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-17700 | 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} }}
{{Assumption |2|p \implies q}}
{{Assumption |3|p}}
{{ModusPonens |4|1,3|q \implies r|1|3}}
{{ModusPonens |5|2,3|q|2|3}}
{{ModusPonen... | :$\vdash \paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$ | {{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} }}
{{Assumption |2|p \implies q}}
{{Assumption |3|p}}
{{ModusPonens |4|1,3|q \implies r|1|3}}
{{ModusPonens |5|2,3|q|2|3}}
{{ModusPonen... | Self-Distributive Law for Conditional/Formulation 2/Forward Implication/Proof 2 | 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_2 | [
"Self-Distributive Law for Conditional"
] | [] | [] |
proofwiki-17701 | Self-Distributive Law for Conditional/Formulation 2/Forward Implication | :$\vdash \paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$ | We apply the Method of Truth Tables.
As can be seen by inspection, the truth value under the main connective is true for all boolean interpretations.
:<nowiki>$\begin{array}{|ccccc|c|ccccccc|}
\hline
(p & \implies & (q & \implies & r)) & \implies & ((p & \implies & q) & \implies & (p & \implies & r)) \\
\hline
\F & \T ... | :$\vdash \paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth value]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|c... | Self-Distributive Law for Conditional/Formulation 2/Forward Implication/Proof by Truth Table | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Forward_Implication/Proof_by_Truth_Table | [
"Self-Distributive Law for Conditional"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-17702 | Self-Distributive Law for Conditional/Formulation 2/Reverse Implication | :$\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$ | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption |1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{SequentIntro |2|1|p \implies \paren {q \implies r}|1|Self-Distributive Law for Conditional: Formulation... | :$\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$ | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption |1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{SequentIntro |2|1|p \implies \paren {q \implies r}|1|[[Self-Distributive Law for Conditional/Formulatio... | Self-Distributive Law for Conditional/Formulation 2/Reverse Implication/Proof 1 | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Reverse_Implication/Proof_1 | [
"Self-Distributive Law for Conditional"
] | [] | [
"Self-Distributive Law for Conditional/Formulation 1/Reverse Implication"
] |
proofwiki-17703 | Self-Distributive Law for Conditional/Formulation 2/Reverse Implication | :$\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$ | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption|1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{Assumption|2|p}}
{{Assumption|3|q}}
{{SequentIntro|4|3|p \implies q|3|True Statement is implied by Every S... | :$\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} }$ | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption|1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{Assumption|2|p}}
{{Assumption|3|q}}
{{SequentIntro|4|3|p \implies q|3|[[True Statement is implied by Every... | Self-Distributive Law for Conditional/Formulation 2/Reverse Implication/Proof 2 | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Reverse_Implication/Proof_2 | [
"Self-Distributive Law for Conditional"
] | [] | [
"True Statement is implied by Every Statement"
] |
proofwiki-17704 | Odd Integers under Addition do not form Group | Let $S$ be the set of odd integers:
:$S = \set {x \in \Z: \exists n \in \Z: x = 2 n + 1}$
Let $\struct {S, +}$ denote the algebraic structure formed by $S$ under the operation of addition.
Then $\struct {S, +}$ is not a group. | It is to be demonstrated that $\struct {S, +}$ does not satisfy the group axioms.
Let $a$ and $b$ be odd integers.
Then $a = 2 n + 1$ and $b = 2 m + 1$ for some $m, n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = a + b
| r = 2 n + 1 + 2 m + 1
| c =
}}
{{eqn | r = 2 \paren {n + m + 1}
| c =
}}
{{end-eqn}... | Let $S$ be the [[Definition:Set|set]] of [[Definition:Odd Integer|odd integers]]:
:$S = \set {x \in \Z: \exists n \in \Z: x = 2 n + 1}$
Let $\struct {S, +}$ denote the [[Definition:Algebraic Structure with One Operation|algebraic structure]] formed by $S$ under the [[Definition:Binary Operation|operation]] of [[Defini... | It is to be demonstrated that $\struct {S, +}$ does not satisfy the [[Axiom:Group Axioms|group axioms]].
Let $a$ and $b$ be [[Definition:Odd Integer|odd integers]].
Then $a = 2 n + 1$ and $b = 2 m + 1$ for some $m, n \in \Z$.
Then:
{{begin-eqn}}
{{eqn | l = a + b
| r = 2 n + 1 + 2 m + 1
| c =
}}
{{eqn... | Odd Integers under Addition do not form Group | https://proofwiki.org/wiki/Odd_Integers_under_Addition_do_not_form_Group | https://proofwiki.org/wiki/Odd_Integers_under_Addition_do_not_form_Group | [
"Odd Integers",
"Integer Addition"
] | [
"Definition:Set",
"Definition:Odd Integer",
"Definition:Algebraic Structure/One Operation",
"Definition:Operation/Binary Operation",
"Definition:Addition/Integers",
"Definition:Group"
] | [
"Axiom:Group Axioms",
"Definition:Odd Integer",
"Definition:Even Integer"
] |
proofwiki-17705 | Number of Digits in Number | Let $n \in \Z_{>0}$ be a strictly positive integer.
Let $b \in \Z_{>1}$ be an integer greater than $1$.
Let $n$ be expressed in base $b$.
Then the number of digits $d$ in this expression for $n$ is:
:$d = 1 + \floor {\log_b n}$
where:
:$\floor {\, \cdot \,}$ denotes the floor function
:$\log_b$ denotes the logarithm to... | Let $n$ have $d$ digits when expressed in base $b$.
Then $n$ is expressed as:
:$n = \sqbrk {n_{d - 1} n_{d - 2} \dotsm d_1 d_0}$
where:
:$n = \ds \sum_{k \mathop = 0}^{d - 1} n_k b^k$
Thus:
:$b^{d - 1} \le n < b^d$
Thus we have:
{{begin-eqn}}
{{eqn | l = b^{d - 1}
| o = \le
| m = n
| mo= <
| r =... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Let $b \in \Z_{>1}$ be an [[Definition:Integer|integer]] greater than $1$.
Let $n$ be expressed in [[Definition:Number Base|base]] $b$.
Then the number of [[Definition:Digit|digits]] $d$ in this expression for $n$ is:
:$d ... | Let $n$ have $d$ [[Definition:Digit|digits]] when expressed in [[Definition:Number Base|base]] $b$.
Then $n$ is expressed as:
:$n = \sqbrk {n_{d - 1} n_{d - 2} \dotsm d_1 d_0}$
where:
:$n = \ds \sum_{k \mathop = 0}^{d - 1} n_k b^k$
Thus:
:$b^{d - 1} \le n < b^d$
Thus we have:
{{begin-eqn}}
{{eqn | l = b^{d - 1}
... | Number of Digits in Number | https://proofwiki.org/wiki/Number_of_Digits_in_Number | https://proofwiki.org/wiki/Number_of_Digits_in_Number | [
"Number Theory"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Integer",
"Definition:Number Base",
"Definition:Digit",
"Definition:Floor Function",
"Definition:General Logarithm"
] | [
"Definition:Digit",
"Definition:Number Base",
"Integer equals Floor iff Number between Integer and One More",
"Category:Number Theory"
] |
proofwiki-17706 | Power of 2^10 Minus Power of 10^3 is Divisible by 24 | Let $n \in \Z_{\ge 0}$ be a non-negative integer.
Then $2^{10 n} - 10^{3 n}$ is divisible by $24$.
That is:
:$2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$ | {{begin-eqn}}
{{eqn | l = 2^{10 n}
| r = \paren {2^{10} }^n
| c = Power of Power
}}
{{eqn | r = 1024^n
| c = as $2^{10} = 1024$
}}
{{eqn | r = \paren {1000 + 24}^n
| c = rewriting $1024$ as the sum of a power of $10$ and some integer
}}
{{eqn | r = \sum_{k \mathop = 0}^n 1000^{n - k} \, 24^k
... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Non-Negative Integer|non-negative integer]].
Then $2^{10 n} - 10^{3 n}$ is [[Definition:Divisor of Integer|divisible]] by $24$.
That is:
:$2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$ | {{begin-eqn}}
{{eqn | l = 2^{10 n}
| r = \paren {2^{10} }^n
| c = [[Power of Power]]
}}
{{eqn | r = 1024^n
| c = as $2^{10} = 1024$
}}
{{eqn | r = \paren {1000 + 24}^n
| c = rewriting $1024$ as the sum of a power of $10$ and some integer
}}
{{eqn | r = \sum_{k \mathop = 0}^n 1000^{n - k} \, 24^k... | Power of 2^10 Minus Power of 10^3 is Divisible by 24/Proof 1 | https://proofwiki.org/wiki/Power_of_2^10_Minus_Power_of_10^3_is_Divisible_by_24 | https://proofwiki.org/wiki/Power_of_2^10_Minus_Power_of_10^3_is_Divisible_by_24/Proof_1 | [
"Modulo Arithmetic",
"Power of 2^10 Minus Power of 10^3 is Divisible by 24"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Exponent Combination Laws/Power of Power",
"Binomial Theorem",
"Definition:Summation",
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-17707 | Power of 2^10 Minus Power of 10^3 is Divisible by 24 | Let $n \in \Z_{\ge 0}$ be a non-negative integer.
Then $2^{10 n} - 10^{3 n}$ is divisible by $24$.
That is:
:$2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$ | For $n = 0$ both powers are $1$, and $1 - 1 = 0$ is divisible by $24$.
For $n > 1$:
{{begin-eqn}}
{{eqn | l = 2^{10 n} - 10^{3 n}
| r = 2^{3 n} \paren {2^{7 n} - 5^{3 n} }
}}
{{eqn | o = \equiv
| r = 0
| rr= \pmod 8
| c = because $2^3 \divides 2^{3 n}$
}}
{{eqn | l = 2^{10 n} - 10^{3 n}
| ... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Non-Negative Integer|non-negative integer]].
Then $2^{10 n} - 10^{3 n}$ is [[Definition:Divisor of Integer|divisible]] by $24$.
That is:
:$2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$ | For $n = 0$ both [[Definition:Integer Power|powers]] are $1$, and $1 - 1 = 0$ is [[Definition:Divisor of Integer|divisible]] by $24$.
For $n > 1$:
{{begin-eqn}}
{{eqn | l = 2^{10 n} - 10^{3 n}
| r = 2^{3 n} \paren {2^{7 n} - 5^{3 n} }
}}
{{eqn | o = \equiv
| r = 0
| rr= \pmod 8
| c = because $2... | Power of 2^10 Minus Power of 10^3 is Divisible by 24/Proof 2 | https://proofwiki.org/wiki/Power_of_2^10_Minus_Power_of_10^3_is_Divisible_by_24 | https://proofwiki.org/wiki/Power_of_2^10_Minus_Power_of_10^3_is_Divisible_by_24/Proof_2 | [
"Modulo Arithmetic",
"Power of 2^10 Minus Power of 10^3 is Divisible by 24"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Power (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Congruence of Powers",
"Chinese Remainder Theorem"
] |
proofwiki-17708 | Power of 2^10 Minus Power of 10^3 is Divisible by 24 | Let $n \in \Z_{\ge 0}$ be a non-negative integer.
Then $2^{10 n} - 10^{3 n}$ is divisible by $24$.
That is:
:$2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$ | {{begin-eqn}}
{{eqn | l = 2^{10 n} - 10^{3 n}
| r = \paren {2^{10} }^n - \paren {10^3}^n
| c = Power of Power
}}
{{eqn | r = \paren {2^{10} - 10^3} \sum_{j \mathop = 0}^{n - 1} \paren {2^{10} }^{n - j - 1} \paren {10^3}^j
| c = Difference of Two Powers
}}
{{eqn | r = 24 k
| c = where $\ds k = \s... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Non-Negative Integer|non-negative integer]].
Then $2^{10 n} - 10^{3 n}$ is [[Definition:Divisor of Integer|divisible]] by $24$.
That is:
:$2^{10 n} - 10^{3 n} \equiv 0 \pmod {24}$ | {{begin-eqn}}
{{eqn | l = 2^{10 n} - 10^{3 n}
| r = \paren {2^{10} }^n - \paren {10^3}^n
| c = [[Power of Power]]
}}
{{eqn | r = \paren {2^{10} - 10^3} \sum_{j \mathop = 0}^{n - 1} \paren {2^{10} }^{n - j - 1} \paren {10^3}^j
| c = [[Difference of Two Powers]]
}}
{{eqn | r = 24 k
| c = where $\d... | Power of 2^10 Minus Power of 10^3 is Divisible by 24/Proof 3 | https://proofwiki.org/wiki/Power_of_2^10_Minus_Power_of_10^3_is_Divisible_by_24 | https://proofwiki.org/wiki/Power_of_2^10_Minus_Power_of_10^3_is_Divisible_by_24/Proof_3 | [
"Modulo Arithmetic",
"Power of 2^10 Minus Power of 10^3 is Divisible by 24"
] | [
"Definition:Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Exponent Combination Laws/Power of Power",
"Difference of Two Powers",
"Definition:Integer"
] |
proofwiki-17709 | Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the 2-sequence space equipped with Euclidean norm.
Let $c_{00}$ be the space of almost-zero sequences.
Then $c_{00}$ is everywhere dense in $\struct {\ell^2, \norm {\, \cdot \,}_2}$ | Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^2$.
By definition of $\ell^2$:
:$\ds \sum_{i \mathop = 0}^\infty \size {x_i}^2 < \infty$
Let $\ds s_n := \sum_{i \mathop = 0}^n \size {x_i}^2$ be a sequence of partial sums of $\ds s = \sum_{i \mathop = 0}^\infty \size {x_i}^2$.
We have that $s$ is a converge... | Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the [[Definition:P-Sequence Space|2-sequence space]] equipped with [[Definition:Euclidean Norm|Euclidean norm]].
Let $c_{00}$ be the [[Definition:Space of Almost-Zero Sequences|space of almost-zero sequences]].
Then $c_{00}$ is [[Definition:Everywhere Dense in Normed ... | Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^2$.
By [[Definition:P-Sequence Space|definition]] of $\ell^2$:
:$\ds \sum_{i \mathop = 0}^\infty \size {x_i}^2 < \infty$
Let $\ds s_n := \sum_{i \mathop = 0}^n \size {x_i}^2$ be a [[Definition:Sequence|sequence]] of [[Definition:Partial Sum|partial sums]] ... | Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_Everywhere_Dense_in_2-Sequence_Space | https://proofwiki.org/wiki/Space_of_Almost-Zero_Sequences_is_Everywhere_Dense_in_2-Sequence_Space | [
"Normed Vector Spaces",
"Denseness"
] | [
"Definition:P-Sequence Space",
"Definition:Euclidean Norm",
"Definition:Space of Almost-Zero Sequences",
"Definition:Everywhere Dense/Normed Vector Space"
] | [
"Definition:P-Sequence Space",
"Definition:Sequence",
"Definition:Series/Sequence of Partial Sums",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Space of Almost-Zero Sequences",
"Definition:Everywhere Dense/Normed Vector Space"
] |
proofwiki-17710 | Linear Function on Stationary Stochastic Model is Stationary | 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 $L_t$ be a linear function of $\sequence {s_n}$:
:$L_t = l_1 z_t + l_2 z_{t - 1} + \dotsb + l_n z_{t - n + 1}... | Follows by definition of stationarity.
{{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 $T$:
... | Follows by definition of [[Definition:Strictly Stationary Stochastic Process|stationarity]].
{{qed}} | Linear Function on Stationary Stochastic Model is Stationary | https://proofwiki.org/wiki/Linear_Function_on_Stationary_Stochastic_Model_is_Stationary | https://proofwiki.org/wiki/Linear_Function_on_Stationary_Stochastic_Model_is_Stationary | [
"Stationary Stochastic Processes"
] | [
"Definition:Strictly Stationary Stochastic Process",
"Definition:Time Series",
"Definition:Sequence",
"Definition:Successive Values of Time Series/Equispaced",
"Definition:Linear Function",
"Definition:Strictly Stationary Stochastic Process"
] | [
"Definition:Strictly Stationary Stochastic Process"
] |
proofwiki-17711 | Ordering of Integers is Reversed by Negation | Let $x, y \in \Z$ such that $x > y$.
Then:
:$-x < -y$ | From the formal definition of integers, $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.
Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for some $x, y \in \Z$.
We have:
{{begin-eqn}}
{{eqn | l = x
| o = >
| r = y
| c =
}}
{{eqn | ll= \leadsto
| l = \e... | Let $x, y \in \Z$ such that $x > y$.
Then:
:$-x < -y$ | From the [[Definition:Integer/Formal Definition|formal definition of integers]], $\eqclass {a, b} {}$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Ordered Pair|ordered pairs]] of [[Definition:Natural Numbers|natural numbers]].
Let $x = \eqclass {a, b} {}$ and $y = \eqclass {c, d} {}$ for s... | Ordering of Integers is Reversed by Negation | https://proofwiki.org/wiki/Ordering_of_Integers_is_Reversed_by_Negation | https://proofwiki.org/wiki/Ordering_of_Integers_is_Reversed_by_Negation | [
"Orderings on Integers"
] | [] | [
"Definition:Integer/Formal Definition",
"Definition:Equivalence Class",
"Definition:Ordered Pair",
"Definition:Natural Numbers",
"Negative of Integer"
] |
proofwiki-17712 | Strict Ordering on Integers is Well-Defined | Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers.
Let:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| r = \eqclass {a', b'} {}
| c =
}}
{{eqn | l = \eqclass {c, d} {}
| r = \eqclass {c', d'} {}
| c =
}}
{{end-eqn}}
Then:
{{begin-eqn}}
{{eqn | l = \eqc... | This is a direct application of the Extension Theorem for Total Orderings.
{{qed}} | Let $\eqclass {a, b} {}$ denote an [[Definition:Integer|integer]], as defined by the [[Definition:Integer/Formal Definition|formal definition of integers]].
Let:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| r = \eqclass {a', b'} {}
| c =
}}
{{eqn | l = \eqclass {c, d} {}
| r = \eqclass {c', d'} {... | This is a direct application of the [[Extension Theorem for Total Orderings]].
{{qed}} | Strict Ordering on Integers is Well-Defined | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Well-Defined | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Well-Defined | [
"Orderings on Integers"
] | [
"Definition:Integer",
"Definition:Integer/Formal Definition"
] | [
"Extension Theorem for Total Orderings"
] |
proofwiki-17713 | Characterization of Stationary Gaussian Process | Let $S$ be a Gaussian stochastic process giving rise to a time series $T$.
Let the the mean of $S$ be fixed.
Let the autocovariance matrix of $S$ also be fixed.
Then $S$ is stationary. | From Characterization of Multivariate Normal Distribution, the normal distribution is completely characterized by its expectation and its variance.
The result follows.
{{qed}} | Let $S$ be a [[Definition:Gaussian Process|Gaussian]] [[Definition:Stochastic Process|stochastic process]] giving rise to a [[Definition:Time Series|time series]] $T$.
Let the the [[Definition:Mean of Stochastic Process|mean]] of $S$ be fixed.
Let the [[Definition:Autocovariance Matrix|autocovariance matrix]] of $S$ ... | From [[Characterization of Multivariate Normal Distribution]], the [[Definition:Normal Distribution|normal distribution]] is completely characterized by its [[Definition:Expectation of Continuous Random Variable|expectation]] and its [[Definition:Variance of Continuous Random Variable|variance]].
The result follows.
{... | Characterization of Stationary Gaussian Process | https://proofwiki.org/wiki/Characterization_of_Stationary_Gaussian_Process | https://proofwiki.org/wiki/Characterization_of_Stationary_Gaussian_Process | [
"Stochastic Processes"
] | [
"Definition:Gaussian Process",
"Definition:Stochastic Process",
"Definition:Time Series",
"Definition:Mean of Stochastic Process",
"Definition:Autocovariance Matrix",
"Definition:Stationary Stochastic Process"
] | [
"Characterization of Multivariate Normal Distribution",
"Definition:Normal Distribution",
"Definition:Expectation/Continuous",
"Definition:Variance/Continuous"
] |
proofwiki-17714 | Sufficient Conditions for Weak Stationarity of Order 2 | Let $S$ be a stochastic process giving rise to a time series $T$.
Let the mean of $S$ be fixed.
Let the autocovariance matrix of $S$ be of the form:
:$\boldsymbol \Gamma_n = \begin {pmatrix}
\gamma_0 & \gamma_1 & \gamma_2 & \cdots & \gamma_{n - 1} \\
\gamma_1 & \gamma_0 & \gamma_1 & \cdots & \gamma_{n - 2} \\
\gamma_2 ... | Follows from the definition of weak stationarity. | Let $S$ be a [[Definition:Stochastic Process|stochastic process]] giving rise to a [[Definition:Time Series|time series]] $T$.
Let the [[Definition:Mean of Stochastic Process|mean]] of $S$ be fixed.
Let the [[Definition:Autocovariance Matrix|autocovariance matrix]] of $S$ be of the form:
:$\boldsymbol \Gamma_n = \beg... | Follows from the definition of [[Definition:Weakly Stationary Stochastic Process|weak stationarity]]. | Sufficient Conditions for Weak Stationarity of Order 2 | https://proofwiki.org/wiki/Sufficient_Conditions_for_Weak_Stationarity_of_Order_2 | https://proofwiki.org/wiki/Sufficient_Conditions_for_Weak_Stationarity_of_Order_2 | [
"Stationary Stochastic Processes"
] | [
"Definition:Stochastic Process",
"Definition:Time Series",
"Definition:Mean of Stochastic Process",
"Definition:Autocovariance Matrix",
"Definition:Weakly Stationary Stochastic Process"
] | [
"Definition:Weakly Stationary Stochastic Process"
] |
proofwiki-17715 | Second Order Weakly Stationary Gaussian Stochastic Process is Strictly Stationary | Let $S$ be a Gaussian stochastic process giving rise to a time series $T$.
Let $S$ be weakly stationary of order $2$.
Then $S$ is strictly stationary. | By definition of a Gaussian process, the probability distribution of $T$ be a multivariate normal distribution.
By definition, a normal distribution is characterized completely by its expectation and its variance.
That is, its $1$st and $2$nd moments.
The result follows.
{{qed}} | Let $S$ be a [[Definition:Gaussian Process|Gaussian]] [[Definition:Stochastic Process|stochastic process]] giving rise to a [[Definition:Time Series|time series]] $T$.
Let $S$ be [[Definition:Weakly Stationary Stochastic Process|weakly stationary of order $2$]].
Then $S$ is [[Definition:Strictly Stationary Stochasti... | By definition of a [[Definition:Gaussian Process|Gaussian process]], the [[Definition:Probability Distribution|probability distribution]] of $T$ be a [[Definition:Multivariate Distribution|multivariate]] [[Definition:Normal Distribution|normal distribution]].
