nyuuzyou/wikis huggingface dataset:
Hypothetical Syllogism/Formulation 2/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables to the propositions in turn.
As can be seen for all boolean interpretations by inspection, where the truth values under the main connective on the (that is, the rightmost $\land$) is $\T$, that under the instance of $r$ on the is also $\T$:
$\begin{array}{|ccccccccc||c|} \hline
((p & \implies & q) & \land & (q & \implies & r)) & \land & p & r \\
\hline
\F & \T & \F & \T & \F & \T & \F & \F & \F & \F \\
\F & \T & \F & \T & \F & \T & \T & \F & \F & \T \\
\F & \T & \T & \F & \T & \F & \F & \F & \F & \F \\
\F & \T & \T & \T & \T & \T & \T & \F & \F & \T \\
\T & \F & \F & \F & \F & \T & \F & \F & \T & \F \\
\T & \F & \F & \F & \F & \T & \T & \F & \T & \T \\
\T & \T & \T & \F & \T & \F & \F & \F & \T & \F \\
\T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\
\hline
\end{array}$
Hence the result.
Category:Truth Table Proofs
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 3

== Theorem ==
:$\vdash \paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$
== Proof 1 ==
== Proof by Truth Table ==
== Sources ==
* : $\S 3.6$: Reference Formulae: $RF \, 19$
* : $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T26}$
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 3/Proof 1

== Theorem ==
== Proof ==
Let us use the following abbreviations
Expanding the abbreviations leads us back to:
:$\paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$
== Sources ==
* : $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T26}$
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 3/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations.
:$\begin{array}{|ccccccc|c|ccc|} \hline
((p & \implies & q) & \land & (q & \implies & r)) & \implies & (p & \implies & r) \\
\hline
\F & \T & \F & \T & \F & \T & \F & \T & \F & \T & \F \\
\F & \T & \F & \T & \F & \T & \T & \T & \F & \T & \T \\
\F & \T & \T & \T & \T & \F & \F & \T & \F & \T & \F \\
\F & \T & \T & \T & \T & \T & \T & \T & \F & \T & \T \\
\T & \F & \F & \F & \F & \T & \F & \T & \T & \F & \F \\
\T & \F & \F & \T & \F & \T & \T & \T & \T & \T & \T \\
\T & \T & \T & \F & \T & \F & \F & \T & \T & \F & \F \\
\T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\
\hline
\end{array}$
== Sources ==
* : $\S1.2$: Some Remarks on the Use of the Connectives and, or, implies: Exercise $2$
* : $\S \text{I}: 7$: Determining the Truth Values of Complex Propositions
* : Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...: Ponderable $1.1.4$
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 4

== Theorem ==
:$\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T4}$
* : Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(2) \ \text{(iii)}$
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 4/Proof 1

== Theorem ==
== Proof ==
Let us use the following abbreviations
Expanding the abbreviations leads us back to:
:$\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }$
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 5

== Theorem ==
:$\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 4.7$: The Derivation of Formulae: $D \, 2$
* : $\text{I}$: 'NOT' and 'IF': $\S 5$: Theorem $\text{T5}$
* : Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Exercise $1 \ \text{(a)}$
Category:Hypothetical Syllogism
Hypothetical Syllogism/Formulation 5/Proof 1

== Theorem ==
== Proof ==
Let us use the following abbreviations
From Hypothetical Syllogism: Formulation 3 we have:
:$(1): \quad \vdash \paren {\paren {p \implies q} \land \paren {q \implies r} } \implies \paren {p \implies r}$
Expanding the abbreviations leads us back to:
:$\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Category:Hypothetical Syllogism
Hölder's Inequality for Finite Sums/Euclidean Plane

== Hölder's Inequality for Sums: Euclidean Plane ==
In the context of a Euclidean plane, Hölder's Inequality for Sums takes the following form:
:$x_1 y_1 + x_2 y_2 \le \paren {\size {x_1}^p + \size {x_2}^p}^{1 / p} \paren {\size {y_1}^q + \size {y_2}^q}^{1 / q}$
where:
:$p, q > 1$
:$\dfrac 1 p + \dfrac 1 q = 1$
== Sources ==
* 
* 
Category:Hölder's Inequality for Sums
Hölder's Inequality for Integrals

