Function of Two Discrete Random Variables

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$. Let $g: \R^2 \to \R$ be a real-valued function. Then $Z = g \left({X, Y}\right)$, defined as: :$\forall \omega \in \Omega: Z \left({\omega}\right) = g \left({X \left({\omega}\right), Y \left({\omega}\right)}\right)$ is also a discrete random variable.


Expectation of Function of Joint Probability Mass Distribution

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\left({\Omega, \Sigma, \Pr}\right)$. Let $E \left({X}\right)$ be the expectation of $X$. Let $g: \R^2 \to \R$ be a real-valued function Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$. :$\displaystyle E \left({g \left({X, Y}\right)}\right) = \sum_{x \mathop \in \Omega_X} \sum_{y \mathop \in \Omega_Y} g \left({x, y}\right) p_{X, Y} \left({x, y}\right)$ whenever the sum is absolutely convergent.


Linearity of Expectation Function

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $\expect X$ denote the expectation of $X$. Then: :$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \, \expect X + \beta \, \expect Y$


Condition for Independence from Product of Expectations

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $\expect X$ denote the expectation of $X$. Then $X$ and $Y$ are independent {{iff}}: :$\expect {\map g x \map h y} = \expect {\map g x} \expect {\map h y}$ for all functions $g, h: \R \to \R$ for which the latter two expectations exist.


PGF of Sum of Independent Discrete Random Variables

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be independent discrete random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $Z$ be a discrete random variable such that $Z = X + Y$. Then: :$\map {\Pi_Z} s = \map {\Pi_X} s \, \map {\Pi_Y} s$ where $\map {\Pi_Z} s$ is the probability generating function of $Z$.


Brahmagupta Theorem

If a cyclic quadrilateral has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. Specifically: Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular, crossing at $M$. Let $EF$ be a line passing through $M$ and crossing opposite sides $BC$ and $AD$ of $ABCD$. Then $EF$ is perpendicular to $BC$ {{iff}} $F$ is the midpoint of $AD$.


Clavius's Law

If, from the negation of a proposition $p,$ we can derive $p$, we may conclude $p$:


Viète's Formulas

Let $P$ be a polynomial of degree $n$ with real or complex coefficients: {{begin-eqn}} {{eqn | l = \map P x        | r = \sum_{i \mathop = 0}^n a_i x^i }} {{eqn | r = a_n x^n + a_{n - 1} x^{n - 1} + \dotsb + a_1 x + a_0 }} {{end-eqn}} where $a_n \ne 0$. Let $z_1, \ldots, z_k$ be real or complex roots of $P$, not assumed distinct. Let $P$ be expressible in the form: :$\displaystyle \map P x = a_n \prod_{k \mathop = 1}^n \paren {x - z_k}$ Then: {{begin-eqn}} {{eqn | l = \paren {-1}^k \dfrac {a_{n - k} } {a_n}       | r = e_k \paren {\set {z_1, \ldots, z_n} }       | c = that is, the elementary symmetric function on $\set {z_1, \ldots, z_n}$ }} {{eqn | r = \sum_{1 \mathop \le i_1 \mathop < \dotsb \mathop < i_k \mathop \le n} z_{i_1} \dotsm z_{i_k}        | c = for $k = 1, 2, \ldots, n$. }} {{end-eqn}}


Inscribed Angle Theorem

An inscribed angle is equal to half the angle that is subtended by that arc. :300px Thus, in the figure above: :$\angle ABC = \frac 1 2 \angle ADC$ {{EuclidSaid}} :''In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.'' {{EuclidPropRef|III|20}}


