nyuuzyou/wikis huggingface dataset:
Equivalence of Definitions of Cosecant of Angle

== Theorem ==
Let $\theta$ be an angle.
=== Definition from Triangle ===
=== Definition from Circle ===
== Proof ==
=== Definition from Triangle implies Definition from Circle ===
Let $\csc \theta$ be defined as $\dfrac {\text{Hypotenuse}} {\text{Opposite}}$ in a right triangle.
Consider the triangle $\triangle OAB$.
By construction, $\angle OAB$ is a right angle.
From Parallelism implies Equal Alternate Angles:
:$\angle OBA = \theta$
Thus:
That is:
:$\csc \theta = OB$
=== Definition from Circle implies Definition from Triangle ===
Let $\csc \theta$ be defined as the length of $OB$ in the triangle $\triangle OAB$.
Compare $\triangle OAB$ with $\triangle ABC$ in the diagram above.
From Parallelism implies Equal Alternate Angles:
:$\angle OBA = \theta$
We have that:
:$\angle CAB = \angle OBA = \theta$
:$\angle ABC = \angle OAB$ which is a right angle
Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAB$ and $\triangle ABC$ are similar.
By definition of similarity:
That is:
:$\dfrac {\text{Hypotenuse} } {\text{Opposite} } = \csc \theta$
Category:Cosecant Function
Equivalence of Definitions of Cosine of Angle

== Theorem ==
Let $\theta$ be an angle.
=== Definition from Triangle ===
=== Definition from Circle ===
== Proof ==
=== Definition from Triangle implies Definition from Circle ===
Let $\cos \theta$ be defined as $\dfrac {\text {Adjacent}} {\text {Hypotenuse}}$ in a right triangle.
Consider the triangle $\triangle OAP$.
By construction, $\angle OAP$ is a right angle.
From Parallelism implies Equal Alternate Angles:
:$\angle OPA = \theta$
Thus:
That is:
:$\cos \theta = AP$
=== Definition from Circle implies Definition from Triangle ===
Let $\cos \theta$ be defined as the length of $AP$ in the triangle $\triangle OAP$.
Compare $\triangle OAP$ with $\triangle ABC$ in the diagram above.
From Parallelism implies Equal Alternate Angles:
:$\angle OPA = \theta$
We have that:
:$\angle CAB = \angle OPA = \theta$
:$\angle ABC = \angle OAP$ which is a right angle
Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAP$ and $\triangle ABC$ are similar.
By definition of similarity:
That is:
:$\dfrac {\text {Adjacent} } {\text {Hypotenuse} } = \cos \theta$
Category:Cosine Function
Equivalence of Definitions of Cotangent of Angle

== Theorem ==
Let $\theta$ be an angle.
=== Definition from Triangle ===
=== Definition from Circle ===
== Proof ==
=== Definition from Triangle implies Definition from Circle ===
Let $\cot \theta$ be defined as $\dfrac {\text{Adjacent}} {\text{Opposite}}$ in a right triangle.
Consider the triangle $\triangle OAB$.
By construction, $\angle OAB$ is a right angle.
From Parallelism implies Equal Alternate Angles:
:$\angle OBA = \theta$
Thus:
That is:
:$\cot \theta = AB$
=== Definition from Circle implies Definition from Triangle ===
Let $\cot \theta$ be defined as the length of $AB$ in the triangle $\triangle OAB$.
Compare $\triangle OAB$ with $\triangle ABC$ in the diagram above.
From Parallelism implies Equal Alternate Angles:
:$\angle OBA = \theta$
We have that:
:$\angle CAB = \angle OBA = \theta$
:$\angle ABC = \angle OAB$ which is a right angle
Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAB$ and $\triangle ABC$ are similar.
By definition of similarity:
That is:
:$\dfrac {\text{Adjacent} } {\text{Opposite} } = \cot \theta$
Category:Cotangent Function
Equivalence of Definitions of Countable Set

