Tarski's Undefinability Theorem

Let $\ZZ$ be the standard structure $\struct {\Z, +, \cdot, s, <, 0}$ for the language of arithmetic. Let $\operatorname {Th}_\ZZ$ be the sentences which are true in $\ZZ$. Let $\Theta$ be the set of Gödel numbers of those sentences in $\operatorname {Th}_\ZZ$. $\Theta$ is not definable in $\operatorname {Th}_\ZZ$.


Equivalence of Definitions of Limit Point

{{TFAE|def = Limit Point of Set|view = limit point}} That is, let $T = \struct {S, \tau}$ be a topological space. Let $H \subseteq S$. Let $H^{\complement}$ denote the relative complement of $H$ in $S$. Then the following conditions are equivalent for any point $x \in S$: :$(1): \quad$ Every open neighborhood $U$ of $x$ satisfies $H \cap \paren {U \setminus \set x} \ne \O$. :$(2): \quad x$ belongs to the closure of $H$ but is not an isolated point of $H$. :$(3): \quad x$ is an adherent point of $H$ but is not an isolated point of $H$. :$(4): \quad H^{\complement} \cup \set x$ is ''not'' a neighborhood of $x$.


Denseness Preserved in Coarser Topology

Let $T = \struct {S, \tau}$ be a topological space. Let $T' = \struct {S, \tau'}$ be another topological space on $S$ such that $\tau'$ is coarser than $\tau$. Let $H \subseteq S$ be everywhere dense in $T$. Then $H$ is also everywhere dense in $T'$.


Rationals are Dense in Compact Complement Topology

Let $T = \struct {\R, \tau^*}$ be the compact complement topology on $\R$. Let $\Q$ be the set of rational numbers. Then $\Q$ is everywhere dense in $T$.


Compact Complement Topology is Separable

Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$. Then $T$ is a separable space.


Either-Or Topology is Locally Connected

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a locally connected space.


Either-Or Topology is not Locally Arc-Connected

Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is not a locally arc-connected space.


Fort Space is Scattered

Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$. Then $T$ is a scattered space.


Dense-in-itself Subset of T1 Space is Infinite

Let $T = \struct {S, \tau_p}$ be a topological space which is $T_1$ (Fréchet). Let $H \subseteq T$ be dense-in-itself. Then $H$ is infinite.


Equal Chords in Circle

In a circle, equal chords are equally distant from the center, and chords that are equally distant from the center are equal in length. {{:Euclid:Proposition/III/14}}


Relative Lengths of Chords of Circles

Of chords in a circle, the diameter is the greatest, and of the rest the nearer to the center is always greater than the more remote. {{:Euclid:Proposition/III/15}}


Construction of Tangent from Point to Circle

From a given point outside a given circle, it is possible to draw a tangent to that circle. {{:Euclid:Proposition/III/17}}


Construction of Circle from Segment


Equal Angles in Equal Circles

In equal circles, equal angles stand on equal arcs, whether at the center or at the circumference of those circles. {{:Euclid:Proposition/III/26}}


Angles on Equal Arcs are Equal

In equal circles, angles standing on equal arcs are equal to one another, whether at the center or at the circumference of those circles. {{:Euclid:Proposition/III/27}}


Bisection of Arc

It is possible to bisect an arc of a circle. {{:Euclid:Proposition/III/30}}


Relative Sizes of Angles in Segments

In a circle: : the angle in a semicircle is right : the angle in a segment greater than a semicircle is acute : the angle in a segment less than a semicircle is obtuse. Further: : the angle of a segment greater than a semicircle is obtuse : the angle of a segment less than a semicircle is acute. {{:Euclid:Proposition/III/31}}


Angles made by Chord with Tangent

Let $EF$ be a tangent to a circle $ABCD$, touching it at $B$. Let $BD$ be a chord of $ABCD$. Then: : the angle in segment $BCD$ equals $\angle DBE$ and:  : the angle in segment $BAD$ equals $\angle DBF$. {{:Euclid:Proposition/III/32}}


Construction of Segment on Given Line Admitting Given Angle

On any given line segment, it is possible to describe a segment of a circle which admits an angle equal to any given rectilineal angle. {{:Euclid:Proposition/III/33}}


Tangent Secant Theorem

Let $D$ be a point outside a circle $ABC$. Let $DB$ be tangent to the circle $ABC$. Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$. Then $DB^2 = AD \cdot DC$. {{:Euclid:Proposition/III/36}}


Identity Element of Natural Number Multiplication is One

Let $1$ be the element one of $\N$. Then $1$ is the identity element of multiplication: :$\forall n \in \N: n \times 1 = n = 1 \times n$


Secant Secant Theorem

Let $C$ be a point external to a circle $ABED$. Let $CA$ and $CB$ be straight lines which cut the circle at $D$ and $E$ respectively. Then: : $CA \cdot CD = CB \cdot CE$


Converse of Tangent Secant Theorem

Let $D$ be a point outside a circle $ABC$. Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$. Let $DB$ intersect the circle at $B$ such that $DB^2 = AD \cdot DC$. Then $DB$ is tangent to the circle $ABC$. {{:Euclid:Proposition/III/37}}


Fitting Chord Into Circle

Into a given circle, it is possible to fit a chord equal to a given line segment which is not greater than the diameter of the circle. {{:Euclid:Proposition/IV/1}}


Construction of Isosceles Triangle whose Base Angle is Twice Apex

It is possible to construct an isosceles triangle such that each of the angles at the base is twice that at the apex. {{:Euclid:Proposition/IV/10}}


Inscribing Regular Pentagon in Circle

In a given circle, it is possible to inscribe a regular pentagon. {{:Euclid:Proposition/IV/11}}


Inscribing Circle in Regular Pentagon

In any given regular pentagon it is possible to inscribe a circle. {{:Euclid:Proposition/IV/13}}


Inscribing Regular Hexagon in Circle

In a given circle, it is possible to inscribe a regular hexagon. {{:Euclid:Proposition/IV/15}}


Multiplication of Numbers is Left Distributive over Addition

{{:Euclid:Proposition/V/1}} That is, if $ma, mb, mc$ etc. be any equimultiples of $a, b, c$ etc., then: :$m a + m b + m c + \cdots = m \paren {a + b + c + \cdots }$


Multiplication of Numbers is Right Distributive over Addition

{{:Euclid:Proposition/V/2}} That is: :$ma + na + pa + \cdots = \paren {m + n + p + \cdots} a$


Multiples of Terms in Equal Ratios

Let $a, b, c, d$ be quantities. Let $a : b = c : d$ where $a : b$ denotes the ratio between $a$ and $b$. Then for any numbers $m$ and $n$: :$m a : n b = m c : n d$ {{:Euclid:Proposition/V/4}}


Ratios of Equal Magnitudes

{{:Euclid:Proposition/V/7}} That is: :$a = b \implies a : c = b : c$ :$a = b \implies c : a = c : b$


Relative Sizes of Ratios on Unequal Magnitudes

{{:Euclid:Proposition/V/8}} That is: :$a > b \implies a : c > b : c$ :$a > b \implies c : a < c : b$


Magnitudes with Same Ratios are Equal

{{:Euclid:Proposition/V/9}} That is: :$a : c = b : c \implies a = b$ :$c : a = c : b \implies a = b$


Relative Sizes of Magnitudes on Unequal Ratios

{{:Euclid:Proposition/V/10}} That is: :$a : c > b : c \implies a > b$ :$c : b > c : a \implies b < a$


Equality of Ratios is Transitive

{{:Euclid:Proposition/V/11}} That is: :$A : B = C : D, C : D = E : F \implies A : B = E : F$


Sum of Components of Equal Ratios

{{:Euclid:Proposition/V/12}} That is: :$a_1 : b_1 = a_2 : b_2 = a_3 : b_3 = \cdots \implies \left({a_1 + a_2 + a_3 + \cdots}\right) : \left({b_1 + b_2 + b_3 + \cdots}\right)$


Relative Sizes of Proportional Magnitudes

{{:Euclid:Proposition/V/13}} That is: :$a : b = c : d, c : d > e : f \implies a : b > e : f$


Relative Sizes of Components of Ratios

{{:Euclid:Proposition/V/14}} That is, if $a : b = c : d$ then: :$a > c \implies b > d$ :$a = c \implies b = d$ :$a < c \implies b < d$


Ratio Equals its Multiples

{{:Euclid:Proposition/V/15}} That is: :$a : b \implies ma = mb$


Proportional Magnitudes are Proportional Alternately

{{:Euclid:Proposition/V/16}} That is: :$a : b = c : d \implies a : c = b : d$


Magnitudes Proportional Compounded are Proportional Separated

{{:Euclid:Proposition/V/17}} That is: :$a : b = c : d \implies \left({a - b}\right) : b = \left({c - d}\right) : d$


Magnitudes Proportional Separated are Proportional Compounded

{{:Euclid:Proposition/V/18}} That is: :$a : b = c : d \implies \left({a + b}\right) : b = \left({c + d}\right) : d$


Proportional Magnitudes have Proportional Remainders

{{:Euclid:Proposition/V/19}} That is: :$a : b = c : d \implies \left({a - c}\right) : \left({b - d}\right) = a : b$ where $a : b$ denotes the ratio of $a$ to $b$.


Relative Sizes of Successive Ratios

{{:Euclid:Proposition/V/20}} That is, let: :$a : b = d : e$ :$b : c = e : f$ Then: :$a > c \implies d > f$ :$a = c \implies d = f$ :$a < c \implies d < f$


Relative Sizes of Elements in Perturbed Proportion

{{:Euclid:Proposition/V/21}} That is, let: :$a : b = e : f$ :$b : c = d : e$ Then: :$a > c \implies d > f$ :$a = c \implies d = f$ :$a < c \implies d < f$


Equality of Ratios Ex Aequali

{{:Euclid:Proposition/V/22}} That is, if: :$a : b = d : e$ :$b : c = e : f$ then: :$a : c = d : f$


Equality of Ratios in Perturbed Proportion

{{:Euclid:Proposition/V/23}} That is, if: :$a : b = e : f$ :$b : c = d : e$ then: :$a : c = d : f$


Sum of Antecedent and Consequent of Proportion

{{:Euclid:Proposition/V/25}} That is, if $a : b = c : d$ and $a$ is the greatest and $d$ is the least, then: :$a + d > b + c$


Areas of Triangles and Parallelograms Proportional to Base

{{:Euclid:Proposition/VI/1}} Let $ABC$ and $ACD$ be triangles. Let $EC, CF$ be parallelograms under the same height. Then: :$AC : CD = \triangle ABC : \triangle ACD = \Box EC : \Box CF$ where: :$AC : CD$ denotes the ratio of the length of $AC$ to that of $CD$ :$\triangle ABC : \triangle ACD$ denotes the ratio of the area of $\triangle ABC$ to that of $\triangle ACD$ :$\Box EC : \Box CF$ denotes the ratio of the area of parallelogram $EC$ to that of parallelogram $CF$.


Angle Bisector Theorem

Let $\triangle ABC$ be a triangle. Let $D$ lie on the base $BC$ of $\triangle ABC$. Then the following are equivalent: :$(1): \quad AD$ is the angle bisector of $\angle BAC$ :$(2): \quad BD : DC = AB : AC$ where $BD : DC$ denotes the ratio between the lengths $BD$ and $DC$. {{:Euclid:Proposition/VI/3}}


Equiangular Triangles are Similar

Let two triangles have the same corresponding angles. Then their corresponding sides are proportional. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/4}}


Triangles with One Equal Angle and Two Sides Proportional are Similar

Let two triangles have two corresponding sides which are proportional. Let the angles adjacent to both of these sides be equal. Then all of their corresponding angles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/6}}


Triangles with One Equal Angle and Two Other Sides Proportional are Similar

Let two triangles be such that one of the angles of one triangle equals one of the angles of the other. Let two corresponding sides which are adjacent to one of the other angles, be proportional. Let the third angle in both triangles be either both acute or both not acute. Then all of the corresponding angles of these triangles are equal. Thus, by definition, such triangles are similar. {{:Euclid:Proposition/VI/7}}


Zeckendorf's Theorem

Every positive integer has a unique Zeckendorf representation. That is: Let $n$ be a positive integer. Then there exists a unique increasing sequence of integers $\left\langle{c_i}\right\rangle$ such that: :$\forall i \in \N: c_i \ge 2$ :$c_{i + 1} > c_i + 1$ :$\displaystyle n = \sum_{i \mathop = 0}^k F_{c_i}$ where $F_m$ is the $m$th Fibonacci number. For any given $n$, such a $\left\langle{c_i}\right\rangle$ is unique.