By definition, a [[Definition:Normal Distribution|normal di... | Second Order Weakly Stationary Gaussian Stochastic Process is Strictly Stationary | https://proofwiki.org/wiki/Second_Order_Weakly_Stationary_Gaussian_Stochastic_Process_is_Strictly_Stationary | https://proofwiki.org/wiki/Second_Order_Weakly_Stationary_Gaussian_Stochastic_Process_is_Strictly_Stationary | [
"Stationary Stochastic Processes"
] | [
"Definition:Gaussian Process",
"Definition:Stochastic Process",
"Definition:Time Series",
"Definition:Weakly Stationary Stochastic Process",
"Definition:Strictly Stationary Stochastic Process"
] | [
"Definition:Gaussian Process",
"Definition:Probability Distribution",
"Definition:Joint Distribution",
"Definition:Normal Distribution",
"Definition:Normal Distribution",
"Definition:Expectation/Discrete",
"Definition:Variance/Discrete",
"Definition:Moment (Probability Theory)/Discrete"
] |
proofwiki-17716 | Ordering on Integers is Transitive | Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers.
Then:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = \le
| r = \eqclass {c, d} {}
| c =
}}
{{eqn | lo= \land
| l = \eqclass {c, d} {}
| o = \le
| r = \eqclass {e, f} {}
| c =
}}
{... | By the formal definition of integers, we have that $a, b, c, d, e, f$ are all natural numbers.
To eliminate confusion between integer ordering and the ordering on the natural numbers, let $a \preccurlyeq b$ denote that the natural number $a$ is less than or equal to the natural number $b$.
We have:
{{begin-eqn}}
{{eqn ... | Let $\eqclass {a, b} {}$ denote an [[Definition:Integer|integer]], as defined by the [[Definition:Integer/Formal Definition|formal definition of integers]].
Then:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = \le
| r = \eqclass {c, d} {}
| c =
}}
{{eqn | lo= \land
| l = \eqclass {c, d} {}... | By the [[Definition:Integer/Formal Definition|formal definition of integers]], we have that $a, b, c, d, e, f$ are all [[Definition:Natural Number|natural numbers]].
To eliminate confusion between [[Definition:Ordering on Integers|integer ordering]] and the [[Definition:Ordering on Natural Numbers|ordering on the natu... | Ordering on Integers is Transitive | https://proofwiki.org/wiki/Ordering_on_Integers_is_Transitive | https://proofwiki.org/wiki/Ordering_on_Integers_is_Transitive | [
"Orderings on Integers"
] | [
"Definition:Integer",
"Definition:Integer/Formal Definition",
"Definition:Ordering on Integers",
"Definition:Integer",
"Definition:Transitive Relation"
] | [
"Definition:Integer/Formal Definition",
"Definition:Natural Numbers",
"Definition:Ordering on Integers",
"Definition:Ordering on Natural Numbers",
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Category:Orderings on Integers"
] |
proofwiki-17717 | Strict Ordering on Integers is Asymmetric | Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers.
Then:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = <
| r = \eqclass {c, d} {}
| c =
}}
{{eqn | lo= \implies
| l = \eqclass {c, d} {}
| o = \nless
| r = \eqclass {a, b} {}
| c =
... | By the formal definition of integers, we have that $a, b, c, d, e, f$ are all natural numbers.
To eliminate confusion between integer ordering and the ordering on the natural numbers, let $a \prec b$ denote that the natural number $a$ is less than the natural number $b$.
We have:
{{begin-eqn}}
{{eqn | l = \eqclass {a, ... | Let $\eqclass {a, b} {}$ denote an [[Definition:Integer|integer]], as defined by the [[Definition:Integer/Formal Definition|formal definition of integers]].
Then:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = <
| r = \eqclass {c, d} {}
| c =
}}
{{eqn | lo= \implies
| l = \eqclass {c, d} {... | By the [[Definition:Integer/Formal Definition|formal definition of integers]], we have that $a, b, c, d, e, f$ are all [[Definition:Natural Number|natural numbers]].
To eliminate confusion between [[Definition:Strict Ordering on Integers|integer ordering]] and the [[Definition:Ordering on Natural Numbers|ordering on t... | Strict Ordering on Integers is Asymmetric | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Asymmetric | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Asymmetric | [
"Orderings on Integers"
] | [
"Definition:Integer",
"Definition:Integer/Formal Definition",
"Definition:Strict Ordering on Integers",
"Definition:Integer",
"Definition:Asymmetric Relation"
] | [
"Definition:Integer/Formal Definition",
"Definition:Natural Numbers",
"Definition:Strict Ordering on Integers",
"Definition:Ordering on Natural Numbers",
"Definition:Natural Numbers",
"Definition:Natural Numbers"
] |
proofwiki-17718 | Strict Ordering on Integers is Transitive | Let $\eqclass {a, b} {}$ denote an integer, as defined by the formal definition of integers.
Then:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = <
| r = \eqclass {c, d} {}
| c =
}}
{{eqn | lo= \land
| l = \eqclass {c, d} {}
| o = <
| r = \eqclass {e, f} {}
| c =
}}
{{eqn... | By the formal definition of integers, we have that $a, b, c, d, e, f$ are all natural numbers.
To eliminate confusion between integer ordering and the ordering on the natural numbers, let $a \prec b$ denote that the natural number $a$ is less than the natural number $b$.
We have:
{{begin-eqn}}
{{eqn | l = \eqclass {a, ... | Let $\eqclass {a, b} {}$ denote an [[Definition:Integer|integer]], as defined by the [[Definition:Integer/Formal Definition|formal definition of integers]].
Then:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = <
| r = \eqclass {c, d} {}
| c =
}}
{{eqn | lo= \land
| l = \eqclass {c, d} {}
... | By the [[Definition:Integer/Formal Definition|formal definition of integers]], we have that $a, b, c, d, e, f$ are all [[Definition:Natural Number|natural numbers]].
To eliminate confusion between [[Definition:Strict Ordering on Integers|integer ordering]] and the [[Definition:Ordering on Natural Numbers|ordering on t... | Strict Ordering on Integers is Transitive | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Transitive | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Transitive | [
"Orderings on Integers"
] | [
"Definition:Integer",
"Definition:Integer/Formal Definition",
"Definition:Strict Ordering on Integers",
"Definition:Integer",
"Definition:Transitive Relation"
] | [
"Definition:Integer/Formal Definition",
"Definition:Natural Numbers",
"Definition:Strict Ordering on Integers",
"Definition:Ordering on Natural Numbers",
"Definition:Natural Numbers",
"Definition:Natural Numbers"
] |
proofwiki-17719 | Strict Ordering on Integers is Trichotomy | Let $\eqclass {a, b} {}$ and $\eqclass {c, d} {}$ be integers, as defined by the formal definition of integers.
Then exactly one of the following holds:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = <
| r = \eqclass {c, d} {}
| c =
}}
{{eqn | l = \eqclass {a, b} {}
| o = =
| r = \eqc... | By the formal definition of integers, we have that $a, b, c, d, e, f$ are all natural numbers.
To eliminate confusion between integer ordering and the ordering on the natural numbers, let $a \preccurlyeq b$ denote that the natural number $a$ is less than or equal to the natural number $b$.
We have:
{{begin-eqn}}
{{eqn ... | Let $\eqclass {a, b} {}$ and $\eqclass {c, d} {}$ be [[Definition:Integer|integers]], as defined by the [[Definition:Integer/Formal Definition|formal definition of integers]].
Then exactly one of the following holds:
{{begin-eqn}}
{{eqn | l = \eqclass {a, b} {}
| o = <
| r = \eqclass {c, d} {}
| c = ... | By the [[Definition:Integer/Formal Definition|formal definition of integers]], we have that $a, b, c, d, e, f$ are all [[Definition:Natural Number|natural numbers]].
To eliminate confusion between [[Definition:Strict Ordering on Integers|integer ordering]] and the [[Definition:Ordering on Natural Numbers|ordering on t... | Strict Ordering on Integers is Trichotomy | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Trichotomy | https://proofwiki.org/wiki/Strict_Ordering_on_Integers_is_Trichotomy | [
"Orderings on Integers"
] | [
"Definition:Integer",
"Definition:Integer/Formal Definition",
"Definition:Strict Ordering on Integers",
"Definition:Trichotomy"
] | [
"Definition:Integer/Formal Definition",
"Definition:Natural Numbers",
"Definition:Strict Ordering on Integers",
"Definition:Ordering on Natural Numbers",
"Definition:Natural Numbers",
"Definition:Natural Numbers",
"Strict Ordering on Integers is Asymmetric",
"Strict Ordering on Integers is Asymmetric"... |
proofwiki-17720 | Negative of Integer | Let $x \in \Z$ be an integer.
Let $x = \eqclass {a, b} {}$ be defined from the formal definition of integers, where $\eqclass {a, b} {}$ is an equivalence class of ordered pairs of natural numbers.
Then:
:$-x = \eqclass {b, a} {}$ | Follows from Inverse for Integer Addition.
{{finish|The whole area of construction of integers from the inverse completion needs to be reviewed}}
Category:Integers
Category:Examples of Inverse Elements
9uetksvhenii1xxcqn40gg4t1y8huzv | Let $x \in \Z$ be an [[Definition:Integer|integer]].
Let $x = \eqclass {a, b} {}$ be defined from the [[Definition:Integer/Formal Definition|formal definition of integers]], where $\eqclass {a, b} {}$ is an [[Definition:Equivalence Class|equivalence class]] of [[Definition:Ordered Pair|ordered pairs]] of [[Definition:... | Follows from [[Inverse for Integer Addition]].
{{finish|The whole area of construction of integers from the inverse completion needs to be reviewed}}
[[Category:Integers]]
[[Category:Examples of Inverse Elements]]
9uetksvhenii1xxcqn40gg4t1y8huzv | Negative of Integer | https://proofwiki.org/wiki/Negative_of_Integer | https://proofwiki.org/wiki/Negative_of_Integer | [
"Integers",
"Examples of Inverse Elements"
] | [
"Definition:Integer",
"Definition:Integer/Formal Definition",
"Definition:Equivalence Class",
"Definition:Ordered Pair",
"Definition:Natural Numbers"
] | [
"Inverse for Integer Addition",
"Category:Integers",
"Category:Examples of Inverse Elements"
] |
proofwiki-17721 | Euler's Integral Theorem | :$H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$
where:
:$H_n$ denotes the $n$th harmonic number
:$\gamma$ denotes the Euler-Mascheroni constant
:$\map \OO {\dfrac 1 n}$ denotes big-$\OO$ of $\dfrac 1 n$. | Recall the definition of the floor function:
{{Definition:Floor Function/Definition 3}}
For all $n \in \N_{>0}$:
{{begin-eqn}}
{{eqn | l = H_n - \ln n
| r = 1 + \int_1 ^n \dfrac {\floor u} {u^2} \rd u - \ln n
| c = Integral Expression of Harmonic Number
}}
{{eqn | r = 1 + \int_1 ^n \dfrac {\floor u} {u^2} \... | :$H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$
where:
:$H_n$ denotes the $n$th [[Definition:Harmonic Number|harmonic number]]
:$\gamma$ denotes the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni constant]]
:$\map \OO {\dfrac 1 n}$ denotes [[Definition:Big-O Notation/Sequence|big-$\OO$]] of $\dfrac 1 n$. | Recall the definition of the [[Definition:Floor Function|floor function]]:
{{Definition:Floor Function/Definition 3}}
For all $n \in \N_{>0}$:
{{begin-eqn}}
{{eqn | l = H_n - \ln n
| r = 1 + \int_1 ^n \dfrac {\floor u} {u^2} \rd u - \ln n
| c = [[Integral Expression of Harmonic Number]]
}}
{{eqn | r = 1 +... | Euler's Integral Theorem/Proof 1 | https://proofwiki.org/wiki/Euler's_Integral_Theorem | https://proofwiki.org/wiki/Euler's_Integral_Theorem/Proof_1 | [
"Euler's Integral Theorem",
"Integral Calculus",
"Analytic Number Theory"
] | [
"Definition:Harmonic Numbers",
"Definition:Euler-Mascheroni Constant",
"Definition:Big-O Notation/Sequence"
] | [
"Definition:Floor Function",
"Integral Expression of Harmonic Number",
"Integral Operator is Linear",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Integral Operator is Positive",
"Definition:Cauchy Sequence",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Euler-Mascher... |
proofwiki-17722 | Euler's Integral Theorem | :$H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$
where:
:$H_n$ denotes the $n$th harmonic number
:$\gamma$ denotes the Euler-Mascheroni constant
:$\map \OO {\dfrac 1 n}$ denotes big-$\OO$ of $\dfrac 1 n$. | Recall the definition of the floor function:
{{Definition:Floor Function/Definition 3}}
Hence:
:$0 \le x - \floor x < 1$
For all $n \in \N_{>0}$:
{{begin-eqn}}
{{eqn | l = H_n - \ln n - \gamma
| r = H_n - \ln n - \lim_{k \mathop \to +\infty} \paren {H_k - \ln k}
| c = {{Defof|Euler-Mascheroni Constant}} and... | :$H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$
where:
:$H_n$ denotes the $n$th [[Definition:Harmonic Number|harmonic number]]
:$\gamma$ denotes the [[Definition:Euler-Mascheroni Constant|Euler-Mascheroni constant]]
:$\map \OO {\dfrac 1 n}$ denotes [[Definition:Big-O Notation/Sequence|big-$\OO$]] of $\dfrac 1 n$. | Recall the definition of the [[Definition:Floor Function|floor function]]:
{{Definition:Floor Function/Definition 3}}
Hence:
:$0 \le x - \floor x < 1$
For all $n \in \N_{>0}$:
{{begin-eqn}}
{{eqn | l = H_n - \ln n - \gamma
| r = H_n - \ln n - \lim_{k \mathop \to +\infty} \paren {H_k - \ln k}
| c = {{Defof... | Euler's Integral Theorem/Proof 2 | https://proofwiki.org/wiki/Euler's_Integral_Theorem | https://proofwiki.org/wiki/Euler's_Integral_Theorem/Proof_2 | [
"Euler's Integral Theorem",
"Integral Calculus",
"Analytic Number Theory"
] | [
"Definition:Harmonic Numbers",
"Definition:Euler-Mascheroni Constant",
"Definition:Big-O Notation/Sequence"
] | [
"Definition:Floor Function",
"Existence of Euler-Mascheroni Constant",
"Integral Expression of Harmonic Number",
"Sum of Integrals on Adjacent Intervals for Continuous Functions",
"Integral Operator is Linear",
"Primitive of Power",
"Existence of Euler-Mascheroni Constant/Proof 1",
"Definition:Decreas... |
proofwiki-17723 | Ordering on Positive Integers is Equivalent to Ordering on Natural Numbers | Let $u, v \in \Z_{>0}$ be natural numbers.
Consider the mapping $\phi: \N_{>0} \to \Z_{>0}$ defined as:
:$\forall u \in \N_{>0}: \map \phi u = u'$
where $u' \in \Z$ denotes the (strictly) positive integer $\eqclass {b + u, b} {}$.
Let $u', v' \in \Z_{>0}$ be strictly positive integers.
Then:
:$u > v \iff u' > v'$ | Let $u' = \eqclass {b + u, b} {}$.
Let $v' = \eqclass {c + v, c} {}$.
Then:
{{begin-eqn}}
{{eqn | l = u'
| o = >
| r = v'
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \eqclass {b + u, b} {}
| o = >
| r = \eqclass {c + v, c} {}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = b ... | Let $u, v \in \Z_{>0}$ be [[Definition:Natural Number|natural numbers]].
Consider the [[Definition:Mapping|mapping]] $\phi: \N_{>0} \to \Z_{>0}$ defined as:
:$\forall u \in \N_{>0}: \map \phi u = u'$
where $u' \in \Z$ denotes the [[Definition:Strictly Positive Integer|(strictly) positive integer]] $\eqclass {b + u, b... | Let $u' = \eqclass {b + u, b} {}$.
Let $v' = \eqclass {c + v, c} {}$.
Then:
{{begin-eqn}}
{{eqn | l = u'
| o = >
| r = v'
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \eqclass {b + u, b} {}
| o = >
| r = \eqclass {c + v, c} {}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = ... | Ordering on Positive Integers is Equivalent to Ordering on Natural Numbers | https://proofwiki.org/wiki/Ordering_on_Positive_Integers_is_Equivalent_to_Ordering_on_Natural_Numbers | https://proofwiki.org/wiki/Ordering_on_Positive_Integers_is_Equivalent_to_Ordering_on_Natural_Numbers | [
"Orderings on Integers"
] | [
"Definition:Natural Numbers",
"Definition:Mapping",
"Definition:Strictly Positive/Integer",
"Definition:Strictly Positive/Integer"
] | [] |
proofwiki-17724 | Product of Absolute Values of Integers | Let $a, b \in \Z$ be integers.
Let $\size a$ denote the absolute value of $a$:
:$\size a = \begin {cases} a & : a \ge 0 \\ -a & : a < 0 \end {cases}$
Then:
:$\size a \times \size b = \size {a \times b}$ | From Integers form Ordered Integral Domain, $\Z$ is an ordered integral domain.
The result follows from Product of Absolute Values on Ordered Integral Domain.
{{qed}} | Let $a, b \in \Z$ be [[Definition:Integer|integers]].
Let $\size a$ denote the [[Definition:Absolute Value|absolute value]] of $a$:
:$\size a = \begin {cases} a & : a \ge 0 \\ -a & : a < 0 \end {cases}$
Then:
:$\size a \times \size b = \size {a \times b}$ | From [[Integers form Ordered Integral Domain]], $\Z$ is an [[Definition:Ordered Integral Domain|ordered integral domain]].
The result follows from [[Product of Absolute Values on Ordered Integral Domain]].
{{qed}} | Product of Absolute Values of Integers | https://proofwiki.org/wiki/Product_of_Absolute_Values_of_Integers | https://proofwiki.org/wiki/Product_of_Absolute_Values_of_Integers | [
"Absolute Value Function",
"Integers"
] | [
"Definition:Integer",
"Definition:Absolute Value"
] | [
"Integers form Ordered Integral Domain",
"Definition:Ordered Integral Domain",
"Product of Absolute Values on Ordered Integral Domain"
] |
proofwiki-17725 | Cardinality of Set of Self-Mappings on Finite Set | Let $S$ be a finite set.
Let the cardinality of $S$ be $n$.
The cardinality of the set of all mappings from $S$ to itself (that is, the total number of self-maps on $S$) is:
:$\card {S^S} = n^n$ | This is a specific example of Cardinality of Set of All Mappings where $S = T$.
{{qed}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let the [[Definition:Cardinality|cardinality]] of $S$ be $n$.
The [[Definition:Cardinality|cardinality]] of the [[Definition:Set of All Mappings|set of all mappings]] from $S$ to itself (that is, the total number of [[Definition:Self-Map|self-maps]] on $S$) is:
:$\c... | This is a specific example of [[Cardinality of Set of All Mappings]] where $S = T$.
{{qed}} | Cardinality of Set of Self-Mappings on Finite Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_Self-Mappings_on_Finite_Set | https://proofwiki.org/wiki/Cardinality_of_Set_of_Self-Mappings_on_Finite_Set | [
"Cardinality of Set of All Mappings",
"Finite Sets"
] | [
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Cardinality",
"Definition:Set of All Mappings",
"Definition:Self-Map"
] | [
"Cardinality of Set of All Mappings"
] |
proofwiki-17726 | Displacement of Particle under Force | Let $P$ be a particle of constant mass $m$.
Let the position of $P$ at time $t$ be specified by the position vector $\mathbf r$.
Let a force applied to $P$ be represented by the vector $\mathbf F$.
Then the motion of $P$ can be given by the differential equation:
:$\mathbf F = m \dfrac {\d^2 \mathbf r} {\d t^2}$
or usi... | {{begin-eqn}}
{{eqn | l = \mathbf F
| r = \map {\dfrac \d {\d t} } {m \mathbf v}
| c = Newton's Second Law of Motion
}}
{{eqn | r = \map {\dfrac \d {\d t} } {m \dfrac {\d \mathbf r} {\d t} }
| c = {{Defof|Velocity}}
}}
{{eqn | r = m \map {\dfrac \d {\d t} } {\dfrac {\d \mathbf r} {\d t} }
| c = ... | Let $P$ be a [[Definition:Particle|particle]] of constant [[Definition:Mass|mass]] $m$.
Let the [[Definition:Position|position]] of $P$ at [[Definition:Time|time]] $t$ be specified by the [[Definition:Position Vector|position vector]] $\mathbf r$.
Let a [[Definition:Force|force]] applied to $P$ be represented by the ... | {{begin-eqn}}
{{eqn | l = \mathbf F
| r = \map {\dfrac \d {\d t} } {m \mathbf v}
| c = [[Newton's Second Law of Motion]]
}}
{{eqn | r = \map {\dfrac \d {\d t} } {m \dfrac {\d \mathbf r} {\d t} }
| c = {{Defof|Velocity}}
}}
{{eqn | r = m \map {\dfrac \d {\d t} } {\dfrac {\d \mathbf r} {\d t} }
| ... | Displacement of Particle under Force | https://proofwiki.org/wiki/Displacement_of_Particle_under_Force | https://proofwiki.org/wiki/Displacement_of_Particle_under_Force | [
"Classical Mechanics"
] | [
"Definition:Particle",
"Definition:Mass",
"Definition:Position",
"Definition:Time",
"Definition:Position Vector",
"Definition:Force",
"Definition:Vector",
"Definition:Motion",
"Definition:Differential Equation",
"Definition:Derivative/Notation/Newton Notation"
] | [
"Newton's Laws of Motion/Second Law",
"Derivative of Constant Multiple"
] |
proofwiki-17727 | 1-Sequence Space is Separable | $\ell^1$ space is a separable space. | Let $D$ be the set of all finitely supported sequences with rational terms:
:$D = \set {\sequence {q_i}_{i \mathop \in \N} : n \in \N : i \le n : q_i \in \Q}$
We have that:
:Rational Numbers are Countably Infinite
:A finite set is countable
:$D$ is a union of finite sets indexed by $n$, which is countable
By Countable ... | [[Definition:P-Sequence Space|$\ell^1$ space]] is a [[Definition:Separable Normed Vector Space|separable space]]. | Let $D$ be the [[Definition:Set|set]] of all [[Definition:Finite Set|finitely]] [[Definition:Support of Mapping to Algebraic Structure|supported]] [[Definition:Sequence|sequences]] with [[Definition:Rational Number|rational]] terms:
:$D = \set {\sequence {q_i}_{i \mathop \in \N} : n \in \N : i \le n : q_i \in \Q}$
We... | 1-Sequence Space is Separable | https://proofwiki.org/wiki/1-Sequence_Space_is_Separable | https://proofwiki.org/wiki/1-Sequence_Space_is_Separable | [
"P-Sequence Spaces",
"Examples of Separable Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:Separable Space/Normed Vector Space"
] | [
"Definition:Set",
"Definition:Finite Set",
"Definition:Support of Mapping to Algebraic Structure",
"Definition:Sequence",
"Definition:Rational Number",
"Rational Numbers are Countably Infinite",
"Definition:Finite Set",
"Definition:Countable Set/Definition 3",
"Definition:Finite Set",
"Definition:... |
proofwiki-17728 | Primitive of Root of Function under Half its Derivative | Let $f$ be a real function which is integrable.