Existence and Uniqueness Theorem for 1st Order IVPs

Let $x' = \map f {t, x}$, $\map x {t_0} = x_0$ be an explicit ODE of dimension $n$. Let there exist an open ball $V = \sqbrk {t_0 - \ell_0, t_0 + \ell_0} \times \map {\overline B} {x_0, \epsilon}$ of $\tuple {t_0, x_0}$ in phase space $\R \times \R^n$ such that $f$ is Lipschitz continuous on $V$. {{explain|Notation needs to be explained: $\sqbrk {t_0 - \ell_0, t_0 + \ell_0}$ looks as though it should be an interval (and so needs to be written in Wirth interval notation $\closedint {t_0 - \ell_0} {t_0 + \ell_0}$ so as to abide by house style rules), and $\tuple {t_0, x_0}$ is probably an ordered pair. It's not clear enough. The immediate confusion arises because as $\closedint {t_0 - \ell_0} {t_0 + \ell_0}$ is a closed interval it is counter-intuitive for it to be one of the factors of an open ball expressed as a Cartesian product.}} Then there exists $\ell < \ell_0$ such that there exists a unique solution $\map x t$ defined for $t \in \closedint {t_0 - \ell_0} {t_0 + \ell_0}$. {{explain|what is an "IVP"?}}


Maximum Modulus Principle

A non-constant analytic function $f$ in an open connected set $D$ does not have any interior maximum points. That is, for each $z \in D$ and $\delta > 0$, there exists some $\omega \in B_\delta \left({z}\right) \cap D$, such that: :$\left\vert{f \left({\omega}\right)}\right\vert > \left\vert{f \left({z}\right)}\right\vert$.


Euler Polyhedron Formula

For any convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces:  :$V - E + F = 2$


Line Joining Centers of Two Circles Touching Internally

Let two circles touch internally. Then the straight line joining their centers passes through the point where they touch. {{:Euclid:Proposition/III/11}}


Properties of Matrix Exponential

In the following: : $\mathrm A$ and $\mathrm B$ are constant square matrices : $P$ is a nonsingular matrix : $t, s \in \R$ The matrix exponential $e^{\mathrm A t}$ has the following properties:


Lyapunov's Stability Theorem

Let $V$ be a Lyapunov function on an open set containing an equilibrium point $x_0$. {{Disambiguate|Definition:Open Set}} Then $x_0$ is stable. If $V$ is strict, then $x_0$ is asymptotically stable. {{explain|strict}}


Bendixson-Dulac Theorem

Suppose there exists a continuously differentiable function $\alpha \left({x, y}\right)$ on a simply connected domain. {{Explain|What ''is'' the domain? Reals, complex, or what?}} Suppose that: :$\nabla \cdot \left({\alpha F}\right)$  is either always positive or always negative. Then the two-dimensional autonomous system: :$ \left({x, y}\right)' = F \left({x, y}\right)$ does not have a periodic solution.


General Vector Solution of Fundamental Matrix

Let $\map \Phi t$ be a fundamental matrix of the system $x' = \map A t x$. Then: :$\map \Phi t c$ is a general solution of $x' = \map A t x$.


General Fundamental Matrix

Let $\map \Phi t$ be a fundamental matrix of the system $x' = \map A t x$. Then: :$\map \Phi t C$ is a general fundamental matrix of $x' = \map A t x$, where $C$ is ''any'' nonsingular matrix.


Condition for Composite Mapping on Left

Let $A, B, C$ be sets. Suppose that $C$ is non-empty. Let $f: A \to B$ and $g: A \to C$ be mappings. Let $\RR: B \to C$ be a relation such that $g = \RR \circ f$ is the composite of $f$ and $\RR$. Then $\RR$ may be a mapping {{iff}}: :$\forall x, y \in A: \map f x = \map f y \implies \map g x = \map g y$ That is: :$\forall x, y \in A: \map f x = \map f y \implies \map g x = \map g y$ {{iff}}: :$\exists h: B \to C$ such that $h$ is a mapping and $h \circ f = g$


Condition for Composite Mapping on Right

Let $A, B, C$ be sets. Let $f: B \to A$ and $g: C \to A$ be mappings. Let $\mathcal R: C \to B$ be a relation such that $g = f \circ \mathcal R$ is the composite of $\mathcal R$ and $f$. Then $\mathcal R$ may be a mapping {{iff}}: :$\Img g \subseteq \Img f$ That is: :$\Img g \subseteq \Img f$ {{iff}}: :$\exists h: C \to B$ such that $h$ is a mapping and $f \circ h = g$