Sum of Non-Consecutive Fibonacci Numbers

Let $S$ be a non-empty set of distinct non-consecutive Fibonacci numbers not containing $F_0$ or $F_1$. Let the largest element of $S$ be $F_j$. Then: :$\displaystyle \sum_{F_i \mathop \in S} F_i < F_{j + 1}$ That is, the sum of all the elements of $S$ is strictly less than the next largest Fibonacci number. That is, given some increasing sequence $\left\langle {c_i}\right\rangle$ satisfying $c_i \ge 2$ and $c_{i + 1} \ge c_i + 1$: :$\displaystyle F_{c_k + 1} > \sum_{i \mathop = 0}^k F_{c_i}$


Construction of Third Proportional Straight Line

Given any two straight lines of given length $a$ and $b$, it is possible to construct a third straight line of length $c$ such that $a : b = b : c$. {{:Euclid:Proposition/VI/11}}


Construction of Fourth Proportional Straight Line

Given three straight lines of lengths $a, b, c$, it is possible to construct a fourth straight line of length $d$ such that $a : b = c : d$. {{:Euclid:Proposition/VI/12}}


Construction of Mean Proportional

Given any two straight lines of length $a$ and $b$ it is possible to find a straight line of length $c$ such that $a : c = c : b$. {{:Euclid:Proposition/VI/13}}


Sides of Equal and Equiangular Parallelograms are Reciprocally Proportional

{{:Euclid:Proposition/VI/14}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''.


Sides of Equiangular Triangles are Reciprocally Proportional

{{:Euclid:Proposition/VI/15}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''.


Rectangles Contained by Proportional Straight Lines

{{:Euclid:Proposition/VI/16}} Note: in the above, ''equal'' is to be taken to mean ''of equal area''.


Construction of Similar Polygon

On any given straight line it is possible to construct a polygon similar to any given polygon. {{:Euclid:Proposition/VI/18}}


Triangles with Two Equal Angles are Similar

Two triangles which have two corresponding angles which are equal are similar.


Ratio of Areas of Similar Triangles

{{:Euclid:Proposition/VI/19}} That is, the ratio of the areas of the similar triangles is the square of the ratio of the corresponding sides.


Similarity of Polygons is Equivalence Relation

Let $A, B, C$ be polygons. If $A$ and $B$ are both similar to $C$, then $A$ is similar to $B$. {{:Euclid:Proposition/VI/21}} It is also worth noting that: :$A$ is similar to $A$, and so similarity between polygons is reflexive. :If $A$ is similar to $B$, then $B$ is similar to $A$, and so similarity between polygons is symmetric. Hence the relation of similarity between polygons is an equivalence relation.


Integral Multiple of an Algebraic Number

Let $K$ be a number field and $\alpha \in K$. Then there exists a positive $n \in \Z$ such that $n \alpha \in \mathcal O_K$. In this context, $ \mathcal O_K$ denotes the algebraic integers in $K$.


Schönemann-Eisenstein Theorem

Let $\map f x = a_d x^d + a_{d - 1} x^{d - 1} + \dotsb + a_0 \in \Z \sqbrk x$ be a polynomial over the ring of integers $\Z$. Let $p$ be a prime such that: :$(1): \quad p \divides a_i \iff i \ne d$ :$(2): \quad p^2 \nmid a_0$ where $p \divides a_i$ signifies that $p$ is a divisor of $a_i$. Then $f$ is irreducible in $\Q \sqbrk x$.


Isolated Point of Closure of Subset is Isolated Point of Subset

Let $\struct {T, \tau}$ be a topological space. Let $H \subseteq T$ be a subset of $T$. Let $\map \cl H$ denote the closure of $H$. Let $x \in \map \cl H$ be an isolated point of $\map \cl H$. Then $x$ is also an isolated point of $H$.


Disjoint Open Sets remain Disjoint with one Closure

Let $T = \left({S, \tau}\right)$ be a topological space. Let $A, B \in \tau$ such that $A \cap B = \varnothing$. Then: : $A^- \cap B = \varnothing$ where $A^-$ denotes the closure of $A$.


Gauss's Lemma on Primitive Rational Polynomials

Let $\Q$ be the field of rational numbers. Let $\Q \sqbrk X$ be the ring of polynomials over $\Q$ in one indeterminate $X$. Let $\map f X, \map g X \in \Q \sqbrk X$ be primitive polynomials. Then their product $f g$ is also a primitive polynomial.


Preimage of Cover is Cover

Let $\phi: S \to T$ be a mapping between the sets $S$ and $T$. Let $\mathcal U$ be a cover of $T$. Then the set: :$\left\{{\phi^{-1} \left({U}\right): U \in \mathcal U}\right\}$ is a cover of $S$.


Compactness is Preserved under Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is compact, then $T_B$ is also compact.


Sigma-Compactness is Preserved under Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is $\sigma$-compact, then $T_B$ is also $\sigma$-compact.


Countable Compactness is Preserved under Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is countably compact, then $T_B$ is also countably compact.


Sequential Compactness is Preserved under Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is sequentially compact, then $T_B$ is also sequentially compact.


Lindelöf Property is Preserved under Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a continuous surjection. If $T_A$ is a Lindelöf space, then $T_B$ is also a Lindelöf space.


Multiples of Divisors obey Distributive Law

{{:Euclid:Proposition/VII/6}} In modern algebraic language: :$a = \dfrac m n b, c = \dfrac m n d \implies a + c = \dfrac m n \left({b + d}\right)$


First-Countability is Preserved under Open Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a surjective open mapping which is also continuous. If $T_A$ is first-countable, then $T_B$ is also first-countable.


Second-Countability is Preserved under Open Continuous Surjection

Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces. Let $\phi: T_A \to T_B$ be a surjective open mapping which is also continuous. If $T_A$ is second-countable, then $T_B$ is also second-countable.


Arens-Fort Space is Countable

Let $T = \left({S, \tau}\right)$ be the Arens-Fort space. Then $S$ is countably infinite.


Connected iff no Proper Clopen Sets

Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is connected if and only if there exists no proper subset of $S$ which is clopen in $T$.


Element of Principal Ideal Domain is Finite Product of Irreducible Elements

Let $R$ be a principal ideal domain. Let $p \in R$ such that $p \ne 0$ and $p$ is not a unit. Then there exist irreducible elements $p_1, \ldots, p_n$ such that $p = p_1 \cdots p_n$.


Principal Ideal Domain fulfills Ascending Chain Condition

Let $R$ be a principal ideal domain. Then $R$ fulfills the ascending chain condition.


Subspace of Product Space Homeomorphic to Factor Space

Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Suppose that $X$ is non-empty. Then for each $i \in I$ there is a subspace $Y_i \subseteq X$ which is homeomorphic to $\struct {X_i, \tau_i}$. Specifically, for any $z \in X$, let: :$Y_i = \set {x \in X: \forall j \in I \setminus \set i: x_j = z_j}$ and let $\upsilon_i$ be the subspace topology of $Y_i$ relative to $\tau$. Then $\struct {Y_i, \upsilon_i}$ is homeomorphic to $\struct {X_i, \tau_i}$, where the homeomorphism is the restriction of the projection $\pr_i$ to $Y_i$.


Closed Set of Countable Fort Space is G-Delta

Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Let $H \subseteq S$ be closed in $T$. Then $H$ is a $G_\delta$ set.


Fundamental Theorem of Galois Theory

Let $L / K$ be a finite Galois extension. Let $\operatorname{Gal} \left({L / K}\right)$ denote the Galois group of the extension $L / K$. Let $H$ denote a subgroup of $\operatorname{Gal} \left({L / K}\right)$ and $F$ denote an intermediate field. The mappings: :$H \mapsto L_H$, and :$F \mapsto \operatorname{Gal} \left({L / F}\right)$ are inclusion-reversing and inverses. Moreover, these maps induce a bijection between the normal subgroups of $\operatorname{Gal} \left({L / K}\right)$ and the normal, intermediate extensions of $L / K$.


Ring of Algebraic Integers

Let $K / \Q$ be a number field. Let $\Z \sqbrk x$ denote the polynomial ring in one variable over $\Z$. Let $\mathbb A$ denote the set of all elements of $K / \Q$ which are a root of some monic polynomial $P \in \Z \sqbrk x$. That is, let $\mathbb A$ denote the algebraic integers over $K$. Then $\mathbb A$ is a ring, called the '''Ring of Algebraic Integers'''.


Subtraction of Divisors obeys Distributive Law

{{:Euclid:Proposition/VII/7}} In modern algebraic language: :$a = \dfrac 1 n b, c = \dfrac 1 n d \implies a - c = \dfrac 1 n \paren {b - d}$


Minimal Polynomial is Unique

Let $L / K$ be a field extension and $\alpha \in L$ be algebraic over $K$. Then the minimal polynomial of $\alpha$ over $K$ is unique.


Minimal Polynomial is Irreducible

Let $L / K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Then the minimal polynomial in $\alpha$ over $K$ is unique and irreducible.


Equivalence of Definitions of Normal Extension

Let $L / K$ be an algebraic field extension. {{TFAE|def = Normal Extension}}


Abstract Model of Algebraic Extensions

Let $K$ be a field and $\alpha \in \overline{K}$ be an element of the algebraic closure of $K$ which is algebraic over $K$. Let $m_\alpha$ be the minimal polynomial of $\alpha$ over $K$. Then: :$K[\alpha] \cong K[x]/ \langle m_\alpha\rangle$


Union of Local Bases is Basis

Let $T = \left({X, \tau}\right)$ be a topological space. For each $x \in X$, let $\mathcal B_x$ be a local basis at $x$ which consists entirely of open sets. Then $\displaystyle \mathcal B = \bigcup_{x \mathop \in X} \mathcal B_x$ is a basis for the topology $\tau$.


Subtraction of Multiples of Divisors obeys Distributive Law

{{:Euclid:Proposition/VII/8}} In modern algebraic language: :$a = \dfrac m n b, c = \dfrac m n d \implies a - c = \dfrac m n \paren {b - d}$


Proportional Numbers have Proportional Differences

{{:Euclid:Proposition/VII/11}} That is: :$a : b = c : d \implies \left({a - c}\right) : \left({b - d}\right) = a : b$ where $a : b$ denotes the ratio of $a$ to $b$.


Galois Group is Group

Let $L / K$ be a normal extension. Let $\Gal {L / K}$ be the Galois group of $L / K$. Then $\Gal {L / K}$ forms a group under the operation of composition of mappings.


Relation of Ratios to Products

{{:Euclid:Proposition/VII/19}} That is: :$a : b = c : d \iff ad = bc$


Ratios of Fractions in Lowest Terms

Let $a, b, c, d \in \Z_{>0}$ be positive integers. Let $\dfrac a b$ be in canonical form. Let $\dfrac a b = \dfrac c d$. Then: :$a \divides c$ and: :$b \divides d$ where $\divides$ denotes divisibility. {{:Euclid:Proposition/VII/20}}


Gelfond-Schneider Theorem

Let $\alpha$ and $\beta$ be algebraic numbers (possibly complex) such that $\alpha \notin \set {0, 1}$. Let $\beta$ be irrational. Then any value of $\alpha^\beta$ is transcendental.


Associative Algebra has Multiplicative Inverses iff Unitary Division Algebra

Let $\struct {A_R, \oplus}$ be an associative algebra over the ring $A_R$. Then: :$\struct {A_R, \oplus}$ has a unique multiplicative inverse for every non-zero $a \in A_R$ {{iff}}: :$\struct {A_R, \oplus}$ is a unitary division algebra.


