq B \cup \set x$
By definition of [[Definition:Base of Matroid|base]]:
:$B$ is a [[Definition:Maximal Set|maximal]] [[Definition:Independent Subset (Matroid)|indepe... | Union of Matroid Base with Element of Complement is Dependent | https://proofwiki.org/wiki/Union_of_Matroid_Base_with_Element_of_Complement_is_Dependent | https://proofwiki.org/wiki/Union_of_Matroid_Base_with_Element_of_Complement_is_Dependent | [
"Matroid Bases",
"Matroid Dependent Subsets"
] | [
"Definition:Matroid/Dependent Set",
"Definition:Subset/Superset"
] | [
"Set is Subset of Union",
"Definition:Base of Matroid",
"Definition:Maximal/Set",
"Definition:Matroid/Independent Set",
"Category:Matroid Bases",
"Category:Matroid Dependent Subsets"
] |
proofwiki-17654 | Primitive of Power of a x + b/Proof 1 | {{:Primitive of Power of a x + b}}
where $n \ne 1$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \frac 1 a \int u^n \rd u
| c = Primitive of Function of $a x + b$
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
| c = Primitive of Power
}}
{{eqn | r = \frac {\paren {a x + b}^{n + 1} } {\paren {n + 1} a} ... | {{:Primitive of Power of a x + b}}
where $n \ne 1$. | Let $u = a x + b$.
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \frac 1 a \int u^n \rd u
| c = [[Primitive of Function of a x + b|Primitive of Function of $a x + b$]]
}}
{{eqn | r = \frac 1 a \frac {u^{n + 1} } {n + 1} + C
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {\pa... | Primitive of Power of a x + b/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_1 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_1 | [
"Primitive of Power of a x + b"
] | [] | [
"Primitive of Function of a x + b",
"Primitive of Power"
] |
proofwiki-17655 | Primitive of Power of a x + b/Proof 2 | {{:Primitive of Power of a x + b}}
where $n \ne 1$. | Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \int \dfrac {u^n} a \rd u
| c = Integration by Substitution
}}
{{eqn | r = \dfrac 1 a \dfrac {u^{n + 1} } {n + 1}
| c = Primitive of Power
}}
{{eqn | r = \frac {\paren {a x + b}^{n ... | {{:Primitive of Power of a x + b}}
where $n \ne 1$. | Let $u = a x + b$.
Then:
:$\dfrac {\d u} {\d x} = a$
Then:
{{begin-eqn}}
{{eqn | l = \int \paren {a x + b}^n \rd x
| r = \int \dfrac {u^n} a \rd u
| c = [[Integration by Substitution]]
}}
{{eqn | r = \dfrac 1 a \dfrac {u^{n + 1} } {n + 1}
| c = [[Primitive of Power]]
}}
{{eqn | r = \frac {\paren {a ... | Primitive of Power of a x + b/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_2 | https://proofwiki.org/wiki/Primitive_of_Power_of_a_x_+_b/Proof_2 | [
"Primitive of Power of a x + b"
] | [] | [
"Integration by Substitution",
"Primitive of Power"
] |
proofwiki-17656 | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form | $\quad \ds \int \dfrac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\
& \\
\dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\
& \\
\text {undefined} & : x = a \end {cases}$ | First note that if $x = a$ then $a^2 - x^2 = 0$ and so $\dfrac 1 {a^2 - x^2}$ is undefined. | $\quad \ds \int \dfrac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\
& \\
\dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\
& \\
\text {undefined} & : x = a \end {cases}$ | First note that if $x = a$ then $a^2 - x^2 = 0$ and so $\dfrac 1 {a^2 - x^2}$ is undefined. | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Function_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Function_Form | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [] |
proofwiki-17657 | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form | :$\ds \int \frac {\d x} {a^2 - x^2} = \frac 1 a \coth^{-1} \frac x a + C$
where $\size x > a$. | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\dfrac x a > 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
| r... | :$\ds \int \frac {\d x} {a^2 - x^2} = \frac 1 a \coth^{-1} \frac x a + C$
where $\size x > a$. | Let $\size x > a$.
Let:
{{begin-eqn}}
{{eqn | l = u
| r = \coth^{-1} {\frac x a}
| c = {{Defof|Real Inverse Hyperbolic Cotangent}}, which is defined where $\dfrac x a > 1$
}}
{{eqn | ll=\leadsto
| l = x
| r = a \coth u
| c =
}}
{{eqn | ll=\leadsto
| l = \frac {\d x} {\d u}
... | Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form/Proof | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Cotangent_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Inverse_Hyperbolic_Cotangent_Form/Proof | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Derivative of Hyperbolic Cotangent",
"Integration by Substitution",
"Primitive of Constant Multiple of Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Integral of Constant"
] |
proofwiki-17658 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} } } + C
| c = $\artanh \dfrac x a$ in Logarith... | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfrac... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Real Area Hyperbolic Tangent of x over a in Logarithm Form"
] |
proofwiki-17659 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = Partial Fraction Expansion
}}... | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = [[Primitive of Reciproca... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_2 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Difference of Two Squares",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-17660 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = Primitive of $\dfrac 1 {x^2 - a^2}$ for $\size x < a$
}}
{{eq... | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = [[Primitive of Reciprocal of x squared minus a squared/L... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_3 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a",
"Logarithm of Reciprocal"
] |
proofwiki-17661 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a | Let $\size x > a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C$ | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth \dfrac x a$ in Logarit... | Let $\size x > a$.
Then:
:$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C$ | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfr... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_greater_than_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_greater_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form"
] |
proofwiki-17662 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth \dfrac x a$ in Logarit... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\arcoth$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfr... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x greater than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_greater_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant"
] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form"
] |
proofwiki-17663 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form
}}
{{eqn | r = \frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} } } + C
| c = $\artanh \dfrac x a$ in Logarith... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \frac 1 a \artanh {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of $\dfrac 1 {a^2 - x^2}$: $\artanh$ form]]
}}
{{eqn | r = \frac 1 a \paren {\dfrac... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant"
] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Real Area Hyperbolic Tangent of x over a in Logarithm Form"
] |
proofwiki-17664 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = Partial Fraction Expansion
}}... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = \int \frac {\d x} {\paren {a + x} \paren {a - x} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {a + x} } + \int \frac {\d x} {2 a \paren {a - x} }
| c = [[Primitive of Reciproca... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_2 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant"
] | [
"Difference of Two Squares",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-17665 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = Primitive of Constant Multiple of Function
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = Primitive of $\dfrac 1 {x^2 - a^2}$ for $\size x < a$
}}
{{eq... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $x \in \R$ such that $\size x \ne a$. | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {a^2 - x^2}
| r = -\int \frac {\d x} {x^2 - a^2}
| c = [[Primitive of Constant Multiple of Function]]
}}
{{eqn | r = -\frac 1 {2 a} \map \ln {\frac {a - x} {a + x} } + C
| c = [[Primitive of Reciprocal of x squared minus a squared/L... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1/size of x less than a/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_1/size_of_x_less_than_a/Proof_3 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant"
] | [
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a",
"Logarithm of Reciprocal"
] |
proofwiki-17666 | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2 | :$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \ln \size {\dfrac {a + x} {a - x} } + C$ | From the $1$st logarithm form:
:$\ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a\\ & \\ \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\ & \\ \text {undefined} & : \size x = a \end {cases}$
From Primitive of Reciprocal of ... | :$\ds \int \frac {\d x} {a^2 - x^2} = \dfrac 1 {2 a} \ln \size {\dfrac {a + x} {a - x} } + C$ | From the [[Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1|$1$st logarithm form]]:
:$\ds \int \frac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a + x} {a - x} } + C & : \size x < a\\ & \\ \dfrac 1 {2 a} \map \ln {\dfrac {x + a} {x - a} } + C & : \size x > a \\ & \\ \text... | Primitive of Reciprocal of a squared minus x squared/Logarithm Form 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_a_squared_minus_x_squared/Logarithm_Form_2 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form 1",
"Primitive of Reciprocal of a squared minus x squared/Logarithm Form/Lemma"
] |
proofwiki-17667 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$ such that $\size x \ne a$.