Set Union Preserves Subsets

Let $A, B, S, T$ be sets. Then: :$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$


Set Complement inverts Subsets

:$S \subseteq T \iff \map \complement T \subseteq \map \complement S$


Set Difference with Superset is Empty Set

:$S \subseteq T \iff S \setminus T = \O$


Intersection with Subset is Subset

:$S \subseteq T \iff S \cap T = S$


Union with Superset is Superset

:$S \subseteq T \iff S \cup T = T$


Intersection with Complement is Empty iff Subset

:$S \subseteq T \iff S \cap \map \complement T = \O$


Complement Union with Superset is Universe

:$S \subseteq T \iff \map \complement S \cup T = \mathbb U$


Set Difference is not Associative

Let $R, S, T$ be sets. The expression: :$\paren {R \setminus S} \setminus T = R \setminus \paren {S \setminus T}$ holds exactly when $R \cap T = \O$. Here $R \setminus S$ denotes set difference. Thus, set difference is not associative.


Cardinality of Set Union

=== Union of 2 Sets === {{:Cardinality of Set Union/2 Sets}} === Union of 3 Sets === {{:Cardinality of Set Union/3 Sets}}


Group Isomorphism Preserves Identity

Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism. Let: :$e_G$ be the identity of $\struct {G, \circ}$ :$e_H$ be the identity of $\struct {H, *}$. Then: :$\map \phi {e_G} = e_H$


Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group

Let $\Q_{> 0}$ be the set of strictly positive rational numbers, i.e. $\Q_{> 0} = \set {x \in \Q: x > 0}$. The structure $\struct {\Q_{> 0}, \times}$ is a countably infinite abelian group.


Multinomial Theorem

Let $x_1, x_2, \ldots, x_k \in F$, where $F$ is a field. Then: :$\displaystyle \paren {x_1 + x_2 + \cdots + x_m}^n = \sum_{k_1 \mathop + k_2 \mathop + \mathop \cdots \mathop + k_m \mathop = n} \binom n {k_1, k_2, \ldots, k_m} {x_1}^{k_1} {x_2}^{k_2} \cdots {x_m}^{k_m}$ where: :$m \in \Z_{> 0}$ is a positive integer :$n \in \Z_{\ge 0}$ is a non-negative integer :$\dbinom n {k_1, k_2, \ldots, k_m} = \dfrac {n!} {k_1! \, k_2! \, \cdots k_m!}$ denotes a multinomial coefficient. The sum is taken for all non-negative integers $k_1, k_2, \ldots, k_m$ such that $k_1 + k_2 + \cdots + k_m = n$, and with the understanding that wherever $0^0$ may appear  it shall be considered to have a value of $1$. The '''multinomial theorem''' is a generalization of the Binomial Theorem.


Mapping is Constant iff Increasing and Decreasing

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping. Then $\phi$ is a constant mapping {{iff}} $\phi$ is both increasing and decreasing.


Equivalence of Definitions of Euler's Number

{{TFAE|def = Euler's Number}} === Limit of Series === {{:Definition:Euler's Number/Limit of Series}} === Limit of Sequence === {{:Definition:Euler's Number/Limit of Sequence}} === Base of Logarithm === {{:Definition:Euler's Number/Base of Logarithm}} === Exponential Function === {{:Definition:Euler's Number/Exponential Function}}


Powers Drown Logarithms

Let $r \in \R_{>0}$ be a (strictly) positive real number. Then: :$\displaystyle \lim_{x \mathop \to \infty} x^{-r} \ln x = 0$


Equivalence of Well-Ordering Principle and Induction

The Well-Ordering Principle, the Principle of Finite Induction and the Principle of Complete Finite Induction are logically equivalent. That is: :Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: :: $0 \in S$ :: $n \in S \implies n + 1 \in S$ :then $S = \N$. {{iff}}: :Principle of Complete Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: :: $0 \in S$ :: $\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$ :then $S = \N$. {{iff}}: :Well-Ordering Principle: Every non-empty subset of $\N$ has a minimal element.