Then:
:$\ds \int \frac {\map {f'} x} {2 \sqrt {\map f x} } \rd x = \sqrt {\map f x} + C$
where $C$ is an arbitrary constant. | By Integration by Substitution (with appropriate renaming of variables):
:$\ds \int \map g u \rd u = \int \map g {\map f x} \map {f'} x \rd x$
Let $\map u x = \sqrt {\map f x}$
{{begin-eqn}}
{{eqn | l = \map u x
| r = \sqrt {\map f x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
| r... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]].
Then:
:$\ds \int \frac {\map {f'} x} {2 \sqrt {\map f x} } \rd x = \sqrt {\map f x} + C$
where $C$ is an [[Definition:Arbitrary Constant (Calculus)|arbitrary constant]]. | By [[Integration by Substitution]] (with appropriate renaming of variables):
:$\ds \int \map g u \rd u = \int \map g {\map f x} \map {f'} x \rd x$
Let $\map u x = \sqrt {\map f x}$
{{begin-eqn}}
{{eqn | l = \map u x
| r = \sqrt {\map f x}
| c =
}}
{{eqn | ll= \leadsto
| l = \dfrac {\d u} {\d x}
... | Primitive of Root of Function under Half its Derivative | https://proofwiki.org/wiki/Primitive_of_Root_of_Function_under_Half_its_Derivative | https://proofwiki.org/wiki/Primitive_of_Root_of_Function_under_Half_its_Derivative | [
"Primitives"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Primitive (Calculus)/Constant of Integration"
] | [
"Integration by Substitution",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Integration by Substitution",
"Primitive of Constant"
] |
proofwiki-17729 | Primitive of Hyperbolic Secant Function/Arctangent of Half Hyperbolic Tangent Form | :$\ds \int \sech x \rd x = 2 \map \arctan {\tanh \dfrac x 2} + C$ | Let $u = \tanh \dfrac x 2$.
Then:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac {1 - u^2} {1 + u^2} \frac {2 \rd u} {1 - u^2}
| c = Hyperbolic Tangent Half-Angle Substitution
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u^2}
| c = simplifying
}}
{{eqn | r = 2 \arctan u + C
| c = Prim... | :$\ds \int \sech x \rd x = 2 \map \arctan {\tanh \dfrac x 2} + C$ | Let $u = \tanh \dfrac x 2$.
Then:
{{begin-eqn}}
{{eqn | l = \int \sech x \rd x
| r = \int \frac {1 - u^2} {1 + u^2} \frac {2 \rd u} {1 - u^2}
| c = [[Hyperbolic Tangent Half-Angle Substitution]]
}}
{{eqn | r = \int \frac {2 \rd u} {1 + u^2}
| c = simplifying
}}
{{eqn | r = 2 \arctan u + C
| c ... | Primitive of Hyperbolic Secant Function/Arctangent of Half Hyperbolic Tangent Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arctangent_of_Half_Hyperbolic_Tangent_Form | https://proofwiki.org/wiki/Primitive_of_Hyperbolic_Secant_Function/Arctangent_of_Half_Hyperbolic_Tangent_Form | [
"Primitive of Hyperbolic Secant Function",
"Hyperbolic Tangent Half-Angle Substitutions"
] | [] | [
"Hyperbolic Tangent Half-Angle Substitution",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form"
] |
proofwiki-17730 | First Order ODE/y' + 2 x y = 1 | The first order ODE:
:$y' + 2 x y = 1$
has the general solution:
:$y = e^{-{x^2} } \ds \int_a^x e^{t^2} \rd t$
where $a$ is an arbitrary constant. | This is a linear first order ODE in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map p x = 2 x$
:$\map Q x = 1$
From Solution to Linear First Order Ordinary Differential Equation:
:$\ds y = e^{-\int P \rd x} \paren {\int Q e^{\int P \rd x} \rd x + C}$
Thus
{{begin-eqn}}
{{eqn | l = y
| r =... | The [[Definition:First Order ODE|first order ODE]]:
:$y' + 2 x y = 1$
has the [[Definition:General Solution of Differential Equation|general solution]]:
:$y = e^{-{x^2} } \ds \int_a^x e^{t^2} \rd t$
where $a$ is an [[Definition:Arbitrary Constant|arbitrary constant]]. | This is a [[Definition:Linear First Order Ordinary Differential Equation|linear first order ODE]] in the form:
:$\dfrac {\d y} {\d x} + \map P x y = \map Q x$
where:
:$\map p x = 2 x$
:$\map Q x = 1$
From [[Solution to Linear First Order Ordinary Differential Equation]]:
:$\ds y = e^{-\int P \rd x} \paren {\int Q e^{\... | First Order ODE/y' + 2 x y = 1 | https://proofwiki.org/wiki/First_Order_ODE/y'_+_2_x_y_=_1 | https://proofwiki.org/wiki/First_Order_ODE/y'_+_2_x_y_=_1 | [
"Examples of First Order ODEs"
] | [
"Definition:First Order Ordinary Differential Equation",
"Definition:Differential Equation/Solution/General Solution",
"Definition:Arbitrary Constant"
] | [
"Definition:Linear First Order Ordinary Differential Equation",
"Solution to Linear First Order Ordinary Differential Equation",
"Primitive of Power"
] |
proofwiki-17731 | Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 4 | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Then:
:$\mathscr B$ satisfies formulation $1$ of base axiom
{{Axiom:Base Axiom (Matroid)/Formulation 1}}
{{iff}}
:$\mathscr B$ satisfies formulation $4$ of base axiom
{{Axiom:Base Axiom (Matroid)/Formulation 4}} | ==== Necessary Condition ====
Follows immediately from:
:* Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom
:* Matroid Bases Satisfy Formulation 4 Base Axiom
{{qed|lemma}}
==== Sufficient Condition ====
Follows immediately from formulation $4$ and formulation $1$. | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then:
:$\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 1|formulation $1$ of base axiom]]
{{Axiom:Base Axiom (Matroid)/Formulation... | ==== Necessary Condition ====
Follows immediately from:
:* [[Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom]]
:* [[Matroid Bases Satisfy Formulation 4 Base Axiom]]
{{qed|lemma}}
==== Sufficient Condition ====
Follows immediately from [[Axiom:Base Axiom (Matroid)/Formulation 4|formulation $4$]] and [... | Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_1_Iff_Formulation_4 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_1_Iff_Formulation_4 | [
"Equivalence of Definitions of Matroid Base Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty",
"Definition:Set",
"Definition:Subset",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulation 4"
] | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom",
"Matroid Bases Satisfy Formulation 4 Base Axiom",
"Axiom:Base Axiom (Matroid)/Formulation 4",
"Axiom:Base Axiom (Matroid)/Formulation 1"
] |
proofwiki-17732 | Equivalence of Definitions of Matroid Base Axioms/Formulation 3 Iff Formulation 7 | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Then:
:$\mathscr B$ satisfies formulation $3$ of base axiom:
{{Axiom:Base Axiom (Matroid)/Formulation 3}}
{{iff}}
:$\mathscr B$ satisfies formulation $7$ of base axiom:
{{Axiom:Base Axiom (Matroid)/Formulation 7}} | ==== Necessary Condition ====
Let $\mathscr B$ satisfy formulation $3$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 3}}
Let $B_1, B_2 \in \mathscr B$.
From $(\text B 3)$:
:$\exists \text{ a bijection } \pi : B_2 \to B_1 : \forall y \in B_2: \paren {B_2 \setminus \set y } \cup \set {\map \pi y} \in \mathscr ... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then:
:$\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 3|formulation $3$ of base axiom]]:
{{Axiom:Base Axiom (Matroid)/Formulatio... | ==== Necessary Condition ====
Let $\mathscr B$ satisfy [[Axiom:Base Axiom (Matroid)/Formulation 3|formulation $3$ of base axiom]]:
{{:Axiom:Base Axiom (Matroid)/Formulation 3}}
Let $B_1, B_2 \in \mathscr B$.
From $(\text B 3)$:
:$\exists \text{ a bijection } \pi : B_2 \to B_1 : \forall y \in B_2: \paren {B_2 \setmin... | Equivalence of Definitions of Matroid Base Axioms/Formulation 3 Iff Formulation 7 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_3_Iff_Formulation_7 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_3_Iff_Formulation_7 | [
"Equivalence of Definitions of Matroid Base Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty",
"Definition:Set",
"Definition:Subset",
"Axiom:Base Axiom (Matroid)/Formulation 3",
"Axiom:Base Axiom (Matroid)/Formulation 7"
] | [
"Axiom:Base Axiom (Matroid)/Formulation 3",
"Definition:Inverse Mapping",
"Inverse of Bijection is Bijection",
"Definition:Bijection",
"Inverse Element of Bijection",
"Axiom:Base Axiom (Matroid)/Formulation 7",
"Axiom:Base Axiom (Matroid)/Formulation 7",
"Definition:Inverse Mapping",
"Inverse of Bij... |
proofwiki-17733 | Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 7 | Let $S$ be a finite set.
Let $\mathscr B$ be a non-empty set of subsets of $S$.
Then:
:$\mathscr B$ satisfies formulation $1$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 1}}
{{iff}}
:$\mathscr B$ satisfies formulation $7$ of base axiom:
{{:Axiom:Base Axiom (Matroid)/Formulation 7}} | ==== Necessary Condition ====
Follows immediately from:
:* Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom
:* Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom
{{qed|lemma}}
==== Sufficient Condition ====
From Formulation 3 Iff Formulation 7:
:$\mathscr B$ satisfies formulation $7$ of base axiom... | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr B$ be a [[Definition:Non-Empty|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then:
:$\mathscr B$ satisfies [[Axiom:Base Axiom (Matroid)/Formulation 1|formulation $1$ of base axiom]]:
{{:Axiom:Base Axiom (Matroid)/Formulati... | ==== Necessary Condition ====
Follows immediately from:
:* [[Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom]]
:* [[Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom]]
{{qed|lemma}}
==== Sufficient Condition ====
From [[Equivalence of Definitions of Matroid Base Axioms/Formulation 3 Iff For... | Equivalence of Definitions of Matroid Base Axioms/Formulation 1 Iff Formulation 7 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_1_Iff_Formulation_7 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Base_Axioms/Formulation_1_Iff_Formulation_7 | [
"Equivalence of Definitions of Matroid Base Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty",
"Definition:Set",
"Definition:Subset",
"Axiom:Base Axiom (Matroid)/Formulation 1",
"Axiom:Base Axiom (Matroid)/Formulation 7"
] | [
"Matroid Bases Iff Satisfies Formulation 1 of Matroid Base Axiom",
"Matroid Bases Satisfy Formulation 7 of Matroid Base Axiom",
"Equivalence of Definitions of Matroid Base Axioms/Formulation 3 Iff Formulation 7",
"Axiom:Base Axiom (Matroid)/Formulation 7",
"Axiom:Base Axiom (Matroid)/Formulation 3",
"Axio... |
proofwiki-17734 | Number of Permutations with Repetition | Set $S$ be a set of $n$ elements.
Let $\sequence T_m$ be a sequence of $m$ terms of $S$.
Then there are $n^m$ different instances of $\sequence T_m$. | Let $N_m$ denote the set $\set {1, 2, \ldots, m}$.
Let $f: N_m \to S$ be the mapping defined as:
:$\forall k \in N_m: \map f t = t_m$
By definition, $f$ corresponds to one of the specific instances of $\sequence T_m$.
Hence the number of different instances of $\sequence T_m$ is found from Cardinality of Set of All Map... | Set $S$ be a [[Definition:Set|set]] of $n$ [[Definition:Element|elements]].
Let $\sequence T_m$ be a [[Definition:Sequence|sequence]] of $m$ [[Definition:Term|terms]] of $S$.
Then there are $n^m$ different instances of $\sequence T_m$. | Let $N_m$ denote the [[Definition:Set|set]] $\set {1, 2, \ldots, m}$.
Let $f: N_m \to S$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall k \in N_m: \map f t = t_m$
By definition, $f$ corresponds to one of the specific instances of $\sequence T_m$.
Hence the number of different instances of $\sequence T_m... | Number of Permutations with Repetition | https://proofwiki.org/wiki/Number_of_Permutations_with_Repetition | https://proofwiki.org/wiki/Number_of_Permutations_with_Repetition | [
"Combinatorics",
"Number of Permutations with Repetition"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Sequence",
"Definition:Term"
] | [
"Definition:Set",
"Definition:Mapping",
"Cardinality of Set of All Mappings"
] |
proofwiki-17735 | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Corollary | :$\ds \int_0^\infty \dfrac {\d x} {1 + x^2} = \frac \pi 2$ | From Definite Integral to Infinity of Reciprocal of x Squared plus a Squared:
:$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
which holds for for $a \ne 0$.
The result follows by setting $a = 1$.
{{qed}} | :$\ds \int_0^\infty \dfrac {\d x} {1 + x^2} = \frac \pi 2$ | From [[Definite Integral to Infinity of Reciprocal of x Squared plus a Squared]]:
:$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
which holds for for $a \ne 0$.
The result follows by setting $a = 1$.
{{qed}} | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Corollary | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared/Corollary | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared/Corollary | [
"Definite Integral to Infinity of Reciprocal of x Squared plus a Squared",
"Definite Integrals involving x squared plus a squared"
] | [] | [
"Definite Integral to Infinity of Reciprocal of x Squared plus a Squared"
] |
proofwiki-17736 | P-Norm is Norm/P-Sequence Space | The $p$-norm on the $p$-sequence space is a vector space norm. | {{refactor|This should be presented in the context of a general Banach space.|level = advanced}} | The [[Definition:P-Norm|$p$-norm]] on the [[Definition:P-Sequence Space|$p$-sequence space]] is a [[Definition:Norm on Vector Space|vector space norm]]. | {{refactor|This should be presented in the context of a general Banach space.|level = advanced}} | P-Norm is Norm/P-Sequence Space | https://proofwiki.org/wiki/P-Norm_is_Norm/P-Sequence_Space | https://proofwiki.org/wiki/P-Norm_is_Norm/P-Sequence_Space | [
"P-Norms"
] | [
"Definition:P-Norm",
"Definition:P-Sequence Space",
"Definition:Norm/Vector Space"
] | [] |
proofwiki-17737 | P-Norm is Norm/Complex Numbers | The $p$-norm on the complex numbers is a norm. | Let $K \in \C^d$, where $d \in \N_{>0}$. | The [[Definition:Complex P-Norm|$p$-norm]] on the [[Definition:Complex Number|complex numbers]] is a [[Definition:Norm on Vector Space|norm]]. | Let $K \in \C^d$, where $d \in \N_{>0}$. | P-Norm is Norm/Complex Numbers | https://proofwiki.org/wiki/P-Norm_is_Norm/Complex_Numbers | https://proofwiki.org/wiki/P-Norm_is_Norm/Complex_Numbers | [
"P-Norms",
"Examples of Norms"
] | [
"Definition:P-Norm/Complex",
"Definition:Complex Number",
"Definition:Norm/Vector Space"
] | [] |
proofwiki-17738 | P-Norm is Norm/Real Numbers | The $p$-norm on the real numbers is a norm. | We have that $p$-norm is a norm on complex numbers.
Since real numbers are wholly real complex numbers, the same result holds.
{{qed}}
Category:P-Norms
49fea5yjsxkrx7plgbhhne7b5myaxez | The [[Definition:Real P-Norm|$p$-norm]] on the [[Definition:Real Number|real numbers]] is a [[Definition:Norm on Vector Space|norm]]. | We have that [[P-Norm is Norm/Complex Numbers|$p$-norm is a norm on complex numbers]].
Since [[Definition:Real Numbers|real numbers]] are [[Definition:Wholly Real|wholly real]] [[Definition:Complex Number|complex numbers]], the same result holds.
{{qed}}
[[Category:P-Norms]]
49fea5yjsxkrx7plgbhhne7b5myaxez | P-Norm is Norm/Real Numbers | https://proofwiki.org/wiki/P-Norm_is_Norm/Real_Numbers | https://proofwiki.org/wiki/P-Norm_is_Norm/Real_Numbers | [
"P-Norms"
] | [
"Definition:P-Norm/Real",
"Definition:Real Number",
"Definition:Norm/Vector Space"
] | [
"P-Norm is Norm/Complex Numbers",
"Definition:Real Number",
"Definition:Complex Number/Wholly Real",
"Definition:Complex Number",
"Category:P-Norms"
] |
proofwiki-17739 | P-Sequence Space with P-Norm forms Normed Vector Space | A $p$-Sequence Space with a $p$-norm forms a normed vector space. | We have that:
:a $p$-sequence space is a vector space
:the $p$-norm on the $p$-sequence space is a norm
By definition, $\struct {\ell^p, \norm {\, \cdot \,}_p}$ is a normed vector space.
{{qed}} | A [[Definition:P-Sequence Space|$p$-Sequence Space]] with a [[Definition:P-Norm|$p$-norm]] forms a [[Definition:Normed Vector Space|normed vector space]]. | We have that:
:a [[P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space|$p$-sequence space is a vector space]]
:the [[P-Norm is Norm/P-Sequence Space|$p$-norm on the $p$-sequence space is a norm]]
By definition, $\struct {\ell^p, \norm {\, \cdot \,}_p}$ ... | P-Sequence Space with P-Norm forms Normed Vector Space | https://proofwiki.org/wiki/P-Sequence_Space_with_P-Norm_forms_Normed_Vector_Space | https://proofwiki.org/wiki/P-Sequence_Space_with_P-Norm_forms_Normed_Vector_Space | [
"P-Norms",
"Examples of Normed Vector Spaces"
] | [
"Definition:P-Sequence Space",
"Definition:P-Norm",
"Definition:Normed Vector Space"
] | [
"P-Sequence Space with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space",
"P-Norm is Norm/P-Sequence Space",
"Definition:Normed Vector Space"
] |
proofwiki-17740 | Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space | Let $\map {\ell^\infty} \C$ be the space of bounded sequences on $\C$.
Let $\struct {\C, +_\C, \times_\C}$ be the field of complex numbers.
Let $\paren +$ be the pointwise addition on the ring of sequences.
Let $\paren {\, \cdot \,}$ be the pointwise multiplication on the ring of sequences.
Then $\struct {\map {\ell^\i... | Let $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N}, \sequence {c_n}_{n \mathop \in \N} \in \map {\ell^\infty} \C$.
Let $\lambda, \mu \in \C$.
Let $\sequence 0 := \tuple {0, 0, 0, \dots}$ be a complex-valued function.
Let us use real number addition and multiplication.
Define pointwise addition ... | Let $\map {\ell^\infty} \C$ be the [[Definition:Space of Bounded Sequences|space of bounded sequences on $\C$]].
Let $\struct {\C, +_\C, \times_\C}$ be the [[Definition:Field of Complex Numbers|field of complex numbers]].
Let $\paren +$ be the [[Definition:Pointwise Addition on Ring of Sequences|pointwise addition on... | Let $\sequence {a_n}_{n \mathop \in \N}, \sequence {b_n}_{n \mathop \in \N}, \sequence {c_n}_{n \mathop \in \N} \in \map {\ell^\infty} \C$.
Let $\lambda, \mu \in \C$.
Let $\sequence 0 := \tuple {0, 0, 0, \dots}$ be a [[Definition:Complex-Valued Function|complex-valued function]].
Let us use [[Definition:Complex Numb... | Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_on_Ring_of_Sequences_forms_Vector_Space | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_with_Pointwise_Addition_and_Pointwise_Scalar_Multiplication_on_Ring_of_Sequences_forms_Vector_Space | [
"Examples of Vector Spaces",
"Functional Analysis",
"Space of Bounded Sequences"
] | [
"Definition:Space of Bounded Sequences",
"Definition:Field of Complex Numbers",
"Definition:Ring of Sequences/Pointwise Addition",
"Definition:Pointwise Scalar Multiplication on Ring of Sequences",
"Definition:Vector Space"
] | [
"Definition:Complex-Valued Function",
"Definition:Complex Number",
"Definition:Addition/Complex Numbers",
"Definition:Multiplication/Complex Numbers",
"Definition:Ring of Sequences/Pointwise Addition",
"Definition:Pointwise Scalar Multiplication on Ring of Sequences",
"Definition:Ring of Sequences/Addit... |
proofwiki-17741 | Zero Vector has no Direction | A zero vector has no direction. | Let $\mathbf 0$ denote a zero vector.
{{AimForCont}} $\mathbf 0$ has a direction.
Then $\mathbf 0$ can be represented as an arrow in a real vector space $\R^n$ with a Cartesian frame.
Let $\mathbf 0$ be so embedded.
Thus it consists of a line segment between two points with an initial point $A$ and a terminal point $B$... | A [[Definition:Zero Vector Quantity|zero vector]] has no [[Definition:Direction|direction]]. | Let $\mathbf 0$ denote a [[Definition:Zero Vector Quantity|zero vector]].
{{AimForCont}} $\mathbf 0$ has a [[Definition:Direction|direction]].
Then $\mathbf 0$ can be [[Definition:Arrow Representation of Vector Quantity|represented as an arrow]] in a [[Definition:Real Vector Space|real vector space]] $\R^n$ with a [[... | Zero Vector has no Direction | https://proofwiki.org/wiki/Zero_Vector_has_no_Direction | https://proofwiki.org/wiki/Zero_Vector_has_no_Direction | [
"Zero Vectors"
] | [
"Definition:Zero Vector/Vector Quantity",
"Definition:Direction"
] | [
"Definition:Zero Vector/Vector Quantity",
"Definition:Direction",
"Definition:Vector Quantity/Arrow Representation",
"Definition:Real Vector Space",
"Definition:Cartesian Coordinate System",
"Definition:Line/Segment",
"Definition:Point",
"Definition:Initial Point of Vector",
"Definition:Terminal Poi... |
proofwiki-17742 | Power Set is Closed under Countable Unions | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A_n \in \powerset S: n = 1, 2, \ldots: \ds \bigcup_{n \mathop = 1}^\infty A_n \in \powerset S$ | Let $\sequence {A_i}$ be a countably infinite sequence of sets in $\powerset S$.
Consider an element of the union of all the sets in this $\sequence {A_i}$:
:$\ds x \in \bigcup_{i \mathop \in \N} A_i$
By definition of union:
:$\exists i \in \N: x \in A_i$
But $A_i \in \powerset S$ and so by definition $A_i \subseteq S$... | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Then:
:$\forall A_n \in \powerset S: n = 1, 2, \ldots: \ds \bigcup_{n \mathop = 1}^\infty A_n \in \powerset S$ | Let $\sequence {A_i}$ be a [[Definition:Countable Set|countably infinite]] [[Definition:Sequence|sequence]] of [[Definition:Set|sets]] in $\powerset S$.