== Theorem ==
Let $S$ be a set.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition 1 implies Definition 2 ===
Let $S$ be a countable set by Definition 1.
Then there is an injection $f: S \to \N$.
By Law of Excluded Middle, $S$ is finite or infinite.
If $S$ is finite, then it trivially satisfies Definition 2.
If $S$ is infinite, then it is countably infinite by Infinite Set of Natural Numbers is Countably Infinite, and thus satisfies Definition 2.
=== Definition 2 implies Definition 1 ===
Let $S$ be a countable set by Definition 2.
Then $S$ is finite or countably infinite.
If $S$ is countably infinite, then there is a bijection $f: S \to \N$, which is injective by definition.
If $S$ is finite, then for some natural number $n$ there is a bijection $f: S \to \N_n$, where $\N_n$ is the initial segment of $\N$ determined by $n$.
Let $f': S \to \N$ be the extension of $f$ to $\N$.
Then $f'$ is an injection.
Thus in either case $S$ is countable by Definition 1.
== Sources ==
* : $1$: Set Theory and Logic: $\S 7$: Countable and Uncountable Sets: Theorem $7.1$
Category:Countable Sets
Equivalence of Definitions of Countably Compact Space

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
=== Definition 4 ===
=== Definition 5 ===
== Proof ==
=== $(1) \iff (2)$: Countably Compact Space satisfies Countable Finite Intersection Axiom ===
=== $(1) \implies (4)$: Countably Infinite Set in Countably Compact Space has $\omega$-Accumulation Point ===
This is proven in Countably Infinite Set in Countably Compact Space has $\omega$-Accumulation Point.
=== $(4) \implies (3)$ ===
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $S$.
Let $A \subseteq S$ be the range of $\sequence {x_n}$.
If $A$ is finite, then consider the equality:
:$\ds \N = \bigcup_{y \mathop \in A} \set {n \in \N: x_n = y}$
Therefore, there exists a $y \in A$ such that $\set {n \in \N: x_n = y}$ is an infinite set.
Hence, $y$ is an accumulation point of $\sequence {x_n}$.
Otherwise, $A$ is countably infinite.
By assumption, $A$ has an $\omega$-accumulation point $x \in S$.
It follows that $x$ is an accumulation point of $\sequence {x_n}$.
=== $(5) \implies (4)$ ===
=== $(3) \implies (1)$ ===
We use a Proof by Contradiction.
 $S$ has a countably infinite open cover $\set {U_n: n \in \N}$ which does not have a finite subcover.
Then, using the axiom of countable choice, we can obtain a sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$ such that:
:$\ds \forall n \in \N: x_n \notin \bigcup_{k \mathop = 0}^n U_k$
Let $x \in S$.
Then, by the definition of a cover, there exists an $n \in \N$ such that $x \in U_n$.
By construction:
:$U_n \subseteq \set {x_0, x_1, \ldots, x_{n - 1} }$
Hence $x$ is not an accumulation point of $\sequence {x_n}$.
=== $(4) \implies (5)$ ===
Follows directly from Infinite Set has Countably Infinite Subset and the definition of an $\omega$-accumulation point.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $3$: Compactness: Global Compactness Properties
Category:Countably Compact Spaces
Equivalence of Definitions of Countably Infinite Set

== Theorem ==
Let $S$ be a set.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
From Integers are Countably Infinite there is a bijection between $\Z$, the set of integers, and $\N$, the set of natural numbers.
Let $h: \N \to \Z$ be such a bijection.
Let $f: S \to \N$ be a bijection.
From Composite of Bijections is Bijection:
:$h \circ f: S \to \Z$ is a bijection.
Similarly, let $g: S \to \Z$ be a bijection.
By Inverse of Bijection is Bijection, $h^{-1}: \Z \to \N$ is a bijection.
Again from Composite of Bijections is Bijection:
:$h^{-1} \circ g: S \to \N$ is a bijection.
Hence the result.
Category:Countable Sets
Equivalence of Definitions of Cover of Set

== Theorem ==
Let $S$ be a set.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition 1 Implies Definition 2 ===
Let $\CC$ be a set of sets such that:
:$\ds S \subseteq \bigcup \CC$
where $\bigcup \CC$ denotes the union of $\CC$.
By definition of subset:
:$\forall s \in S : s \in \ds \bigcup \CC$
By definition of set union:
:$\forall s \in S : \exists C \in \CC : s \in C$
=== Definition 2 Implies Definition 1 ===
Let $\CC$ be a set of sets such that:
:$\forall s \in S : \exists C \in \CC : s \in C$
From Set is Subset of Union (General Result):
:$\forall C \in \CC : C \subseteq \ds \bigcup \CC$
By definition of subset:
:$\forall s \in S : s \in \ds \bigcup \CC$
By definition of subset:
:$S \subseteq \ds \bigcup \CC$
Category:Covers
Equivalence of Definitions of Curvature