== Theorem ==
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $p, q \in \R_{>0}$ such that $\dfrac 1 p + \dfrac 1 q = 1$.
Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space.
Then their pointwise product $f g$ is $\mu$-integrable, that is:
:$f g \in \map {\LL^1} \mu$
and:
where:
:$\size {f g}$ denotes the absolute value function applied to the pointwise product of $f$ and $g$
:the $\norm {\, \cdot \,}_p$ signify $p$-seminorms.
=== Equality ===
Equality, that is:
=== Generalized Hölder Inequality ===
== Proof ==
Let $x \in X$.
Let:
:$a_x := \dfrac {\size {\map f x} } {\norm f_p}$
and:
:$b_x := \dfrac {\size {\map g x} } {\norm g_q}$
Applying Young's Inequality for Products to $a_x$ and $b_x$:
:$\dfrac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q} \le \dfrac {\size {\map f x}^p} {p \norm f_p^p} + \dfrac {\size {\map g x}^q} { q \norm g_q^q}$
By Integral of Positive Measurable Function is Monotone, integrating both sides of this inequality over x yields:
:$\ds \dfrac {\int \size {\map f x \map g x} \map \mu {\d x} } {\norm f_p \cdot \norm g_q} \le \frac {\norm f_p^p} {p \norm f_p^p} + \frac {\norm g_p^q} {q \norm g_q^q} = \frac 1 p + \frac 1 q = 1$
so:
:$\ds \int \size {\map f x \map g x} \map \mu {\d x} \le \norm f_p \cdot \norm g_q$
If we have equality, then:
:$\ds \int \paren {\frac {\size {\map f x}^p} {p \norm f_p^p} + \frac {\size {\map g x}^q} {q \norm g_q^q} - \frac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q} } \map \mu {\d x} = 0$
As:
:$\dfrac {\size {\map f x}^p} {p \norm f_p^p} + \dfrac {\size {\map g x}^q} {q \norm g_q^q} - \dfrac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q} \ge 0$
it follows from Measurable Function Zero A.E. iff Absolute Value has Zero Integral that:
:$\dfrac {\size {\map g x}^p} {p \norm f_p^p} + \dfrac {\size {\map g x}^q} {q \norm g_q^q} = \dfrac {\size {\map f x \map g x} } {\norm f_p \cdot \norm g_q}$ a.e.
By Young's Inequality for Products, we have equality $b_x = {a_x}^{p - 1}$.
Raising both sides to the $q$th power gives:
:$\dfrac {\size {\map g x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$
as $\paren {p - 1} q = p$. 
== Also known as ==
== Also see ==
* Hölder's Inequality for Sums
== Sources ==
* : $3$: Elementary Analytic Methods: $3.2$ Inequalities: Hölder's Inequality for Integrals: $3.2.10$
* : $\S 36$: Inequalities: $36.14$: Hölder's Inequality for Integrals
* 
* : $6.12$: The 'elementary formula' for expectation
* : $12.2$
Category:Hölder's Inequality for Integrals
Category:Hölder's Inequality
Category:Lebesgue Spaces
Category:Inequalities
Hölder's Inequality for Integrals/Also known as

== Hölder's Inequality for Integrals: Also known as ==
Hölder's Inequality for Integrals is also seen referred to just as Hölder's Inequality.
This allows it to be confused with Hölder's Inequality for Sums, so the full form is used on .
== Sources ==
* 
Category:Hölder's Inequality for Integrals
Hölder's Inequality for Integrals/Equality