Consider an element of the [[Definition:Set Union|union]] of all the [[Definition:Set|sets]] in this $\sequence {A_i}$:
:$\ds x \in \bigcup_{i \mathop \in \N} A_i$
... | Power Set is Closed under Countable Unions | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Countable_Unions | https://proofwiki.org/wiki/Power_Set_is_Closed_under_Countable_Unions | [
"Power Set",
"Set Union",
"Countable Sets"
] | [
"Definition:Set",
"Definition:Power Set"
] | [
"Definition:Countable Set",
"Definition:Sequence",
"Definition:Set",
"Definition:Set Union",
"Definition:Set",
"Definition:Set Union",
"Definition:Subset",
"Definition:Subset",
"Definition:Power Set"
] |
proofwiki-17743 | Set of Elementary Events belonging to k Events is Event | Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $A_1, A_2, \ldots, A_m$ be events in the event space $\Sigma$ of $\EE$.
Let $S$ denote the set of all elementary events of $\EE$ which are elements of exactly $k$ of the events $A_1, A_2, \ldots, A_m$.
Then $S$ is an event of $\Sig... | Let $r_1, r_2, \ldots r_k$ be a set of $k$ elements of $\set {1, 2, \ldots, m}$.
Then:
:$\paren {A_{r_1} \cap A_{r_2} \cap \cdots \cap A_{r_k} } \setminus \paren { A_{r_{k + 1} } \cup A_{r_{k + 2} } \cup \cdots \cup A_{r_m} }$
contains exactly those elements of $\Omega$ which are contained in exactly those events $A_{r... | Let $\EE$ be an [[Definition:Experiment|experiment]] with a [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $A_1, A_2, \ldots, A_m$ be [[Definition:Event|events]] in the [[Definition:Event Space|event space]] $\Sigma$ of $\EE$.
Let $S$ denote the [[Definition:Set|set]] of all [... | Let $r_1, r_2, \ldots r_k$ be a [[Definition:Set|set]] of $k$ [[Definition:Element|elements]] of $\set {1, 2, \ldots, m}$.
Then:
:$\paren {A_{r_1} \cap A_{r_2} \cap \cdots \cap A_{r_k} } \setminus \paren { A_{r_{k + 1} } \cup A_{r_{k + 2} } \cup \cdots \cup A_{r_m} }$
contains exactly those [[Definition:Element|elemen... | Set of Elementary Events belonging to k Events is Event | https://proofwiki.org/wiki/Set_of_Elementary_Events_belonging_to_k_Events_is_Event | https://proofwiki.org/wiki/Set_of_Elementary_Events_belonging_to_k_Events_is_Event | [
"Event Spaces"
] | [
"Definition:Experiment",
"Definition:Probability Space",
"Definition:Event",
"Definition:Event Space",
"Definition:Set",
"Definition:Elementary Event",
"Definition:Element",
"Definition:Event",
"Definition:Event"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Element",
"Definition:Event",
"Elementary Properties of Event Space",
"Definition:Set",
"Definition:Event",
"Definition:Combination",
"Definition:Subset",
"Definition:Elementary Event",
"Definition:Event",
"Definition:Element",
"Definition:... |
proofwiki-17744 | Event Space of Experiment with Final Sample Space has Even Cardinality | Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $\Omega$ be a finite set.
Then the event space $\Sigma$ consists of an even number of subsets of $\Omega$. | Let $A \in \Sigma$ be one of the events of $\EE$.
We have by definition that $\Omega$ is itself an events of $\EE$.
Hence by Set Difference of Events is Event, $\Omega \setminus A$ is also an event of $\EE$.
As $A$ is arbitrary, the same applies to all events of $\EE$.
Thus all events of $\EE$ come in pairs: $A$ and $\... | Let $\EE$ be an [[Definition:Experiment|experiment]] with a [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $\Omega$ be a [[Definition:Finite Set|finite set]].
Then the [[Definition:Event Space|event space]] $\Sigma$ consists of an [[Definition:Even Integer|even number]] of [[... | Let $A \in \Sigma$ be one of the [[Definition:Event|events]] of $\EE$.
We have by definition that $\Omega$ is itself an [[Definition:Event|events]] of $\EE$.
Hence by [[Set Difference of Events is Event]], $\Omega \setminus A$ is also an [[Definition:Event|event]] of $\EE$.
As $A$ is arbitrary, the same applies to a... | Event Space of Experiment with Final Sample Space has Even Cardinality | https://proofwiki.org/wiki/Event_Space_of_Experiment_with_Final_Sample_Space_has_Even_Cardinality | https://proofwiki.org/wiki/Event_Space_of_Experiment_with_Final_Sample_Space_has_Even_Cardinality | [
"Event Spaces"
] | [
"Definition:Experiment",
"Definition:Probability Space",
"Definition:Finite Set",
"Definition:Event Space",
"Definition:Even Integer",
"Definition:Subset"
] | [
"Definition:Event",
"Definition:Event",
"Set Difference of Events is Event",
"Definition:Event",
"Definition:Event",
"Definition:Event",
"Definition:Doubleton"
] |
proofwiki-17745 | Probability of Union of Disjoint Events is Sum of Individual Probabilities | Let $\EE$ be an experiment.
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability measure on $\EE$.
Then:
:$\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$
where $A \cap B$ denotes the '''union''' of $A$ and $B$. | From the Kolmogorov Axioms:
:$\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} = \sum_{i \mathop \ge 1} \map \Pr {A_i}$
where $\set {A_1, A_2, \ldots}$ is a countable set of pairwise disjoint events of $\EE$.
This applies directly to $\map \Pr {A \cup B}$ where $A \cap B = \O$.
{{qed}} | Let $\EE$ be an [[Definition:Experiment|experiment]].
Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Measure|probability measure]] on $\EE$.
Then:
:$\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$
where $A \cap B$ denotes the '''[[Definition:Union of ... | From the [[Axiom:Kolmogorov Axioms|Kolmogorov Axioms]]:
:$\ds \map \Pr {\bigcup_{i \mathop \ge 1} A_i} = \sum_{i \mathop \ge 1} \map \Pr {A_i}$
where $\set {A_1, A_2, \ldots}$ is a [[Definition:Countable Set|countable set]] of [[Definition:Pairwise Disjoint Events|pairwise disjoint events]] of $\EE$.
This applies dir... | Probability of Union of Disjoint Events is Sum of Individual Probabilities/Proof 1 | https://proofwiki.org/wiki/Probability_of_Union_of_Disjoint_Events_is_Sum_of_Individual_Probabilities | https://proofwiki.org/wiki/Probability_of_Union_of_Disjoint_Events_is_Sum_of_Individual_Probabilities/Proof_1 | [
"Probability of Union of Disjoint Events is Sum of Individual Probabilities",
"Addition Law of Probability",
"Unions of Events",
"Disjoint Events"
] | [
"Definition:Experiment",
"Definition:Probability Measure",
"Definition:Event/Occurrence/Union"
] | [
"Axiom:Kolmogorov Axioms",
"Definition:Countable Set",
"Definition:Pairwise Disjoint Events"
] |
proofwiki-17746 | Probability of Union of Disjoint Events is Sum of Individual Probabilities | Let $\EE$ be an experiment.
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability measure on $\EE$.
Then:
:$\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$
where $A \cap B$ denotes the '''union''' of $A$ and $B$. | From the Addition Law of Probability, the union of $A$ and $B$ can be evaluated as:
:$\map \Pr {A \cup B} = \map \Pr A + \map \Pr B - \map \Pr {A \cap B}$
From the definition of disjoint events:
:$\map \Pr {A \cap B} = 0$
Hence the resullt.
{{qed}} | Let $\EE$ be an [[Definition:Experiment|experiment]].
Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Measure|probability measure]] on $\EE$.
Then:
:$\forall A, B \in \Sigma: A \cap B = \O \implies \map \Pr {A \cup B} = \map \Pr A + \map \Pr B$
where $A \cap B$ denotes the '''[[Definition:Union of ... | From the [[Addition Law of Probability]], the [[Definition:Union of Events|union]] of $A$ and $B$ can be evaluated as:
:$\map \Pr {A \cup B} = \map \Pr A + \map \Pr B - \map \Pr {A \cap B}$
From the definition of [[Definition:Disjoint Events|disjoint events]]:
:$\map \Pr {A \cap B} = 0$
Hence the resullt.
{{qed}} | Probability of Union of Disjoint Events is Sum of Individual Probabilities/Proof 2 | https://proofwiki.org/wiki/Probability_of_Union_of_Disjoint_Events_is_Sum_of_Individual_Probabilities | https://proofwiki.org/wiki/Probability_of_Union_of_Disjoint_Events_is_Sum_of_Individual_Probabilities/Proof_2 | [
"Probability of Union of Disjoint Events is Sum of Individual Probabilities",
"Addition Law of Probability",
"Unions of Events",
"Disjoint Events"
] | [
"Definition:Experiment",
"Definition:Probability Measure",
"Definition:Event/Occurrence/Union"
] | [
"Addition Law of Probability",
"Definition:Event/Occurrence/Union",
"Definition:Disjoint Events"
] |
proofwiki-17747 | Discrete Uniform Distribution gives rise to Probability Measure | Let $\EE$ be an experiment.
Let the probability space $\struct {\Omega, \Sigma, \Pr}$ be defined as:
:$\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$
:$\Sigma = \powerset \Omega$
:$\forall A \in \Sigma: \map \Pr A = \dfrac 1 n \card A$
where:
:$\powerset \Omega$ denotes the power set of $\Omega$
:$\card A$ denot... | From Power Set of Sample Space is Event Space we have that $\Sigma$ is an event space.
{{qed|lemma}}
We check the axioms defining a probability measure:
{{begin-axiom}}
{{axiom | n = \text I
| q = \forall A \in \Sigma
| ml= \map \Pr A
| mo= \ge
| mr= 0
}}
{{axiom | n = \text {II}
... | Let $\EE$ be an [[Definition:Experiment|experiment]].
Let the [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$ be defined as:
:$\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$
:$\Sigma = \powerset \Omega$
:$\forall A \in \Sigma: \map \Pr A = \dfrac 1 n \card A$
where:
:$\po... | From [[Power Set of Sample Space is Event Space]] we have that $\Sigma$ is an [[Definition:Event Space|event space]].
{{qed|lemma}}
We check the axioms defining a [[Definition:Probability Measure|probability measure]]:
{{begin-axiom}}
{{axiom | n = \text I
| q = \forall A \in \Sigma
| ml= \map \Pr A
... | Discrete Uniform Distribution gives rise to Probability Measure | https://proofwiki.org/wiki/Discrete_Uniform_Distribution_gives_rise_to_Probability_Measure | https://proofwiki.org/wiki/Discrete_Uniform_Distribution_gives_rise_to_Probability_Measure | [
"Discrete Uniform Distribution"
] | [
"Definition:Experiment",
"Definition:Probability Space",
"Definition:Power Set",
"Definition:Cardinality",
"Definition:Probability Measure"
] | [
"Power Set of Sample Space is Event Space",
"Definition:Event Space",
"Definition:Probability Measure",
"Definition:Elementary Event",
"Definition:Cardinality",
"Definition:Set",
"Definition:Element",
"Union of Set of Singletons"
] |
proofwiki-17748 | Probability Measure on Finite Sample Space | Let $\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$ be a finite set.
Let $\Sigma$ be a $\sigma$-algebra on $\Omega$.
Let $p_1, p_2, \ldots, p_n$ be non-negative real numbers such that:
:$p_1 + p_2 + \cdots + p_n = 1$
Let $Q: \Sigma \to \R$ be the mapping defined as:
:$\forall A \in \Sigma: \map Q A = \ds \sum_{i... | Recall the Kolmogorov axioms:
{{:Axiom:Kolmogorov Axioms}}
First we determine that $\Pr$ as defined is actually a probability measure.
By definition, we have that $\map \Pr A$ is the sum of some subset of $\set {p_1, p_2, \ldots, p_n}$.
Thus $0 \le \map \Pr A \le 1$ and Axiom $(1)$ is fulfilled trivially by definition.... | Let $\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$ be a [[Definition:Finite Set|finite set]].
Let $\Sigma$ be a [[Definition:Sigma-Algebra|$\sigma$-algebra]] on $\Omega$.
Let $p_1, p_2, \ldots, p_n$ be [[Definition:Non-Negative Real Number|non-negative real numbers]] such that:
:$p_1 + p_2 + \cdots + p_n = 1$... | Recall the [[Axiom:Kolmogorov Axioms|Kolmogorov axioms]]:
{{:Axiom:Kolmogorov Axioms}}
First we determine that $\Pr$ as defined is actually a [[Definition:Probability Measure|probability measure]].
By definition, we have that $\map \Pr A$ is the sum of some subset of $\set {p_1, p_2, \ldots, p_n}$.
Thus $0 \le \ma... | Probability Measure on Finite Sample Space | https://proofwiki.org/wiki/Probability_Measure_on_Finite_Sample_Space | https://proofwiki.org/wiki/Probability_Measure_on_Finite_Sample_Space | [] | [
"Definition:Finite Set",
"Definition:Sigma-Algebra",
"Definition:Positive/Real Number",
"Definition:Mapping",
"Definition:Probability Space",
"Definition:Probability Measure"
] | [
"Axiom:Kolmogorov Axioms",
"Definition:Probability Measure",
"Axiom:Kolmogorov Axioms",
"Simple Events are Mutually Exclusive",
"Definition:Set",
"Definition:Pairwise Disjoint Events",
"Axiom:Kolmogorov Axioms",
"Axiom:Kolmogorov Axioms"
] |
proofwiki-17749 | Probability Measure on Single-Subset Event Space | Let $\EE$ be an experiment whose sample space is $\Omega$.
Let $\O \subsetneqq A \subsetneqq \Omega$.
Let $\Sigma := \set {\O, A, \Omega \setminus A, \Omega}$ be the event space of $\EE$.
Let $\Pr: \Sigma \to \R$ be a probability measure on $\struct {\Omega, \Sigma}$.
Then $\Pr$ has the form:
{{begin-eqn}}
{{eqn | n = ... | From Event Space from Single Subset of Sample Space, we have that $\Sigma$ is an event space.
Recall the Kolmogorov axioms:
{{:Axiom:Kolmogorov Axioms}}
First we determine that $\Pr$ as defined is actually a probability measure.
Axiom $(1)$ and axiom $(2)$ are fulfilled trivially by definition.
Then we note that, apart... | Let $\EE$ be an [[Definition:Experiment|experiment]] whose [[Definition:Sample Space|sample space]] is $\Omega$.
Let $\O \subsetneqq A \subsetneqq \Omega$.
Let $\Sigma := \set {\O, A, \Omega \setminus A, \Omega}$ be the [[Definition:Event Space|event space]] of $\EE$.
Let $\Pr: \Sigma \to \R$ be a [[Definition:Prob... | From [[Event Space from Single Subset of Sample Space]], we have that $\Sigma$ is an [[Definition:Event Space|event space]].
Recall the [[Axiom:Kolmogorov Axioms|Kolmogorov axioms]]:
{{:Axiom:Kolmogorov Axioms}}
First we determine that $\Pr$ as defined is actually a [[Definition:Probability Measure|probability meas... | Probability Measure on Single-Subset Event Space | https://proofwiki.org/wiki/Probability_Measure_on_Single-Subset_Event_Space | https://proofwiki.org/wiki/Probability_Measure_on_Single-Subset_Event_Space | [
"Probability Theory"
] | [
"Definition:Experiment",
"Definition:Sample Space",
"Definition:Event Space",
"Definition:Probability Measure"
] | [
"Event Space from Single Subset of Sample Space",
"Definition:Event Space",
"Axiom:Kolmogorov Axioms",
"Definition:Probability Measure",
"Axiom:Kolmogorov Axioms",
"Axiom:Kolmogorov Axioms",
"Definition:Pairwise Disjoint Events",
"Definition:Set Union",
"Axiom:Kolmogorov Axioms",
"Axiom:Kolmogorov... |
proofwiki-17750 | Probability of Set Difference of Events | Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be events of $\EE$.
Let $\map \Pr A$ denote the probability of event $A$ occurring.
Then:
:$\map \Pr {A \setminus B} = \map \Pr A - \map \Pr {A \cap B}$ | From Set Difference and Intersection form Partition:
:$A$ is the union of the two disjoint sets $A \setminus B$ and $A \cap B$
So, by the definition of probability measure:
:$\map \Pr A = \map \Pr {A \setminus B} + \map \Pr {A \cap B}$
Hence the result.
{{qed}} | Let $\EE$ be an [[Definition:Experiment|experiment]] with [[Definition:Probability Space|probability space]] $\struct {\Omega, \Sigma, \Pr}$.
Let $A, B \in \Sigma$ be [[Definition:Event|events]] of $\EE$.
Let $\map \Pr A$ denote the [[Definition:Probability|probability]] of [[Definition:Event|event]] $A$ [[Definition... | From [[Set Difference and Intersection form Partition]]:
:$A$ is the [[Definition:Set Union|union]] of the two [[Definition:Disjoint Sets|disjoint sets]] $A \setminus B$ and $A \cap B$
So, by the definition of [[Definition:Probability Measure|probability measure]]:
:$\map \Pr A = \map \Pr {A \setminus B} + \map \Pr {... | Probability of Set Difference of Events | https://proofwiki.org/wiki/Probability_of_Set_Difference_of_Events | https://proofwiki.org/wiki/Probability_of_Set_Difference_of_Events | [
"Probability Theory"
] | [
"Definition:Experiment",
"Definition:Probability Space",
"Definition:Event",
"Definition:Probability",
"Definition:Event",
"Definition:Event/Occurrence"
] | [
"Set Difference and Intersection form Partition",
"Definition:Set Union",
"Definition:Disjoint Sets",
"Definition:Probability Measure"
] |
proofwiki-17751 | Equivalence of Definitions of Matroid Rank Axioms | Let $S$ be a finite set.
Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ to the integers.
{{TFAE|axiom = Rank Axioms (Matroid)|view = rank axioms}}
=== Formulation 1 ===
$\rho$ is said to satisfy the '''rank axioms''' {{iff}}
{{:Axiom:Rank Axioms (Matroid)/Definition 1}}
=== Formulation 2 ===
$\r... | === Formulation 1 implies Formulation 2 ===
{{:Equivalence of Definitions of Matroid Rank Axioms/Formulation 1 Implies Formulation 2}}{{qed|lemma}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\rho : \powerset S \to \Z$ be a [[Definition:Mapping|mapping]] from the [[Definition:Power Set|power set]] of $S$ to the [[Definition:Integer|integers]].
{{TFAE|axiom = Rank Axioms (Matroid)|view = rank axioms}}
=== [[Axiom:Rank Axioms (Matroid)/Definition 1|F... | === [[Equivalence of Definitions of Matroid Rank Axioms/Formulation 1 Implies Formulation 2|Formulation 1 implies Formulation 2]] ===
{{:Equivalence of Definitions of Matroid Rank Axioms/Formulation 1 Implies Formulation 2}}{{qed|lemma}} | Equivalence of Definitions of Matroid Rank Axioms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Rank_Axioms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Rank_Axioms | [
"Equivalence of Definitions of Matroid Rank Axioms",
"Matroid Rank Functions"
] | [
"Definition:Finite Set",
"Definition:Mapping",
"Definition:Power Set",
"Definition:Integer",
"Axiom:Rank Axioms (Matroid)/Definition 1",
"Axiom:Rank Axioms (Matroid)",
"Axiom:Rank Axioms (Matroid)/Definition 2",
"Axiom:Rank Axioms (Matroid)"
] | [
"Equivalence of Definitions of Matroid Rank Axioms/Formulation 1 Implies Formulation 2"
] |
proofwiki-17752 | Length of Element of Arc in Orthogonal Curvilinear Coordinates | Let $\tuple {q_1, q_2, q_3}$ denote a set of orthogonal curvilinear coordinates.
Let the relation between those orthogonal curvilinear coordinates and Cartesian coordinates be expressed as:
{{begin-eqn}}
{{eqn | l = x
| r = \map x {q_1, q_2, q_3}
}}
{{eqn | l = y
| r = \map y {q_1, q_2, q_3}
}}
{{eqn | l = ... | By definition of the metric of $\tuple {q_1, q_2, q_3}$:
{{begin-eqn}}
{{eqn | l = \d s^2
| r = \d x^2 + \d y^2 + \d z^2
| c =
}}
{{eqn | r = \sum_{i, j} {h_{i j} }^2 \rd q_i \rd q_j
| c = for $i, j \in \set {1, 2, 3}$
}}
{{end-eqn}}
From Value of Curvilinear Coordinate Metric:
:$\forall i, j \in \se... | Let $\tuple {q_1, q_2, q_3}$ denote a set of [[Definition:Orthogonal Curvilinear Coordinates|orthogonal curvilinear coordinates]].
Let the relation between those [[Definition:Orthogonal Curvilinear Coordinates|orthogonal curvilinear coordinates]] and [[Definition:Cartesian Coordinates|Cartesian coordinates]] be expres... | By definition of the [[Definition:Metric (Curvilinear Coordinates)|metric]] of $\tuple {q_1, q_2, q_3}$:
{{begin-eqn}}
{{eqn | l = \d s^2
| r = \d x^2 + \d y^2 + \d z^2
| c =
}}
{{eqn | r = \sum_{i, j} {h_{i j} }^2 \rd q_i \rd q_j
| c = for $i, j \in \set {1, 2, 3}$
}}
{{end-eqn}}
From [[Value of C... | Length of Element of Arc in Orthogonal Curvilinear Coordinates | https://proofwiki.org/wiki/Length_of_Element_of_Arc_in_Orthogonal_Curvilinear_Coordinates | https://proofwiki.org/wiki/Length_of_Element_of_Arc_in_Orthogonal_Curvilinear_Coordinates | [
"Orthogonal Curvilinear Coordinates"
] | [
"Definition:Orthogonal Curvilinear Coordinates",
"Definition:Orthogonal Curvilinear Coordinates",
"Definition:Cartesian Coordinate System",
"Definition:Cartesian Coordinate System",
"Definition:Infinitesimal",
"Definition:Curve/Arc",
"Definition:Arc Length",
"Definition:Projection",
"Definition:Curv... | [
"Definition:Metric (Curvilinear Coordinates)",
"Value of Curvilinear Coordinate Metric",
"Definition:Orthogonal Curvilinear Coordinates",
"Definition:Orthogonal Curvilinear Coordinates/Definition 1",
"Definition:Element",
"Definition:Element",
"Definition:Metric (Curvilinear Coordinates)",
"Definition... |
proofwiki-17753 | Laplacian of Function in Orthogonal Curvilinear Coordinates | Let $\map \psi {q_1, q_2, q_3}$ denote a real-valued function embedded in an orthogonal curvilinear coordinate system.