Lower Bound for Subset

Let $\left({S, \preceq}\right)$ be an ordered set. Let $L$ be a lower bound for $S$. Let $\left({T, \preceq}\right)$ be a subset of $\left({S, \preceq}\right)$. Then $L$ is a lower bound for $T$.


Upper Bound for Subset

Let $\left({S, \preceq}\right)$ be an ordered set. Let $U$ be an upper bound for $S$. Let $\left({T, \preceq}\right)$ be a subset of $\left({S, \preceq}\right)$. Then $U$ is an upper bound for $T$.


Common Divisor Divides GCD

Let $a, b \in \Z$ such that not both of $a$ and $b$ are zero. Let $c$ be any common divisor of $a$ and $b$. That is, let $c \in \Z: c \divides a, c \divides b$. Then: :$c \divides \gcd \set {a, b}$ where $\gcd \set {a, b}$ is the greatest common divisor of $a$ and $b$.


Square Modulo 5

Let $x \in \Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^2       | o = \equiv       | r = 0 \pmod 5       | c =  }} {{eqn | l = x^2       | o = \equiv       | r = 1 \pmod 5       | c =  }} {{eqn | l = x^2       | o = \equiv       | r = 4 \pmod 5       | c =  }} {{end-eqn}}


Square Modulo 3

Let $x \in \Z$ be an integer. Then one of the following holds: {{begin-eqn}} {{eqn | l = x^2       | o = \equiv       | r = 0 \pmod 3       | c =  }} {{eqn | l = x^2       | o = \equiv       | r = 1 \pmod 3       | c =  }} {{end-eqn}}


Additive Group of Integers is Countably Infinite Abelian Group

The set of integers under addition $\struct {\Z, +}$ forms a countably infinite abelian group.


Division Laws for Groups

Let $G$ be a group. Let $a, b, x \in G$. Then: : $(1): \quad a x = b \iff x = a^{-1} b$ : $(2): \quad x a = b \iff x = b a^{-1}$


Center is Intersection of Centralizers

The center of a group is the intersection of all the centralizers of the elements of that group: :$\displaystyle \map Z G = \bigcap_{g \mathop \in G} \map {C_G} g$


Equivalent Statements for Congruence Modulo Subgroup

Let $G$ be a group. Let $H$ be a subgroup of $G$.


Congruence Modulo Subgroup is Equivalence Relation

Let $G$ be a group, and let $H$ be a subgroup of $G$. Let $x, y \in G$.


Group Homomorphism Preserves Identity

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism. Let: :$e_G$ be the identity of $G$ :$e_H$ be the identity of $H$. Then: :$\map \phi {e_G} = e_H$


Group Homomorphism Preserves Inverses

Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a group homomorphism. Let: :$e_G$ be the identity of $G$ :$e_H$ be the identity of $H$ Then: :$\forall x \in G: \map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$


Subset Product with Normal Subgroup is Subgroup

Let $G$ be a group whose identity is $e$. Let: :$(1): \quad H$ be a subgroup of $G$ :$(2): \quad N$ be a normal subgroup of $G$. Let $H N$ denote subset product. Then $H N$ and $N H$ are both subgroups of $G$.


Coset by Identity

Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$.


Element of Group is in its own Coset

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


Element in Coset iff Product with Inverse in Subgroup

==== Element in Left Coset iff Product with Inverse in Subgroup ==== {{:Element in Left Coset iff Product with Inverse in Subgroup}} ==== Element in Right Coset iff Product with Inverse in Subgroup ==== {{:Element in Right Coset iff Product with Inverse in Subgroup}}


==== Left Cosets are Equal iff Product with Inverse in Subgroup ==== {{:Left Cosets are Equal iff Product with Inverse in Subgroup}} ==== Right Cosets are Equal iff Product with Inverse in Subgroup ==== {{:Right Cosets are Equal iff Product with Inverse in Subgroup}}