Commutator on Algebra is Alternating Bilinear Mapping

Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. Then the commutator on $\left({A_R, \oplus}\right)$ is an alternating bilinear mapping: :$\forall a, b \in A_R: \left[{a, b}\right] = -\left[{b, a}\right]$


Axiom of Subsets Equivalents

The Axiom of Specification states that: :$\forall z: \forall P \left({y}\right): \exists x: \forall y: \left({y \in x \iff \left({y \in z \land P \left({y}\right)}\right)}\right)$ We will prove that this statement is equivalent to the following statements: :$\forall z: \forall A: \left({\left({z \cap A}\right) \in U}\right)$ :$\forall z: \forall A: \left({A \subseteq z \implies A \in U}\right)$ In the above statements, the universe is $U$.


Artin's Theorem on Alternative Algebras

Let $A = \struct {A_R, \oplus}$ be an algebra over the ring $R$ such that $A$ is ''not'' a boolean algebra. Then $A$ is alternative {{iff}}: :$\forall a, b \in A: \paren {a \oplus a} \oplus b = a \oplus \paren {a \oplus b}$ :$\forall a, b \in A: \paren {b \oplus a} \oplus a = b \oplus \paren {a \oplus a}$


Real Numbers form Algebra

The set of real numbers $\R$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An associative algebra. :$(2): \quad$ A commutative algebra. :$(3): \quad$ A normed division algebra. :$(4): \quad$ A nicely normed $*$-algebra whose $*$ operator is the identity mapping. :$(5): \quad$ A real $*$-algebra.


Real Numbers form Vector Space

The set of real numbers $\R$, with the operations of addition and multiplication, forms a vector space.


Burali-Forti Paradox

The existence of the set of all ordinals leads to a contradiction.


Complex Numbers form Algebra

The set of complex numbers $\C$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An associative algebra. :$(2): \quad$ A commutative algebra. :$(3): \quad$ A normed division algebra. :$(4): \quad$ A nicely normed $*$-algebra. However, $\C$ is not a real algebra.


Equality implies Substitution

Let $P \left({x}\right)$ denote a Well-Formed Formula which contains $x$ as a free variable. Then the following are tautologies: :$\forall x: \left({P \left({ x }\right) \iff \exists y: \left({y = x \land P \left({y}\right)}\right)}\right)$ :$\forall x: \left({P \left({ x }\right) \iff \forall y: \left({y = x \implies P \left({y}\right)}\right)}\right)$ Note that when $y$ is substituted for $x$ in either formula, it is false in general; compare Confusion of Bound Variables.


Fundamental Law of Universal Class

:$\forall x: x \in \Bbb U$ where: : $\Bbb U$ denotes the universal class : $x$ denotes a set.


Cayley-Dickson Construction forms Star-Algebra

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A'_F, \oplus'}\right) = \left({A, \oplus}\right)^2$ be the algebra formed from $A$ by the Cayley-Dickson construction. Then $A'$ is also a $*$-algebra.


Empty Set is Small

:$\O \in U$ where $U$ is the universal class.


Algebra from Cayley-Dickson Construction Never Real

Let $A$ be a $*$-algebra. Let $A'$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is not a real algebra.


Cayley-Dickson Construction from Real Algebra is Commutative

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A$ is a real algebra {{iff}} $A'$ is a commutative algebra.


Cayley-Dickson Construction from Commutative Associative Algebra is Associative

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then: : $A'$ is an associative algebra {{iff}}: : $A$ is both a commutative algebra and an associative algebra.


Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is a nicely normed alternative algebra {{iff}} $A$ is a nicely normed associative algebra.


Element of Universe

Let $A$ be a class, which may be either a set or a proper class. Then: :$\forall A: \paren {A \in U \iff \exists x: x = A}$ where $U$ is the universal class.


Conjunction with Tautology

: $p \land \top \dashv \vdash p$


Disjunction with Tautology

: $p \lor \top \dashv \vdash \top$


Conjunction with Contradiction

:$p \land \bot \dashv \vdash \bot$


Disjunction with Contradiction

: $p \lor \bot \dashv \vdash p$


Contradiction is Negation of Tautology

A contradiction implies and is implied by the negation of a tautology: :$\bot \dashv \vdash \neg \top$ That is, a falsehood can not be true, and a non-truth is a falsehood.


Tautology is Negation of Contradiction

A tautology implies and is implied by the negation of a contradiction: :$\top \dashv \vdash \neg \bot$ That is, a truth can not be false, and a non-falsehood must be a truth.


Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra. Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction. Then $A'$ is a nicely normed algebra {{iff}} $A$ is also a nicely normed algebra.


Quaternions form Algebra

The set of quaternions $\Bbb H$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An associative algebra, but '''not''' a commutative algebra. :$(2): \quad$ A normed division algebra. :$(3): \quad$ A nicely normed $*$-algebra.


Octonions form Algebra

The set of octonions $\Bbb O$ forms an algebra over the field of real numbers. This algebra is: :$(1): \quad$ An alternative algebra, but '''not''' an associative algebra. :$(2): \quad$ A normed division algebra. :$(3): \quad$ A nicely normed $*$-algebra.


Division Algebra has No Zero Divisors

Let $A = \left({A_F, \oplus}\right)$ be an algebra over a field $F$. Then $A$ is a division algebra {{iff}} it has no zero divisors. That is: :$\forall a, b \in A_F: a \oplus b = \mathbf 0_A \implies a = \mathbf 0_A \lor b = \mathbf 0_A$ If the product of two elements of $A$ is zero, then at least one of those elements must itself be zero. Some sources use this as the definition of a division algebra and from it deduce: :$\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$


Normed Division Algebra is Division Algebra

Let $A = \struct {A_F, \oplus}$ be a normed divison algebra over a field $F$. Let the unit of $A$ be $1_A$, and the zero of $A$ be $0_A$. Then $A$ is a unitary division algebra. Also: :$\norm {1_A} = 1$ where $\norm {1_A}$ denotes the norm of $1_A$.


Zermelo's Theorem (Set Theory)

Every set of cardinals is well-ordered with respect to $\le$.


Cardinals form Equivalence Classes

Let $\map \Card S$ denote the cardinal of the set $S$. Then $\map \Card S$ induces an equivalence class which contains all sets which have the same cardinality as $S$.


Integers form Commutative Ring with Unity

The integers $\struct {\Z, +, \times}$ form a commutative ring with unity under addition and multiplication.


Ring of Square Matrices over Ring is Ring

Let $R$ be a ring. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. Then $\struct {\map {\MM_R} n, +, \times}$ is a ring.


Ring of Square Matrices over Ring with Unity

Let $R$ be a ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_R} n, +, \times}$ denote the ring of square matrices of order $n$ over $R$. Then $\struct {\map {\MM_R} n, +, \times}$ is a ring with unity.


Ring of Square Matrices over Field is Ring with Unity

Let $F$ be a field. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\map {\MM_F} n, +, \times}$ denote the ring of square matrices of order $n$ over $F$. Then $\struct {\map {\MM_F} n, +, \times}$ is a ring with unity, but is not a commutative ring.


Matrix Multiplication is not Commutative/Order 2 Square Matrices

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\map {\MM_R} 2$ denote the $2 \times 2$ matrix space over $R$. The operation of (conventional) matrix multiplication is not commutative over $\map {\MM_R} 2$.


Ring Zero is Unique

Let $\struct {R, +, \circ}$ be a ring. Then the ring zero of $R$ is unique.


Ring Negative is Unique

Let $\struct {R, +, \circ}$ be a ring. Let $a \in R$. Then the ring negative $-a$ of $a$ is unique.


Negative of Ring Negative

Let $\struct {R, +, \circ}$ be a ring. Let $a \in R$ and let $-a$ be the ring negative of $a$. Then: :$-\paren {-a} = a$


Even Perfect Number is Triangular

All perfect numbers which are even are triangular.


Field has no Proper Zero Divisors

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Then $\struct {F, +, \times}$ has no proper zero divisors. That is: :$a \times b = 0_F \implies a = 0_F \lor b = 0_F$


Multiplicative Inverse in Field is Unique

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ such that $a \ne 0_F$. Then the multiplicative inverse $a^{-1}$ of $a$ is unique.


Inverse of Multiplicative Inverse

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$. Let $a \in F$ such that $a \ne 0_F$. Let $a^{-1}$ be the multiplicative inverse of $a$. Then $\paren {a^{-1} }^{-1} = a$.


Integral Multiple Distributes over Ring Addition

Let $\struct {R, +, \times}$ be a ring, or a field. Let $a, b \in R$ and $m, n \in \Z$. Then: :$(1): \quad \paren {m + n} \cdot a = \paren {m \cdot a} + \paren {n \cdot a}$ :$(2): \quad m \cdot \paren {a + b} = \paren {m \cdot a} + \paren {m \cdot b}$ where $m \cdot a$ is as defined in integral multiple.


Integral Multiple of Integral Multiple

Let $\struct {F, +, \times}$ be a field. Let $a \in F$ and $m, n \in \Z$. Then: :$\paren {m n} \cdot a = m \cdot \paren {n \cdot a}$ where $n \cdot a$ is as defined in integral multiple.


Product of Integral Multiples

Let $\struct {F, +, \times}$ be a field. Let $a, b \in F$ and $m, n \in \Z$. Then: :$\paren {m \cdot a} \times \paren {n \cdot b} = \paren {m n} \cdot \paren {a \times b}$ where $m \cdot a$ is as defined in Integral Multiple.


Characteristic of Field by Annihilator

Let $\struct {F, +, \times}$ be a field. Then of the following two cases, exactly one applies:


Annihilator of Ring Always Contains Zero

Let $\struct {R, +, \times}$ be a ring. Let $\map {\mathrm {Ann} } R$ be the annihilator of $R$. Then $0 \in \map {\mathrm {Ann} } R$.


Non-Trivial Annihilator Contains Positive Integer

Let $\left({R, +, \times}\right)$ be a ring with unity. Let $A = \operatorname{Ann} \left({R}\right)$ be the annihilator of $R$. Let $a \in A$ such that $a \ne 0$. Then $A$ contains at least one strictly positive integer.


Area of Sector

Let $\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$. Let $BAC$ be the sector of $\CC$ whose angle between $AB$ and $AC$ is $\theta$. :300px Then the area $\AA$ of sector $BAC$ is given by: :$\AA = \dfrac {r^2 \theta} 2$ where: :$r = AB$ is the length of the radius of the circle :$\theta$ is measured in radians.


Rational Numbers form Prime Field

The field of rational numbers $\struct {\Q, +, \times}$ is a prime field. That is, the only subset of $\Q$ which sustains both addition and multiplication are $\Q$ and $\set 0$, and vacuously $\O$.


Field of Integers Modulo Prime is Prime Field

Let $p$ be a prime number. Let $\struct {\Z_p, +, \times}$ be the field of integers modulo $p$. Then $\struct {\Z_p, +, \times}$ is a prime field.


Field has Prime Subfield

Let $\struct {F, +, \times}$ be a field. Then $F$ has a subfield which is a prime field.


Constant Mapping to Identity is Homomorphism/Rings

Let $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ be rings with zeroes $0_1$ and $0_2$ respectively. Let $\zeta$ be the zero homomorphism from $R_1$ to $R_2$, that is: :$\forall x \in R_1: \map \zeta x = 0_2$ Then $\zeta$ is a ring homomorphism whose image is $\set {0_2}$ and whose kernel is $R_1$.


Constant Mapping to Identity is Homomorphism/Groups

Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups with identities $e_1$ and $e_2$ respectively. Let $\phi_e: \struct {G_1, \circ_1} \to \struct {G_2, \circ_2}$ be the constant mapping defined as: :$\forall x \in G_1: \map {\phi_e} x = e_2$ Then $\phi_e$ is a group homomorphism whose image is $\set {e_2}$ and whose kernel is $G_1$.