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1}}
=== $2$nd Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2}} | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form
}}
{{eqn | r = -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth {\dfrac x a}$ in Logar... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $x \in \R$ such that $\size x \ne a$.
=== [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1|$1$st Logarithm Form]] ===
{{:Primitive of Reciprocal of x squared minus... | Let $x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth {\frac x a} + C
| c = [[Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form|Primitive of Reciprocal of $x^2 - a^2$ in $\arcoth$ form]]
}}
{{eqn | r = -\frac 1 a \paren {\df... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_1 | [
"Primitive of Reciprocal of x squared minus a squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] | [
"Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form",
"Logarithm of Reciprocal",
"Integration by Substitution",
"Logarithm of Reciprocal"
] |
proofwiki-17668 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$ such that $\size x \ne a$.
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1}}
=== $2$nd Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2}} | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = Difference of Two Squares
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {2 a... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $x \in \R$ such that $\size x \ne a$.
=== [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1|$1$st Logarithm Form]] ===
{{:Primitive of Reciprocal of x squared minus... | {{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = \int \frac {\d x} {\paren {x - a} \paren {x + a} }
| c = [[Difference of Two Squares]]
}}
{{eqn | r = \int \frac {\d x} {2 a \paren {x - a} } - \int \frac {\d x} {2 a \paren {x + a} }
| c = [[Primitive of Reciprocal of x squared minus a squ... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_2 | [
"Primitive of Reciprocal of x squared minus a squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] | [
"Difference of Two Squares",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 2/Partial Fraction Expansion",
"Primitive of Constant Multiple of Function",
"Primitive of Reciprocal",
"Difference of Logarithms"
] |
proofwiki-17669 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form | Let $a \in \R_{>0}$ be a strictly positive real constant.
Let $x \in \R$ such that $\size x \ne a$.
=== $1$st Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1}}
=== $2$nd Logarithm Form ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2}} | From the $1$st logarithm form:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$
From Primitive of Reciproca... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real]] [[Definition:Constant|constant]].
Let $x \in \R$ such that $\size x \ne a$.
=== [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1|$1$st Logarithm Form]] ===
{{:Primitive of Reciprocal of x squared minus... | From the [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1|$1$st logarithm form]]:
$\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a \\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2/Proof 3 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_2/Proof_3 | [
"Primitive of Reciprocal of x squared minus a squared"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Constant",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 2"
] | [
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1",
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form/Lemma"
] |
proofwiki-17670 | Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form | $\quad \ds \int \dfrac {\d x} {x^2 - a^2} = \begin {cases} -\dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\
& \\
-\dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\
& \\
\text {undefined} & : x = a \end {cases}$ | First note that if $x = a$ then $a^2 - x^2 = 0$ and so $\dfrac 1 {x^2 - a^2}$ is undefined. | $\quad \ds \int \dfrac {\d x} {x^2 - a^2} = \begin {cases} -\dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\
& \\
-\dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\
& \\
\text {undefined} & : x = a \end {cases}$ | First note that if $x = a$ then $a^2 - x^2 = 0$ and so $\dfrac 1 {x^2 - a^2}$ is undefined. | Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Function Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Function_Form | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Inverse_Hyperbolic_Function_Form | [
"Primitive of Reciprocal of x squared minus a squared"
] | [] | [] |
proofwiki-17671 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1 | $\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a\\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | === Case where $\size x < a$ ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a}} | $\quad \ds \int \frac {\d x} {x^2 - a^2} = \begin {cases} \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C & : \size x < a\\
& \\
\dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C & : \size x > a \\
& \\
\text {undefined} & : \size x = a \end {cases}$ | === [[Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a|Case where $\size x < a$]] ===
{{:Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a}} | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_1 | [
"Primitive of Reciprocal of a squared minus x squared"
] | [] | [
"Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a"
] |
proofwiki-17672 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \artanh {\frac x a} + C
| c = Primitive of $\dfrac 1 {x^2 - a^2}$: $\artanh$ form
}}
{{eqn | r = -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {a + x} {a - x} } } + C
| c = $\artanh \dfrac x a$ in Logari... | Let $\size x < a$.
Then:
:$\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C$ | Let $\size x < a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \artanh {\frac x a} + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form|Primitive of $\dfrac 1 {x^2 - a^2}$: $\artanh$ form]]
}}
{{eqn | r = -\frac 1 a \paren {\dfr... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_1/size_of_x_less_than_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_1/size_of_x_less_than_a | [
"Primitive of Reciprocal of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Tangent Form",
"Real Area Hyperbolic Tangent of x over a in Logarithm Form",
"Logarithm of Reciprocal"
] |
proofwiki-17673 | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x greater than a | Let $\size x > a$.
Then:
:$\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C$ | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth \frac x a + C
| c = Primitive of $\dfrac 1 {x^2 - a^2}$: $\arcoth$ form
}}
{{eqn | r = -\frac 1 a \paren {\dfrac 1 2 \map \ln {\dfrac {x + a} {x - a} } } + C
| c = $\arcoth \dfrac x a$ in Logarith... | Let $\size x > a$.
Then:
:$\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {x - a} {x + a} } + C$ | Let $\size x > a$.