== Theorem ==
=== Reciprocal of Radius of Osculating Circle ===
=== Whewell Form ===
=== Cartesian Form ===
=== Parametric Form ===
== Proof ==
=== Whewell Form to Cartesian Form ===
=== Whewell Form to Parametric Polar Form ===
Category:Curvature
Equivalence of Definitions of Curvature/Whewell Form to Cartesian Form

== Theorem ==
=== Whewell Form ===
=== Cartesian Form ===
== Proof ==
Consider the curvature of a curve $C$ at a point $P$ expressed as a Whewell equation:
:$\kappa = \dfrac {\d \psi} {\d s}$
where:
:$\psi$ is the turning angle of $C$
:$s$ is the arc length of $C$.
The derivative of the tangent of the turning angle $\psi$ at a point $P$ $\psi$ is also the derivative of the tangent to $C$ at $P$, again $\psi$.
That is:
We also have that:
Then:
which is the Cartesian form of curvature as required.
Category:Curvature
Equivalence of Definitions of Curvature/Whewell Form to Parametric Polar Form

== Theorem ==
=== Whewell Form ===
=== Parametric Polar Form ===
== Proof ==
Consider the curvature of a curve $C$ at a point $P$ expressed as a Whewell equation:
:$\kappa = \dfrac {\d \psi} {\d s}$
where:
:$\psi$ is the turning angle of $C$
:$s$ is the arc length of $C$.
Let us consider $C$ expressed in cartesian form:
Then:
In Whewell form:
Let:
We have:
:$\map \kappa t = \dfrac {\d \psi} {\d g} \dfrac {\d g} {\d t} \dfrac {\d t} {\d s}$
Then:
Category:Curvature
Equivalence of Definitions of Cyclic Group

== Theorem ==
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== $(1)$ implies $(2)$ ===
Let $G$ be a Cyclic Group by definition 1.
Then by definition:
:$(1): \quad g \in G$
:$(2): \quad$ every element of $G$ is expressible as a power of $g$.
From $(1)$, it follows from that:
:$\gen g \subseteq G$
From $(2)$ it follows that:
:$G \subseteq \gen g$
Thus:
:$G = \gen g$
and $G$ is a Cyclic Group by definition 2.
=== $(2)$ implies $(1)$ ===
Let $G$ be a Cyclic Group by definition 2.
Then by definition:
:$G = \gen g$
Thus as $g^1 \in \gen g$
:$g \in G$
and by definition of generator:
:$\forall h \in \gen g: h = g^n$
for some $n \in \Z$.
Thus $G$ is a Cyclic Group by definition 1.
== Sources ==
* : $\S 39.1$ Cyclic groups
Category:Cyclic Groups
Equivalence of Definitions of Deficient Number

== Theorem ==
The following definitions of a deficient number are equivalent:
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
== Proof ==
By definition of abundance:
:$\map A n = \map {\sigma_1} n - 2 n$
By definition of divisor sum function:
:$\map {\sigma_1} n$ is the sum of all the divisors of $n$.
Thus $\map {\sigma_1} n - n$ is the aliquot sum of $n$.
Hence the result.
Category:Deficient Numbers
Equivalence of Definitions of Derivative

== Theorem ==
Let $I$ be an open real interval.
Let $f: I \to \R$ be a real function defined on $I$.
Let $\xi \in I$ be a point in $I$.
=== Definition 1 ===
</onlyinclude>
=== Definition 2 ===
== Proof ==
== Sources ==
* : $\S 2.1$, Appendix $A$: Alternate Form of the Derivative
Category:Differential Calculus
Category:Analysis
Equivalence of Definitions of Dipper Semigroup