Identity Mapping is Automorphism/Groups

Let $\struct {G, \circ}$ be a group whose identity is $e$. Then $I_G: \struct {G, \circ} \to \struct {G, \circ}$ is a group automorphism. Its kernel is $\set e$.


Identity Mapping is Automorphism/Rings

Let $\struct {R, +, \circ}$ be a ring whose zero is $0$. Then $I_R: \struct {R, +, \circ} \to \struct {R, +, \circ}$ is a ring automorphism. Its kernel is $\set 0$.


Field Homomorphism Preserves Unity

Let $\phi: \struct {F_1, +_1, \times_1} \to \struct {F_2, +_2, \times_2}$ be a field homomorphism. Let: :$1_{F_1}$ be the unity of $F_1$ :$1_{F_2}$ be the unity of $F_2$. Then: :$\map \phi {1_{F_1} } = 1_{F_2}$


Ring Homomorphism from Field is Monomorphism or Zero Homomorphism

Let $\struct {F, +_F, \circ}$ be a field whose zero is $0_F$. Let $\struct {S, +_S, *}$ be a ring whose zero is $0_S$. Let $\phi: F \to S$ be a ring homomorphism. Then either: :$(1): \quad \phi$ is a monomorphism (that is, $\phi$ is injective) or :$(2): \quad \phi$ is the zero homomorphism (that is, $\forall a \in F: \map \phi a = 0_S$).


Zero and Unity of Subfield

Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$. Let $\struct {K, +, \times}$ be a subfield of $F$.


Galois Field has Non-Zero Characteristic

Let $\GF$ be a Galois field. Then the characteristic of $\GF$ is non-zero.


Power to Characteristic of Field is Monomorphism

Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let the characteristic of $F$ be $p$ where $p \ne 0$. Let $\phi: F \to F$ be the mapping on $F$ defined as: :$\forall x \in F: \map \phi x = x^p$ Then $\phi$ is a (field) monomorphism.


Scalar Product with Identity

:$\lambda \circ e = 0_R \circ x = e$


Scalar Product with Inverse

:$\lambda \circ \struct {-x} = \struct {-\lambda} \circ x = -\struct {\lambda \circ x}$


Scalar Product with Sum

:$\ds \lambda \circ \paren {\sum_{k \mathop = 1}^m x_k} = \sum_{k \mathop = 1}^m \paren {\lambda \circ x_k}$


Product with Sum of Scalar

:$\ds \paren {\sum_{k \mathop = 1}^m \lambda_k} \circ x = \sum_{k \mathop = 1}^m \paren {\lambda_k \circ x}$


Scalar Product with Product

:$\lambda \circ \paren {n \cdot x} = n \cdot \paren {\lambda \circ x} = \paren {n \cdot \lambda} \circ x$


Expression of Vector as Linear Combination from Basis is Unique

Let $V$ be a vector space of dimension $n$. Let $\mathcal B = \set {\mathbf x_1, \mathbf x_2, \ldots, \mathbf x_n}$ be a basis for $V$. Let $\mathbf x \in V$ be any vector of $V$. Then $\mathbf x$ can be expressed as a unique linear combination of elements of $\mathcal B$. {{explain|The definition we have of linear combination doesn't really work for this. We probably need to expand that definition.}}


Homogeneous Linear Equations with More Unknowns than Equations

Let $\alpha_{ij}$ be elements of a field $F$, where $1 \le i \le m, 1 \le j \le n$. Let $n > m$. Then there exist $x_1, x_2, \ldots, x_n \in F$ not all zero, such that: :$\displaystyle \forall i: 1 \le i \le m: \sum_{j \mathop = 1}^n \alpha_{ij} x_j = 0$ Alternatively, this can be expressed as: If $n > m$, the following system of homogeneous linear equations: {{begin-eqn}} {{eqn | l = 0       | r = \alpha_{11} x_1 + \alpha_{12} x_2 + \cdots + \alpha_{1n} x_n }} {{eqn | l = 0       | r = \alpha_{21} x_1 + \alpha_{22} x_2 + \cdots + \alpha_{2n} x_n }} {{eqn | o = \cdots }} {{eqn | l = 0       | r = \alpha_{m1} x_1 + \alpha_{m2} x_2 + \cdots + \alpha_{mn} x_n }} {{end-eqn}} has at least one solution such that not all of $x_1, \ldots, x_n$ is zero.


Generator of Vector Space Contains Basis

:$G$ contains a basis for $E$.


Cardinality of Linearly Independent Set is No Greater than Dimension

:$H$ has at most $n$ elements.


Cardinality of Generator of Vector Space is not Less than Dimension

Let $V$ be a vector space over a field $F$. Let $\BB$ be a generator for $V$ containing $m$ elements. Then: :$\map {\dim_F} V \le m$ where $\map {\dim_F} V$ is the dimension of $V$.


Trichotomy Law for Real Numbers

The real numbers obey the Trichotomy Law. That is, $\forall a, b \in \R$, exactly one of the following holds: {{begin-axiom}} {{axiom | n = 1         | lc= $a$ is greater than $b$:         | m = a > b }} {{axiom | n = 2         | lc= $a$ is equal to $b$:         | m = a = b }} {{axiom | n = 3         | lc= $a$ is less than $b$:         | m = a < b }} {{end-axiom}}


=== Positive Factor === {{:Real Number Ordering is Compatible with Multiplication/Positive Factor}} === Negative Factor === {{:Real Number Ordering is Compatible with Multiplication/Negative Factor}}


Real Number Ordering is Compatible with Addition

:$\forall a, b, c \in \R: a < b \implies a + c < b + c$


Real Number Inequalities can be Added

Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$. Then: :$a + c > b + d$


Positive Real Number Inequalities can be Multiplied

Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$. Let $b > 0$ and $d > 0$. Then $a c > b d$. If $b < 0$ or $d < 0$ the inequality does not hold.


Real Ordering Incompatible with Subtraction

Let $a, b, c, d \in R$ be real numbers such that $a > b$ and $c > d$. Then it does not necessarily hold that: :$a - c > b - d$


Real Ordering Incompatible with Division

Let $a, b, c, d \in \R$ be real numbers such that $a > b$ and $c > d$. Then it does not necessarily hold that: :$\dfrac a c > \dfrac b d$


Positive Real Number Inequalities can be Multiplied/Disproof for Negative Parameters

Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$. Let $b > 0$ and $d > 0$. From Positive Real Number Inequalities can be Multiplied, $a c > b d$ holds. However, if $b < 0$ or $d < 0$ the inequality does ''not'' hold.


Reverse Triangle Inequality/Real and Complex Fields/Corollary

Let $x$ and $y$ be elements of either the real numbers $\R$ or the complex numbers $\C$. Then: :$\size {x - y} \ge \size x - \size y$ where $\size x$ denotes either the absolute value of a real number or the complex modulus of a complex number.


Combination Theorem for Sequences/Real/Sum Rule

:$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$


Combination Theorem for Sequences/Real/Multiple Rule

:$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n} = \lambda l$


Combination Theorem for Sequences/Real/Combined Sum Rule

:$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$


Combination Theorem for Sequences/Real/Product Rule

:$\displaystyle \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$


Combination Theorem for Sequences/Real/Quotient Rule

:$\displaystyle \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \frac l m$ provided that $m \ne 0$.


Squeeze Theorem/Sequences/Real Numbers

Let $\sequence {x_n}$, $\sequence {y_n}$ and $\sequence {z_n}$ be sequences in $\R$. Let $\sequence {y_n}$ and $\sequence {z_n}$ both be convergent to the following limit: :$\displaystyle \lim_{n \mathop \to \infty} y_n = l, \lim_{n \mathop \to \infty} z_n = l$ Suppose that: :$\forall n \in \N: y_n \le x_n \le z_n$ Then: :$x_n \to l$ as $n \to \infty$ that is: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Thus, if $\sequence {x_n}$ is always between two other sequences that both converge to the same limit, $\sequence {x_n} $ is said to be '''sandwiched''' or '''squeezed''' between those two sequences and itself must therefore converge to that same limit.


Squeeze Theorem/Sequences/Complex Numbers

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$ which is null, that is: :$a_n \to 0$ as $n \to \infty$ Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$. Suppose $\left \langle {a_n} \right \rangle$ dominates $\left \langle {z_n} \right \rangle$. That is, suppose that: : $\forall n \in \N: \left|{z_n}\right| \le a_n$ Then $\left \langle {z_n} \right \rangle$ is a null sequence.


Squeeze Theorem/Sequences

There are two versions of this result: * one for sequences in the set of complex numbers $\C$, and more generally for sequences in a metric space. * one for sequences in the set of real numbers $\R$, and more generally in linearly ordered spaces (which is stronger). {{explain|stronger in what sense?}} === Sequences of Real Numbers === {{:Squeeze Theorem/Sequences/Real Numbers}} === Sequences of Complex Numbers === {{:Squeeze Theorem/Sequences/Complex Numbers}} === Sequences in a Linearly Ordered Space === {{:Squeeze Theorem/Sequences/Linearly Ordered Space}} === Sequences in a Metric Space === {{:Squeeze Theorem/Sequences/Metric Spaces}}


Squeeze Theorem/Functions

Let $a$ be a point on an open real interval $I$. Let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$. Suppose that: :$\forall x \ne a \in I: \map g x \le \map f x \le \map h x$ :$\displaystyle \lim_{x \mathop \to a} \ \map g x = \lim_{x \mathop \to a} \ \map h x = L$ Then: :$\displaystyle \lim_{x \mathop \to a} \ \map f x = L$


Weierstrass's Theorem

There exists a real function $f: \closedint 0 1 \to \closedint 0 1$ such that: :$(1): \quad f$ is continuous :$(2): \quad f$ is nowhere differentiable.


Let $X$ be one of the standard number fields $\Q, \R, \C$. Let $f$ and $g$ be functions which are continuous on an open subset $S \subseteq X$. Let $\lambda, \mu \in X$ be arbitrary numbers in $X$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Continuous Functions/Sum Rule}} === Multiple Rule === {{:Combination Theorem for Continuous Functions/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Continuous Functions/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Continuous Functions/Product Rule}} === Quotient Rule === {{:Combination Theorem for Continuous Functions/Quotient Rule}}


Combination Theorem for Limits of Functions/Sum Rule

:$\displaystyle \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$


Combination Theorem for Limits of Functions/Multiple Rule

:$\displaystyle \lim_{x \mathop \to c} \ \paren {\lambda f \paren x} = \lambda l$


Combination Theorem for Limits of Functions/Combined Sum Rule

:$\displaystyle \lim_{x \mathop \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$


Combination Theorem for Limits of Functions/Product Rule

:$\displaystyle \lim_{x \mathop \to c} \ \paren {\map f x \map  g x} = l m$


Combination Theorem for Limits of Functions/Quotient Rule

:$\displaystyle \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$ provided that $m \ne 0$.


Combination Theorem for Continuous Functions/Sum Rule

:$f + g$ is continuous on $S$.


Combination Theorem for Continuous Functions/Product Rule

:$f g$ is continuous on $S$


Combination Theorem for Continuous Functions/Multiple Rule

:$\lambda f$ is continuous on $S$.


Combination Theorem for Continuous Functions/Combined Sum Rule

:$\lambda f + \mu g$ is continuous on $S$.


Combination Theorem for Continuous Functions/Quotient Rule

:$\dfrac f g$ is continuous on $S \setminus \set {x \in S: \map g x = 0}$ that is, on all the points $x$ of $S$ where $\map g x \ne 0$.


Brouwer's Fixed Point Theorem/One-Dimensional Version

Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$. Then: :$\exists \xi \in \closedint a b: \map f \xi = \xi$ That is, a continuous real function from a closed real interval to itself fixes some point of that interval.


Brouwer's Fixed Point Theorem/Smooth Mapping

A smooth mapping $f$ of the closed unit ball $B^n \subset \R^n$ into itself has a fixed point: :$\forall f \in \map {C^\infty} {B^n \to B^n}: \exists x \in B^n: \map f x = x$


Power Rule for Derivatives/Natural Number Index

Let $n \in \N$. Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = x^n$. Then: :$f' \left({x}\right) = n x^{n-1}$ everywhere that $f \left({x}\right) = x^n$ is defined. When $x = 0$ and $n = 0$, $f^{\prime} \left({x}\right)$ is undefined.