Then:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^2 - a^2}
| r = -\frac 1 a \arcoth \frac x a + C
| c = [[Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form|Primitive of $\dfrac 1 {x^2 - a^2}$: $\arcoth$ form]]
}}
{{eqn | r = -\frac 1 a \paren {\dfr... | Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x greater than a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_1/size_of_x_greater_than_a | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_x_squared_minus_a_squared/Logarithm_Form_1/size_of_x_greater_than_a | [
"Primitive of Reciprocal of x squared minus a squared"
] | [] | [
"Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Cotangent Form",
"Real Area Hyperbolic Cotangent of x over a in Logarithm Form",
"Reciprocal of Logarithm"
] |
proofwiki-17674 | Power Set is Closed under Complement | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A \in \powerset S: \relcomp S A \in \powerset S$
where $\relcomp S A$ denotes the complement of $A$ relative to $S$. | {{begin-eqn}}
{{eqn | l = A
| o = \in
| r = \powerset S
| c =
}}
{{eqn | ll= \leadsto
| l = A
| o = \subseteq
| r = S
| c = {{Defof|Power Set}}
}}
{{eqn | ll= \leadsto
| l = S \setminus A
| o = \subseteq
| r = S
| c = Set Difference is Subset
}}
{{eqn |... | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then:
:$\forall A \in \powerset S: \relcomp S A \in \powerset S$
where $\relcomp S A$ denotes the [[Definition:Relative Complement|complement of $A$ relative to $S$]]. | {{begin-eqn}}
{{eqn | l = A
| o = \in
| r = \powerset S
| c =
}}
{{eqn | ll= \leadsto
| l = A
| o = \subseteq
| r = S
| c = {{Defof|Power Set}}
}}
{{eqn | ll= \leadsto
| l = S \setminus A
| o = \subseteq
| r = S
| c = [[Set Difference is Subset]]
}}
{{e... | Power Set is Closed under Complement | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Complement | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Complement | [
"Power Set",
"Relative Complement"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Relative Complement"
] | [
"Set Difference is Subset"
] |
proofwiki-17675 | Symmetric Difference with Intersection forms Boolean Ring | Let $S$ be a set.
Let:
:$\symdif$ denote the symmetric difference operation
:$\cap$ denote the set intersection operation
:$\powerset S$ denote the power set of $S$.
Then $\struct {\powerset S, \symdif, \cap}$ is a Boolean ring. | From Symmetric Difference with Intersection forms Ring:
:$\struct {\powerset S, \symdif, \cap}$ is a commutative ring with unity.
From Set Intersection is Idempotent, $\cap$ is an idempotent operation on $S$.
Hence the result by definition of Boolean ring.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let:
:$\symdif$ denote the [[Definition:Symmetric Difference|symmetric difference operation]]
:$\cap$ denote the [[Definition:Set Intersection|set intersection operation]]
:$\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Then $\struct {\powerset S, \symdif, \c... | From [[Symmetric Difference with Intersection forms Ring]]:
:$\struct {\powerset S, \symdif, \cap}$ is a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
From [[Set Intersection is Idempotent]], $\cap$ is an [[Definition:Idempotent Operation|idempotent operation]] on $S$.
Hence the result by d... | Symmetric Difference with Intersection forms Boolean Ring | https://proofwiki.org/wiki/Symmetric_Difference_with_Intersection_forms_Boolean_Ring | https://proofwiki.org/wiki/Symmetric_Difference_with_Intersection_forms_Boolean_Ring | [
"Boolean Rings",
"Set Intersection",
"Symmetric Difference",
"Power Set"
] | [
"Definition:Set",
"Definition:Symmetric Difference",
"Definition:Set Intersection",
"Definition:Power Set",
"Definition:Boolean Ring"
] | [
"Symmetric Difference with Intersection forms Ring",
"Definition:Commutative and Unitary Ring",
"Set Intersection is Idempotent",
"Definition:Idempotence/Operation",
"Definition:Boolean Ring"
] |
proofwiki-17676 | Parallelism is Reflexive Relation | Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$.
Then $\parallel$ is a reflexive relation on $S$. | By definition of parallel lines, the contemporary definition is for a straight line to be declared parallel to itself.
Hence for a straight line $l$:
:$l \parallel l$
Thus $\parallel$ is seen to be reflexive. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is [[Definition:Parallel Lines|parallel]] to $l_2$.
Then $\parallel$ is a [[Definition:Reflexive Relation|reflexive relation]] on... | By definition of [[Definition:Parallel Lines|parallel lines]], the contemporary definition is for a [[Definition:Straight Line|straight line]] to be declared [[Definition:Parallel Lines|parallel]] to itself.
Hence for a [[Definition:Straight Line|straight line]] $l$:
:$l \parallel l$
Thus $\parallel$ is seen to be [[... | Parallelism is Reflexive Relation | https://proofwiki.org/wiki/Parallelism_is_Reflexive_Relation | https://proofwiki.org/wiki/Parallelism_is_Reflexive_Relation | [
"Parallel Lines",
"Examples of Reflexive Relations"
] | [
"Definition:Set",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Parallel (Geometry)/Lines",
"Definition:Reflexive Relation"
] | [
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Reflexive Relation"
] |
proofwiki-17677 | Parallelism is Symmetric Relation | Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$.
Then $\parallel$ is a symmetric relation on $S$. | Let $l_1 \parallel l_2$.
By definition of parallel lines, $l_1$ does not meet $l_2$ when produced indefinitely.
Hence $l_2$ similarly does not meet $l_1$ when produced indefinitely.
That is:
:$l_2 \parallel l_1$
Thus $\parallel$ is seen to be symmetric. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is [[Definition:Parallel Lines|parallel]] to $l_2$.
Then $\parallel$ is a [[Definition:Symmetric Relation|symmetric relation]] on... | Let $l_1 \parallel l_2$.
By definition of [[Definition:Parallel Lines|parallel lines]], $l_1$ does not meet $l_2$ when [[Definition:Production|produced]] indefinitely.
Hence $l_2$ similarly does not meet $l_1$ when [[Definition:Production|produced]] indefinitely.
That is:
:$l_2 \parallel l_1$
Thus $\parallel$ is se... | Parallelism is Symmetric Relation | https://proofwiki.org/wiki/Parallelism_is_Symmetric_Relation | https://proofwiki.org/wiki/Parallelism_is_Symmetric_Relation | [
"Parallel Lines",
"Examples of Symmetric Relations"
] | [
"Definition:Set",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Parallel (Geometry)/Lines",
"Definition:Symmetric Relation"
] | [
"Definition:Parallel (Geometry)/Lines",
"Definition:Production",
"Definition:Production",
"Definition:Symmetric Relation"
] |
proofwiki-17678 | Parallelism is Equivalence Relation/Transitivity | Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$.
Then $\parallel$ is a transitive relation on $S$. | From Parallelism is Transitive Relation:
:$l_1 \parallel l_2$ and $l_2 \parallel l_3$ implies $l_1 \parallel l_3$.
Thus $\parallel$ is seen to be transitive. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is [[Definition:Parallel Lines|parallel]] to $l_2$.
Then $\parallel$ is a [[Definition:Transitive Relation|transitive relation]] ... | From [[Parallelism is Transitive Relation]]:
:$l_1 \parallel l_2$ and $l_2 \parallel l_3$ implies $l_1 \parallel l_3$.
Thus $\parallel$ is seen to be [[Definition:Transitive Relation|transitive]]. | Parallelism is Equivalence Relation/Transitivity | https://proofwiki.org/wiki/Parallelism_is_Equivalence_Relation/Transitivity | https://proofwiki.org/wiki/Parallelism_is_Equivalence_Relation/Transitivity | [
"Parallel Lines",
"Examples of Transitive Relations"
] | [
"Definition:Set",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Parallel (Geometry)/Lines",
"Definition:Transitive Relation"
] | [
"Parallelism is Transitive Relation",
"Definition:Transitive Relation"
] |
proofwiki-17679 | 1-Seminorm on Continuous on Closed Interval Real-Valued Functions is Norm | Let $C \closedint a b$ be the space of real-valued functions continuous on $\closedint a b$.