== Theorem ==
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $p, q \in \R_{>0}$ such that $\dfrac 1 p + \dfrac 1 q = 1$.
Let $f \in \map {\LL^p} \mu, f: X \to \R$, and $g \in \map {\LL^q} \mu, g: X \to \R$, where $\LL$ denotes Lebesgue space.
Then equality in Hölder's Inequality for Integrals, that is:
:$\ds \int \size {f g} \rd \mu = \norm f_p \cdot \norm g_q$
holds , for $\mu$-almost all $x \in X$:
:$\dfrac {\size {\map f x}^p} {\norm f_p^p} = \dfrac {\size {\map g x}^q} {\norm g_q^q}$
:$\dfrac {\size {\map f x}^{p - 1} } {\size {\map g x} } = c$
for some $c \in \R$.
== Proof ==
== Sources ==
* : $3$: Elementary Analytic Methods: $3.2$ Inequalities: Hölder's Inequality for Integrals: $3.2.10$
* : $\S 36$: Inequalities: $36.14$: Hölder's Inequality for Integrals
* : $12.2$
Category:Hölder's Inequality for Integrals
Hölder's Inequality for Integrals/General

== Theorem ==
Let $\struct {X, \Sigma, \mu}$ be a measure space.
For $i = 1, \ldots, n$ let $p_i \in \R_{>0}$ such that:
:$\ds \sum_{i \mathop = 1}^n \frac 1 {p_i} = 1$
Let $f_i \in \map {\LL^{p_i} } \mu, f_i: X \to \R$, where $\LL$ denotes Lebesgue space.
Then their pointwise product $\ds \prod_{i \mathop = 1}^n f_i$ is integrable, that is:
:$\ds \prod_{i \mathop = 1}^n f_i \in \map {\LL^1} \mu$
and:
:$\ds \norm {\prod_{i \mathop = 1}^n f_i}_1 = \int \size {\prod_{i \mathop = 1}^n f_i} \rd \mu \le \prod_{i \mathop = 1}^n \norm {f_i}_{p_i}$
where the various instances of $\norm {\, \cdot \,}$ signify $p$-seminorms.
== Proof ==
We use the Principle of Mathematical Induction.
Let it be assumed that the result holds for $i = n - 1$.
We show that the result holds for $i = n$. 
Define:
:$q_n := \dfrac {p_n} {p_n - 1}$
and for $i = 1, \ldots, n - 1$, define:
:$r_i := p_i \cdot \paren {1 - \dfrac 1 {p_n} }$
Then:
:$\dfrac 1 {p_n} + \dfrac 1 {q_n} = 1$
:$\ds \sum_{i \mathop = 1}^{n - 1} \dfrac 1 {r_i} = 1$
and:
:$q_n \cdot r_i = p_i$
Applying Hölder's Inequality for Integrals to $\ds f := \prod_{i \mathop = 1}^{n - 1} f_i$ and $g := f_n$, we find:
Category:Hölder's Inequality for Integrals
Hölder's Inequality for Sums

== Theorem ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
=== Formulation $1$ ===
=== Formulation $2$ ===
=== Finite Form ===
Hölder's Inequality for Sums can also be seen presented in the less general form:
=== Condition for Equality ===
== Parameter Inequalities ==
== Also known as ==
== Also see ==
* Hölder's Inequality for Integrals
* Minkowski's Inequality for Sums
== Historical Note ==
== Sources ==
* 
* 
Category:Hölder's Inequality for Sums
Category:Hölder's Inequality
Category:Analysis
Category:Inequalities
Hölder's Inequality for Sums/Also known as

== Hölder's Inequality for Sums: Also known as ==
Hölder's Inequality for Sums is also seen referred to just as Hölder's Inequality.
This allows it to be confused with Hölder's Inequality for Integrals, so the full form is used on .
== Sources ==
* : $\S 36$: Inequalities: Hölder's Inequality
* 
* 
* 
* 
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Equality

== Theorem ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
=== Formulation $1$ ===
Let $\mathbf x$ and $\mathbf y$ denote the vectors consisting of the sequences:
:$\mathbf x = \sequence {x_n} \in {\ell^p}_\GF$
:$\mathbf y = \sequence {y_n} \in {\ell^q}_\GF$
where ${\ell^p}_\GF$ denotes the $p$-sequence space in $\GF$.
Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.
==== Formulation $1$: Condition for Equality ====
=== Formulation $2$ ===
Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be (possibly infinite) convergent sequences in $\GF$.
==== Formulation $2$: Condition for Equality ====
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Finite