Power Rule for Derivatives/Integer Index

Let $n \in \Z$. Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$. Then: :$\map {f'} x = n x^{n - 1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.


Power Rule for Derivatives/Fractional Index

Let $n \in \N_{>0}$. Let $f: \R \to \R$ be the real function defined as $\map f x = x^{1 / n}$. Then: :$\map {f'} x = n x^{n - 1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.


Power Rule for Derivatives/Rational Index

Let $n \in \Q$. Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = x^n$. Then: :$\map {f'} x = n x^{n-1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.


Power Rule for Derivatives/Real Number Index

Let $n \in \R$. Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$. Then: :$\map {f'} x = n x^{n-1}$ everywhere that $\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.


Well-Ordering Minimal Elements are Unique

Let $\left({S,\preceq}\right)$ be a well-ordered set. Then every non-empty subset of $S$ has a unique minimal element.


Fundamental Theorem of Calculus/First Part

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Let $F$ be a real function which is defined on $\closedint a b$ by: :$\displaystyle \map F x = \int_a^x \map f t \rd t$ Then $F$ is a primitive of $f$ on $\closedint a b$.


Fundamental Theorem of Calculus/Second Part

Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Then: :$(1): \quad f$ has a primitive on $\closedint a b$ :$(2): \quad$ If $F$ is any primitive of $f$ on $\closedint a b$, then: :::$\displaystyle \int_a^b \map f t \rd t = \map F b - \map F a = \bigintlimits {\map F t} a b$


Natural Logarithm of 1 is 0

:$\ln 1 = 0$


Derivative of Natural Logarithm Function

Let $\ln x$ be the natural logarithm function. Then: :$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$


Derivative of General Logarithm Function

Let $a \in \R_{>0}$ such that $a \ne 1$ Let $\log_a x$ be the logarithm function to base $a$. Then: :$\map {\dfrac \d {\d x} } {\log_a x} = \dfrac {\log_a e} x$


Logarithm is Strictly Increasing

:$\ln x: x > 0$ is strictly increasing.


Logarithm Tends to Infinity

:$\ln x \to +\infty$ as $x \to +\infty$


Logarithm Tends to Negative Infinity

:$\ln x \to -\infty$ as $x \to 0^+$


Second Derivative of Natural Logarithm Function

Let $\ln x$ be the natural logarithm function. Then: :$\map {\dfrac {\d^2} {\d x^2} } {\ln x} = -\dfrac 1 {x^2}$


Exponential of Zero and One

=== Exponential of Zero === {{:Exponential of Zero}} === Exponential of One === {{:Exponential of One}}


Exponential is Strictly Increasing

:The function $\map f x = \exp x$ is strictly increasing.


Exponential Tends to Zero and Infinity

:$\exp x \to +\infty$ as $x \to +\infty$ :$\exp x \to 0$ as $x \to -\infty$ Thus the exponential function has domain $\R$ and image $\openint 0 \to$.


Exponential of Natural Logarithm

: $\forall x > 0: \exp \left({\ln x}\right) = x$ : $\forall x \in \R: \ln \left({\exp x}\right) = x$


Exponent Combination Laws/Product of Powers

:$a^x a^y = a^{x + y}$


Exponent Combination Laws/Power of Product

:$\paren {a b}^x = a^x b^x$


Exponent Combination Laws/Negative Power

: $a^{-x} = \dfrac 1 {a^x}$


Exponent Combination Laws/Power of Power

Let $x, y \in \R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then: :$\paren {a^x}^y = a^{x y}$


Exponent Combination Laws/Quotient of Powers

:$\dfrac{a^x} {a^y} = a^{x - y}$


Exponent Combination Laws/Power of Quotient

:$\paren {\dfrac a b}^x = \dfrac {a^x} {b^x}$


Exponent Combination Laws/Negative Power of Quotient

: $\left({\dfrac a b}\right)^{-x} = \left({\dfrac b a}\right)^x$


Binomial Theorem/General Binomial Theorem

Let $\alpha \in \R$ be a real number. Let $x \in \R$ be a real number such that $\size x < 1$. Then: {{begin-eqn}} {{eqn | l = \paren {1 + x}^\alpha       | r = \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n       | c =  }} {{eqn | r = \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n       | c =  }} {{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n       | c =  }} {{eqn | r = 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots       | c =  }} {{end-eqn}} where: :$\alpha^{\underline n}$ denotes the falling factorial :$\dbinom \alpha n$ denotes a binomial coefficient.


Real Sine Function is Continuous

:$\sin x$ is continuous on $\R$.


Sine Function is Absolutely Convergent

:$\sin x$ is absolutely convergent for all $x \in \R$.


Sine of Zero is Zero

:$\sin 0 = 0$


Sine Function is Odd

:$\map \sin {-z} = -\sin z$ That is, the sine function is odd.


Cosine Function is Continuous

:$\cos x$ is continuous on $\R$.


Cosine Function is Absolutely Convergent

:$\cos x$ is absolutely convergent for all $x \in \R$.


Cosine of Zero is One

:$\cos 0 = 1$


Cosine of Integer Multiple of Pi

:$\forall n \in \Z: \cos n \pi = \paren {-1}^n$


Cosine Function is Even

:$\map \cos {-z} = \cos z$ That is, the cosine function is even.


Image of Interval by Derivative

Let $f$ be a real function that is everywhere differentiable. Let $I \subseteq \Dom f$ be a real interval. Then: :$f' \sqbrk I$ is a real interval where $f'$ denotes the derivative of $f$


Intermediate Value Theorem for Derivatives

Let $I$ be an open interval. Let $f : I \to \R$ be everywhere differentiable. Then $f'$ satisfies the Intermediate Value Property.


Sine of Half-Integer Multiple of Pi

:$\forall n \in \Z: \map \sin {n + \dfrac 1 2} \pi = \paren {-1}^n$


Zeroes of Sine and Cosine

:$(1): \quad \forall n \in \Z: x = \paren {n + \dfrac 1 2} \pi \implies \cos x = 0$ :$(2): \quad \forall n \in \Z: x = n \pi \implies \sin x = 0$


Sine of Integer Multiple of Pi

:$\forall n \in \Z: \sin n \pi = 0$


Cosine of Half-Integer Multiple of Pi

:$\forall n \in \Z: \map \cos {n + \dfrac 1 2} \pi = 0$


Rational Square Root of Integer is Integer

Let $n$ be an integer. Suppose that $\sqrt n$ is a rational number. Then $\sqrt n$ is an integer.


Definite Integral of Even Function

Let $f$ be an even function with a primitive on the closedinterval $\closedint {-a} a$, where $a > 0$. Then: :$\displaystyle \int_{-a}^a \map f x \rd x = 2 \int_0^a \map f x \rd x$


Definite Integral of Odd Function

Let $f$ be an odd function with a primitive on the open interval $\closedint {-a} a$, where $a > 0$. Then: :$\displaystyle \int_{-a}^a \map f x \rd x = 0$


Union of Equivalence Classes is Whole Set

Let $\RR \subseteq S \times S$ be an equivalence on a set $S$. Then the set of $\RR$-classes constitutes the whole of $S$.


Ratio of Consecutive Fibonacci Numbers

For $n \in \N$, let $f_n$ be the $n$th Fibonacci number. Then: :$\displaystyle \lim_{n \mathop \to \infty} \frac {f_{n + 1} } {f_n} = \phi$ where $\phi = \dfrac {1 + \sqrt 5} 2$ is the golden mean.


Cardinality of Image of Injection

Let $f: S \rightarrowtail T$ be an injection. Let $A \subseteq S$ be a finite subset of $S$. Then: :$\card {f \paren A} = \card A$ where $\card A$ denotes the cardinality of $A$.


Composition of Mappings is not Commutative

The composition of mappings is '''not''' in general a commutative binary operation: :$f_2 \circ f_1 \ne f_1 \circ f_2$


Intersection of Injective Image with Relative Complement

Let $f: S \to T$ be a mapping. Then $f$ is an injection {{iff}}: :$\forall A \subseteq S: f \sqbrk A \cap f \sqbrk {\relcomp S A} = \O$


Injection to Image is Bijection

Let $f: S \rightarrowtail T$ be an injection. Let $X \subseteq T$ be the image of $f$. Then the restriction $f {\restriction_{S \times X}}: S \to X$ of $f$ to the image of $f$ is a bijection of $S$ onto $X$.


Cardinality of Set of Induced Equivalence Classes of Surjection

Let $f: S \to T$ be a mapping. Let $\mathcal R_f \subseteq S \times S$ be the relation induced by $f$: :$\tuple {s_1, s_2} \in \mathcal R_f \iff \map f {s_1} = \map f {s_2}$ Let $f$ be a surjection. Then there are $\card T$ different $\mathcal R_f$-classes.


Power of Elements is Subgroup

Let $G$ be an abelian group whose identity is $e$. Then for any $n \in \Z$, the set $G^n = \set {x^n: x \in G}$ is a subgroup of $G$. Moreover, if: :$(1): \quad G$ is finite and: :$(2): \quad n \ne 1$ is a divisor of the order of $G$ then $G^n$ is a proper subgroup of $G$.


Index Laws/Sum of Indices/Semigroup

Let $\struct {S, \circ}$ be a semigroup. For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$: :$a^n = \begin{cases} a & : n = 1 \\ a^x \circ a & : n = x + 1 \end{cases}$ That is: : $a^n = \underbrace {a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \paren a$ Then: :$\forall m, n \in \N_{>0}: a^{n + m} = a^n \circ a^m$


Index Laws/Product of Indices/Semigroup

Let $\struct {S, \circ}$ be a semigroup. For $a \in S$, let $\circ^n a = a^n$ be the $n$th power of $a$. Then: :$\forall m, n \in \N_{>0}: a^{n m} = \paren {a^n}^m = \paren {a^m}^n$


Semigroup is Subsemigroup of Itself

Let $\struct {S, \circ}$ be a semigroup. Then $\struct {S, \circ}$ is a subsemigroup of itself.


Identity of Monoid is Unique

Let $\struct {S, \circ}$ be a monoid. Then $S$ has a unique identity.


Identity of Group is Unique

Let $\struct {G, \circ}$ be a group which has an identity element $e \in G$. Then $e$ is unique.


Unity of Integral Domain is Unique

Let $\struct {D, +, \times}$ be an integral domain. Then the unity of $\struct {D, +, \times}$ is unique.


Zero of Integral Domain is Unique

Let $\struct {D, +, \times}$ be an integral domain. Then the zero of $\struct {D, +, \times}$ is unique.


Negative in Integral Domain is Unique

Let $\struct {D, +, \times}$ be an integral domain. Let $a \in R$. Then the negative $-a$ of $a$ is unique.


Inverse in Group is Unique

Let $\struct {G, \circ}$ be a group. Then every element $x \in G$ has exactly one inverse: :$\forall x \in G: \exists_1 x^{-1} \in G: x \circ x^{-1} = e^{-1} = x \circ x$ where $e$ is the identity element of $\struct {G, \circ}$.


Product Inverse in Ring is Unique

Let $\struct {R, +, \circ}$ be a ring with unity. Let $x \in R$ be a unit of $R$. Then the product inverse $x^{-1}$ of $x$ is unique.


Units of Gaussian Integers form Group

Let $U_\C$ be the set of units of the Gaussian integers: :$U_\C = \set {1, i, -1, -i}$ where $i$ is the imaginary unit: $i = \sqrt {-1}$. Let $\struct {U_\C, \times}$ be the algebraic structure formed by $U_\C$ under the operation of complex multiplication. Then $\struct {U_\C, \times}$ forms a cyclic group under complex multiplication.


Permutation Group is Subgroup of Symmetric Group

Let $S$ be a set. Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$, where $\circ$ denotes the composition operation. Let $\struct {H, \circ}$ be a set of permutations of $S$ which forms a group under $\circ$. Then $\struct {H, \circ}$ is a subgroup of $\struct {\map \Gamma S, \circ}$.