Let $x \in C \closedint a b$ be a continuous real valued function.
Let $\ds \norm x_1 := \int_a^b \size {\map x t} \rd t$ be the 1-seminorm.
Then $\norm {\, \cdot \,}_1$ is a norm on $C \closedint a b$. | === Positive definiteness ===
Let $x \in C \closedint a b$.
Then $\forall t \in \closedint 0 1 : \size {\map x t} \ge 0 $.
Hence:
:$\ds \int_a^b \size {\map x t} \rd t = \norm x_1 \ge 0$.
Suppose $\forall t \in \closedint a b : \map x t = 0$.
Then $\norm x_1 = 0$.
Therefore:
:$\paren {x = 0} \implies \paren {\norm x_1 ... | Let $C \closedint a b$ be the [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|space of real-valued functions continuous on $\closedint a b$]].
Let $x \in C \closedint a b$ be a [[Definition:Continuous Real-Valued Vector Function|continuous real valued function]].
Let $\ds \norm x_1 := \int_a... | === Positive definiteness ===
Let $x \in C \closedint a b$.
Then $\forall t \in \closedint 0 1 : \size {\map x t} \ge 0 $.
Hence:
:$\ds \int_a^b \size {\map x t} \rd t = \norm x_1 \ge 0$.
Suppose $\forall t \in \closedint a b : \map x t = 0$.
Then $\norm x_1 = 0$.
Therefore:
:$\paren {x = 0} \implies \paren {\n... | 1-Seminorm on Continuous on Closed Interval Real-Valued Functions is Norm | https://proofwiki.org/wiki/1-Seminorm_on_Continuous_on_Closed_Interval_Real-Valued_Functions_is_Norm | https://proofwiki.org/wiki/1-Seminorm_on_Continuous_on_Closed_Interval_Real-Valued_Functions_is_Norm | [
"Examples of Norms"
] | [
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Continuous Real-Valued Vector Function",
"Definition:P-Seminorm",
"Definition:Norm/Vector Space"
] | [
"Definition:Assumption",
"Definition:Continuous Real Function/Closed Interval",
"Definition:Assumption",
"Definition:Continuous Real Function/Closed Interval",
"Reverse Triangle Inequality/Normed Vector Space",
"Definition:Contradiction"
] |
proofwiki-17680 | Approximation/Examples/22 over 7 | $\dfrac {22} 7$ is a convenient approximation to $\pi$:
:$\dfrac {22} 7 = 3 \cdotp \dot 14285 \dot 7$ | {{begin-eqn}}
{{eqn | l = \dfrac {22} 7
| r = 3 \cdotp \dot 14285 \dot 7
| c =
}}
{{eqn | l = \pi
| o = \approx
| r = 3 \cdotp 14159265
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {22} 7 - \pi
| o = \approx
| r = 0 \cdotp 0012645
| c =
}}
{{eqn | ll= \leadsto
... | $\dfrac {22} 7$ is a convenient [[Definition:Approximation|approximation]] to $\pi$:
:$\dfrac {22} 7 = 3 \cdotp \dot 14285 \dot 7$ | {{begin-eqn}}
{{eqn | l = \dfrac {22} 7
| r = 3 \cdotp \dot 14285 \dot 7
| c =
}}
{{eqn | l = \pi
| o = \approx
| r = 3 \cdotp 14159265
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {22} 7 - \pi
| o = \approx
| r = 0 \cdotp 0012645
| c =
}}
{{eqn | ll= \leadsto
... | Approximation/Examples/22 over 7 | https://proofwiki.org/wiki/Approximation/Examples/22_over_7 | https://proofwiki.org/wiki/Approximation/Examples/22_over_7 | [
"Approximations to Pi",
"Examples of Approximations"
] | [
"Definition:Approximation"
] | [] |
proofwiki-17681 | Perpendicularity is Symmetric Relation | Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$.
Then $\perp$ is a symmetric relation on $S$. | Let $l_1 \perp l_2$.
By definition of perpendicular lines, $l_1$ meets $l_2$ at a right angle.
Hence $l_2$ similarly meets $l_1$ at a right angle.
That is:
:$l_2 \perp l_1$
Thus $\parallel$ is seen to be symmetric. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is [[Definition:Perpendicular Lines|perpendicular]] to $l_2$.
Then $\perp$ is a [[Definition:Symmetric Relation|symmetric relation]] ... | Let $l_1 \perp l_2$.
By definition of [[Definition:Perpendicular Lines|perpendicular lines]], $l_1$ meets $l_2$ at a [[Definition:Right Angle|right angle]].
Hence $l_2$ similarly meets $l_1$ at a [[Definition:Right Angle|right angle]].
That is:
:$l_2 \perp l_1$
Thus $\parallel$ is seen to be [[Definition:Symmetric ... | Perpendicularity is Symmetric Relation | https://proofwiki.org/wiki/Perpendicularity_is_Symmetric_Relation | https://proofwiki.org/wiki/Perpendicularity_is_Symmetric_Relation | [
"Perpendiculars",
"Examples of Symmetric Relations"
] | [
"Definition:Set",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Right Angle/Perpendicular",
"Definition:Symmetric Relation"
] | [
"Definition:Right Angle/Perpendicular",
"Definition:Right Angle",
"Definition:Right Angle",
"Definition:Symmetric Relation"
] |
proofwiki-17682 | Perpendicularity is Antireflexive Relation | Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$.
Then $\perp$ is an antireflexive relation on $S$. | By definition of perpendicular lines, for $l_1$ to be perpendicular to itself would mean it would have to meet itself in a right angle.
This it does not do.
So $l_1 \not \perp l_1$.
Thus $\perp$ is seen to be antireflexive. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is [[Definition:Perpendicular Lines|perpendicular]] to $l_2$.
Then $\perp$ is an [[Definition:Antireflexive Relation|antireflexive re... | By definition of [[Definition:Perpendicular Lines|perpendicular lines]], for $l_1$ to be [[Definition:Perpendicular Lines|perpendicular]] to itself would mean it would have to meet itself in a [[Definition:Right Angle|right angle]].
This it does not do.
So $l_1 \not \perp l_1$.