== Hölder's Inequality for Sums: Finite Form ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$ and $\sequence {y_k}_{1 \mathop \le k \mathop \le n}$ be finite sequences in $\GF$.
Then:
:$\ds \sum \limits_{k \mathop = 1}^n \size {x_k y_k} \le \paren {\sum_{k \mathop = 1}^n \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop = 1}^n \size {y_k}^q}^{1 / q}$
where the summations are finite.
=== Euclidean Plane ===
== Proof ==
== Parameter Inequalities ==
== Also known as ==
== Sources ==
* : $3$: Elementary Analytic Methods: $3.2$ Inequalities: Hölder's Inequality for Sums: $3.2.8$
* : $\S 36$: Inequalities: Hölder's Inequality: $36.8$
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Finite/Proof

== Hölder's Inequality for Finite Sums: Proof ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
Let $n \in \N_{>0}$ be a non-zero natural number.
Let $\sequence {x_k}_{1 \mathop \le k \mathop \le n}$ and $\sequence {y_k}_{1 \mathop \le k \mathop \le n}$ be finite sequences in $\GF$.
Then:
== Proof ==
Let $\sequence {x_k}_{k \mathop \in \N}$ and $\sequence {y_k}_{k \mathop \in \N}$ be infinite sequences in $\GF$ such that:
:$\forall m > n: x_m = y_m = 0$
Then we have:
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Formulation 1

== Theorem ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
Let $\mathbf x$ and $\mathbf y$ denote the vectors consisting of the sequences:
:$\mathbf x = \sequence {x_n} \in {\ell^p}_\GF$
:$\mathbf y = \sequence {y_n} \in {\ell^q}_\GF$
where ${\ell^p}_\GF$ denotes the $p$-sequence space in $\GF$.
Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.
Then:
:$\mathbf x \mathbf y \in {\ell^1}_\GF$
and:
:$\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$
where:
:$\mathbf x \mathbf y := \sequence {x_n y_n}_{n \mathop \in \N}$
:$\norm {\mathbf x \mathbf y}_1$ is the $1$-norm, also known as the taxicab norm.
=== Condition for Equality ===
== Proof ==
Define:
Then:
and similarly:
:$\norm {\mathbf v}_q = 1$
Then:
By the comparison test, it follows that:
Therefore:
Hence the result.
== Parameter Inequalities ==
== Also known as ==
== Also see ==
* Minkowski's Inequality for Sums
== Historical Note ==
== Sources ==
* : $1.1$: Basic Definitions
* : $\S 1.2$: Normed and Banach spaces. Normed spaces
* 
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Formulation 1/Equality

== Theorem ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let:
:$\mathbf x = \sequence {x_n} \in \ell^p$
:$\mathbf y = \sequence {y_n} \in \ell^q$
where $\ell^p$ denotes the $p$-sequence space.
Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$.
Hölder's Inequality for Sums states that:
:$\mathbf x \mathbf y \in \ell^1$
and:
:$\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$
We have that:
:$\norm {\mathbf x \mathbf y}_1 = \norm {\mathbf x}_p \norm {\mathbf y}_q$
:
:$\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
for some real constant $c$.
== Proof ==
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Formulation 2

== Theorem ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be sequences in $\GF$ such that $\ds \sum_{k \mathop \in \N} \size {x_k}^p$ and $\ds \sum_{k \mathop \in \N} \size {y_k}^q$ are convergent.
Then:
:$\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$
=== Finite Form ===
=== Condition for Equality ===
== Proof ==
Let:
We note that $X, Y \in \R_{\ge 0}$.
Let:
We have:
Then:
Hence the result.
== Parameter Inequalities ==
== Also known as ==
== Also see ==
* Minkowski's Inequality for Sums
== Historical Note ==
== Sources ==
* 
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Formulation 2/Equality

== Theorem ==
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
:$\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be (possibly infinite) convergent sequences in $\GF$.
Hölder's Inequality for Sums states that:
:$\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$
We have that:
:$\ds \sum_{k \mathop \in \N} \size {x_k y_k} = \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$
:
:$\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
for some real constant $c$.
== Proof ==
== Sources ==
* : $3$: Elementary Analytic Methods: $3.2$ Inequalities: Hölder's Inequality for Sums: $3.2.8$
* : $\S 36$: Inequalities: Hölder's Inequality
Category:Hölder's Inequality for Sums
Hölder's Inequality for Sums/Parameter Inequalities