Then the Laplacian of $\psi$ can be expressed as:
:$\nabla^2 \psi = \dfrac 1 {h_1 h_2 h_3} \paren {\map {\dfrac \partial {\partial q_1} } {\dfrac {h_2 h_3} {h_1} \dfrac {\partial \psi} {\partial q_1} }... | {{ProofWanted|a coherent understanding of exactly what it means would be useful here}} | Let $\map \psi {q_1, q_2, q_3}$ denote a [[Definition:Real-Valued Function|real-valued function]] embedded in an [[Definition:Orthogonal Curvilinear Coordinates|orthogonal curvilinear coordinate system]].
Then the [[Definition:Laplacian|Laplacian]] of $\psi$ can be expressed as:
:$\nabla^2 \psi = \dfrac 1 {h_1 h_2 h... | {{ProofWanted|a coherent understanding of exactly what it means would be useful here}} | Laplacian of Function in Orthogonal Curvilinear Coordinates | https://proofwiki.org/wiki/Laplacian_of_Function_in_Orthogonal_Curvilinear_Coordinates | https://proofwiki.org/wiki/Laplacian_of_Function_in_Orthogonal_Curvilinear_Coordinates | [
"Orthogonal Curvilinear Coordinates"
] | [
"Definition:Real-Valued Function",
"Definition:Orthogonal Curvilinear Coordinates",
"Definition:Laplacian"
] | [] |
proofwiki-17754 | Shortest Distance between Two Points is Straight Line | The shortest distance between $2$ points is a straight line. | Let $s$ be the length of a curve between $2$ points $A$ and $B$.
The problem becomes one of finding the curve for which $\ds \int_a^B \rd s$ is a minimum.
{{ProofWanted|In due course as the work progresses}}
Hence such a curve has the equation:
:$y = m x + c$
which defines a straight line. | The shortest [[Definition:Distance between Points|distance]] between $2$ [[Definition:Point|points]] is a [[Definition:Straight Line|straight line]]. | Let $s$ be the [[Definition:Arc Length|length]] of a [[Definition:Curve|curve]] between $2$ [[Definition:Point|points]] $A$ and $B$.
The problem becomes one of finding the [[Definition:Curve|curve]] for which $\ds \int_a^B \rd s$ is a [[Definition:Absolute Minimum|minimum]].
{{ProofWanted|In due course as the work pr... | Shortest Distance between Two Points is Straight Line | https://proofwiki.org/wiki/Shortest_Distance_between_Two_Points_is_Straight_Line | https://proofwiki.org/wiki/Shortest_Distance_between_Two_Points_is_Straight_Line | [
"Euclidean Geometry"
] | [
"Definition:Distance between Points",
"Definition:Point",
"Definition:Line/Straight Line"
] | [
"Definition:Arc Length",
"Definition:Line/Curve",
"Definition:Point",
"Definition:Line/Curve",
"Definition:Minimum Value of Real Function/Absolute",
"Definition:Line/Curve",
"Definition:Equation of Geometric Figure",
"Definition:Line/Straight Line"
] |
proofwiki-17755 | Equivalence of Definitions of Matroid Circuit Axioms | Let $S$ be a finite set.
Let $\mathscr C$ be a non-empty set of subsets of $S$.
{{TFAE|axiom=Circuit Axioms (Matroid)|view=Matroid Circuit Axioms}}
=== Formulation 1 ===
$\mathscr C$ satisfies the circuit axioms:
{{:Axiom:Circuit Axioms (Matroid)/Formulation 1}}
=== Formulation 2 ===
$\mathscr C$ satisfies the circuit ... | === Formulation 1 implies Formulation 2 ===
{{:Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2}}{{qed|lemma}} | Let $S$ be a [[Definition:Finite Set|finite set]].
Let $\mathscr C$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
{{TFAE|axiom=Circuit Axioms (Matroid)|view=Matroid Circuit Axioms}}
=== [[Axiom:Circuit Axioms (Matroid)/Formulation 1|Formulation 1]] ===
$... | === [[Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2|Formulation 1 implies Formulation 2]] ===
{{:Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2}}{{qed|lemma}} | Equivalence of Definitions of Matroid Circuit Axioms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Matroid_Circuit_Axioms | [
"Matroid Circuits",
"Equivalence of Definitions of Matroid Circuit Axioms"
] | [
"Definition:Finite Set",
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Subset",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 1",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Axiom:Circuit Axioms (Matroid)/Formulation 2",
"Axiom:Circuit A... | [
"Equivalence of Definitions of Matroid Circuit Axioms/Formulation 1 Implies Formulation 2"
] |
proofwiki-17756 | Rank of Matroid Circuit is One Less Than Cardinality | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $C \subseteq S$ be a circuit of $M$.
Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.
Then:
:$\map \rho C = \card C -1$ | By definition of a circuit:
:$C$ is dependent
By matroid axiom $(\text I 1)$:
:$C \ne \O$
Let $x \in C$. | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $C \subseteq S$ be a [[Definition:Circuit (Matroid)|circuit]] of $M$.
Let $\rho: \powerset S \to \Z$ denote the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Then:
:$\map \rho C = \card C -1$ | By definition of a [[Definition:Circuit (Matroid)|circuit]]:
:$C$ is [[Definition:Dependent Subset (Matroid)|dependent]]
By [[Axiom:Matroid Axioms|matroid axiom $(\text I 1)$]]:
:$C \ne \O$
Let $x \in C$. | Rank of Matroid Circuit is One Less Than Cardinality | https://proofwiki.org/wiki/Rank_of_Matroid_Circuit_is_One_Less_Than_Cardinality | https://proofwiki.org/wiki/Rank_of_Matroid_Circuit_is_One_Less_Than_Cardinality | [
"Matroid Circuits",
"Matroid Rank Functions",
"Rank of Matroid Circuit is One Less Than Cardinality"
] | [
"Definition:Matroid",
"Definition:Circuit (Matroid)",
"Definition:Rank Function (Matroid)"
] | [
"Definition:Circuit (Matroid)",
"Definition:Matroid/Dependent Set",
"Axiom:Matroid Axioms"
] |
proofwiki-17757 | Bound for Cardinality of Matroid Circuit | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $C \subseteq S$ be a circuit of $M$.
Let $\rho: \powerset S \to \Z$ denote the rank function of $M$.
Then:
:$\card C \le \map \rho S + 1$ | By definition of a circuit:
:$C$ is dependent
By matroid axiom $(\text I 1)$:
:$C \ne \O$
Let $x \in C$.
From Set Difference is Subset and Set Difference with Disjoint Set:
:$C \setminus \set x \subsetneq C$
From Proper Subset of Matroid Circuit is Independent and matroid axiom $(\text I 1)$:
:$C \setminus \set x \in \... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $C \subseteq S$ be a [[Definition:Circuit (Matroid)|circuit]] of $M$.
Let $\rho: \powerset S \to \Z$ denote the [[Definition:Rank Function (Matroid)|rank function]] of $M$.
Then:
:$\card C \le \map \rho S + 1$ | By definition of a [[Definition:Circuit (Matroid)|circuit]]:
:$C$ is [[Definition:Dependent Subset (Matroid)|dependent]]
By [[Axiom:Matroid Axioms|matroid axiom $(\text I 1)$]]:
:$C \ne \O$
Let $x \in C$.
From [[Set Difference is Subset]] and [[Set Difference with Disjoint Set]]:
:$C \setminus \set x \subsetneq C$
... | Bound for Cardinality of Matroid Circuit | https://proofwiki.org/wiki/Bound_for_Cardinality_of_Matroid_Circuit | https://proofwiki.org/wiki/Bound_for_Cardinality_of_Matroid_Circuit | [
"Matroid Circuits"
] | [
"Definition:Matroid",
"Definition:Circuit (Matroid)",
"Definition:Rank Function (Matroid)"
] | [
"Definition:Circuit (Matroid)",
"Definition:Matroid/Dependent Set",
"Axiom:Matroid Axioms",
"Set Difference is Subset",
"Set Difference with Disjoint Set",
"Proper Subset of Matroid Circuit is Independent",
"Axiom:Matroid Axioms",
"Cardinality of Set Difference with Subset",
"Cardinality of Singleto... |
proofwiki-17758 | Matroid with No Circuits Has Single Base | Let $M = \struct {S, \mathscr I}$ be a matroid with no circuits.
Then:
:$S$ is the only base on $M$. | From Dependent Subset Contains a Circuit:
:$M$ has no dependent subsets
By definition of dependent subsets:
:Every subset of $S$ is independent
In particular:
:$S \in \mathscr I$
By definition of independent subsets:
:$X \in \mathscr I \implies X \subseteq S$
Hence $S$ is a base on $M$ by definition.
Let $X$ be a base ... | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]] with no [[Definition:Circuit (Matroid)|circuits]].
Then:
:$S$ is the only [[Definition:Base of Matroid|base]] on $M$. | From [[Dependent Subset Contains a Circuit]]:
:$M$ has no [[Definition:Dependent Subset (Matroid)|dependent subsets]]
By definition of [[Definition:Dependent Subset (Matroid)|dependent subsets]]:
:Every [[Definition:Subset|subset]] of $S$ is [[Definition:Independent Subset (Matroid)|independent]]
In particular:
:$S \... | Matroid with No Circuits Has Single Base | https://proofwiki.org/wiki/Matroid_with_No_Circuits_Has_Single_Base | https://proofwiki.org/wiki/Matroid_with_No_Circuits_Has_Single_Base | [
"Matroid Circuits"
] | [
"Definition:Matroid",
"Definition:Circuit (Matroid)",
"Definition:Base of Matroid"
] | [
"Dependent Subset Contains a Circuit",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Definition:Subset",
"Definition:Matroid/Independent Set",
"Definition:Matroid/Independent Set",
"Definition:Base of Matroid",
"Definition:Base of Matroid",
"Definition:Base of Matroid",
... |
proofwiki-17759 | Proper Subset of Matroid Circuit is Independent | Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $C \subseteq S$ be a circuit of $M$.
Then every proper subset $A$ of $C$ is independent. | By definition of a circuit of $M$:
:$C$ is a minimum dependent subset of $M$
By definition of the minimum dependent subset of $M$:
:every proper subset $A$ of $C$ is not a dependent subset.
By definition of a dependent subset:
:every proper subset $A$ of $C$ is an independent subset.
{{qed}} | Let $M = \struct {S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $C \subseteq S$ be a [[Definition:Circuit (Matroid)|circuit]] of $M$.
Then every [[Definition:Proper Subset|proper subset]] $A$ of $C$ is [[Definition:Independent Subset (Matroid)|independent]]. | By definition of a [[Definition:Circuit (Matroid)|circuit]] of $M$:
:$C$ is a [[Definition:Minimal Set|minimum]] [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$
By definition of the [[Definition:Minimal Set|minimum]] [[Definition:Dependent Subset (Matroid)|dependent subset]] of $M$:
:every [[Definiti... | Proper Subset of Matroid Circuit is Independent | https://proofwiki.org/wiki/Proper_Subset_of_Matroid_Circuit_is_Independent | https://proofwiki.org/wiki/Proper_Subset_of_Matroid_Circuit_is_Independent | [
"Matroid Circuits"
] | [
"Definition:Matroid",
"Definition:Circuit (Matroid)",
"Definition:Proper Subset",
"Definition:Matroid/Independent Set"
] | [
"Definition:Circuit (Matroid)",
"Definition:Minimal/Set",
"Definition:Matroid/Dependent Set",
"Definition:Minimal/Set",
"Definition:Matroid/Dependent Set",
"Definition:Proper Subset",
"Definition:Matroid/Dependent Set",
"Definition:Matroid/Dependent Set",
"Definition:Proper Subset",
"Definition:Ma... |
proofwiki-17760 | Supremum Norm is Norm/Space of Bounded Sequences | The supremum norm on the vector space of bounded sequences is a norm. | === {{NormAxiomVector|1|nolink}} ===
Let $x \in \ell^\infty$.
By definition of supremum norm:
:$\ds \norm {\mathbf x}_\infty = \sup_{n \mathop \in \N} \size {x_n}$
The complex modulus of $x_n$ is real and non-negative.
Hence, $\norm {\mathbf x}_\infty \ge 0$.
Suppose $\norm {\mathbf x}_\infty = 0$.
Then:
{{begin-eqn}}
... | The [[Definition:Supremum Norm|supremum norm]] on the [[Definition:Vector Space of Bounded Sequences|vector space of bounded sequences]] is a [[Definition:Norm on Vector Space|norm]]. | === {{NormAxiomVector|1|nolink}} ===
Let $x \in \ell^\infty$.
By definition of [[Definition:Supremum Norm|supremum norm]]:
:$\ds \norm {\mathbf x}_\infty = \sup_{n \mathop \in \N} \size {x_n}$
The [[Definition:Complex Modulus|complex modulus]] of $x_n$ is [[Definition:Real Number|real]] and [[Complex Modulus is Non... | Supremum Norm is Norm/Space of Bounded Sequences | https://proofwiki.org/wiki/Supremum_Norm_is_Norm/Space_of_Bounded_Sequences | https://proofwiki.org/wiki/Supremum_Norm_is_Norm/Space_of_Bounded_Sequences | [
"Supremum Norm is Norm"
] | [
"Definition:Supremum Norm",
"Definition:Space of Bounded Sequences/Vector Space",
"Definition:Norm/Vector Space"
] | [
"Definition:Supremum Norm",
"Definition:Complex Modulus",
"Definition:Real Number",
"Complex Modulus is Non-Negative",
"Complex Modulus equals Zero iff Zero"
] |
proofwiki-17761 | Parallelogram Law for Vector Subtraction | Let $\mathbf u$ and $\mathbf v$ be vectors.
Consider a parallelogram, two of whose adjacent sides represent $\mathbf y$ and $\mathbf v$ (in magnitude and direction).
:400px
Then the diagonal of the parallelogram connecting the terminal points of $\mathbf u$ and $\mathbf v$ represents the magnitude and direction of $\ma... | We can construct a parallelogram as follows:
:400px
and the construction is apparent. | Let $\mathbf u$ and $\mathbf v$ be [[Definition:Vector (Linear Algebra)|vectors]].
Consider a [[Definition:Parallelogram|parallelogram]], two of whose adjacent sides represent $\mathbf y$ and $\mathbf v$ (in [[Definition:Magnitude|magnitude]] and [[Definition:Direction|direction]]).
:[[File:ParallelogramLaw-Differenc... | We can construct a [[Definition:Parallelogram|parallelogram]] as follows:
:[[File:Vector-difference-parallelogram.png|400px]]
and the construction is apparent. | Parallelogram Law for Vector Subtraction | https://proofwiki.org/wiki/Parallelogram_Law_for_Vector_Subtraction | https://proofwiki.org/wiki/Parallelogram_Law_for_Vector_Subtraction | [
"Parallelogram Law",
"Vector Subtraction"
] | [
"Definition:Vector/Linear Algebra",
"Definition:Quadrilateral/Parallelogram",
"Definition:Magnitude",
"Definition:Direction",
"File:ParallelogramLaw-Difference.png",
"Definition:Diameter of Quadrilateral",
"Definition:Quadrilateral/Parallelogram",
"Definition:Terminal Point of Vector",
"Definition:M... | [
"Definition:Quadrilateral/Parallelogram",
"File:Vector-difference-parallelogram.png"
] |
proofwiki-17762 | Space of Bounded Sequences with Supremum Norm forms Normed Vector Space | The vector space of bounded sequences with the supremum norm forms a normed vector space. | We have that:
:Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space
:Supremum norm on the space of bounded sequences is a norm
By definition, $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is a normed vector space.
{{qed}} | The [[Definition:Vector Space of Bounded Sequences|vector space of bounded sequences]] with the [[Definition:Supremum Norm|supremum norm]] forms a [[Definition:Normed Vector Space|normed vector space]]. | We have that:
:[[Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space]]
:[[Supremum Norm is Norm/Space of Bounded Sequences|Supremum norm on the space of bounded sequences is a norm]]
By definition, $\struct {\ell^\infty, \norm {\, \cdot \,}_\... | Space of Bounded Sequences with Supremum Norm forms Normed Vector Space | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_with_Supremum_Norm_forms_Normed_Vector_Space | https://proofwiki.org/wiki/Space_of_Bounded_Sequences_with_Supremum_Norm_forms_Normed_Vector_Space | [
"Examples of Normed Vector Spaces"
] | [
"Definition:Space of Bounded Sequences/Vector Space",
"Definition:Supremum Norm",
"Definition:Normed Vector Space"
] | [
"Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space",
"Supremum Norm is Norm/Space of Bounded Sequences",
"Definition:Normed Vector Space"
] |
proofwiki-17763 | Riesz's Lemma | Let $X$ be a normed vector space.
Let $Y$ be a proper closed linear subspace of $X$.
Let $\alpha \in \openint 0 1$.
Then there exists $x_\alpha \in X$ such that:
:$\norm {x_\alpha} = 1$
with:
:$\norm {x_\alpha - y} > \alpha$
for all $y \in Y$. | Since $Y < X$:
:$X \setminus Y$ is non-empty.
Since $Y$ is closed:
:$X \setminus Y$ is open.
Let $x \in X \setminus Y$.
Then there exists $\epsilon > 0$ such that:
:$\map {B_\epsilon} x \subset X \setminus Y$
So, for all $y \in Y$, we must have:
:$\norm {x - y} \ge \epsilon$
That is:
:$\inf \set {\norm {x - y} \col... | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $Y$ be a [[Definition:Proper Subset|proper]] [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $\alpha \in \openint 0 1$.
Then there exists $x_\alpha \in X$ such that:
:$\norm {x_\alpha} = 1$
with:
:$\norm {x_\alpha... | Since $Y < X$:
:$X \setminus Y$ is [[Definition:Non-Empty Set|non-empty]].
Since $Y$ is [[Definition:Closed Set in Normed Vector Space|closed]]:
:$X \setminus Y$ is [[Definition:Open Set in Normed Vector Space|open]].
Let $x \in X \setminus Y$.
Then there exists $\epsilon > 0$ such that:
:$\map {B_\epsilon} x \... | Riesz's Lemma/Proof 1 | https://proofwiki.org/wiki/Riesz's_Lemma | https://proofwiki.org/wiki/Riesz's_Lemma/Proof_1 | [
"Riesz's Lemma",
"Functional Analysis"
] | [
"Definition:Normed Vector Space",
"Definition:Proper Subset",
"Definition:Closed Linear Subspace"
] | [
"Definition:Non-Empty Set",
"Definition:Closed Set/Normed Vector Space",
"Definition:Open Set/Normed Vector Space",
"Definition:Infimum of Mapping/Real-Valued Function",
"Definition:Closed under Mapping",
"Definition:Linear Combination"
] |
proofwiki-17764 | Riesz's Lemma | Let $X$ be a normed vector space.
Let $Y$ be a proper closed linear subspace of $X$.
Let $\alpha \in \openint 0 1$.
Then there exists $x_\alpha \in X$ such that:
:$\norm {x_\alpha} = 1$
with:
:$\norm {x_\alpha - y} > \alpha$
for all $y \in Y$. | Consider the normed quotient vector space $X / Y$ with quotient mapping $\pi$.
From Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1, we have:
:$\norm \pi_{\map B {X, X/Y} } = 1$
Since $\alpha \in \openint 0 1$, there exists $x_\alpha \in X$ with $\norm {x_\alpha} = 1$ and:
:$\norm {\map \pi {x... | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $Y$ be a [[Definition:Proper Subset|proper]] [[Definition:Closed Linear Subspace|closed linear subspace]] of $X$.
Let $\alpha \in \openint 0 1$.
Then there exists $x_\alpha \in X$ such that:
:$\norm {x_\alpha} = 1$
with:
:$\norm {x_\alpha... | Consider the [[Definition:Normed Quotient Vector Space|normed quotient vector space]] $X / Y$ with [[Definition:Quotient Mapping|quotient mapping]] $\pi$.
From [[Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1]], we have:
:$\norm \pi_{\map B {X, X/Y} } = 1$
Since $\alpha \in \openint 0 1$, t... | Riesz's Lemma/Proof 2 | https://proofwiki.org/wiki/Riesz's_Lemma | https://proofwiki.org/wiki/Riesz's_Lemma/Proof_2 | [
"Riesz's Lemma",
"Functional Analysis"
] | [
"Definition:Normed Vector Space",
"Definition:Proper Subset",
"Definition:Closed Linear Subspace"
] | [
"Definition:Normed Quotient Vector Space",
"Definition:Quotient Mapping",
"Operator Norm of Quotient Mapping in Quotient Normed Vector Space is 1",
"Definition:Norm/Bounded Linear Transformation",
"Definition:Quotient Norm"
] |
proofwiki-17765 | Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact | Let $X$ be a normed vector space.
Let $\Bbb S = \map {\Bbb S_1} 0$ be the unit sphere centred at $0$ in $X$.
Then $X$ is finite dimensional {{iff}} $\Bbb S$ is compact. | === Necessary Condition ===
{{:Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact/Necessary Condition}}{{qed|lemma}} | Let $X$ be a [[Definition:Normed Vector Space|normed vector space]].
Let $\Bbb S = \map {\Bbb S_1} 0$ be the [[Definition:Sphere in Normed Vector Space|unit sphere]] [[Definition:Sphere/Normed Vector Space/Center|centred]] at $0$ in $X$.
Then $X$ is [[Definition:Finite Dimensional Vector Space|finite dimensional]] {... | === [[Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact/Necessary Condition|Necessary Condition]] ===
{{:Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact/Necessary Condition}}{{qed|lemma}} | Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact | https://proofwiki.org/wiki/Normed_Vector_Space_is_Finite_Dimensional_iff_Unit_Sphere_is_Compact | https://proofwiki.org/wiki/Normed_Vector_Space_is_Finite_Dimensional_iff_Unit_Sphere_is_Compact | [
"Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact",
"Compact Normed Vector Spaces",
"Finite Dimensional Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Sphere/Normed Vector Space",
"Definition:Sphere/Normed Vector Space/Center",
"Definition:Dimension of Vector Space/Finite",
"Definition:Compact Space/Normed Vector Space"
] | [
"Normed Vector Space is Finite Dimensional iff Unit Sphere is Compact/Necessary Condition"
] |
proofwiki-17766 | Total Force on Point Charge from 2 Point Charges | Let $q_1$, $q_2$ and $q_3$ be point charges.
Let $\mathbf F_{ij}$ denote the electrostatic force exerted on $q_j$ by $q_i$.
Let $\mathbf F_i$ denote the electrostatic force exerted on $q_i$ by the combined action of the other two point charges.
Then the electrostatic force $\mathbf F_1$ exerted on $q_1$ by the combined... | :500px
By definition, the electrostatic force $\mathbf F_{21}$ and $\mathbf F_{31}$ are vector quantities.
Hence their resultant can be found by using the Parallelogram Law.
The result follows from Coulomb's Law of Electrostatics.
{{qed}} | Let $q_1$, $q_2$ and $q_3$ be [[Definition:Point Charge|point charges]].