Thus $\perp$ is seen to be [[Definitio... | Perpendicularity is Antireflexive Relation | https://proofwiki.org/wiki/Perpendicularity_is_Antireflexive_Relation | https://proofwiki.org/wiki/Perpendicularity_is_Antireflexive_Relation | [
"Perpendiculars",
"Examples of Antireflexive Relations"
] | [
"Definition:Set",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Right Angle/Perpendicular",
"Definition:Antireflexive Relation"
] | [
"Definition:Right Angle/Perpendicular",
"Definition:Right Angle/Perpendicular",
"Definition:Right Angle",
"Definition:Antireflexive Relation"
] |
proofwiki-17683 | Larger Set has Larger Set Difference | :$\card {V \setminus U} < \card {U \setminus V}$ | We have:
{{begin-eqn}}
{{eqn | l = \card {U \setminus V}
| r = \card U - \card {U \cap V}
| c = Cardinality of Set Difference
}}
{{eqn | o = >
| r = \card V - \card {U \cap V}
| c = As $\card V < \card U$
}}
{{eqn | r = \card {V \setminus U}
| c = Cardinality of Set Difference
}}
{{end-eq... | :$\card {V \setminus U} < \card {U \setminus V}$ | We have:
{{begin-eqn}}
{{eqn | l = \card {U \setminus V}
| r = \card U - \card {U \cap V}
| c = [[Cardinality of Set Difference]]
}}
{{eqn | o = >
| r = \card V - \card {U \cap V}
| c = As $\card V < \card U$
}}
{{eqn | r = \card {V \setminus U}
| c = [[Cardinality of Set Difference]]
}}
... | Larger Set has Larger Set Difference | https://proofwiki.org/wiki/Larger_Set_has_Larger_Set_Difference | https://proofwiki.org/wiki/Larger_Set_has_Larger_Set_Difference | [
"Set Difference",
"Cardinality"
] | [] | [
"Cardinality of Set Difference",
"Cardinality of Set Difference",
"Category:Set Difference",
"Category:Cardinality"
] |
proofwiki-17684 | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 1 | :$\exists B_3 \in \mathscr B$:
::$V \subseteq B_3$
::$\card{B_1 \cap B_3} > \card{B_1 \cap B_2}$ | === Lemma 2 ===
{{:Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 2}}{{qed|lemma}}
From Larger Set has Larger Set Difference:
:$(1):\quad \card {V \setminus U} < \card {U \setminus V}$
We have:
{{begin-eqn}}
{{eqn | l = \card {B_1}
| r = \card{ \paren{B_1 \cap B_2} \cup \paren{B_1 \setminus... | :$\exists B_3 \in \mathscr B$:
::$V \subseteq B_3$
::$\card{B_1 \cap B_3} > \card{B_1 \cap B_2}$ | === [[Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 2|Lemma 2]] ===
{{:Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 2}}{{qed|lemma}}
From [[Larger Set has Larger Set Difference]]:
:$(1):\quad \card {V \setminus U} < \card {U \setminus V}$
We have:
{{begin-eqn}}
{{eqn... | Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 1 | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Lemma_1 | https://proofwiki.org/wiki/Matroid_Bases_Iff_Satisfies_Formulation_1_of_Matroid_Base_Axiom/Lemma_1 | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom"
] | [] | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom/Lemma 2",
"Larger Set has Larger Set Difference",
"Set Difference Union Intersection",
"Set Difference and Intersection are Disjoint",
"Set Difference Union Intersection",
"Set Difference and Intersection are Disjoint",
"Set Difference Uni... |
proofwiki-17685 | Perpendicularity is Antitransitive Relation | Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$.
Then $\perp$ is an antitransitive relation on $S$. | Let $l_1 \perp l_2$ and $l_2 \perp l_3$.
Then $l_1$ and $l_3$ are parallel, and not perpendicular.
Thus $\perp$ is seen to be antitransitive. | Let $S$ be the [[Definition:Set|set]] of [[Definition:Straight Line|straight lines]] in [[Definition:The Plane|the plane]].
For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is [[Definition:Perpendicular Lines|perpendicular]] to $l_2$.
Then $\perp$ is an [[Definition:Antitransitive Relation|antitransitive ... | Let $l_1 \perp l_2$ and $l_2 \perp l_3$.
Then $l_1$ and $l_3$ are [[Definition:Parallel Lines|parallel]], and not [[Definition:Perpendicular Lines|perpendicular]].
Thus $\perp$ is seen to be [[Definition:Antitransitive Relation|antitransitive]]. | Perpendicularity is Antitransitive Relation | https://proofwiki.org/wiki/Perpendicularity_is_Antitransitive_Relation | https://proofwiki.org/wiki/Perpendicularity_is_Antitransitive_Relation | [
"Perpendiculars",
"Examples of Antitransitive Relations"
] | [
"Definition:Set",
"Definition:Line/Straight Line",
"Definition:Plane Surface/The Plane",
"Definition:Right Angle/Perpendicular",
"Definition:Antitransitive Relation"
] | [
"Definition:Parallel (Geometry)/Lines",
"Definition:Right Angle/Perpendicular",
"Definition:Antitransitive Relation"
] |
proofwiki-17686 | Equivalence of Formulations of Axiom of Choice/Formulation 2 implies Formulation 1 | The following formulation of the Axiom of Choice: | Suppose that Formulation 2 holds.
That is, the Cartesian product of a non-empty family of non-empty sets is non-empty.
Let $\CC$ be a non-empty set of non-empty sets.
$\CC$ may be converted into an indexed set by using $\CC$ itself as the indexing set and using the identity mapping on $\CC$ to do the indexing.
Then the... | The following formulation of the [[Axiom:Axiom of Choice|Axiom of Choice]]: | Suppose that [[Axiom:Axiom of Choice/Formulation 2|Formulation 2]] holds.
That is, the [[Definition:Cartesian Product|Cartesian product]] of a [[Definition:Non-Empty Set|non-empty]] [[Definition:Indexed Family of Sets|family]] of [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]] is [[Definition:Non-Empty ... | Equivalence of Formulations of Axiom of Choice/Formulation 2 implies Formulation 1 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_2_implies_Formulation_1 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_2_implies_Formulation_1 | [
"Equivalence of Formulations of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Axiom:Axiom of Choice"
] | [
"Axiom:Axiom of Choice/Formulation 2",
"Definition:Cartesian Product",
"Definition:Non-Empty Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Non-Empty Set",
"Definition:Set of Sets",
"Definition:Indexing Set/Inde... |
proofwiki-17687 | Equivalence of Formulations of Axiom of Choice/Formulation 1 implies Formulation 3 | The following formulation of the Axiom of Choice: | Let $\SS$ be the set:
:$\SS = \set {s: \O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O}$
Let $c$ be a choice function on $\SS$ and consider the image set $c \sqbrk \SS$:
:$c \sqbrk \SS = \set {\map c s: \O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O}$
By the definition of choice function:... | The following formulation of the [[Axiom:Axiom of Choice|Axiom of Choice]]: | Let $\SS$ be the set:
:$\SS = \set {s: \O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O}$
Let $c$ be a [[Definition:Choice Function|choice function]] on $\SS$ and consider the [[Definition:Image of Mapping|image set]] $c \sqbrk \SS$:
:$c \sqbrk \SS = \set {\map c s: \O \notin s \land \forall t, u \in s... | Equivalence of Formulations of Axiom of Choice/Formulation 1 implies Formulation 3 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_1_implies_Formulation_3 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_1_implies_Formulation_3 | [
"Equivalence of Formulations of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Axiom:Axiom of Choice"
] | [
"Definition:Choice Function",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Choice Function"
] |
proofwiki-17688 | Equivalence of Formulations of Axiom of Choice/Formulation 3 implies Formulation 1 | The following formulation of the Axiom of Choice: | Let $\BB$ be a non-empty indexed family of non-empty sets indexed by $\II$.