== Hölder's Inequality for Sums: Parameter Inequalities ==
Statements of Hölder's Inequality for Sums will commonly insist that $p, q > 1$.
However, we note that from Positive Real Numbers whose Reciprocals Sum to 1 we have that if:
:$p, q > 0$
and:
:$\dfrac 1 p + \dfrac 1 q = 1$ 
it follows directly that $p, q > 1$.
== Sources ==
* : $3$: Elementary Analytic Methods: $3.2$ Inequalities: Hölder's Inequality for Sums: $3.2.8$
* : $\S 36$: Inequalities: Hölder's Inequality: $36.9$
* 
Category:Hölder's Inequality for Sums
Hölder Mean for Exponent -1 is Harmonic Mean

== Theorem ==
Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.
Then:
:$\map {M_{-1} } {x_1, x_2, \ldots, x_n} = \dfrac 1 {\dfrac 1 n \paren {\dfrac 1 {x_1} + \dfrac 1 {x_2} + \cdots + \dfrac 1 {x_n} } }$
which is the harmonic mean of $x_1, x_2, \ldots, x_n$.
== Proof ==
Recall the definition of the Hölder mean with exponent $p$:
:$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$
Then:
which is the harmonic mean by definition.
== Sources ==
* : $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Generalized Mean: $3.1.20$
Category:Hölder Mean
Category:Harmonic Mean
Hölder Mean for Exponent 1 is Arithmetic Mean

== Theorem ==
Let $x_1, x_2, \ldots, x_n \in \R_{\ge 0}$ be real numbers.
For $p \in \R_{\ne 0}$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$.
Then:
:$\map {M_1} {x_1, x_2, \ldots, x_n} = \dfrac {x_1 + x_2 + \cdots + x_n} n$
which is the arithmetic mean of $x_1, x_2, \ldots, x_n$.
== Proof ==
Recall the definition of the Hölder mean with exponent $p$:
:$\ds \map {M_p} {x_1, x_2, \ldots, x_n} = \paren {\frac 1 n \sum_{k \mathop = 1}^n {x_k}^p}^{1 / p}$
Then:
which is the arithmetic mean by definition.
== Sources ==
* : $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Generalized Mean: $3.1.19$
Category:Hölder Mean
Category:Arithmetic Mean
ISBN-10/Examples/0-19-853453-1

== Example of ISBN-$10$ ==
The book whose ISBN-$10$ code begins:
:$0$-$19$-$853453$
has a check digit of $1$.
== Proof ==
The check digit $d$ of an ISBN-$10$ is calculated by definition as:
:$d = \ds \sum_{k \mathop = 1}^9 k d_k \pmod {11}$
which may or may not need to be translated into $X$ if it equals $10$.
So, we have:
== Sources ==
* : Chapter $6$: Error-correcting codes
Category:Examples of International Standard Book Numbers
ISBN-10 is Error-Correcting Code/Transmission Error

== Theorem ==
ISBN-$10$ is an error-correcting code in the following sense:
If an error has been made in any one of the first $9$ digits, the check digit will be wrong.
== Proof ==
Let $S$ denote an ISBN-$10$ whose $k$th digit is $d_k$.
Let $d_S$ denote the check digit of $S$.
Let $S'$ denote the ISBN-$10$ $S$ whose $n$th digit has been transmitted incorrectly, as $d'_n$.
Let $d'_S$ denote the check digit calculated on $S'$ according to the algorithm via which calculated $d_S$ on $S$.
It will be demonstrated that $d'_S \ne d_S$.
By definition of ISBN-$10$:
:$d_S = \ds \sum_{k \mathop = 1}^9 k d_k \pmod {11}$
where $d_k$ is assigned the symbol $X$ if it calculates to $10$.
Then:
:$d'_S = \ds \sum_{k \mathop = 1}^9 k d_k - n d_n + n d'_n \pmod {11}$
and so:
:$d'_S - d_S = n \paren {d'_n - d_n} \pmod {11}$
It follows that:
:$d'_S - d_S \ne 0$
and so the check digit on $d_S$ is the incorrect check digit for $S'$.
== Sources ==
* : Chapter $6$: Error-correcting codes
Category:ISBN-10 is Error-Correcting Code
Ideal Contained in Finite Union of Prime Ideals