Units of Field of Complex Numbers form Group

:$\C^\times = \C \setminus \set 0$ where $\C^\times$ denotes the group of units of $\C$.


Determinant of Block Diagonal Matrix

Let $\mathbf A$ be a block diagonal matrix of order $n$. Let $\mathbf A_1,\ldots,\mathbf{A}_k$ be the square matrices on the diagonal, i.e.: :$\displaystyle \mathbf A = \begin{bmatrix} \mathbf A_1 & 0 & \cdots & 0 \\ 0 & \mathbf A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \mathbf A_k  \end{bmatrix}$ Then the determinant $\det \left({\mathbf A}\right)$ of $\mathbf A$ satisfies: :$\displaystyle \det \left({\mathbf A}\right) = \prod_{i \mathop = 1}^k \det \left({\mathbf A_i}\right)$


Coset Space forms Partition

=== Left Coset Space forms Partition === {{:Left Coset Space forms Partition}} === Right Coset Space forms Partition === {{:Right Coset Space forms Partition}}


Element of Group is in Unique Coset of Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $x \in G$.


General Linear Group to Determinant is Homomorphism

Let $\GL {n, \R}$ be the general linear group over the field of real numbers. Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers. Let $\det: \GL {n, \R} \to \struct {\R_{\ne 0}, \times}$ be the mapping: :$\mathbf A \mapsto \map \det {\mathbf A}$ where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$. Then $\det$ is a group homomorphism.


Logarithm on Positive Real Numbers is Group Isomorphism

Let $\struct {\R_{>0}, \times}$ be the multiplicative group of positive real numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $b$ be any real number such that $b > 1$. Let $\log_b: \struct {\R_{>0}, \times} \to \struct {\R, +}$ be the mapping: :$x \mapsto \map {\log_b} x$ where $\log_b$ is the logarithm to base $b$. Then $\log_b$ is a group isomorphism.


Kernel is Trivial iff Monomorphism/Group

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a group homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then $\phi$ is a group monomorphism {{iff}} $\map \ker \phi$ is trivial.


Kernel is Trivial iff Monomorphism/Ring

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then $\phi$ is a ring monomorphism {{iff}} $\map \ker \phi = 0_{R_1}$.


First Isomorphism Theorem/Groups

Let $\phi: G_1 \to G_2$ be a group homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then: :$\Img \phi \cong G_1 / \map \ker \phi$ where $\cong$ denotes group isomorphism.


First Isomorphism Theorem/Rings

Let $\phi: R \to S$ be a ring homomorphism. Let $\map \ker \phi$ be the kernel of $\phi$. Then: :$\Img \phi \cong R / \map \ker \phi$ where $\cong$ denotes ring isomorphism.


Second Isomorphism Theorem/Groups

Let $G$ be a group, and let: :$(1): \quad H$ be a subgroup of $G$ :$(2): \quad N$ be a normal subgroup of $G$. Then: :$\displaystyle \frac H {H \cap N} \cong \frac {H N} N$ where $\cong$ denotes group isomorphism.


Second Isomorphism Theorem/Rings

Let $R$ be a ring, and let: :$S$ be a subring of $R$ :$J$ be an ideal of $R$. Then: :$(1): \quad S + J$ is a subring of $R$ :$(2): \quad J$ is an ideal of $S + J$ :$(3): \quad S \cap J$ is an ideal of $S$ :$(4): \quad \dfrac S {S \cap J} \cong \dfrac {S + J} J$ where $\cong$ denotes group isomorphism.


Third Isomorphism Theorem/Groups

Let $G$ be a group, and let: : $H, N$ be normal subgroups of $G$ : $N$ be a subset of $H$. Then: :$(1): \quad H / N$ is a normal subgroup of $G / N$ :::where $H / N$ denotes the quotient group of $H$ by $N$ :$(2): \quad \dfrac {G / N} {H / N} \cong \dfrac G H$ :::where $\cong$ denotes group isomorphism.


Third Isomorphism Theorem/Rings

Let $R$ be a ring, and let: :$J, K$ be ideals of $R$ :$J$ be a subset of $K$. Then: :$(1): \quad K / J$ is an ideal of $R / J$ :where $K / J$ denotes the quotient ring of $K$ by $J$ :$(2): \quad \dfrac {R / J} {K / J} \cong \dfrac R K$ :where $\cong$ denotes ring isomorphism.


Continuous Mapping on Union of Open Sets

Let $T = \left({X, \tau}\right)$ and $S = \left({Y, \sigma}\right)$ be topological spaces. Let $I$ be an indexing set. For all $i \in I$, let $C_i$ be open in $T$. Let $f: X \to Y$ be a mapping such that the restriction $f \restriction_{C_i}$ is continuous for all $i \in I$. Then $f$ is continuous on $C = \displaystyle \bigcup_{i \mathop \in I} C_i$, i.e., $f \restriction_C$ is continuous.


Inertia Principle

Let $\left \langle {a_n}\right \rangle$ be a sequence in $\R$. Let $a_n \to l$ as $n \to \infty$, and let $c < l$ where $c \in \R$. Then $\exists N \in \N$ such that: : $\forall n \in \N, n \ge N: c < a_n$


Composite of Homomorphisms is Homomorphism/Algebraic Structure

Let: : $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$ : $\left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be algebraic structures. Let: : $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be homomorphisms. Then the composite of $\phi$ and $\psi$ is also a homomorphism.


Composite of Homomorphisms is Homomorphism/R-Algebraic Structure

Let: : $\left({S_1, *_1}\right)_R$ : $\left({S_2, *_2}\right)_R$ : $\left({S_3, *_3}\right)_R$ be $R$-algebraic structures with the same number of operations. Let: : $\phi: \left({S_1, *_1}\right)_R \to \left({S_2, *_2}\right)_R$ : $\psi: \left({S_2, *_2}\right)_R \to \left({S_3, *_3}\right)_R$ be homomorphisms. Then the composite of $\phi$ and $\psi$ is also a homomorphism.


Subgroup is Subgroup of Normalizer

Let $G$ be a group. A subgroup $H \le G$ is a subgroup of its normalizer: :$H \le G \implies H \le \map {N_G} H$


Inclusion Mappings to Topological Sum from Components

Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces. Let $\struct {Z, \tau_3}$ be the topological sum of $X$ and $Y$ where $\tau_3$ is the topology generated by $\tau_1$ and $\tau_2$. Then $\tau_3$ is the finest topology on $Z$ in which the inclusion mappings from $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ to $\struct {Z, \tau_3}$ are continuous.


Principal Ultrafilter is All Sets Containing Cluster Point

Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF \subset \powerset S$ be a principal ultrafilter on $S$. Let its cluster point be $x$. Then $\FF$ is the set of all subsets of $S$ which contain $x$.


Equivalence of Definitions of T4 Space

{{TFAE|def = T4 Space|view = $T_4$ space}} Let $T = \struct {S, \tau}$ be a topological space.


Equivalence of Definitions of T5 Space

{{TFAE|def = T5 Space|view = $T_5$ space}} Let $T = \struct {S, \tau}$ be a topological space.


Product Space is T2 iff Factor Spaces are T2

Let $\mathbb S = \family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_2$ (Hausdorff) space {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space.


Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.


Product Space is T3 iff Factor Spaces are T3

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_3$ space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_3$ space.


Product Space is T3 1/2 iff Factor Spaces are T3 1/2

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$ with $S_\alpha \ne \O$ for every $\alpha \in I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_{3 \frac 1 2}$ space {{iff}} each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_{3 \frac 1 2}$ space.


Factor Spaces are T4 if Product Space is T4

Let $\mathbb S = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_4$ space. Then each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_4$ space.


Factor Spaces are T5 if Product Space is T5

Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_5$ space. Then each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_5$ space.


T4 Property is not Hereditary

Let $T = \struct {S, \tau}$ be a topological space which is a $T_4$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then it does not necessarily follow that $T_H$ is a $T_4$ space.


Derivative of Arcsecant Function

:$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {+1} {x \sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {-1} {x \sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x < \pi \ (\text {that is: $x < -1$})  \\ \end{cases}$


Paracompactness is Preserved under Projections

Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate. If $\struct {S, \tau}$ is paracompact, then each of $\struct {S_\alpha, \tau_\alpha}$ is also paracompact.


Connected Space is Connected Between Two Points

Let $T$ be a topological space which is connected. Then $T$ is connected between two points.


Equality is Reflexive

:$\forall a: a = a$


Closed Set in Metric Space is G-Delta

Let $\left({X, d}\right)$ be a metric space, and let $F \subset X$ be a closed set.  Then $F$ is a $G_\delta$ set of X.


Totally Disconnected Space is Totally Pathwise Disconnected

Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected. Then $T$ is a totally pathwise disconnected space.


Second-Countable T3 Space is T5

Let $T = \struct {S, \tau}$ be a $T_3$ space which is also second-countable. Then $T$ is a $T_5$ space.


T3 Space is Fully T4 iff Paracompact

Let $T = \struct {S, \tau}$ be a $T_3$ space. Then $T$ is fully $T_4$ {{iff}} $T$ is paracompact.


Mean Value Theorem for Integrals

Let $f$ be a continuous real function on the closed interval $\closedint a b$. Then there exists a real number $k \in \closedint a b$ such that: :$\displaystyle \int_a^b \map f x \rd x = \map f k \paren {b - a}$


Cover of Doubletons of Infinite Particular Point Space has no Locally Finite Refinement

Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Let $\CC$ be the open cover of $T$ defined as: :$\CC = \set {\set {x, p}: x \in S, x \ne p}$ Then $\CC$ has no open refinement which is locally finite.


Path-Connected iff Path-Connected to Point

Let $T = \left({S, \tau}\right)$ be a topological space. Let $U \subseteq S$ be a non-empty subset of $T$. Then $U$ is path-connected {{iff}}: :$\exists p \in U: \forall q \in U: \exists f: \left[{0 \,.\,.\, 1}\right] \to U: f$ continuous, $f \left({0}\right) = p$ and $f \left({1}\right) = q$


Uncountable Fort Space is not Separable

Let $T = \struct {S, \tau_p}$ be a Fort space on an uncountable set $S$. Then $T$ is not a separable space.


Countable Fort Space is Separable

Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a separable space.


Countable Fort Space is Second-Countable

Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a second-countable space.


Fort Space is Regular

Let $T = \struct {S, \tau_p}$ be a Fort space. Then $T$ is a regular space.


Countable Fort Space is Metrizable

Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a metrizable space.


Fortissimo Space is not Pseudocompact

Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not a pseudocompact space.


Double Pointed Fortissimo Space is not Sigma-Compact

Let $T = \struct {S, \tau}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is not $\sigma$-compact.


Neighborhood of Origin in Arens-Fort Space is not Compact

Let $T = \struct {S, \tau}$ be the Arens-Fort space. Let $N_p$ be a neighborhood of the point $p = \tuple {0, 0}$. Then $N_p$ is not compact.


Arens-Fort Space is not Extremally Disconnected

Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not extremally disconnected.


Modified Fort Space is Totally Disconnected

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is totally disconnected.


Sets in Modified Fort Space are Disconnected

Let $T = \struct {S, \tau_{a, b}}$ be a modified Fort space. Let $H$ be a subset of $S$ with more than one point. Then $H$ is disconnected.


Modified Fort Space is not Totally Separated

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not totally separated.


Modified Fort Space is Scattered

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is scattered.


Isolated Points in Subsets of Modified Fort Space

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Let $H \subseteq S$ contain more than two points. Then $H$ contains an isolated point.


Modified Fort Space is Sequentially Compact

Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is sequentially compact.


Tangent Function is Odd

:$\map \tan {-x} = -\tan x$ That is, the tangent function is odd.


Tangent of Sum

:$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$


Natural Logarithm Function is Differentiable

The (real) natural logarithm function is differentiable.