Consider sets of the following form:
:$\CC = \set {\tuple {B_i, x}: i \in \II, B_i \in \BB, x \in B_i}$
That is, it is the set of ordered pairs of which the first coordinate is a set $B_i \in \BB$ and the second coordinate is an element of $B_i... | The following formulation of the [[Axiom:Axiom of Choice|Axiom of Choice]]: | Let $\BB$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Indexed Family of Sets|indexed family]] of [[Definition:Non-Empty Set|non-empty sets]] [[Definition:Indexing Set|indexed]] by $\II$.
Consider [[Definition:Set|sets]] of the following form:
:$\CC = \set {\tuple {B_i, x}: i \in \II, B_i \in \BB, x \in B... | Equivalence of Formulations of Axiom of Choice/Formulation 3 implies Formulation 1 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_3_implies_Formulation_1 | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Axiom_of_Choice/Formulation_3_implies_Formulation_1 | [
"Equivalence of Formulations of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Axiom:Axiom of Choice"
] | [
"Definition:Non-Empty Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Non-Empty Set",
"Definition:Indexing Set",
"Definition:Set",
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair",
"Definition:Set",
"Definition:Coordinate Sy... |
proofwiki-17689 | C^k Norm is Norm | Let $I = \closedint a b$ be a closed real interval.
Let $\struct {\map {C^k} I, +, \, \cdot \,}_\R$ be the vector space of real-valued functions, k-times differentiable on $I$.
Let $x \in \map {C^k} I$ be a real-valued function of differentiability class $k$.
Let $\norm {\, \cdot \,}_{\map {C^k} I}$ be the $C^k$ norm o... | === Positive definiteness ===
Let $x \in \map {C^k} I$.
Then:
{{begin-eqn}}
{{eqn | l = \norm x_{\map {C^k} I}
| r = \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty
}}
{{eqn | o = \ge
| r = \sum_{i \mathop = 0}^k 0
| c = Supremum Norm is Norm, {{NormAxiomVector|1}}
}}
{{eqn | r = 0
}}
{{end-eqn}}
... | Let $I = \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\struct {\map {C^k} I, +, \, \cdot \,}_\R$ be the [[Definition:Space of Continuous Functions of Differentiability Class k|vector space of real-valued functions, k-times differentiable on]] $I$.
Let $x \in \map {C^k} I$ be a [... | === Positive definiteness ===
Let $x \in \map {C^k} I$.
Then:
{{begin-eqn}}
{{eqn | l = \norm x_{\map {C^k} I}
| r = \sum_{i \mathop = 0}^k \norm {x^{\paren i} }_\infty
}}
{{eqn | o = \ge
| r = \sum_{i \mathop = 0}^k 0
| c = [[Supremum Norm is Norm]], {{NormAxiomVector|1}}
}}
{{eqn | r = 0
}}
{{end... | C^k Norm is Norm | https://proofwiki.org/wiki/C^k_Norm_is_Norm | https://proofwiki.org/wiki/C^k_Norm_is_Norm | [
"Examples of Norms"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:Real-Valued Function",
"Definition:Differentiability Class",
"Definition:C^k Norm",
"Definition:Norm/Vector Space"
] | [
"Supremum Norm is Norm",
"Sum of Nonnegative Real Numbers is Zero iff Every Element is Zero",
"Supremum Norm is Norm",
"Supremum Norm is Norm"
] |
proofwiki-17690 | Existence of Minimal Polynomial for Square Matrix over Field | Let $K$ be a field.
Let $n$ be a natural number.
Let $K^{n \times n}$ be the set of $n \times n$ matrices over $K$.
Let $A \in K^{n \times n}$.
Then the minimal polynomial of $A$ exists and has degree at most $n^2$. | By Matrices over Field form Vector Space:
:$K^{n \times n}$ forms a vector space under usual matrix addition and scalar multiplication.
By Dimension of Vector Space of Matrices:
:$K^{n \times n}$ has dimension $n^2$.
Consider the collection of vectors:
:$I, A, A^2, \ldots, A^{n^2}$
Since this is a collection of $n^2 ... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $n$ be a [[Definition:Natural Number|natural number]].
Let $K^{n \times n}$ be the set of $n \times n$ [[Definition:Matrix|matrices]] over $K$.
Let $A \in K^{n \times n}$.
Then the [[Definition:Minimal Polynomial|minimal polynomial]] of $A$ exists an... | By [[Matrices over Field form Vector Space]]:
:$K^{n \times n}$ forms a [[Definition:Vector Space|vector space]] under usual [[Definition:Matrix Addition|matrix addition]] and scalar multiplication.
By [[Dimension of Vector Space of Matrices]]:
:$K^{n \times n}$ has [[Definition:Dimension (Linear Algebra)|dimension]... | Existence of Minimal Polynomial for Square Matrix over Field | https://proofwiki.org/wiki/Existence_of_Minimal_Polynomial_for_Square_Matrix_over_Field | https://proofwiki.org/wiki/Existence_of_Minimal_Polynomial_for_Square_Matrix_over_Field | [
"Linear Algebra",
"Minimal Polynomials"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Natural Numbers",
"Definition:Matrix",
"Definition:Minimal Polynomial",
"Definition:Degree of Polynomial"
] | [
"Matrices over Field form Vector Space",
"Definition:Vector Space",
"Definition:Matrix Addition",
"Dimension of Vector Space of Matrices",
"Definition:Dimension (Linear Algebra)",
"Definition:Dimension (Linear Algebra)",
"Size of Linearly Independent Subset is at Most Size of Finite Generator",
"Defin... |
proofwiki-17691 | Homeomorphic Topologies on Same Set may not be Identical | Let $S$ be a set.
Let $\tau_1$ and $\tau_2$ both be topologies on $S$ such that the topological spaces $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are homeomorphic.
Then it is not necessarily the case that $\struct {S, \tau_1} = \struct {S, \tau_2}$. | Let $p, q \in S$ such that $p \ne q$.
Let $\tau_p$ and $\tau_q$ be the particular point topologies on $S$ by $p$ and $q$ respectively.
From Homeomorphic Non-Comparable Particular Point Topologies, $\struct {S, \tau_p}$ and $\struct {S, \tau_q}$ are homeomorphic.
But $\struct {S, \tau_p} \ne \struct {S, \tau_q}$, as $\s... | Let $S$ be a [[Definition:Set|set]].
Let $\tau_1$ and $\tau_2$ both be [[Definition:Topology|topologies]] on $S$ such that the [[Definition:Topological Space|topological spaces]] $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are [[Definition:Homeomorphic Topological Spaces|homeomorphic]].
Then it is not necessarily... | Let $p, q \in S$ such that $p \ne q$.
Let $\tau_p$ and $\tau_q$ be the [[Definition:Particular Point Topology|particular point topologies]] on $S$ by $p$ and $q$ respectively.
From [[Homeomorphic Non-Comparable Particular Point Topologies]], $\struct {S, \tau_p}$ and $\struct {S, \tau_q}$ are [[Definition:Homeomorph... | Homeomorphic Topologies on Same Set may not be Identical | https://proofwiki.org/wiki/Homeomorphic_Topologies_on_Same_Set_may_not_be_Identical | https://proofwiki.org/wiki/Homeomorphic_Topologies_on_Same_Set_may_not_be_Identical | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Set",
"Definition:Topology",
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Particular Point Topology",
"Homeomorphic Non-Comparable Particular Point Topologies",
"Definition:Homeomorphism/Topological Spaces"
] |
proofwiki-17692 | Topologies on Set with More than One Element may not be Homeomorphic | Let $S$ be a set which contains at least $2$ elements.