== Theorem ==
Let $m \in \N$ be a natural number.
Let $n \in \N_{>0}$ be a non-zero natural number.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
We have established in Dipper Semigroup is Commutative Semigroup that $\struct {\N_{< \paren {m + n} }, \oplus_{m, n} }$ is a (commutative) semigroup.
Let $\phi_{m, n}$ be the canonical surjection from $\N$ onto $\map D {m, n}$.
It will be established that the restriction of $\phi_{m, n}$ to $\N_{< \paren {m + n} }$ is an isomorphism from the semigroup $\struct {\N_{< \paren {m + n} }, +_{m, n} }$ onto $\struct {\map D {m, n}, \oplus_{m, n} }$.
From Dipper Relation is Equivalence Relation we have that:
:$\RR_{m, n}$ is an equivalence relation
By definition of canonical surjection:
:$\phi_{m, n}: \N_{< \paren {m + n} } \to \map D {m, n}: \map {\phi_{m, n} } x = \eqclass x {\RR_{m, n} }$
By definition $\phi_{m, n}$ is indeed a surjection.
Then we have:
Hence $\phi_{m, n}$ is an injection.
Thus $\phi_{m, n}$ is by definition a bijection.
It remains to show that $\phi_{m, n}$ has the morphism property.
Let $x, y \in \N_{< \paren {m + n} }$ be arbitrary.
We have:
== Examples ==
== Sources ==
* : Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.6 \ \text {(b)}$
Category:Equivalence of Definitions of Dipper Semigroup
Category:Dipper Semigroups
Category:Semigroup Isomorphisms
Equivalence of Definitions of Dipper Semigroup/Examples/m = 0

== Example of Use of Equivalence of Definitions of Dipper Semigroup ==
Let $n \in \Z$ be integers such that $n > 0$.
Let $\N_{<n}$ denote the initial segment of the natural numbers:
:$\N_{<n} := \set {0, 1, \ldots, n - 1}$
Let $+_n$ be the operation on $\N_{<n}$ defined as:
:$\forall a, b \in \N_{<n}: a +_n b = a + b - k n$
where $k$ is the largest integer satisfying:
:$k n \le a + b$
Let $\RR_n$ be the relation on $\N$ defined as:
:$\forall x, y \in \N: x \mathrel {\RR_n} y \iff n \divides \size {x - y}$
Let $\map D n := \N / \RR_n$ be the quotient set of $\N$ induced by $\RR_n$.
Let $\oplus_n$ be the operation induced on $\map D n$ by addition on $\N$.
Then $\struct {\N_{<n}, +_n}$ is isomorphic to $\struct {\map D n, \oplus_n}$.
== Proof ==
This is an instance of Equivalence of Definitions of Dipper Semigroup where $m = 0$.
== Sources ==
* : Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.6 \ \text {(b)}$
Category:Equivalence of Definitions of Dipper Semigroup
Equivalence of Definitions of Disconnected Space

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
=== Definition $1$ ===
=== Definition $2$ ===
== Proof ==
By definition $1$, $T$ is disconnected $T$ is not connected.
By definition of connected topological space:
Hence precisely a disconnected space by definition $2$.
Category:Disconnected Spaces
Equivalence of Definitions of Distance to Nearest Integer Function

== Theorem ==
The following definitions of the distance to nearest integer function $\norm \cdot: \R \to \closedint 0 {\dfrac 1 2}$ are equivalent:
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
Let $\alpha \in \R$, $n \in \Z$.
From Real Number is between Floor Functions:
:$\floor \alpha \le \alpha < \floor \alpha + 1$
For any $n < \floor \alpha \le \alpha$:
For any $n > \floor \alpha + 1 > \alpha$:
Thus:
which shows that the definitions are indeed equivalent.
Category:Real Analysis
Equivalence of Definitions of Distributive Lattice

== Theorem ==
Let $\struct {S, \vee, \wedge, \preceq}$ be a lattice.
=== Definition 1===
=== Definition 2===
== Proof ==
In what follows it is shown that if one of the axioms distributive lattice axioms holds then they all hold.
=== 1 is equivalent to 1' ===
By applying Meet is Commutative several times, we have:
which (after renaming variables as appropriate) establishes the equivalence.
=== 2 is equivalent to 2' ===
By applying Join is Commutative several times, we have:
which (after renaming variables as appropriate) establishes the equivalence.
=== 1 implies 2 ===
Suppose that $(1)$ holds, and hence $(1')$ as well.
=== 2 implies 1 ===
By inspection, aided by Dual Pairs (Order Theory), we see that $(2)$ is dual to $(1)$.
Thus by Global Duality, $(2)$ implies $(1)$ as soon as $(1)$ implies $(2)$.
That direction was already established above.
== Also see ==
* Definition:Distributive Lattice
Category:Distributive Lattices
Equivalence of Definitions of Division Ring