== Theorem ==
Let $A$ be a commutative ring with unity.
Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be prime ideals.
Let $\mathfrak a \subseteq \ds \bigcup_{i \mathop = 1}^n \mathfrak p_i$ be an ideal contained in their union.
Then $\mathfrak a \subseteq \mathfrak p_i$ for some $i \in \{1, \ldots, n\}$.
== Proof ==
The proof goes by induction on $n$.
For $n = 1$, the statement is trivial.
Let $n > 1$.
Suppose that the statement holds for $n - 1$ prime ideals.
It is to be shown that the statement holds for $n$ prime ideals.
 that $\mathfrak a \nsubseteq \mathfrak p_i$ for all $i \in \set {1, \ldots, n}$.
By the induction hypothesis, for all $i$ we have:
:$\mathfrak a \nsubseteq \ds \bigcup_{j \mathop \ne i} \mathfrak p_j$
For all $i$, let $x_i \in \mathfrak a \setminus \ds \bigcup_{j \mathop \ne i} \mathfrak p_j$.
Suppose there exists $i$ such that $x_i \notin \mathfrak p_i$.
Then:
:$x_i \in \mathfrak a \setminus \ds \bigcup_j \mathfrak p_j$
and we conclude that:
:$\mathfrak a \nsubseteq \ds \bigcup_j \mathfrak p_j$
which is a contradiction.
The other possibility is that for all $i$:
:$x_i \in \mathfrak p_i$
Let:
:$y = \ds \sum_{i \mathop = 1}^n \prod_{\substack{j \mathop = 1 \\ j \mathop\neq i}}^n x_j$
be defined as a summation of products.
All terms except the $i$th are in $\mathfrak p_i$.
Since Sum of an Element in an Ideal and an Element not in the Ideal is not in the Ideal:
:$\forall i:\quad y \notin \mathfrak p_i$
Thus:
:$y \in \mathfrak a \setminus \bigcup_j \mathfrak p_j$
This also leads to a contradiction.
== Also see ==
=== Generalization ===
Let $G,H,K$ be groups.
If $G \subseteq H \cup K$, then $G \subseteq H$ or $G \subseteq K$.
=== Proof ===
 that neither $G \subseteq H$ nor $G \subseteq K$.
Then there exists $x \in G \setminus H$.
By assumption $G \subseteq H \cup K$, then $x \in H \cup K$, so $x \in K$.
Similarly, there exists $y \in G \setminus K$.
By assumption $G \subseteq H \cup K$, then $y \in H \cup K$, so $y \in H$.
Since $x,y \in G$, we have $xy \in G$, so $xy \in H \cup K$.
 $xy \in H$.
Since $xy \in H, y \in H$, we have $x = (xy)y^{-1} \in H$,
which is a contradiction.
== Sources ==
* : Chapter $1$: Rings and Ideals: $\S$ Operations on Ideals: Proposition $1.11$
* : Exercises to $\S16$ Regular sequences and the Koszul complex: Exercise $16.8$
Category:Ideal Theory
Ideal Contains Extension of Contraction

== Theorem ==
Let $A$ and $B$ be commutative rings with unity.
Let $f : A \to B$ be a ring homomorphism.
Let $\mathfrak b \subseteq B$ be an ideal.
Then $\mathfrak b$ contains the extension of its contraction by $f$:
:$\mathfrak b^{ce} \subseteq \mathfrak b$
== Proof ==
Since $\mathfrak b^{ce}$ is generated by $f \sqbrk {\mathfrak b ^c}$ according to , it is contained in $\mathfrak b$
Category:Ideal Theory
Category:Commutative Algebra
Category:Ring Theory