Let $\tau_1$ and $\tau_2$ be topologies on $S$.
Then it is not necessarily the case that $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are homeomorphic. | Let $\tau_1$ be the indiscrete topology on $S$.
Let $\tau_2$ be the discrete topology on $S$.
Then $\struct {S, \tau_1}$ has $2$ elements: $S$ and $\O$.
Let $a, b \in S$ such that $a \ne b$.
Then $\set a \in \tau_2$ and $\set b \in \tau 2$, as well as $S$ and $\O$.
So there cannot be a bijection between $\struct {S, \t... | Let $S$ be a [[Definition:Set|set]] which contains at least $2$ [[Definition:Element|elements]].
Let $\tau_1$ and $\tau_2$ be [[Definition:Topology|topologies]] on $S$.
Then it is not necessarily the case that $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are [[Definition:Homeomorphic Topological Spaces|homeomorphi... | Let $\tau_1$ be the [[Definition:Indiscrete Topology|indiscrete topology]] on $S$.
Let $\tau_2$ be the [[Definition:Discrete Topology|discrete topology]] on $S$.
Then $\struct {S, \tau_1}$ has $2$ [[Definition:Element|elements]]: $S$ and $\O$.
Let $a, b \in S$ such that $a \ne b$.
Then $\set a \in \tau_2$ and $\set... | Topologies on Set with More than One Element may not be Homeomorphic | https://proofwiki.org/wiki/Topologies_on_Set_with_More_than_One_Element_may_not_be_Homeomorphic | https://proofwiki.org/wiki/Topologies_on_Set_with_More_than_One_Element_may_not_be_Homeomorphic | [
"Homeomorphisms (Topological Spaces)"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Topology",
"Definition:Homeomorphism/Topological Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Discrete Topology",
"Definition:Element",
"Definition:Bijection"
] |
proofwiki-17693 | Clopen Sets in Indiscrete Topology | The only subsets of $S$ which are both closed and open in $T$ are $S$ and $\O$. | By definition of indiscrete topological space, the only open sets in $\struct {S, \tau}$ are $S$ and $\O$.
From Open and Closed Sets in Topological Space, both $S$ and $\O$ are both closed and open in $\struct {S, \tau}$ are $S$ and $\O$.
Hence the result.
{{qed}} | The only [[Definition:Subset|subsets]] of $S$ which are both [[Definition:Closed Set (Topology)|closed]] and [[Definition:Open Set (Topology)|open]] in $T$ are $S$ and $\O$. | By definition of [[Definition:Indiscrete Topology|indiscrete topological space]], the only [[Definition:Open Set (Topology)|open sets]] in $\struct {S, \tau}$ are $S$ and $\O$.
From [[Open and Closed Sets in Topological Space]], both $S$ and $\O$ are both [[Definition:Closed Set (Topology)|closed]] and [[Definition:Op... | Clopen Sets in Indiscrete Topology | https://proofwiki.org/wiki/Clopen_Sets_in_Indiscrete_Topology | https://proofwiki.org/wiki/Clopen_Sets_in_Indiscrete_Topology | [
"Indiscrete Topology",
"Clopen Sets"
] | [
"Definition:Subset",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Indiscrete Topology",
"Definition:Open Set/Topology",
"Open and Closed Sets in Topological Space",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-17694 | Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets | {{TFAE|def = Connected Topological Space}}
Let $T = \struct {S, \tau}$ be a topological space. | === Definition by No Clopen Sets implies Definition by Separation ===
Let $T$ be connected by having no clopen sets.
{{AimForCont}} $T$ admits a separation, $A \mid B$ say.
Then both $A$ and $B$ are clopen sets of $T$, neither of which is either $S$ or $\O$.
From this contradiction it follows that $T$ can admit no sepa... | {{TFAE|def = Connected Topological Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | === Definition by No Clopen Sets implies Definition by Separation ===
Let $T$ be [[Definition:Connected Topological Space/Definition 4|connected by having no clopen sets]].
{{AimForCont}} $T$ admits a [[Definition:Separation (Topology)|separation]], $A \mid B$ say.
Then both $A$ and $B$ are [[Definition:Clopen Set|c... | Equivalence of Definitions of Connected Topological Space/No Separation iff No Clopen Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Separation_iff_No_Clopen_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Connected_Topological_Space/No_Separation_iff_No_Clopen_Sets | [
"Equivalence of Definitions of Connected Topological Space"
] | [
"Definition:Topological Space"
] | [
"Definition:Connected Topological Space/Definition 4",
"Definition:Separation (Topology)",
"Definition:Clopen Set",
"Definition:Contradiction",
"Definition:Separation (Topology)",
"Definition:Connected Topological Space/Definition 1",
"Definition:Clopen Set",
"Definition:Clopen Set",
"Definition:Sep... |
proofwiki-17695 | Dependent Subset of Independent Set Union Singleton Contains Singleton | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $X$ be an independent subset of $M$.
Let $x \in S$.
Let $C$ be a dependent subset of $M$ such that:
:$C \subseteq X \cup \set x$.
Then:
:$x \in C$ | From the contrapositive statement of Superset of Dependent Set is Dependent:
:$C \nsubseteq X$
From the contrapositive statement of Set Difference with Superset is Empty Set:
:$C \setminus X \ne \O$
From Set Difference over Subset:
:$C \setminus X \subseteq \paren {X \cup \set x} \setminus X = \set x$
From Power Set of... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $X$ be an [[Definition:Independent Subset (Matroid)|independent subset]] of $M$.
Let $x \in S$.
Let $C$ be a [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$ such that:
:$C \subseteq X \cup \set x$.
Then:
:$x \in C$ | From the [[Definition:Contrapositive Statement|contrapositive statement]] of [[Superset of Dependent Set is Dependent]]:
:$C \nsubseteq X$
From the [[Definition:Contrapositive Statement|contrapositive statement]] of [[Set Difference with Superset is Empty Set]]:
:$C \setminus X \ne \O$
From [[Set Difference over Subs... | Dependent Subset of Independent Set Union Singleton Contains Singleton | https://proofwiki.org/wiki/Dependent_Subset_of_Independent_Set_Union_Singleton_Contains_Singleton | https://proofwiki.org/wiki/Dependent_Subset_of_Independent_Set_Union_Singleton_Contains_Singleton | [
"Matroid Dependent Subsets",
"Matroid Independent Subsets"
] | [
"Definition:Matroid",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Dependent Set"
] | [
"Definition:Contrapositive Statement",
"Superset of Dependent Set is Dependent",
"Definition:Contrapositive Statement",
"Set Difference with Superset is Empty Set",
"Set Difference over Subset",
"Power Set of Singleton",
"Set Difference is Subset",
"Singleton of Element is Subset",
"Category:Matroid... |
proofwiki-17696 | Matroid Unique Circuit Property/Corollary | Let $B$ be a base of $M$.