Countable Basis of Real Number Line

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\BB$ be the set of subsets of $\R$ defined as: :$\BB = \set {\openint a b: a, b \in \Q,\ a < b}$ That is, $\BB$ is the set of open intervals of $\R$ whose endpoints are rational numbers. Then $\BB$ forms a countable basis of $\struct {\R, \tau_d}$


Second-Countable Space is Separable

Let $T = \struct {S, \tau}$ be a second-countable topological space. Then $T$ is also a separable space.


Real Number Line is Second-Countable

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is second-countable.


Real Number Line is not Countably Compact

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is not countably compact.


Real Number Line is Sigma-Compact

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is $\sigma$- compact.


Closed Subset of Real Number Line is G-Delta

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $H \subseteq \R$ be a closed subset of $\R$. Then $H$ is a $G_\delta$ set.


Real Number Line is Paracompact

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is paracompact.


Real Number Line with Off-Center Distance Function is Quasimetric Space

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\tau_d$ can be given by a quasimetric defined as: :$\map d {x, y} = \begin {cases} y - x & : y \ge x \\ 2 \paren {x - y} & : y < x \end {cases}$ Thus $\struct {\R, \tau_d}$ is a quasimetric space.


Sub-basis for Uniformity on Real Number Line

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $a, b \in \R$ such that $a < b$. Let $S_{a b}$ be the set of subsets of $\R$ defined as: :$S_{a b} = \set {\tuple {x, y}: x, y < b \text{ or } x, y > a}$ Then $S_{ab}$ is a basis for a uniformity $U$ which generates the usual topology on $\R$. Note that $U$ is clearly not the usual metric uniformity. {{MissingLinks|Lots of assumed definitions here which might not be properly specified.}}


NAND with Equal Arguments

:$p \uparrow p \dashv \vdash \neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation.


NAND is Commutative

:$p \uparrow q \dashv \vdash q \uparrow p$


NAND is not Associative

:$p \uparrow \left({q \uparrow r}\right) \not \vdash \left({p \uparrow q}\right) \uparrow r$


NOR with Equal Arguments

:$p \downarrow p \dashv \vdash \neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator.


NOR is Commutative

:$p \downarrow q \dashv \vdash q \downarrow p$


NOR is not Associative

:$p \downarrow \left({q \downarrow r}\right) \not \vdash \left({p \downarrow q}\right) \downarrow r$


Representation of Ternary Expansions

Let $x \in \R$ be a real number. Let $x$ be represented in base $3$ notation. While it may be possible for $x$ to have two different such representations, for example: :$\dfrac 1 3 = 0.100000 \ldots_3 = 0.022222 \ldots_3$ it is not possible for $x$ be written in more than one way without using the digit $1$.


Equivalence of Definitions of Cantor Set

{{TFAE|def = Cantor Set|view = the Cantor Set $\CC$}}:


Cantor Set is Closed in Real Number Space

Let $\CC$ be the Cantor set. Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$. Then $\CC$ is a closed subset of $\struct {\R, \tau_d}$.


Word Metric is Metric

Let $\left({G, \circ}\right)$ be a group. Let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \in S \iff x\in S$). Let $d_S$ be the associated word metric. Then $d_S$ is a metric on $G$.


Cantor Space is Compact

Let $\CC$ be the Cantor set. Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$. Then $\CC$ is a compact subset of $\struct {\R, \tau_d}$.


Cantor Space is Complete Metric Space

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is a complete metric space.


Cantor Space satisfies all Separation Axioms

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ satisfies all the separation axioms.


Cantor Space is Second-Countable

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is a second-countable space.


Cantor Space is Dense-in-itself

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is dense-in-itself.


Cantor Space is not Scattered

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is not scattered.


Cantor Space is Perfect

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $\CC$ is a perfect set of the real number space $\R$ under the usual (Euclidean) topology $\tau_d$.


Cantor Space is Nowhere Dense

Let $T = \left({\mathcal C, \tau_d}\right)$ be the Cantor space. Then $T$ is nowhere dense in $\left[{0 \,.\,.\, 1}\right]$.


Cantor Space is Meager in Closed Unit Interval

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is meager in $\closedint 0 1$.


Cantor Set is Uncountable

The Cantor set $\mathcal C$ is uncountable.


Equality is Symmetric

:$\forall a, b: a = b \implies b = a$


Biconditional is Commutative/Formulation 1/Proof 2

: $p \iff q \dashv \vdash q \iff p$


Derivative of Hyperbolic Cosine Function

:$\map {\dfrac \d {\d x} } {\cosh u} = \sinh u \dfrac {\d u} {\d x}$


Cantor Space is Totally Separated

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is totally separated.


Cantor Space is not Extremally Disconnected

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is not extremally disconnected.


Cantor Space as Countably Infinite Product

Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the discrete space of the two points $0$ and $2$. Let $\displaystyle A = \prod_{n \mathop = 1}^\infty A_n$. Let $\struct {A, \tau}$ be the product space where $\tau$ is the Tychonoff topology on $A$. Then $A$ is homeomorphic to the Cantor space.


Cantor Space is not Locally Connected

Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is not locally connected.


Local Connectedness is not Preserved under Infinite Product

The property of local connectedness is not preserved under the operation of forming an infinite product space.


Difference of Squares of Hyperbolic Cosine and Sine

:$\cosh^2 x - \sinh^2 x = 1$ where $\cosh$ and $\sinh$ are hyperbolic cosine and hyperbolic sine.


Derivative of Arccosecant Function

:$\dfrac {\map \d {\arccsc x} } {\d x} = \dfrac {-1} {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {-1} {x \sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {+1} {x \sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc x < 0 \ (\text {that is: $x < -1$})  \\ \end{cases}$


Power of Sum Modulo Prime

Let $p$ be a prime number. Then: :$\paren {a + b}^p \equiv a^p + b^p \pmod p$


Volume of Right Circular Cone

The volume $V$ of a right circular cone is given by: :$V = \dfrac 1 3 \pi r^2 h$ where: : $r$ is the radius of the base : $h$ is the height of the cone, that is, the distance between the apex and the center of the base.


Intersection of Topologies is Topology

Let $\left({\tau_i}\right)_{i \in I}$ be an arbitrary indexed set of topologies for a set $S$. Then $\tau := \displaystyle \bigcap_{i \mathop \in I} {\tau_i}$ is also a topology for $S$.


Power Set is Closed under Set Difference

Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A \setminus B \in \powerset S$ where $A \setminus B$ denotes the set difference of $A$ and $B$.


Powers of Semigroup Element Commute

Let $\struct {S, \odot}$ be a semigroup. Let $a \in S$. Let $m, n \in \Z_{>0}$. Then: :$\forall m, n \in \Z_{>0}: a^n \odot a^m = a^m \odot a^n$


Equivalence of Definitions of Derivative

{{TFAE|def = Derivative of Real Function at Point}} 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 $I$.


Derivative of x to the x

:$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$


Commutativity of Powers in Semigroup

:$\forall m, n \in \N_{>0}: a \circ b = b \circ a \implies a^m \circ b^n = b^n \circ a^m$ but it is not necessarily the case that: :$\forall m, n \in \N_{>0}: a^m \circ b^n = b^n \circ a^m \implies a \circ b = b \circ a$


Power of Product of Commutative Elements in Semigroup

:$\forall n \in \N_{>1}: \paren {x \circ y}^n = x^n \circ y^n \iff x \circ y = y \circ x$


Sum of Arcsecant and Arccosecant

Let $x \in \R$ be a real number such that $\left|{x}\right| \ge 1$. Then: : $\operatorname{arcsec} x + \operatorname{arccsc} x = \dfrac \pi 2$ where $\operatorname{arcsec}$ and $\operatorname{arccsc}$ denote arcsecant and arccosecant respectively.


Equation of Straight Line Tangent to Circle

Let $\tuple {a, b}$ be the center of a circle $\mathcal C$. Let $P_n = \tuple {x_n, y_n}$ be any point on $\mathcal C$. The equation of a non-vertical tangent line $\mathcal T$ to $\mathcal C$ is given by: :$y - y_n = \dfrac {a - x_n} {y_n - b} \paren {x - x_n}$ The equations of the vertical tangent lines to $\mathcal C$ are: :$x = r - a$ for $P = \tuple {r - a, b}$ :$x = a - r$ for $P = \tuple {a - r, b}$


Index Laws for Monoids/Negative Index

:$\forall n \in \Z: \paren {a^n}^{-1} = a^{-n} = \paren {a^{-1} }^n$


Infinite Limit Theorem

Let $f$ be a real function of $x$ of the form  :$\map f x = \dfrac {\map g x} {\map h x}$ Further, let $g$ and $h$ be continuous on some open interval $\mathbb I$, where $c$ is a constant in $\mathbb I$. If: :$(1): \quad \map g c \ne 0$ :$(2): \quad \map h c = 0$ :$(3): \quad \forall x \in \mathbb I: x \ne c \implies \map h x \ne 0$ then the limit of $f$ as $x \to c$ will not exist, and: :$\displaystyle \lim_{x \mathop \to c^+} \map f x = +\infty$ or $-\infty$ :$\displaystyle \lim_{x \mathop \to c^-} \map f x = +\infty$ or $-\infty$


Index Laws for Monoids/Product of Indices

:$\forall m, n \in \Z: a^{n m} = \paren {a^m}^n = \paren {a^n}^m$


Index Laws for Monoids/Sum of Indices

:$\forall m, n \in \Z: a^{n + m} = a^n \circ a^m$


Powers of Group Elements/Negative Index

:$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$


Powers of Group Elements/Sum of Indices

:$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$


Powers of Group Elements/Product of Indices

:$\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$


Group Element Commutes with Inverse

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $x \in G$. Then: :$x \circ x^{-1} = x^{-1} \circ x$ That is, $x$ commutes with its inverse $x^{-1}$.


Commutativity of Powers in Monoid

:$\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$


Power of Product of Commutative Elements in Monoid

:$\forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$


Commutativity of Powers in Group

: $\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$


Power of Product of Commutative Elements in Group

:$a \circ b = b \circ a \iff \forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$


Primitive of Tangent Function/Cosine Form

:$\displaystyle \int \tan x \rd x = -\ln \size {\cos x} + C$ where $\cos x \ne 0$.


Primitive of Reciprocal

:$\ds \int \frac {\d x} x = \ln \size x + C$ for $x \ne 0$.


Powers of Commuting Elements of Semigroup Commute

:$\forall m, n \in \N_{>0}: \paren {\circ^m a} \circ \paren {\circ^n b} = \paren {\circ^n b} \circ \paren {\circ^m a}$


Right Operation All Elements Right Zeroes

Let $\struct {S, \rightarrow}$ be an algebraic structure in which the operation $\rightarrow$ is the right operation. Then no matter what $S$ is, $\struct {S, \rightarrow}$ is a semigroup all of whose elements are right zeroes. Thus it can be seen that any right zero in a semigroup is not necessarily unique.


Left Operation All Elements Left Zeroes

Let $\struct {S, \leftarrow}$ be an algebraic structure in which the operation $\leftarrow$ is the left operation. Then no matter what $S$ is, $\struct {S, \leftarrow}$ is a semigroup all of whose elements are left zeroes. Thus it can be seen that any left zero in a semigroup is not necessarily unique.


More than One Right Zero then No Left Zero

Let $\struct {S, \circ}$ be an algebraic structure. If $\struct {S, \circ}$ has more than one left zero, then it has no right zero. Likewise, if $\struct {S, \circ}$ has more than one right zero, then it has no left zero.


Left and Right Zero are the Same

Let $\struct {S, \circ}$ be an algebraic structure. Let $z_L \in S$ be a left zero, and $z_R \in S$ be a right zero. Then $z_L = z_R$, that is, both the left and right zero are the same, and are therefore a zero $z$. Furthermore, $z$ is the ''only'' left and right zero for $\circ$.


Primitive of Cotangent Function

:$\displaystyle \int \cot x \rd x = \ln \size {\sin x} + C$ where $\sin x \ne 0$.