Let $\mathbf F_{ij}$ denote the [[Definition:Electrostatic Force|electrostatic force]] exerted on $q_j$ by $q_i$.
Let $\mathbf F_i$ denote the [[Definition:Electrostatic Force|electrostatic force]] exerted on $q_i$ by the combined action of the ... | :[[File:Two-charges-on-another.png|500px]]
By definition, the [[Definition:Electrostatic Force|electrostatic force]] $\mathbf F_{21}$ and $\mathbf F_{31}$ are [[Definition:Vector Quantity|vector quantities]].
Hence their [[Definition:Resultant of Vectors|resultant]] can be found by using the [[Parallelogram Law]].
T... | Total Force on Point Charge from 2 Point Charges | https://proofwiki.org/wiki/Total_Force_on_Point_Charge_from_2_Point_Charges | https://proofwiki.org/wiki/Total_Force_on_Point_Charge_from_2_Point_Charges | [
"Total Force on Point Charge from 2 Point Charges",
"Electrostatic Force",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Electrostatic Force",
"Definition:Electrostatic Force",
"Definition:Point Charge",
"Definition:Electrostatic Force",
"Definition:Vector Sum",
"Definition:Displacement",
"Definition:Distance between Points",
"Definition:Vacuum Permittivity"
] | [
"File:Two-charges-on-another.png",
"Definition:Electrostatic Force",
"Definition:Vector Quantity",
"Definition:Vector Sum",
"Parallelogram Law",
"Coulomb's Law of Electrostatics"
] |
proofwiki-17767 | Total Force on Point Charge from Multiple Point Charges | Let $q_1, q_2, \ldots, q_n$ be point charges.
For all $i$ in $\set {1, 2, \ldots, n}$ where $i \ne j$, let $\mathbf F_{i j}$ denote the force exerted on $q_j$ by $q_i$.
For all $i$ in $\set {1, 2, \ldots, n}$, let $\mathbf F_i$ denote the force exerted on $q_i$ by the combined action of all the other point charges.
The... | The proof proceeds by induction.
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the proposition:
:$\ds \mathbf F_i = \dfrac 1 {4 \pi \varepsilon_0} \sum_{\substack {1 \mathop \le j \mathop \le n \\ i \mathop \ne j} } \dfrac {q_i q_j} {r_{j i}^3} \mathbf r_{j i}$
$\map P 2$ is the case:
$\mathbf F_1 = \dfrac 1 {4 \pi \va... | Let $q_1, q_2, \ldots, q_n$ be [[Definition:Point Charge|point charges]].
For all $i$ in $\set {1, 2, \ldots, n}$ where $i \ne j$, let $\mathbf F_{i j}$ denote the [[Definition:Force|force]] exerted on $q_j$ by $q_i$.
For all $i$ in $\set {1, 2, \ldots, n}$, let $\mathbf F_i$ denote the [[Definition:Force|force]] exe... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 2}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \mathbf F_i = \dfrac 1 {4 \pi \varepsilon_0} \sum_{\substack {1 \mathop \le j \mathop \le n \\ i \mathop \ne j} } \dfrac {q_i q_j} {r_{j i}^3} \mathbf... | Total Force on Point Charge from Multiple Point Charges | https://proofwiki.org/wiki/Total_Force_on_Point_Charge_from_Multiple_Point_Charges | https://proofwiki.org/wiki/Total_Force_on_Point_Charge_from_Multiple_Point_Charges | [
"Electrostatic Force",
"Point Charges"
] | [
"Definition:Point Charge",
"Definition:Force",
"Definition:Force",
"Definition:Point Charge",
"Definition:Electrostatic Force",
"Definition:Point Charge",
"Definition:Summation",
"Definition:Vector Sum",
"Definition:Displacement",
"Definition:Distance between Points",
"Definition:Vacuum Permitti... | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Coulomb's Law of Electrostatics",
"Principle of Mathematical Induction"
] |
proofwiki-17768 | Scalar Multiplication by Zero gives Zero Vector | Let $\mathbf a$ be a vector quantity.
Let $0 \mathbf a$ denote the scalar product of $\mathbf a$ with $0$.
Then:
:$0 \mathbf a = \bszero$
where $\bszero$ denotes the zero vector. | By definition of scalar product:
:$\size {0 \mathbf a} = 0 \size {\mathbf a}$
where $\size {\mathbf a}$ denotes the magnitude of $\mathbf a$.
Thus:
:$\size {0 \mathbf a} = 0$
That is: $0 \mathbf a$ is a vector quantity whose magnitude is zero.
Hence, by definition, $0 \mathbf a$ is the zero vector.
{{qed}} | Let $\mathbf a$ be a [[Definition:Vector Quantity|vector quantity]].
Let $0 \mathbf a$ denote the [[Definition:Scalar Multiplication on Vector Quantity|scalar product]] of $\mathbf a$ with $0$.
Then:
:$0 \mathbf a = \bszero$
where $\bszero$ denotes the [[Definition:Zero Vector Quantity|zero vector]]. | By definition of [[Definition:Scalar Multiplication on Vector Quantity|scalar product]]:
:$\size {0 \mathbf a} = 0 \size {\mathbf a}$
where $\size {\mathbf a}$ denotes the [[Definition:Magnitude|magnitude]] of $\mathbf a$.
Thus:
:$\size {0 \mathbf a} = 0$
That is: $0 \mathbf a$ is a [[Definition:Vector Quantity|vect... | Scalar Multiplication by Zero gives Zero Vector | https://proofwiki.org/wiki/Scalar_Multiplication_by_Zero_gives_Zero_Vector | https://proofwiki.org/wiki/Scalar_Multiplication_by_Zero_gives_Zero_Vector | [
"Zero Vectors",
"Scalar Multiplication"
] | [
"Definition:Vector Quantity",
"Definition:Scalar Multiplication/Vector Quantity",
"Definition:Zero Vector/Vector Quantity"
] | [
"Definition:Scalar Multiplication/Vector Quantity",
"Definition:Magnitude",
"Definition:Vector Quantity",
"Definition:Magnitude",
"Definition:Zero (Number)",
"Definition:Zero Vector"
] |
proofwiki-17769 | Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm forms Normed Vector Space | Space of Continuously Differentiable on Closed Interval Real-Valued Functions with $C^1$ norm forms a normed vector space. | Let $I := \closedint a b$ be a closed real interval.
We have that:
:Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space
:$\map {C^1} I$ norm on the space of continuously differentiable on closed interval real-valued... | [[Definition:Space of Continuous Functions of Differentiability Class k|Space of Continuously Differentiable on Closed Interval Real-Valued Functions]] with [[Definition:C^k Norm|$C^1$ norm]] forms a [[Definition:Normed Vector Space|normed vector space]]. | Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
We have that:
:[[Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space]]
:[[C^k Norm is Norm|$\map {C^1} I$ norm on the space... | Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm forms Normed Vector Space | https://proofwiki.org/wiki/Space_of_Continuously_Differentiable_on_Closed_Interval_Real-Valued_Functions_with_C^1_Norm_forms_Normed_Vector_Space | https://proofwiki.org/wiki/Space_of_Continuously_Differentiable_on_Closed_Interval_Real-Valued_Functions_with_C^1_Norm_forms_Normed_Vector_Space | [
"Examples of Normed Vector Spaces"
] | [
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:C^k Norm",
"Definition:Normed Vector Space"
] | [
"Definition:Real Interval/Closed",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"C^k Norm is Norm",
"Definition:Normed Vector Space"
] |
proofwiki-17770 | Ring of Endomorphisms is Ring with Unity | Let $\struct {G, \oplus}$ be an abelian group.
Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$.
Let $\struct {\mathbb G, \oplus, *}$ denote the '''ring of endomorphisms''' on $\struct {G, \oplus}$.
Then $\struct {\mathbb G, \oplus, *}$ is a ring with unity. | By Structure Induced by Group Operation is Group, $\struct {\mathbb G, \oplus}$ is an abelian group.
{{explain|The abelian nature of $\struct {\mathbb G, \oplus}$ needs to be demonstrated -- we have to invoke a result either that demonstrates that commutativity is preserved, or generate a similar result to the above bu... | Let $\struct {G, \oplus}$ be an [[Definition:Abelian Group|abelian group]].
Let $\mathbb G$ be the [[Definition:Set|set]] of all [[Definition:Group Endomorphism|group endomorphisms]] of $\struct {G, \oplus}$.
Let $\struct {\mathbb G, \oplus, *}$ denote the '''[[Definition:Ring of Endomorphisms|ring of endomorphisms]]... | By [[Structure Induced by Group Operation is Group]], $\struct {\mathbb G, \oplus}$ is an [[Definition:Abelian Group|abelian group]].
{{explain|The abelian nature of $\struct {\mathbb G, \oplus}$ needs to be demonstrated -- we have to invoke a result either that demonstrates that commutativity is preserved, or generat... | Ring of Endomorphisms is Ring with Unity | https://proofwiki.org/wiki/Ring_of_Endomorphisms_is_Ring_with_Unity | https://proofwiki.org/wiki/Ring_of_Endomorphisms_is_Ring_with_Unity | [
"Rings of Endomorphisms"
] | [
"Definition:Abelian Group",
"Definition:Set",
"Definition:Group Endomorphism",
"Definition:Ring of Endomorphisms",
"Definition:Ring with Unity"
] | [
"Structure Induced by Group Operation is Group",
"Definition:Abelian Group",
"Set of Homomorphisms to Abelian Group is Subgroup of All Mappings",
"Definition:Subgroup",
"Definition:Associative Operation",
"Definition:Composition of Mappings",
"Definition:Associative Operation",
"Composition of Mapping... |
proofwiki-17771 | Ring of Endomorphisms is not necessarily Commutative Ring | Let $\struct {G, \oplus}$ be an abelian group.
Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$.
Let $\struct {\mathbb G, \oplus, *}$ denote the '''ring of endomorphisms''' on $\struct {G, \oplus}$.
Then $\struct {\mathbb G, \oplus, *}$ is not necessarily a commutative ring with unity. | From Ring of Endomorphisms is Ring with Unity, we have that $\struct {\mathbb G, \oplus, *}$ is a ring with unity.
It remains to show that the operation $*$ is not necessarily commutative.
{{ProofWanted}}
Category:Rings of Endomorphisms
gky1myybkc97s7laxmal3ixd7dd0zsx | Let $\struct {G, \oplus}$ be an [[Definition:Abelian Group|abelian group]].
Let $\mathbb G$ be the [[Definition:Set|set]] of all [[Definition:Group Endomorphism|group endomorphisms]] of $\struct {G, \oplus}$.
Let $\struct {\mathbb G, \oplus, *}$ denote the '''[[Definition:Ring of Endomorphisms|ring of endomorphisms]]... | From [[Ring of Endomorphisms is Ring with Unity]], we have that $\struct {\mathbb G, \oplus, *}$ is a [[Definition:Ring with Unity|ring with unity]].
It remains to show that the operation $*$ is not necessarily [[Definition:Commutative Operation|commutative]].
{{ProofWanted}}
[[Category:Rings of Endomorphisms]]
gky1... | Ring of Endomorphisms is not necessarily Commutative Ring | https://proofwiki.org/wiki/Ring_of_Endomorphisms_is_not_necessarily_Commutative_Ring | https://proofwiki.org/wiki/Ring_of_Endomorphisms_is_not_necessarily_Commutative_Ring | [
"Rings of Endomorphisms"
] | [
"Definition:Abelian Group",
"Definition:Set",
"Definition:Group Endomorphism",
"Definition:Ring of Endomorphisms",
"Definition:Commutative and Unitary Ring"
] | [
"Ring of Endomorphisms is Ring with Unity",
"Definition:Ring with Unity",
"Definition:Commutative/Operation",
"Category:Rings of Endomorphisms"
] |
proofwiki-17772 | Set of Endomorphisms of Non-Abelian Group is not Ring | Let $\struct {G, \oplus}$ be a group which is non-abelian.
Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$.
Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the operation defined as:
:$\forall u, v \in \mathbb G: u * v = u \circ v$
where $u \circ v$ is defined as composition of mappin... | In order to be a ring, it is necessary that the additive operation $\oplus$ is commutative.
However, as $\struct {G, \oplus}$ is specifically defined as being non-abelian, a fortiori $\oplus$ is not commutative.
Hence the result.
{{qed}} | Let $\struct {G, \oplus}$ be a [[Definition:Group|group]] which is non-[[Definition:Abelian Group|abelian]].
Let $\mathbb G$ be the [[Definition:Set|set]] of all [[Definition:Group Endomorphism|group endomorphisms]] of $\struct {G, \oplus}$.
Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the [[Definition:Binary... | In order to be a [[Definition:Ring (Abstract Algebra)|ring]], it is [[Definition:Necessary Condition|necessary]] that the [[Definition:Ring Addition|additive operation]] $\oplus$ is [[Definition:Commutative Operation|commutative]].
However, as $\struct {G, \oplus}$ is specifically defined as being non-[[Definition:Abe... | Set of Endomorphisms of Non-Abelian Group is not Ring | https://proofwiki.org/wiki/Set_of_Endomorphisms_of_Non-Abelian_Group_is_not_Ring | https://proofwiki.org/wiki/Set_of_Endomorphisms_of_Non-Abelian_Group_is_not_Ring | [
"Rings of Endomorphisms"
] | [
"Definition:Group",
"Definition:Abelian Group",
"Definition:Set",
"Definition:Group Endomorphism",
"Definition:Operation/Binary Operation",
"Definition:Composition of Mappings",
"Definition:Algebraic Structure/Two Operations",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Conditional/Necessary Condition",
"Definition:Ring (Abstract Algebra)/Addition",
"Definition:Commutative/Operation",
"Definition:Abelian Group",
"Definition:A Fortiori",
"Definition:Commutative/Operation"
] |
proofwiki-17773 | Set of Positive Integers does not form Ring | Let $\Z_{\ge 0}$ denote the set of positive integers.
Then the algebraic structure $\struct {\Z_{\ge 0}, +, \times}$ does not form a ring. | For $\struct {\Z_{\ge 0}, +, \times}$ to be a ring, it is necessary for the algebraic structure $\struct {\Z_{\ge 0}, +}$ to form a group.
But from {{Corollary|Natural Numbers under Addition do not form Group}}:
:$\struct {\Z_{\ge 0}, +}$ is not a group.
{{qed}} | Let $\Z_{\ge 0}$ denote the [[Definition:Set|set]] of [[Definition:Positive Integer|positive integers]].
Then the [[Definition:Algebraic Structure with Two Operations|algebraic structure]] $\struct {\Z_{\ge 0}, +, \times}$ does not form a [[Definition:Ring (Abstract Algebra)|ring]]. | For $\struct {\Z_{\ge 0}, +, \times}$ to be a [[Definition:Ring (Abstract Algebra)|ring]], it is [[Definition:Necessary Condition|necessary]] for the [[Definition:Algebraic Structure with One Operation|algebraic structure]] $\struct {\Z_{\ge 0}, +}$ to form a [[Definition:Group|group]].
But from {{Corollary|Natural Nu... | Set of Positive Integers does not form Ring | https://proofwiki.org/wiki/Set_of_Positive_Integers_does_not_form_Ring | https://proofwiki.org/wiki/Set_of_Positive_Integers_does_not_form_Ring | [
"Integers",
"Examples of Rings"
] | [
"Definition:Set",
"Definition:Positive/Integer",
"Definition:Algebraic Structure/Two Operations",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Conditional/Necessary Condition",
"Definition:Algebraic Structure/One Operation",
"Definition:Group",
"Definition:Group"
] |
proofwiki-17774 | Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm is Banach Space | Let $I := \closedint a b$ be a closed real interval.
Let $\map C I$ be the space of real-valued functions continuous on $I$.
Let $\map {C^1} I$ be the space of real-valued functions, continuously differentiable on $I$.
Let $\norm {\, \cdot \,}_{1, \infty}$ be the $\CC^1$ norm.
$\struct {\map {C^1} I, \norm {\, \cdot \,... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {\map {C^1} I, \norm {\, \cdot \,}_{1, \infty} }$:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m}_{1, \infty} < \epsilon$ | Let $I := \closedint a b$ be a [[Definition:Closed Real Interval|closed real interval]].
Let $\map C I$ be the [[Definition:Space of Real-Valued Functions Continuous on Closed Interval|space of real-valued functions continuous on $I$]].
Let $\map {C^1} I$ be the [[Definition:Space of Continuous Functions of Different... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence in Normed Vector Space|Cauchy sequence]] in $\struct {\map {C^1} I, \norm {\, \cdot \,}_{1, \infty} }$:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m}_{1, \infty} < \epsilon$ | Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm is Banach Space | https://proofwiki.org/wiki/Space_of_Continuously_Differentiable_on_Closed_Interval_Real-Valued_Functions_with_C^1_Norm_is_Banach_Space | https://proofwiki.org/wiki/Space_of_Continuously_Differentiable_on_Closed_Interval_Real-Valued_Functions_with_C^1_Norm_is_Banach_Space | [
"Banach Spaces"
] | [
"Definition:Real Interval/Closed",
"Definition:Space of Real-Valued Functions Continuous on Closed Interval",
"Definition:Space of Continuous Functions of Differentiability Class k",
"Definition:C^k Norm",
"Space of Continuously Differentiable on Closed Interval Real-Valued Functions with C^1 Norm forms Nor... | [
"Definition:Cauchy Sequence/Normed Vector Space",
"Definition:Cauchy Sequence",
"Definition:Cauchy Sequence/Normed Vector Space"
] |
proofwiki-17775 | Rational Numbers whose Denominators are not Divisible by 4 do not form Ring | Let $S$ be the set defined as:
:$S = \set {\dfrac m n : m, n \in \Z, m \perp n, 4 \nmid n}$
That is, $S$ is defined as the set of rational numbers such that, when expressed in canonical form, their denominators are not divisible by $4$.
Then the algebraic structure $\struct {S, +, \times}$ is not a ring. | For $\struct {S, +, \times}$ to be a ring, it is a necessary condition that $\struct {S, \times}$ is a semigroup.
For $\struct {S, \times}$ to be a semigroup, it is a necessary condition that $\struct {S, \times}$ is closed.
That is:
:$\forall x, y \in S: x \times y \in S$
Let $x = \dfrac 1 2$ and $y = \dfrac 3 2$.
Bot... | Let $S$ be the [[Definition:Set|set]] defined as:
:$S = \set {\dfrac m n : m, n \in \Z, m \perp n, 4 \nmid n}$
That is, $S$ is defined as the [[Definition:Set|set]] of [[Definition:Rational Number|rational numbers]] such that, when expressed in [[Definition:Canonical Form of Rational Number|canonical form]], their [[... | For $\struct {S, +, \times}$ to be a [[Definition:Ring (Abstract Algebra)|ring]], it is a [[Definition:Necessary Condition|necessary condition]] that $\struct {S, \times}$ is a [[Definition:Semigroup|semigroup]].
For $\struct {S, \times}$ to be a [[Definition:Semigroup|semigroup]], it is a [[Definition:Necessary Condi... | Rational Numbers whose Denominators are not Divisible by 4 do not form Ring | https://proofwiki.org/wiki/Rational_Numbers_whose_Denominators_are_not_Divisible_by_4_do_not_form_Ring | https://proofwiki.org/wiki/Rational_Numbers_whose_Denominators_are_not_Divisible_by_4_do_not_form_Ring | [
"Rational Numbers",
"Examples of Rings"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Rational Number",
"Definition:Rational Number/Canonical Form",
"Definition:Fraction/Denominator",
"Definition:Divisor (Algebra)/Integer",
"Definition:Algebraic Structure/Two Operations",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Conditional/Necessary Condition",
"Definition:Semigroup",
"Definition:Semigroup",
"Definition:Conditional/Necessary Condition",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Rational Number",
"Definition:Rational Number/Canon... |
proofwiki-17776 | Order 2 Square Matrices with Zero Diagonals do not form Ring | Let $S$ be the set of square matrices of order $2$ whose diagonal elements are zero.
Then the algebraic structure $\struct {S, +, \times}$ is not a ring.
Note that $\times$ denotes conventional matrix multiplication. | For $\struct {S, +, \times}$ to be a ring, it is a necessary condition that $\struct {S, \times}$ is a semigroup.
For $\struct {S, \times}$ to be a semigroup, it is a necessary condition that $\struct {S, \times}$ is closed.
That is:
:$\forall x, y \in S: x \times y \in S$
Let $x = \begin {pmatrix} 0 & 1 \\ 1 & 0 \end ... | Let $S$ be the [[Definition:Set|set]] of [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Square Matrix|order $2$]] whose [[Definition:Diagonal Element|diagonal elements]] are [[Definition:Zero (Number)|zero]].
Then the [[Definition:Algebraic Structure with Two Operations|algebraic structure]] $\... | For $\struct {S, +, \times}$ to be a [[Definition:Ring (Abstract Algebra)|ring]], it is a [[Definition:Necessary Condition|necessary condition]] that $\struct {S, \times}$ is a [[Definition:Semigroup|semigroup]].
For $\struct {S, \times}$ to be a [[Definition:Semigroup|semigroup]], it is a [[Definition:Necessary Condi... | Order 2 Square Matrices with Zero Diagonals do not form Ring | https://proofwiki.org/wiki/Order_2_Square_Matrices_with_Zero_Diagonals_do_not_form_Ring | https://proofwiki.org/wiki/Order_2_Square_Matrices_with_Zero_Diagonals_do_not_form_Ring | [
"Square Matrices",
"Examples of Rings"
] | [
"Definition:Set",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Matrix/Order",
"Definition:Main Diagonal/Diagonal Elements",
"Definition:Zero (Number)",
"Definition:Algebraic Structure/Two Operations",
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Product (Conventional)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Conditional/Necessary Condition",
"Definition:Semigroup",
"Definition:Semigroup",
"Definition:Conditional/Necessary Condition",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Square Ma... |
proofwiki-17777 | Symmetric Difference with Union does not form Ring | Let $S$ be a set.
Let:
:$\symdif$ denote the symmetric difference operation
:$\cup$ denote the set union operation
:$\powerset S$ denote the power set of $S$.
Then $\struct {\powerset S, \symdif, \cup}$ does not form a ring. | For $\struct {S, \symdif, \cup}$ to be a ring, it is a necessary condition that $\cup$ be distributive over $*$.
Also, the identity element for set union and symmetric difference must be different.
However:
:$(1): \quad$ the identity for union and symmetric difference is $\O$ for both operations
:$(2): \quad$ set union... | Let $S$ be a [[Definition:Set|set]].
Let:
:$\symdif$ denote the [[Definition:Symmetric Difference|symmetric difference operation]]
:$\cup$ denote the [[Definition:Set Union|set union operation]]
:$\powerset S$ denote the [[Definition:Power Set|power set]] of $S$.
Then $\struct {\powerset S, \symdif, \cup}$ does not ... | For $\struct {S, \symdif, \cup}$ to be a [[Definition:Ring (Abstract Algebra)|ring]], it is a [[Definition:Necessary Condition|necessary condition]] that $\cup$ be [[Definition:Distributive Operation|distributive]] over $*$.