== Theorem ==
A division ring is a ring with unity $\struct {R, +, \circ}$ with the following properties:
=== Definition 1 ===
=== Definition 2 ===
=== Definition 3 ===
== Proof ==
In the following, let:
:$0_R$ denote the zero of $R$
:$1_R$ denote the unity of $R$
:$R^*$ denote the set of elements of $R$ without $0_R$.
=== $(1)$ implies $(2)$ ===
Let $\struct {R, +, \circ}$ be a division ring by definition 1.
Then by definition:
:$\forall x \in R^*: \exists! x^{-1} \in R^*: x^{-1} \circ x = x \circ x^{-1} = 1_R$
That is, $x^{-1}$ is the (unique) product inverse of $x$.
Thus, by definition, $x$ is a unit of $R$.
Thus $\struct {R, +, \circ}$ is a division ring by definition 2.
=== $(2)$ implies $(1)$ ===
Let $\struct {R, +, \circ}$ be a division ring by definition 2.
Then by definition:
:$\forall x \in \R^*: x$ is a unit.
Thus, by definition, $x$ has a product inverse $x^{-1}$.
From Product Inverse in Ring is Unique it follows that:
:$\forall x \in R^*: \exists! x^{-1} \in R^*: x^{-1} \circ x = x \circ x^{-1} = 1_R$
Thus $\struct {R, +, \circ}$ is a division ring by definition 1.
=== $(2)$ is equivalent to $(3)$ ===
By definition, a unit of $R$ is an element of $R$ which has a product inverse.
Also by definition, a proper element of $R$ is a non-zero element of $R$ which does not have a product inverse.
Hence:
:a ring with unity whose non-zero elements are all units
and:
:a ring with unity whose non-zero elements are all not proper elements
are the same thing.
Category:Division Rings
Equivalence of Definitions of Dominate (Set Theory)

== Theorem ==
Let $S, T$ be sets.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition 1 implies Definition 2 ===
Let $f: S \to T$ be an injection.
By Injection to Image is Bijection, $f$ is a bijection from $S$ to the image of $f$.
=== Definition 2 implies Definition 1 ===
Let $T' \subseteq T$ such that there exists a bijection $f: S \to T'$.
Let $i: T' \to T$ be the inclusion of $T'$ in $T$.
Then by Composite of Injections is Injection, $i \circ f: S \to T$ is an injection.
== Sources ==
* : Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability
Category:Set Theory
Equivalence of Definitions of Dot Product

== Theorem ==
Let $\mathbf a$ and $\mathbf b$ be vectors in the real Euclidean space $\R^n$.
=== General Context ===
=== Definition by Cosine ===
== Proof ==
=== General Context implies Definition by Cosine ===
Let $\mathbf a \cdot \mathbf b$ be a dot product in its general context.
From Cosine Formula for Dot Product:
:$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \cos \theta$
where $\theta$ is the angle between $\mathbf v$ and $\mathbf w$.
Thus $\cdot$ is a dot product by cosine definition.
=== Definition by Cosine implies General Context ===
Let $\mathbf a \cdot \mathbf b$ be a dot product by cosine definition.
Let $\mathbf a$ and $\mathbf b$ be expressed in the form:
Then we have:
Thus $\cdot$ is a dot product in its general context.
== Sources ==
* : Chapter $\text {II}$: The Products of Vectors: $2$. The Scalar Product: $(2.10)$
* : Chapter $\text I$: Vector Algebra: $\S 2$.
* : Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product: $(1.3)$
Category:Dot Product
Equivalence of Definitions of Ellipse