Inner Limit in Hausdorff Space by Open Neighborhoods

Let $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in a Hausdorff topological space $\left({\mathcal X, \tau}\right)$. Let $x \in \mathcal X$. Let $\mho \left({x}\right) := \left\{ {V \in \tau:\ x \in V}\right\}$ denote the set of open neighborhoods of $x$. Let $\mathcal N_\infty$ denote the set of cofinite subsets of $\N$: : $\mathcal N_\infty := \left\{ {N \subset \N: \N \setminus N \text{ is finite} }\right\}$ Then the inner limit of $\left \langle{C_n}\right \rangle_{n \in \N}$ is: :$\displaystyle \liminf_n C_n = \left\{ {x \in \mathcal X: \forall V \in \mho \left({x}\right): \exists N \in \mathcal N_\infty: \forall n \in N: C_n \cap V \ne \varnothing}\right\}$ or equivalently: : $\displaystyle \liminf_n C_n = \left\{{x \in \mathcal X: \forall V \in \mho \left({x}\right): \exists N_0 \in \N: \forall n \ge N_0: C_n \cap V \ne \varnothing}\right\}$


Stabilizer of Subspace stabilizes Orthogonal Complement

Let $H$ be a finite-dimensional real or complex Hilbert space (that is, inner product space). Let $t: H \to H$ be a normal operator on $H$. Let $t$ stabilize a subspace $V$. Then $t$ also stabilizes its orthogonal complement $V^\perp$.


Primitive of Secant Function/Secant plus Tangent Form

:$\ds \int \sec x \rd x = \ln \size {\sec x + \tan x} + C$ where $\sec x + \tan x \ne 0$.


Primitive of Cosecant Function/Cosecant plus Cotangent Form

:$\ds \int \csc x \rd x = -\ln \size {\csc x + \cot x} + C$ where $\csc x + \cot x \ne 0$.


Subset of Natural Numbers is Cofinal iff Infinite

Consider the ordered set $\left({\N, \le}\right)$, where $\le$ is the usual ordering on the natural numbers. Let $S \subseteq \N$. Then $S$ is cofinal {{iff}} it is infinite.


Inner Limit in Hausdorff Space by Set Closures

Let $\struct {\XX, \tau}$ be a Hausdorff space. Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $\XX$. Then: :$\displaystyle \liminf_n C_n = \bigcap_{N \mathop \in \NN_\infty^\#} \map \cl {\bigcup_{n \mathop \in N} C_n}$ where: :$\cl$ denotes set closure :$\NN_\infty^\#$ denotes the set of cofinal subsets of $\N$.


Ordinal is Transitive

Every ordinal (by Definition 3) is a transitive set.


Foundational Relation has no Relational Loops

Let $\prec$ be a foundational relation on $A$ and let $x_1, x_2, \ldots, x_n \in A$. Then: : $\neg \left({x_1 \prec x_2 \land x_3 \prec x_4 \cdots \land x_n \prec x_1}\right)$ That is, there are no relational loops within $A$.


Epsilon is Foundational

Let $\Epsilon$ denote the epsilon relation. Then $\Epsilon$ is a foundational relation on every class $A$.


Inner Limit in Normed Spaces by Open Balls

Let $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in a normed vector space $\left({\mathcal X, \left\Vert{\cdot}\right\Vert}\right)$. Then the inner limit of $\left \langle{C_n}\right \rangle_{n \in \N}$ is: : $\displaystyle \liminf_n \ C_n = \left\{{x: \forall \epsilon > 0: \exists N \in \mathcal N_\infty: \forall n \in N: x \in C_n + \epsilon B}\right\}$ where $B$ denotes the open unit ball of the space. {{explain|What are $N$ and $\mathcal N_\infty$ in this context? Also, what point is at the center of $B$? And what does $C_n + \epsilon B$ mean? For the latter one suspects $\cup$, but this needs to be checked. This page might need to be rewritten from a new perspective, as the original author was touchy about symbols used and departed ProofWiki in a rage when his notation was changed.}}


Inner Limit is Closed Set

Let $\struct {S, \tau}$ be a Hausdorff topological space. Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $S$. Then the inner limit $\liminf_n C_n$ is a closed set.


Local Basis of Topological Vector Space

Let $\struct {\XX, \tau}$ be a topological vector space. Let $0_\XX$ denote the zero vector of $\XX$. Then there exists a local basis $\BB$ of $0_\XX$ with the following properties: :$(1): \quad \forall W \in \BB: \exists V \in \BB$ such that $V + V \subseteq W$ (where the addition $V + V$ is meant in the sense of the Minkowski sum) :$(2): \quad$ Every $W \in \BB$ is star-shaped (balanced) :$(3): \quad$ Every $W \in \BB$ is absorbent. :$(4): \quad \displaystyle \bigcap_\BB W = \set {0_\XX}$.


No Membership Loops

For any proper classes or sets $A_1, A_2, \ldots, A_n$: :$\neg \paren {A_1 \in A_2 \land A_2 \in A_3 \land \cdots \land A_n \in A_1}$


Composite of Isomorphisms is Isomorphism/Algebraic Structure

Let: : $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$ : $\left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be algebraic structures. Let: : $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be isomorphisms. Then the composite of $\phi$ and $\psi$ is also an isomorphism.


Composite of Isomorphisms is Isomorphism/R-Algebraic Structure

Let: : $\left({S_1, \ast_1}\right)_R$ : $\left({S_2, \ast_2}\right)_R$ : $\left({S_3, \ast_3}\right)_R$ be $R$-algebraic structures with the same number of operations. Let: : $\phi: \left({S_1, \ast_1}\right)_R \to \left({S_2, \ast_2}\right)_R$ : $\psi: \left({S_2, \ast_2}\right)_R \to \left({S_3, \ast_3}\right)_R$ be isomorphisms. Then the composite of $\phi$ and $\psi$ is also an isomorphism.


Element of Transitive Class

Let $B$ be a transitive class. Then: :$A \in B \implies A \subsetneq B$ where $\subsetneq$ denotes a proper subset).


Restriction of Foundational Relation is Foundational

Let $S$ be a set. Let $\RR \subseteq S \times S$ be a foundational relation on $S$. Let $T \subseteq S$ be a subset of $S$. Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$. Then $\RR {\restriction_T}$ is a foundational relation on $T$.


Restriction of Strict Well-Ordering is Strict Well-Ordering

Let $R$ be a strict well-ordering of $A$. Let $B \subseteq A$. Then $R$ is a strict well-ordering of $B$.


Alternative Definition of Ordinal

A set $S$ is an ordinal {{iff}} $S$ is transitive and is strictly well-ordered by the $\in$-relation.


Ordinal Class is Ordinal

The Ordinal Class $\On$ is an ordinal.


Transitive Set is Proper Subset of Ordinal iff Element of Ordinal

Let $A$ be an ordinal. Let $B$ be a transitive set. Then: :$B \subsetneq A \iff B \in A$


Ordinal Membership is Trichotomy

Let $A$ and $B$ be ordinals. Then: :$\left({A = B}\right) \lor \left({A \in B}\right) \lor \left({B \in A}\right)$ where $\lor$ denotes logical or.


Second Derivative of Concave Real Function is Non-Positive

Let $f$ be a real function which is twice differentiable on the open interval $\openint a b$. Then $f$ is concave on $\openint a b$ {{iff}} its second derivative $f'' \le 0$ on $\openint a b$.


Ordinal is Member of Ordinal Class

Let $A$ be an ordinal. Then: :$A \in \On \lor A = \On$ where $\On$ denote the class of ordinals .


{{:De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection}}


Let $S$ and $T$ be sets. Let $\mathcal P \left({T}\right)$ be the power set of $T$. Let $\mathbb T \subseteq \mathcal P \left({T}\right)$. Then: ==== Difference with Intersection ==== {{:De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection}} ==== Difference with Union ==== {{:De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Union}}


{{:De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection}}


Complement of Preimage equals Preimage of Complement

Let $f: S \to T$ be a mapping. Let $T_1$ be a subset of $T$. Then: :$\relcomp S {f^{-1} \sqbrk {T_1} } = f^{-1} \sqbrk {\relcomp T {T_1} }$ where: :$\complement_S$ (in this context) denotes relative complement :$f^{-1} \sqbrk {T_1}$ denotes preimage.


One-to-Many Image of Set Difference/Corollary 1

Let $\RR \subseteq S \times T$ be a relation which is one-to-many. Let $A \subseteq B \subseteq S$. Then: :$\relcomp {\RR \sqbrk B} {\RR \sqbrk A} = \RR \sqbrk {\relcomp B A}$ where $\complement$ (in this context) denotes relative complement.


One-to-Many Image of Set Difference/Corollary 2

Let $\mathcal R \subseteq S \times T$ be a relation which is one-to-many. Let $A$ be a subset of $S$. Then: :$\relcomp {\Img {\mathcal R} } {\mathcal R \sqbrk A} = \mathcal R \sqbrk {\relcomp S A}$ where $\complement$ (in this context) denotes relative complement. In the language of direct image mappings this can be presented as: :$\forall A \in \powerset S: \map {\paren {\complement_{\Img {\mathcal R} } \circ \mathcal R^\to} } A = \map {\paren {\mathcal R^\to \circ \complement_S} } A$


Integer Power Function is Bijective iff Index is Odd

Let $n \in \Z_{\ge 0}$ be a positive integer. Let $f_n: \R \to \R$ be the real function defined as: :$\map {f_n} x = x^n$ Then $f_n$ is a bijection {{iff}} $n$ is odd.


Set of Infinite Sequences is Uncountable

Let $S$ be a set which contains more than one element. Let $S^\infty$ denote the set of all sequences of elements of $S$. Then $S^\infty$ is uncountable.


Union of Topologies is not necessarily Topology

Let $\left({\tau_i}\right)_{i \in I}$ be an arbitrary indexed set of topologies for a set $S$. Then $\tau := \displaystyle \bigcup_{i \mathop \in I} {\tau_i}$ is not necessarily also a topology for $S$.


Niven's Theorem

Consider the angles $\theta$ in the range $0 \le \theta \le \dfrac \pi 2$. The only values of $\theta$ such that both $\dfrac \theta \pi$ and $\sin \theta$ are rational are: :$\theta = 0: \sin \theta = 0$ :$\theta = \dfrac \pi 6: \sin \theta = \dfrac 1 2$ :$\theta = \dfrac \pi 2: \sin \theta = 1$


Sorgenfrey Line is Topology

The Sorgenfrey Line is a topological space.


Sorgenfrey Line is Hausdorff

Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $T$ is Hausdorff.


Sorgenfrey Line is Perfectly Normal

Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $T$ is perfectly normal.


Sorgenfrey Line is Expansion of Real Line

Let $\R = \struct {\R, d}$ be the metric space defined in Real Number Line is Metric Space. Let $T = \struct {\R, \tau}$ be the Sorgenfrey line. Then $T$ is an expansion of $\R$ as a topological space.


Sorgenfrey Line satisfies all Separation Axioms

Let $T = \left({\R, \tau}\right)$ be the Sorgenfrey line. Then $T$ satisfies all separation axioms.


Argument of Product equals Sum of Arguments

Let $z_1, z_2 \in \C$ be complex numbers. Let $\arg$ be the argument operator. Then: :$\map \arg {z_1 z_2} = \map \arg {z_1} + \map \arg {z_2} + 2 k \pi$ where $k$ can be $0$, $1$ or $-1$.


Convergence of Sequence in Discrete Space

Let $T = \left({S, \tau}\right)$ be a discrete topological space. Let $H = \left \langle{x_n}\right \rangle_{n \in \N}$ be a sequence in $S$. Then $H$ converges in $T$ to a limit iff: :$\exists k \in \N: \forall m \in \N: m > k: x_m = x_k$ That is, iff the sequence reaches some value of $S$ and "stays there".


Discrete Subspace of Fortissimo Space

Let $T = \left({S, \tau_p}\right)$ be a Fortissimo space. Let $T' = \left({S \setminus \left\{{p}\right\}, \tau_p}\right)$ be the topological subspace induced on $T$ by the subset $S \setminus \left\{{p}\right\}$. Then $T'$ is a discrete topological space.