Also, the [[Definition:Identity Element|identity element]] for [[Definition:Set Union|set unio... | Symmetric Difference with Union does not form Ring | https://proofwiki.org/wiki/Symmetric_Difference_with_Union_does_not_form_Ring | https://proofwiki.org/wiki/Symmetric_Difference_with_Union_does_not_form_Ring | [
"Set Union",
"Symmetric Difference",
"Power Set",
"Examples of Rings"
] | [
"Definition:Set",
"Definition:Symmetric Difference",
"Definition:Set Union",
"Definition:Power Set",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Conditional/Necessary Condition",
"Definition:Distributive Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Set Union",
"Definition:Symmetric Difference",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
... |
proofwiki-17778 | Set of Order m times n Matrices does not form Ring | Let $m, n \in \N_{>0}$ be non-zero natural numbers such that $m > n$.
Let $S$ be the set of all matrices of order $m \times n$.
Then the algebraic structure $\struct {S, +, \times}$ is not a ring.
Note that $\times$ denotes conventional matrix multiplication. | For $\struct {S, +, \times}$ to be a ring, it is a necessary condition that $\struct {S, \times}$ is a semigroup.
For $\struct {S, \times}$ to be a semigroup, it is a necessary condition that $\struct {S, \times}$ is closed.
That is:
:$\forall x, y \in S: x \times y \in S$
Let $\mathbf A$ and $\mathbf B$ be elements of... | Let $m, n \in \N_{>0}$ be non-[[Definition:Zero (Number)|zero]] [[Definition:Natural Number|natural numbers]] such that $m > n$.
Let $S$ be the [[Definition:Set|set]] of all [[Definition:Matrix|matrices]] of [[Definition:Order of Matrix|order $m \times n$]].
Then the [[Definition:Algebraic Structure with Two Operati... | For $\struct {S, +, \times}$ to be a [[Definition:Ring (Abstract Algebra)|ring]], it is a [[Definition:Necessary Condition|necessary condition]] that $\struct {S, \times}$ is a [[Definition:Semigroup|semigroup]].
For $\struct {S, \times}$ to be a [[Definition:Semigroup|semigroup]], it is a [[Definition:Necessary Condi... | Set of Order m times n Matrices does not form Ring | https://proofwiki.org/wiki/Set_of_Order_m_times_n_Matrices_does_not_form_Ring | https://proofwiki.org/wiki/Set_of_Order_m_times_n_Matrices_does_not_form_Ring | [
"Matrices",
"Examples of Rings"
] | [
"Definition:Zero (Number)",
"Definition:Natural Numbers",
"Definition:Set",
"Definition:Matrix",
"Definition:Matrix/Order",
"Definition:Algebraic Structure/Two Operations",
"Definition:Ring (Abstract Algebra)",
"Definition:Matrix Product (Conventional)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Conditional/Necessary Condition",
"Definition:Semigroup",
"Definition:Semigroup",
"Definition:Conditional/Necessary Condition",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Element",
"Definition:Matrix Product (Conventional)... |
proofwiki-17779 | Set of Order 3 Vectors under Cross Product does not form Ring | Let $S$ be the set of all vectors in a vector space of dimension $3$.
Let $\times$ denote the cross product operation.
Then the algebraic structure $\struct {S, +, \times}$ is not a ring. | For $\struct {S, +, \times}$ to be a ring, it is a necessary condition that $\struct {S, \times}$ is a semigroup.
For $\struct {S, \times}$ to be a semigroup, it is a necessary condition that $\times$ is associative on $S$.
However, from Vector Cross Product is not Associative, this is not the case here.
The result fol... | Let $S$ be the [[Definition:Set|set]] of all [[Definition:Vector|vectors]] in a [[Definition:Vector Space|vector space]] of [[Definition:Dimension (Linear Algebra)|dimension]] $3$.
Let $\times$ denote the [[Definition:Vector Cross Product|cross product operation]].
Then the [[Definition:Algebraic Structure with Two ... | For $\struct {S, +, \times}$ to be a [[Definition:Ring (Abstract Algebra)|ring]], it is a [[Definition:Necessary Condition|necessary condition]] that $\struct {S, \times}$ is a [[Definition:Semigroup|semigroup]].
For $\struct {S, \times}$ to be a [[Definition:Semigroup|semigroup]], it is a [[Definition:Necessary Condi... | Set of Order 3 Vectors under Cross Product does not form Ring | https://proofwiki.org/wiki/Set_of_Order_3_Vectors_under_Cross_Product_does_not_form_Ring | https://proofwiki.org/wiki/Set_of_Order_3_Vectors_under_Cross_Product_does_not_form_Ring | [
"Vectors",
"Examples of Rings"
] | [
"Definition:Set",
"Definition:Vector",
"Definition:Vector Space",
"Definition:Dimension (Linear Algebra)",
"Definition:Vector Cross Product",
"Definition:Algebraic Structure/Two Operations",
"Definition:Ring (Abstract Algebra)"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Conditional/Necessary Condition",
"Definition:Semigroup",
"Definition:Semigroup",
"Definition:Conditional/Necessary Condition",
"Definition:Associative Operation",
"Vector Cross Product is not Associative"
] |
proofwiki-17780 | Integers under Subtraction do not form Semigroup | Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction.
Then $\struct {\Z, -}$ is not a semigroup. | It is to be demonstrated that $\struct {\Z, -}$ does not satisfy the semigroup axioms.
We then have Subtraction on Numbers is Not Associative.
So, for example:
:$3 - \paren {2 - 1} = 2 \ne \paren {3 - 2} - 1 = 0$
Thus it has been demonstrated that $\struct {\Z, -}$ does not satisfy {{Semigroup-axiom|1}}.
Hence the resu... | Let $\struct {\Z, -}$ denote the [[Definition:Algebraic Structure with One Operation|algebraic structure]] formed by the set of [[Definition:Integer|integers]] under the [[Definition:Binary Operation|operation]] of [[Definition:Integer Subtraction|subtraction]].
Then $\struct {\Z, -}$ is not a [[Definition:Semigroup|... | It is to be demonstrated that $\struct {\Z, -}$ does not satisfy the [[Axiom:Semigroup Axioms|semigroup axioms]].
We then have [[Subtraction on Numbers is Not Associative]].
So, for example:
:$3 - \paren {2 - 1} = 2 \ne \paren {3 - 2} - 1 = 0$
Thus it has been demonstrated that $\struct {\Z, -}$ does not satisfy {{S... | Integers under Subtraction do not form Semigroup | https://proofwiki.org/wiki/Integers_under_Subtraction_do_not_form_Semigroup | https://proofwiki.org/wiki/Integers_under_Subtraction_do_not_form_Semigroup | [
"Integer Subtraction",
"Examples of Semigroups"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Integer",
"Definition:Operation/Binary Operation",
"Definition:Subtraction/Integers",
"Definition:Semigroup"
] | [
"Axiom:Semigroup Axioms",
"Subtraction on Numbers is Not Associative"
] |
proofwiki-17781 | Cardinality of Maximal Independent Subset Equals Rank of Set | Let $M = \struct{S, \mathscr I}$ be a matroid.
Let $A \subseteq S$.
Let $X$ be a maximal independent subset of $A$.
Then:
:$\card X = \map \rho A$
where $\rho$ is the rank function on $M$. | From Independent Subset is Contained in Maximal Independent Subset:
:$\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \card Y = \map \rho A$
By definition of a maximal independent Subset of $A$:
:$X = Y$
The result follows.
{{qed}}
Category:Matroid Independent Subsets
Category:Matroid Rank Functions
n60ritlugze1... | Let $M = \struct{S, \mathscr I}$ be a [[Definition:Matroid|matroid]].
Let $A \subseteq S$.
Let $X$ be a [[Definition:Maximal Set|maximal]] [[Definition:Independent Subset (Matroid)|independent subset]] of $A$.
Then:
:$\card X = \map \rho A$
where $\rho$ is the [[Definition:Rank Function (Matroid)|rank function]] on... | From [[Independent Subset is Contained in Maximal Independent Subset]]:
:$\exists Y \in \mathscr I : X \subseteq Y \subseteq A : \card Y = \map \rho A$
By definition of a [[Definition:Maximal Set|maximal]] [[Definition:Independent Subset (Matroid)|independent Subset]] of $A$:
:$X = Y$
The result follows.
{{qed}}
[[C... | Cardinality of Maximal Independent Subset Equals Rank of Set | https://proofwiki.org/wiki/Cardinality_of_Maximal_Independent_Subset_Equals_Rank_of_Set | https://proofwiki.org/wiki/Cardinality_of_Maximal_Independent_Subset_Equals_Rank_of_Set | [
"Matroid Independent Subsets",
"Matroid Rank Functions"
] | [
"Definition:Matroid",
"Definition:Maximal/Set",
"Definition:Matroid/Independent Set",
"Definition:Rank Function (Matroid)"
] | [
"Independent Subset is Contained in Maximal Independent Subset",
"Definition:Maximal/Set",
"Definition:Matroid/Independent Set",
"Category:Matroid Independent Subsets",
"Category:Matroid Rank Functions"
] |
proofwiki-17782 | Motion of Body with Constant Mass | Let $B$ be a body with constant mass $m$ undergoing a force $\mathbf F$.
Then the equation of motion of $B$ is given by:
:$\mathbf F = m \mathbf a$
where $\mathbf a$ is the acceleration of $B$. | {{begin-eqn}}
{{eqn | l = \mathbf F
| r = \map {\dfrac \d {\d t} } {m \mathbf v}
| c = Newton's Second Law of Motion
}}
{{eqn | r = m \dfrac {\d \mathbf v} {\d t} + \mathbf v \dfrac {\d m} {\d t}
| c =
}}
{{eqn | r = m \dfrac {\d \mathbf v} {\d t} + 0
| c = Derivative of Constant
}}
{{eqn | r =... | Let $B$ be a [[Definition:Body|body]] with [[Definition:Constant|constant]] [[Definition:Mass|mass]] $m$ undergoing a [[Definition:Force|force]] $\mathbf F$.
Then the equation of motion of $B$ is given by:
:$\mathbf F = m \mathbf a$
where $\mathbf a$ is the [[Definition:Acceleration|acceleration]] of $B$. | {{begin-eqn}}
{{eqn | l = \mathbf F
| r = \map {\dfrac \d {\d t} } {m \mathbf v}
| c = [[Newton's Second Law of Motion]]
}}
{{eqn | r = m \dfrac {\d \mathbf v} {\d t} + \mathbf v \dfrac {\d m} {\d t}
| c =
}}
{{eqn | r = m \dfrac {\d \mathbf v} {\d t} + 0
| c = [[Derivative of Constant]]
}}
{{e... | Motion of Body with Constant Mass | https://proofwiki.org/wiki/Motion_of_Body_with_Constant_Mass | https://proofwiki.org/wiki/Motion_of_Body_with_Constant_Mass | [
"Dynamics"
] | [
"Definition:Body",
"Definition:Constant",
"Definition:Mass",
"Definition:Force",
"Definition:Acceleration"
] | [
"Newton's Laws of Motion/Second Law",
"Derivative of Constant"
] |
proofwiki-17783 | Like Vector Quantities are Multiples of Each Other | Let $\mathbf a$ and $\mathbf b$ be like vector quantities.
Then:
:$\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$
where:
:$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$
:$\dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$ denotes the scalar product of $\mathbf b$ by $\dfrac {... | By the definition of like vector quantities:
:$\mathbf a$ and $\mathbf b$ are '''like vector quantities''' {{iff}} they have the same direction.
By definition of unit vector:
:$\dfrac {\mathbf a} {\size {\mathbf a} } = \dfrac {\mathbf b} {\size {\mathbf b} }$
as both are in the same direction, and both have length $1$.... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Like Vector Quantities|like vector quantities]].
Then:
:$\mathbf a = \dfrac {\size {\mathbf a} } {\size {\mathbf b} } \mathbf b$
where:
:$\size {\mathbf a}$ denotes the [[Definition:Magnitude|magnitude]] of $\mathbf a$
:$\dfrac {\size {\mathbf a} } {\size {\mathbf b} } ... | By the definition of [[Definition:Like Vector Quantities|like vector quantities]]:
:$\mathbf a$ and $\mathbf b$ are '''like [[Definition:Vector Quantity|vector quantities]]''' {{iff}} they have the same [[Definition:Direction|direction]].
By definition of [[Definition:Unit Vector|unit vector]]:
:$\dfrac {\mathbf a} {... | Like Vector Quantities are Multiples of Each Other | https://proofwiki.org/wiki/Like_Vector_Quantities_are_Multiples_of_Each_Other | https://proofwiki.org/wiki/Like_Vector_Quantities_are_Multiples_of_Each_Other | [
"Scalar Multiplication"
] | [
"Definition:Like Vector Quantities",
"Definition:Magnitude",
"Definition:Scalar Multiplication/Vector Quantity"
] | [
"Definition:Like Vector Quantities",
"Definition:Vector Quantity",
"Definition:Direction",
"Definition:Unit Vector",
"Definition:Direction",
"Definition:Vector Length",
"Definition:Scalar Division/Vector Quantity"
] |
proofwiki-17784 | Direct Product Norm is Norm | Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be normed vector spaces.
Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations.
Let $\norm {\tuple {x, y} }$ be the direct product norm.
Then $\norm {\tuple {x, y} }$ is a norm o... | === Positive Definiteness ===
Let $\tuple {x , y} \in V$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\tuple {x, y} }
| r = \map \max {\norm x, \norm y}
| c = {{defof|Direct Product Norm}}
}}
{{eqn | o = \ge
| r = 0
| c = Norm Axiom $N1$: Positive Definiteness
}}
{{end-eqn}}
Suppose $\norm {\tuple ... | Let $\struct {X, \norm {\, \cdot \,}}$ and $\struct {Y, \norm {\, \cdot \,}}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $V = X \times Y$ be a [[Definition:Direct Product of Vector Spaces/Finite Case|direct product of vector spaces]] $X$ and $Y$ together with [[Definition:Operation Induced by Dire... | === Positive Definiteness ===
Let $\tuple {x , y} \in V$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\tuple {x, y} }
| r = \map \max {\norm x, \norm y}
| c = {{defof|Direct Product Norm}}
}}
{{eqn | o = \ge
| r = 0
| c = [[Definition:Norm on Vector Space|Norm Axiom]] $N1$: Positive Definiteness... | Direct Product Norm is Norm | https://proofwiki.org/wiki/Direct_Product_Norm_is_Norm | https://proofwiki.org/wiki/Direct_Product_Norm_is_Norm | [
"Examples of Norms",
"Direct Product of Vector Spaces",
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Direct Product of Vector Spaces/Finite Case",
"Definition:Operation Induced by Direct Product",
"Definition:Direct Product Norm",
"Definition:Norm/Vector Space"
] | [
"Definition:Norm/Vector Space",
"Definition:Norm/Vector Space",
"Definition:Norm/Vector Space"
] |
proofwiki-17785 | Vector Quantity as Scalar Product of Unit Vector Quantity | Let $\mathbf a$ be a vector quantity.
Then:
:$\mathbf a = \size {\mathbf a} \mathbf {\hat a}$
where:
:$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$
:$\mathbf {\hat a}$ denotes the unit vector in the direction $\mathbf a$. | {{begin-eqn}}
{{eqn | l = \size {\mathbf {\hat a} }
| r = 1
| c = {{Defof|Unit Vector}}
}}
{{eqn | ll= \leadsto
| l = \mathbf {\hat a} \times \size {\mathbf {\hat a} }
| r = \mathbf {\hat a}
| c =
}}
{{eqn | ll= \leadsto
| l = \size {\mathbf a} \times \size {\mathbf {\hat a} } \time... | Let $\mathbf a$ be a [[Definition:Vector Quantity|vector quantity]].
Then:
:$\mathbf a = \size {\mathbf a} \mathbf {\hat a}$
where:
:$\size {\mathbf a}$ denotes the [[Definition:Magnitude|magnitude]] of $\mathbf a$
:$\mathbf {\hat a}$ denotes the [[Definition:Unit Vector|unit vector]] in the [[Definition:Direction|di... | {{begin-eqn}}
{{eqn | l = \size {\mathbf {\hat a} }
| r = 1
| c = {{Defof|Unit Vector}}
}}
{{eqn | ll= \leadsto
| l = \mathbf {\hat a} \times \size {\mathbf {\hat a} }
| r = \mathbf {\hat a}
| c =
}}
{{eqn | ll= \leadsto
| l = \size {\mathbf a} \times \size {\mathbf {\hat a} } \time... | Vector Quantity as Scalar Product of Unit Vector Quantity | https://proofwiki.org/wiki/Vector_Quantity_as_Scalar_Product_of_Unit_Vector_Quantity | https://proofwiki.org/wiki/Vector_Quantity_as_Scalar_Product_of_Unit_Vector_Quantity | [
"Scalar Multiplication",
"Unit Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Magnitude",
"Definition:Unit Vector",
"Definition:Direction"
] | [] |
proofwiki-17786 | Scalar Product of Magnitude by Unit Vector Quantity | Let $\mathbf a$ be a vector quantity.
Let $m$ be a scalar quantity.
Then:
:$m \mathbf a = m \paren {\size {\mathbf a} \hat {\mathbf a} } = \paren {m \size {\mathbf a} } \hat {\mathbf a}$
where:
:$\size {\mathbf a}$ denotes the magnitude of $\mathbf a$
:$\hat {\mathbf a}$ denotes the unit vector in the direction $\mathb... | {{begin-eqn}}
{{eqn | l = \mathbf a
| r = \size {\mathbf a} \hat {\mathbf a}
| c = Vector Quantity as Scalar Product of Unit Vector Quantity
}}
{{eqn | ll= \leadsto
| l = m \mathbf a
| r = m \paren {\size {\mathbf a} \hat {\mathbf a} }
| c =
}}
{{end-eqn}}
Then:
{{finish|hard to prove som... | Let $\mathbf a$ be a [[Definition:Vector Quantity|vector quantity]].
Let $m$ be a [[Definition:Scalar Quantity|scalar quantity]].
Then:
:$m \mathbf a = m \paren {\size {\mathbf a} \hat {\mathbf a} } = \paren {m \size {\mathbf a} } \hat {\mathbf a}$
where:
:$\size {\mathbf a}$ denotes the [[Definition:Magnitude|magni... | {{begin-eqn}}
{{eqn | l = \mathbf a
| r = \size {\mathbf a} \hat {\mathbf a}
| c = [[Vector Quantity as Scalar Product of Unit Vector Quantity]]
}}
{{eqn | ll= \leadsto
| l = m \mathbf a
| r = m \paren {\size {\mathbf a} \hat {\mathbf a} }
| c =
}}
{{end-eqn}}
Then:
{{finish|hard to pro... | Scalar Product of Magnitude by Unit Vector Quantity | https://proofwiki.org/wiki/Scalar_Product_of_Magnitude_by_Unit_Vector_Quantity | https://proofwiki.org/wiki/Scalar_Product_of_Magnitude_by_Unit_Vector_Quantity | [
"Scalar Multiplication",
"Unit Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Scalar Quantity",
"Definition:Magnitude",
"Definition:Unit Vector",
"Definition:Direction"
] | [
"Vector Quantity as Scalar Product of Unit Vector Quantity"
] |
proofwiki-17787 | Like Unit Vectors are Equal | Let $\mathbf a$ and $\mathbf b$ be like vector quantities.
Then:
:$\mathbf {\hat a} = \mathbf {\hat b}$
where $\mathbf {\hat a}$ and $\mathbf {\hat b}$ denote the unit vectors in the direction of $\mathbf a$ and $\mathbf b$. | By definition of like vector quantities, $\mathbf a$ and $\mathbf b$ have the same direction.
By definition of unit vector, $\mathbf {\hat a}$ and $\mathbf {\hat b}$ are both of magnitude $1$.
Hence the result, by Equality of Vector Quantities.
{{qed}}
Category:Vectors
4dc6u0f04r1xrbj90fu5p7eaj30d0uk | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Like Vector Quantities|like vector quantities]].
Then:
:$\mathbf {\hat a} = \mathbf {\hat b}$
where $\mathbf {\hat a}$ and $\mathbf {\hat b}$ denote the [[Definition:Unit Vector|unit vectors]] in the [[Definition:Direction|direction]] of $\mathbf a$ and $\mathbf b$. | By definition of [[Definition:Like Vector Quantities|like vector quantities]], $\mathbf a$ and $\mathbf b$ have the same [[Definition:Direction|direction]].
By definition of [[Definition:Unit Vector|unit vector]], $\mathbf {\hat a}$ and $\mathbf {\hat b}$ are both of [[Definition:Magnitude|magnitude]] $1$.
Hence the ... | Like Unit Vectors are Equal | https://proofwiki.org/wiki/Like_Unit_Vectors_are_Equal | https://proofwiki.org/wiki/Like_Unit_Vectors_are_Equal | [
"Vectors"
] | [
"Definition:Like Vector Quantities",
"Definition:Unit Vector",
"Definition:Direction"
] | [
"Definition:Like Vector Quantities",
"Definition:Direction",
"Definition:Unit Vector",
"Definition:Magnitude",
"Equality of Vector Quantities",
"Category:Vectors"
] |
proofwiki-17788 | Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors | Let $\mathbf r$ be a vector quantity embedded in space.
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be non-coplanar.
Then $\mathbf r$ can be expressed uniquely as the resultant of $3$ vector quantities which are each parallel to one of $\mathbf a$, $\mathbf b$ and $\mathbf c$. | 400px
Let $\mathbf {\hat a}$, $\mathbf {\hat b}$ and $\mathbf {\hat c}$ be unit vectors in the directions of $\mathbf a$, $\mathbf b$ and $\mathbf c$ respectively.
Let $O$ be a point in space.
Take $\vec {OP} := \mathbf r$.
With $OP$ as its space diagonal, construct a parallelepiped with edges $OA$, $OB$ and $OC$ paral... | Let $\mathbf r$ be a [[Definition:Vector Quantity|vector quantity]] embedded in [[Definition:Ordinary Space|space]].
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be [[Definition:Non-Coplanar Vectors|non-coplanar]].
Then $\mathbf r$ can be expressed [[Definition:Unique|uniquely]] as the [[Definition:Resultant of Vect... | [[File:Resultant-of-3-non-coplanar-vectors.png|400px]]
Let $\mathbf {\hat a}$, $\mathbf {\hat b}$ and $\mathbf {\hat c}$ be [[Definition:Unit Vector|unit vectors]] in the [[Definition:Direction|directions]] of $\mathbf a$, $\mathbf b$ and $\mathbf c$ respectively.