== Theorem ==
The following definitions of an ellipse are equivalent:
=== Equidistance Property ===
=== Focus-Directrix Property ===
== Proof ==
Let $K$ be an ellipse aligned in a cartesian plane in reduced form.
Thus its foci are at $\tuple {\mathop \pm c, 0}$.
Let:
:the major axis of $K$ have length $2 a$
:the minor axis of $K$ have length $2 b$.
From Equation of Ellipse in Reduced Form, the equation of $K$ is:
:$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
This has been derived from the equidistance property of $K$.
From Focus of Ellipse from Major and Minor Axis:
:$a^2 - c^2 = b^2$
and so $K$ can be expressed as:
:$(1): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2 - c^2} = 1$
It remains to be shown that $K$ possesses the focus-directrix property.
:
Let $D$ be the straight line whose equation is $x = \dfrac {a^2} c$.
It will be shown that the locus of points $P = \tuple {x, y}$ such that:
:the distance from $P$ to $F_1$ is $\dfrac c a$ of the distance from $P$ to $D$
is precisely equation $(1)$.
We have that:
:$P F_2 = \epsilon \paren {d - x}$
where:
:$\epsilon = \dfrac c a$
:$d = \dfrac {a^2} c$
Thus:
Category:Ellipses
Equivalence of Definitions of Equivalence Relation

== Theorem ==
Let $\RR$ be a relation on a set $S$.
=== Definition 1 ===
=== Definition 2 ===
== Proof ==
=== Definition 1 implies Definition 2 ===
Let $\RR$ be an equivalence relation by definition 1.
By definition, $\RR$ is reflexive, symmetric and transitive.
From Relation Contains Diagonal Relation iff Reflexive:
:$\Delta_S \subseteq \RR$
From Relation equals Inverse iff Symmetric:
:$\RR^{-1} = \RR$
and so by definition of set equality:
:$\RR^{-1} \subseteq \RR$
From Relation contains Composite with Self iff Transitive:
:$\RR \circ \RR \subseteq \RR$
From Union is Smallest Superset:
:$\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$
Thus $\RR$ is an equivalence relation by definition 2.
=== Definition 2 implies Definition 1 ===
Let $\RR$ be an equivalence relation by definition 2.
That is:
:$\Delta_S \cup \RR^{-1} \cup \RR \circ \RR \subseteq \RR$
By Union is Smallest Superset:
:$(1): \quad \Delta_S \subseteq \RR$
:$(2): \quad \RR^{-1} \subseteq \RR$
:$(3): \quad \RR \circ \RR \subseteq \RR$
From Relation Contains Diagonal Relation iff Reflexive, $(1)$ gives directly that $\RR$ is reflexive.
From Inverse Relation Equal iff Subset, $(2)$ gives that $\RR^{-1} = \RR$.
So from Relation equals Inverse iff Symmetric $\RR$ is symmetric.
From Relation contains Composite with Self iff Transitive, $(3)$ gives that $\RR$ is transitive.
So $\RR$ has been shown to be reflexive, symmetric and transitive.
Thus $\RR$ is an equivalence relation by definition 1.
== Sources ==
* : Chapter $\text{I}$: Sets and Functions: Problem $\text{AA}$: Relations
Category:Equivalence Relations
Equivalence of Definitions of Equivalent Division Ring Norms

== Theorem ==
Let $R$ be a division ring.
Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be norms on $R$.
Let $d_1$ and $d_2$ be the metrics induced by the norms $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively.
=== Topologically Equivalent ===
=== Convergently Equivalent ===
=== Null Sequence Equivalent ===
=== Open Unit Ball Equivalent ===
=== Norm is Power of Other Norm ===
=== Cauchy Sequence Equivalent ===
== Proof ==
=== Topologically Equivalent implies Convergently Equivalent ===
=== Convergently Equivalent implies Null Sequence Equivalent ===
=== Null Sequence Equivalent implies Open Unit Ball Equivalent ===
=== Open Unit Ball Equivalent implies Norm is Power of Other Norm ===
=== Norm is Power of Other Norm implies Topologically Equivalent ===
=== Norm is Power of Other Norm implies Cauchy Sequence Equivalent ===
=== Cauchy Sequence Equivalent implies Open Unit Ball Equivalent ===
== Sources ==
* : $\S 3.1$ Absolute Values on $\Q$: Lemma $3.1.2$ and Problem $66$
* : $\S 1.2$ Normed Fields: Proposition $1.10$ and Exercise $13$
Category:Normed Division Rings
Category:Norm Theory
Category:Equivalence of Definitions of Equivalent Division Ring Norms