Let $x \in S \setminus B$.
Then there exists a unique circuit $C$ such that:
:$x \in C \subseteq B \cup \set x$
That is, $C$ is the fundamental circuit of $x$ in $B$. | From Union of Matroid Base with Element of Complement is Dependent:
:$B \cup \set x$ is dependent.
From Matroid Unique Circuit Property there exists a unique circuit $C$ such that:
:$x \in C \subseteq B \cup \set x$
{{qed}} | Let $B$ be a [[Definition:Base of Matroid|base]] of $M$.
Let $x \in S \setminus B$.
Then there exists a unique [[Definition:Circuit (Matroid)|circuit]] $C$ such that:
:$x \in C \subseteq B \cup \set x$
That is, $C$ is the [[Definition:Fundamental Circuit (Matroid)|fundamental circuit]] of $x$ in $B$. | From [[Union of Matroid Base with Element of Complement is Dependent]]:
:$B \cup \set x$ is [[Definition:Dependent Subset (Matroid)|dependent]].
From [[Matroid Unique Circuit Property]] there exists a unique [[Definition:Circuit (Matroid)|circuit]] $C$ such that:
:$x \in C \subseteq B \cup \set x$
{{qed}} | Matroid Unique Circuit Property/Corollary | https://proofwiki.org/wiki/Matroid_Unique_Circuit_Property/Corollary | https://proofwiki.org/wiki/Matroid_Unique_Circuit_Property/Corollary | [
"Matroid Unique Circuit Property"
] | [
"Definition:Base of Matroid",
"Definition:Circuit (Matroid)",
"Definition:Fundamental Circuit (Matroid)"
] | [
"Union of Matroid Base with Element of Complement is Dependent",
"Definition:Matroid/Dependent Set",
"Matroid Unique Circuit Property",
"Definition:Circuit (Matroid)"
] |
proofwiki-17697 | Matroid Base Substitution From Fundamental Circuit | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $B$ be a base of $M$.
Let $y \in S \setminus B$.
Let $\map C {y,B}$ denote the fundamental circuit of $y$ in $B$.
Let $x \in B$.
Then:
:$\paren{B \setminus \set x} \cup \set y$ is a base of $M$ {{iff}} $x \in \map C {y,B}$
That is, $y$ can be substituted for $x$ in $B... | === Necessary Condition ===
Let $\paren{B \setminus \set x} \cup \set y$ be a base of $M$.
By definition of the fundamental circuit:
:$\map C {y,B} \subseteq B \cup \set y$
and
:$\map C {y,B}$ is dependent
{{AimForCont}} $x \notin \map C {y,B}$.
Then:
:$\map C {y,B} \subseteq \paren{B \setminus \set x} \cup \set y$
By ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $B$ be a [[Definition:Base of Matroid|base]] of $M$.
Let $y \in S \setminus B$.
Let $\map C {y,B}$ denote the [[Definition:Fundamental Circuit (Matroid)|fundamental circuit]] of $y$ in $B$.
Let $x \in B$.
Then:
:$\paren{B \setminus \set x}... | === Necessary Condition ===
Let $\paren{B \setminus \set x} \cup \set y$ be a [[Definition:Base of Matroid|base]] of $M$.
By definition of the [[Definition:Fundamental Circuit (Matroid)|fundamental circuit]]:
:$\map C {y,B} \subseteq B \cup \set y$
and
:$\map C {y,B}$ is [[Definition:Dependent Subset (Matroid)|depend... | Matroid Base Substitution From Fundamental Circuit | https://proofwiki.org/wiki/Matroid_Base_Substitution_From_Fundamental_Circuit | https://proofwiki.org/wiki/Matroid_Base_Substitution_From_Fundamental_Circuit | [
"Matroid Bases",
"Matroid Circuits"
] | [
"Definition:Matroid",
"Definition:Base of Matroid",
"Definition:Fundamental Circuit (Matroid)",
"Definition:Base of Matroid",
"Definition:Substitution (Set Theory)",
"Definition:Base of Matroid"
] | [
"Definition:Base of Matroid",
"Definition:Fundamental Circuit (Matroid)",
"Definition:Matroid/Dependent Set",
"Axiom:Matroid Axioms",
"Definition:Matroid/Independent Set",
"Definition:Contradiction",
"Definition:Matroid/Dependent Set",
"Definition:Base of Matroid",
"Definition:Matroid/Dependent Set"... |
proofwiki-17698 | Socrates is Mortal/Variant | :$(1): \quad$ ''If {{AuthorRef|Socrates}} is a man then {{AuthorRef|Socrates}} is mortal.''
:$(2): \quad$ ''{{AuthorRef|Socrates}} is a man.''
:$(3): \quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.'' | Let $P$ denote the simple statement ''{{AuthorRef|Socrates}} is a man.''.
Let $Q$ denote the simple statement ''{{AuthorRef|Socrates}} is mortal.''.
The argument can then be expressed as:
{{begin-eqn}}
{{eqn | n = 1
| l = P
| o = \implies
| r = Q
| c =
}}
{{eqn | n = 2
| l = P
| o =... | :$(1): \quad$ ''If {{AuthorRef|Socrates}} is a man then {{AuthorRef|Socrates}} is mortal.''
:$(2): \quad$ ''{{AuthorRef|Socrates}} is a man.''
:$(3): \quad$ ''Therefore {{AuthorRef|Socrates}} is mortal.'' | Let $P$ denote the [[Definition:Simple Statement|simple statement]] ''{{AuthorRef|Socrates}} is a man.''.
Let $Q$ denote the [[Definition:Simple Statement|simple statement]] ''{{AuthorRef|Socrates}} is mortal.''.
The argument can then be expressed as:
{{begin-eqn}}
{{eqn | n = 1
| l = P
| o = \implies
... | Socrates is Mortal/Variant | https://proofwiki.org/wiki/Socrates_is_Mortal/Variant | https://proofwiki.org/wiki/Socrates_is_Mortal/Variant | [
"Socrates is Mortal"
] | [] | [
"Definition:Simple Statement",
"Definition:Simple Statement",
"Modus Ponendo Ponens"
] |
proofwiki-17699 | Self-Distributive Law for Conditional/Formulation 2/Forward Implication | :$\vdash \paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$ | {{BeginTableau|\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption |1|p \implies \paren {q \implies r} }}
{{SequentIntro|2|1|\paren {p \implies q} \implies \paren {p \implies r}|1|Self-Distributive Law for Conditional: Formulation 1}}
{{Imp... | :$\vdash \paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$ | {{BeginTableau|\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption |1|p \implies \paren {q \implies r} }}
{{SequentIntro|2|1|\paren {p \implies q} \implies \paren {p \implies r}|1|[[Self-Distributive Law for Conditional/Formulation 1/Forwar... | Self-Distributive Law for Conditional/Formulation 2/Forward Implication/Proof 1 | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Forward_Implication/Proof_1 | [
"Self-Distributive Law for Conditional"
] | [] | [
"Self-Distributive Law for Conditional/Formulation 1/Forward Implication"
] |
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