Let $O$ be a [[Definition:Point|point]] in [[Definit... | Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors | https://proofwiki.org/wiki/Vector_Quantity_can_be_Expressed_as_Sum_of_3_Non-Coplanar_Vectors | https://proofwiki.org/wiki/Vector_Quantity_can_be_Expressed_as_Sum_of_3_Non-Coplanar_Vectors | [
"Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Ordinary Space",
"Definition:Coplanar Vectors/Non-Coplanar",
"Definition:Unique",
"Definition:Vector Sum",
"Definition:Vector Quantity",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Resultant-of-3-non-coplanar-vectors.png",
"Definition:Unit Vector",
"Definition:Direction",
"Definition:Point",
"Definition:Ordinary Space",
"Definition:Space Diagonal",
"Definition:Parallelepiped",
"Definition:Polyhedron/Edge",
"Definition:Parallel (Geometry)/Lines",
"Definition:Parallelepi... |
proofwiki-17789 | Vectors are Equal iff Components are Equal | Two vector quantities are equal {{iff}} they have the same components. | Let $\mathbf a$ and $\mathbf b$ be vector quantities.
Then by Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors, $\mathbf a$ and $\mathbf b$ can be expressed uniquely as components.
So if $\mathbf a$ and $\mathbf b$ then the components of $\mathbf a$ are the same as the components of $\mathbf b$
Suppose... | Two [[Definition:Vector Quantity|vector quantities]] are [[Definition:Equality|equal]] {{iff}} they have the same [[Definition:Component of Vector|components]]. | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]].
Then by [[Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors]], $\mathbf a$ and $\mathbf b$ can be expressed uniquely as [[Definition:Component of Vector|components]].
So if $\mathbf a$ and $\mathbf b$ then the [[Defin... | Vectors are Equal iff Components are Equal | https://proofwiki.org/wiki/Vectors_are_Equal_iff_Components_are_Equal | https://proofwiki.org/wiki/Vectors_are_Equal_iff_Components_are_Equal | [
"Vectors",
"Equality"
] | [
"Definition:Vector Quantity",
"Definition:Equals",
"Definition:Vector Quantity/Component"
] | [
"Definition:Vector Quantity",
"Vector Quantity can be Expressed as Sum of 3 Non-Coplanar Vectors",
"Definition:Vector Quantity/Component",
"Definition:Vector Quantity/Component",
"Definition:Vector Quantity/Component",
"Definition:Vector Quantity/Component"
] |
proofwiki-17790 | Characteristics of Birkhoff-James Orthogonality | Let $\struct {V, \norm {\,\cdot\,} }$ be a normed linear space.
Let $x, y \in V$.
Then $x$ and $y$ are '''Birkhoff-James orthogonal''' {{iff}} either:
:$(1): \quad x = 0$
or:
:$(2): \quad$ there exists a continuous functional $ f$ on $\struct {V, \norm {\,\cdot\,} }$ such that:
::::$\norm f = 1$
::::$\map f x = \norm x... | === Necessary Condition ===
Let $x \perp_B y$.
Let $V' \subset V$ be the subspace spanned by $x$ and $y$.
Define $\overline f$ on $V'$ as:
:$\map {\overline f} {a x + b y} = a \norm x$
for $a$ and $b$ scalars.
Clearly, $\overline f$ is linear and:
:$\map {\overline f} x = \norm x$
:$\map {\overline f} y = 0$
Further:
{... | Let $\struct {V, \norm {\,\cdot\,} }$ be a [[Definition:Normed Linear Space|normed linear space]].
Let $x, y \in V$.
Then $x$ and $y$ are '''[[Definition:Birkhoff-James Orthogonality|Birkhoff-James orthogonal]]''' {{iff}} either:
:$(1): \quad x = 0$
or:
:$(2): \quad$ there exists a [[Definition:Continuous Functional... | === Necessary Condition ===
Let $x \perp_B y$.
Let $V' \subset V$ be the subspace spanned by $x$ and $y$.
Define $\overline f$ on $V'$ as:
:$\map {\overline f} {a x + b y} = a \norm x$
for $a$ and $b$ scalars.
Clearly, $\overline f$ is linear and:
:$\map {\overline f} x = \norm x$
:$\map {\overline f} y = 0$
Fur... | Characteristics of Birkhoff-James Orthogonality | https://proofwiki.org/wiki/Characteristics_of_Birkhoff-James_Orthogonality | https://proofwiki.org/wiki/Characteristics_of_Birkhoff-James_Orthogonality | [
"Normed Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Birkhoff-James Orthogonality",
"Definition:Continuity/Functional"
] | [
"Hahn-Banach Theorem"
] |
proofwiki-17791 | Components of Vector in terms of Direction Cosines | Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space.
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.
Let $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ be the direction cosines of $\mathbf r$ with respect ... | :480px
By definition, the direction cosines are the cosines of the angles that $\mathbf r$ makes with the coordinate axes.
By definition of the components of $\mathbf r$:
:$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$
Thus:
:$\mathbf r = r \cos \alpha \mathbf i + r \cos \beta \mathbf j + r \cos \gamma \mathbf k... | Let $\mathbf r$ be a [[Definition:Vector Quantity|vector quantity]] embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the [[Definition:Unit Vector|unit vectors]] in the [[Definition:Positive Direction|positive directions]] of the [[Definition:X-Axis|$x$... | :[[File:Vector-Components-in-3-Space.png|480px]]
By definition, the [[Definition:Direction Cosines|direction cosines]] are the [[Definition:Cosine|cosines]] of the [[Definition:Angle|angles]] that $\mathbf r$ makes with the [[Definition:Coordinate Axis|coordinate axes]].
By definition of the [[Definition:Component of... | Components of Vector in terms of Direction Cosines | https://proofwiki.org/wiki/Components_of_Vector_in_terms_of_Direction_Cosines | https://proofwiki.org/wiki/Components_of_Vector_in_terms_of_Direction_Cosines | [
"Components of Vector in terms of Direction Cosines",
"Direction Cosines"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis",
"Definition:Direction Cosines",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definit... | [
"File:Vector-Components-in-3-Space.png",
"Definition:Direction Cosines",
"Definition:Cosine",
"Definition:Angle",
"Definition:Axis/Coordinate Axes",
"Definition:Vector Quantity/Component"
] |
proofwiki-17792 | Magnitude of Vector Quantity in terms of Components | Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space.
Let $\mathbf r$ be expressed in terms of its components:
:$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$
where $\mathbf i$, $\mathbf j$ and $\mathbf k$ denote the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis... | Let the initial point of $\mathbf r$ be $\tuple {x_1, y_1, z_1}$.
Let the terminal point of $\mathbf r$ be $\tuple {x_2, y_2, z_2}$.
Thus, by definition of the components of $\mathbf r$, the magnitude of $\mathbf r$ equals the distance between $\tuple {x_1, y_1, z_1}$ and $\tuple {x_2, y_2, z_2}$.
The result follows fr... | Let $\mathbf r$ be a [[Definition:Vector Quantity|vector quantity]] embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
Let $\mathbf r$ be expressed in terms of its [[Definition:Component of Vector|components]]:
:$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$
where $\mathbf i$, $\mathbf j$ and ... | Let the [[Definition:Initial Point of Vector|initial point]] of $\mathbf r$ be $\tuple {x_1, y_1, z_1}$.
Let the [[Definition:Terminal Point of Vector|terminal point]] of $\mathbf r$ be $\tuple {x_2, y_2, z_2}$.
Thus, by definition of the [[Definition:Component of Vector|components of $\mathbf r$]], the [[Definition:... | Magnitude of Vector Quantity in terms of Components | https://proofwiki.org/wiki/Magnitude_of_Vector_Quantity_in_terms_of_Components | https://proofwiki.org/wiki/Magnitude_of_Vector_Quantity_in_terms_of_Components | [
"Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Vector Quantity/Component",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis",
"Definition:Magnitude"
] | [
"Definition:Initial Point of Vector",
"Definition:Terminal Point of Vector",
"Definition:Vector Quantity/Component",
"Definition:Magnitude",
"Definition:Distance between Points",
"Distance Formula/3 Dimensions"
] |
proofwiki-17793 | Components of Zero Vector Quantity are Zero | Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space.
Let $\mathbf r$ be expressed in terms of its components:
:$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$
Let $\mathbf r$ be the zero vector.
Then:
:$x = y = z = 0$ | By definition of the zero vector, the magnitude of $\mathbf r$ is equal to zero.
By Magnitude of Vector Quantity in terms of Components:
:$\size {\mathbf r} = \sqrt {x^2 + y^2 + z^2} = 0$
where $\size {\mathbf r}$ denotes the magnitude of $\mathbf r$.
As each of $x$, $y$ and $z$ are real numbers, each of $x^2$, $y^2$ a... | Let $\mathbf r$ be a [[Definition:Vector Quantity|vector quantity]] embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
Let $\mathbf r$ be expressed in terms of its [[Definition:Component of Vector|components]]:
:$\mathbf r = x \mathbf i + y \mathbf j + z \mathbf k$
Let $\mathbf r$ be the [[Definition... | By definition of the [[Definition:Zero Vector Quantity|zero vector]], the [[Definition:Magnitude|magnitude]] of $\mathbf r$ is equal to [[Definition:Zero (Number)|zero]].
By [[Magnitude of Vector Quantity in terms of Components]]:
:$\size {\mathbf r} = \sqrt {x^2 + y^2 + z^2} = 0$
where $\size {\mathbf r}$ denotes th... | Components of Zero Vector Quantity are Zero | https://proofwiki.org/wiki/Components_of_Zero_Vector_Quantity_are_Zero | https://proofwiki.org/wiki/Components_of_Zero_Vector_Quantity_are_Zero | [
"Zero Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Vector Quantity/Component",
"Definition:Zero Vector/Vector Quantity"
] | [
"Definition:Zero Vector/Vector Quantity",
"Definition:Magnitude",
"Definition:Zero (Number)",
"Magnitude of Vector Quantity in terms of Components",
"Definition:Magnitude",
"Definition:Real Number",
"Definition:Positive/Real Number",
"Definition:Zero (Number)"
] |
proofwiki-17794 | Unit Vector in terms of Direction Cosines | Let $\mathbf r$ be a vector quantity embedded in a Cartesian $3$-space.
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the unit vectors in the positive directions of the $x$-axis, $y$-axis and $z$-axis respectively.
Let $\cos \alpha$, $\cos \beta$ and $\cos \gamma$ be the direction cosines of $\mathbf r$ with respect ... | From Components of Vector in terms of Direction Cosines:
:$(1): \quad \mathbf r = r \cos \alpha \mathbf i + r \cos \beta \mathbf j + r \cos \gamma \mathbf k$
where $r$ denotes the magnitude of $\mathbf r$, that is:
:$r := \size {\mathbf r}$
By Unit Vector in Direction of Vector:
:$\mathbf {\hat r} = \dfrac {\mathbf r} ... | Let $\mathbf r$ be a [[Definition:Vector Quantity|vector quantity]] embedded in a [[Definition:Cartesian 3-Space|Cartesian $3$-space]].
Let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be the [[Definition:Unit Vector|unit vectors]] in the [[Definition:Positive Direction|positive directions]] of the [[Definition:X-Axis|$x$... | From [[Components of Vector in terms of Direction Cosines]]:
:$(1): \quad \mathbf r = r \cos \alpha \mathbf i + r \cos \beta \mathbf j + r \cos \gamma \mathbf k$
where $r$ denotes the [[Definition:Magnitude|magnitude]] of $\mathbf r$, that is:
:$r := \size {\mathbf r}$
By [[Unit Vector in Direction of Vector]]:
:$\m... | Unit Vector in terms of Direction Cosines | https://proofwiki.org/wiki/Unit_Vector_in_terms_of_Direction_Cosines | https://proofwiki.org/wiki/Unit_Vector_in_terms_of_Direction_Cosines | [
"Direction Cosines",
"Unit Vectors"
] | [
"Definition:Vector Quantity",
"Definition:Cartesian 3-Space",
"Definition:Unit Vector",
"Definition:Axis/Positive Direction",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definition:Axis/Z-Axis",
"Definition:Direction Cosines",
"Definition:Axis/X-Axis",
"Definition:Axis/Y-Axis",
"Definit... | [
"Components of Vector in terms of Direction Cosines",
"Definition:Magnitude",
"Unit Vector in Direction of Vector",
"Definition:Scalar Multiplication/Vector Quantity"
] |
proofwiki-17795 | Vectors from Sum and Difference | Let $\mathbf a$ and $\mathbf b$ be vector quantities.
Let $\mathbf c = \mathbf a + \mathbf b$ and $\mathbf d = \mathbf a - \mathbf b$ be given.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = \dfrac 1 2 \paren {\mathbf c + \mathbf d}
}}
{{eqn | l = \mathbf b
| r = \dfrac 1 2 \paren {\mathbf c - \mathbf d}
}... | {{begin-eqn}}
{{eqn | l = \dfrac 1 2 \paren {\mathbf c + \mathbf d}
| r = \dfrac 1 2 \paren {\paren {\mathbf a + \mathbf b} + \paren {\mathbf a - \mathbf b} }
| c =
}}
{{eqn | r = \dfrac 1 2 \paren {\mathbf a + \mathbf b + \mathbf a - \mathbf b}
| c =
}}
{{eqn | r = \dfrac 1 2 \paren {2 \mathbf a}
... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]].
Let $\mathbf c = \mathbf a + \mathbf b$ and $\mathbf d = \mathbf a - \mathbf b$ be given.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf a
| r = \dfrac 1 2 \paren {\mathbf c + \mathbf d}
}}
{{eqn | l = \mathbf b
| r = \dfrac 1 ... | {{begin-eqn}}
{{eqn | l = \dfrac 1 2 \paren {\mathbf c + \mathbf d}
| r = \dfrac 1 2 \paren {\paren {\mathbf a + \mathbf b} + \paren {\mathbf a - \mathbf b} }
| c =
}}
{{eqn | r = \dfrac 1 2 \paren {\mathbf a + \mathbf b + \mathbf a - \mathbf b}
| c =
}}
{{eqn | r = \dfrac 1 2 \paren {2 \mathbf a}
... | Vectors from Sum and Difference | https://proofwiki.org/wiki/Vectors_from_Sum_and_Difference | https://proofwiki.org/wiki/Vectors_from_Sum_and_Difference | [
"Vector Addition",
"Vector Subtraction"
] | [
"Definition:Vector Quantity"
] | [] |
proofwiki-17796 | Direct Product of Banach Spaces is Banach Space | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces.
Let $V = X \times Y$ be a direct product of vector spaces $X$ and $Y$ together with induced component-wise operations.
Let $\norm {\, \cdot \,}_{X \times Y}$ be the direct product norm.
Suppose $X$ and $Y$ are Ban... | Let $\sequence {\tuple {x_n, y_n}}_{n \mathop \in \N}$ be a Cauchy sequence in $V$:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n > N: \norm {\tuple {x_n, y_n} - \tuple {x_m, y_m} }_{X \times Y} < \epsilon$
We have that:
{{begin-eqn}}
{{eqn | l = \norm {x_n - x_m}_X
| o = \le
|... | Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be [[Definition:Normed Vector Space|normed vector spaces]].
Let $V = X \times Y$ be a [[Definition:Direct Product of Vector Spaces/Finite Case|direct product of vector spaces]] $X$ and $Y$ together with [[Definition:Operation Induced by ... | Let $\sequence {\tuple {x_n, y_n}}_{n \mathop \in \N}$ be a [[Definition:Cauchy Sequence|Cauchy sequence]] in $V$:
:$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n > N: \norm {\tuple {x_n, y_n} - \tuple {x_m, y_m} }_{X \times Y} < \epsilon$
We have that:
{{begin-eqn}}
{{eqn | l = \norm {x_n... | Direct Product of Banach Spaces is Banach Space | https://proofwiki.org/wiki/Direct_Product_of_Banach_Spaces_is_Banach_Space | https://proofwiki.org/wiki/Direct_Product_of_Banach_Spaces_is_Banach_Space | [
"Banach Spaces",
"Direct Product of Vector Spaces"
] | [
"Definition:Normed Vector Space",
"Definition:Direct Product of Vector Spaces/Finite Case",
"Definition:Operation Induced by Direct Product",
"Definition:Direct Product Norm",
"Definition:Banach Space",
"Definition:Banach Space"
] | [
"Definition:Cauchy Sequence",
"Definition:Operation Induced by Direct Product",
"Definition:Cauchy Sequence",
"Definition:Banach Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Definition:Convergent Sequence/Normed Vector Space",
"Convergence in Direct Product Norm",
"Definition:Converg... |
proofwiki-17797 | Cartesian Plane Rotated with respect to Another | Let $\mathbf r$ be a position vector embedded in a Cartesian plane $\CC$ with origin $O$.
Let $\CC$ be rotated anticlockwise through an angle $\varphi$ about the axis of rotation $O$.
Let $\CC'$ denote the Cartesian plane in its new position.
Let $\mathbf r$ be kept fixed during this rotation.
Let $\tuple {x, y}$ denot... | :400px
Let $\mathbf r$ be represented by a directed line segment whose initial point coincides with the origin $O$.
Let the terminal point of $\mathbf r$ be identified with the point $P$.
Let $\CC$ be rotated to $\CC'$ through an angle $\varphi$ as shown, keeping $P$ fixed.
We have that:
{{begin-eqn}}
{{eqn | l = x'
... | Let $\mathbf r$ be a [[Definition:Position Vector|position vector]] embedded in a [[Definition:Cartesian Plane|Cartesian plane]] $\CC$ with [[Definition:Origin|origin]] $O$.
Let $\CC$ be [[Definition:Plane Rotation|rotated]] [[Definition:Anticlockwise|anticlockwise]] through an [[Definition:Angle|angle]] $\varphi$ abo... | :[[File:Rotation-of-cartesian-plane.png|400px]]
Let $\mathbf r$ be represented by a [[Definition:Directed Line Segment|directed line segment]] whose [[Definition:Initial Point of Vector|initial point]] coincides with the [[Definition:Origin|origin]] $O$.
Let the [[Definition:Terminal Point of Vector|terminal point]]... | Cartesian Plane Rotated with respect to Another | https://proofwiki.org/wiki/Cartesian_Plane_Rotated_with_respect_to_Another | https://proofwiki.org/wiki/Cartesian_Plane_Rotated_with_respect_to_Another | [
"Coordinate Systems",
"Geometric Rotations",
"Vectors"
] | [
"Definition:Position Vector",
"Definition:Cartesian Plane",
"Definition:Coordinate System/Origin",
"Definition:Rotation (Geometry)/Plane",
"Definition:Anticlockwise",
"Definition:Angle",
"Definition:Rotation (Geometry)/Axis",
"Definition:Cartesian Plane",
"Definition:Rotation (Geometry)/Plane",
"D... | [
"File:Rotation-of-cartesian-plane.png",
"Definition:Directed Line Segment",
"Definition:Initial Point of Vector",
"Definition:Coordinate System/Origin",
"Definition:Terminal Point of Vector",
"Definition:Point",
"Definition:Angle"
] |
proofwiki-17798 | Dot Product of Perpendicular Vectors | Let $\mathbf a$ and $\mathbf b$ be vector quantities such that $\mathbf a \ne \bszero$ and $\mathbf b \ne \bszero$.
Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ and $\mathbf b$.
Then:
:$\mathbf a \cdot \mathbf b = 0$
{{iff}}:
:$\mathbf a$ and $\mathbf b$ are perpendicular. | By definition of dot product:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the angle between $\mathbf a$ and $\mathbf b$.
When $\mathbf a$ and $\mathbf b$ be perpendicular, by definition $\theta = 90 \degrees$.
The result follows by Cosine of Right Angle, which gives ... | Let $\mathbf a$ and $\mathbf b$ be [[Definition:Vector Quantity|vector quantities]] such that $\mathbf a \ne \bszero$ and $\mathbf b \ne \bszero$.
Let $\mathbf a \cdot \mathbf b$ denote the [[Definition:Dot Product|dot product]] of $\mathbf a$ and $\mathbf b$.
Then:
:$\mathbf a \cdot \mathbf b = 0$
{{iff}}:
:$\mathb... | By definition of [[Definition:Dot Product|dot product]]:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the [[Definition:Angle|angle]] between $\mathbf a$ and $\mathbf b$.
When $\mathbf a$ and $\mathbf b$ be [[Definition:Perpendicular|perpendicular]], by definition $\t... | Dot Product of Perpendicular Vectors | https://proofwiki.org/wiki/Dot_Product_of_Perpendicular_Vectors | https://proofwiki.org/wiki/Dot_Product_of_Perpendicular_Vectors | [
"Dot Product"
] | [
"Definition:Vector Quantity",
"Definition:Dot Product",
"Definition:Right Angle/Perpendicular"
] | [
"Definition:Dot Product",
"Definition:Angle",
"Definition:Right Angle/Perpendicular",
"Cosine of Right Angle"
] |
proofwiki-17799 | Dot Product of Orthonormal Basis Vectors | Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be an orthonormal basis of a vector space $V$.
Then:
:$\forall i, j \in \set {1, 2, \ldots, n}: \mathbf e_i \cdot \mathbf e_j = \delta_{i j}$
where:
:$\mathbf e_i \cdot \mathbf e_j$ denotes the dot product of $\mathbf e_i$ and $\mathbf e_j$
:$\delta_{i j}$ de... | By definition of orthonormal basis:
:$(1): \quad \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is an orthogonal basis of $V$
:$(2): \quad \norm {\mathbf e_1} = \norm {\mathbf e_2} = \cdots = \norm {\mathbf e_1} = 1$
From $(1)$ we have by definition that $\mathbf e_i$ and $\mathbf e_j$ are perpendicular whenev... | Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be an [[Definition:Orthonormal Basis of Vector Space|orthonormal basis]] of a [[Definition:Vector Space|vector space]] $V$.
Then:
:$\forall i, j \in \set {1, 2, \ldots, n}: \mathbf e_i \cdot \mathbf e_j = \delta_{i j}$
where:
:$\mathbf e_i \cdot \mathbf e_j$... | By definition of [[Definition:Orthonormal Basis of Vector Space|orthonormal basis]]:
:$(1): \quad \tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is an [[Definition:Orthogonal Basis of Vector Space|orthogonal basis of $V$]]
:$(2): \quad \norm {\mathbf e_1} = \norm {\mathbf e_2} = \cdots = \norm {\mathbf e_1} =... | Dot Product of Orthonormal Basis Vectors | https://proofwiki.org/wiki/Dot_Product_of_Orthonormal_Basis_Vectors | https://proofwiki.org/wiki/Dot_Product_of_Orthonormal_Basis_Vectors | [
"Dot Product"
] | [
"Definition:Orthonormal Basis of Vector Space",
"Definition:Vector Space",
"Definition:Dot Product",
"Definition:Kronecker Delta"
] | [
"Definition:Orthonormal Basis of Vector Space",
"Definition:Orthogonal Basis of Vector Space",
"Definition:Right Angle/Perpendicular",
"Dot Product of Perpendicular Vectors",
"Dot Product of Vector with Itself",
"Definition:Kronecker Delta"
] |
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