==== Left Coset Equals Subgroup iff Element in Subgroup ==== {{:Left Coset Equals Subgroup iff Element in Subgroup}} ==== Right Coset Equals Subgroup iff Element in Subgroup ==== {{:Right Coset Equals Subgroup iff Element in Subgroup}}


Elements in Same Coset iff Product with Inverse in Subgroup

==== Elements in Same Left Coset iff Product with Inverse in Subgroup ==== {{:Elements in Same Left Coset iff Product with Inverse in Subgroup}} ==== Elements in Same Right Coset iff Product with Inverse in Subgroup ==== {{:Elements in Same Right Coset iff Product with Inverse in Subgroup}}


Regular Representation on Subgroup is Bijection to Coset

* The mapping $\lambda_x: H \to x H$, where $\lambda_x$ is the left regular representation of $H$ with respect to $x$, is a bijection from $H$ to $x H$. * The mapping $\rho_x: H \to H x$, where $\rho_x$ is the right regular representation of $H$ with respect to $x$, is a bijection from $H$ to $H x$.


Ring Homomorphism Preserves Zero

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


Image of Set Difference under Mapping

Let $f: S \to T$ be a mapping. The image of the set difference of two subsets of $S$ is a subset of the set difference of the images. That is: Let $S_1$ and $S_2$ be subsets of $S$. Then: :$f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$ where $\setminus$ denotes set difference.


Convergence by Multiple of Error Term

Let $\sequence {s_n}$ be a real sequence. Suppose that $\exists \epsilon \in \R, \epsilon > 0$ such that: :$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \epsilon$ for any $K \in \R, K > 0$, independent of both $\epsilon$ and $N$. Then $\sequence {s_n}$ converges to $l$.


Cauchy Sequence Converges on Real Number Line

Let $\sequence {a_n}$ be a Cauchy sequence in $\R$. Then $\sequence {a_n}$ is convergent. In other words, $\struct {\R, \size {\,\cdot\,}}$ is a Banach space.


Cauchy's Convergence Criterion

Let $\sequence {x_n}$ be a sequence in $\R$. Then $\sequence {x_n}$ is convergent {{iff}} $\sequence {x_n}$ is a Cauchy sequence.


Wedderburn's Theorem

Every finite division ring $D$ is a field.


Preimages All Exist iff Surjection

Let $f: S \to T$ be a mapping. Let $f^{-1}$ be the inverse of $f$. Let $f^{-1} \paren t$ be the preimage of $t \in T$. Then $f^{-1} \paren t$ is empty for no $t \in T$ {{iff}} $f$ is a surjection.


Non-Successor Element of Peano Structure is Unique

Let $\struct {P, s, 0}$ be a Peano structure. Then: :$P \setminus s \sqbrk P$ is a singleton where: :$\setminus$ denotes set difference :$s \sqbrk P$ denotes the image of the mapping $s$. It follows that the non-successor element $0$ is the only element of $P$ with this property.


Equivalence of Formulations of Peano's Axioms

Let $P$ be a set. Let $s: P \to P$ be a mapping. Let $0 \in P$ be a distinguished element. {{TFAE|def = Peano's Axioms|view = Peano's Axioms}}


Product of Row Sum Unity Matrices

Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix. Let $\mathbf B = \left[{b}\right]_{n p}$ be an $n \times p$ matrix. Let the row sum of $\mathbf A$ and $\mathbf B$ be equal to $1$. Then the row sum of their (conventional) product is also $1$.


Restriction of Injection is Injection

Let $f: S \to T$ be an injection. Let $X \subseteq S$ be a subset of $S$. Let $f \sqbrk X$ denote the image of $X$ under $f$. Let $Y \subseteq T$ be a subset of $T$ such that $f \sqbrk X \subseteq Y$. The restriction $f \restriction_{X \times Y}$ of $f$ to $X \times Y$ is an injection from $X$ to $Y$.


Absolute Value is Many-to-One

Let $f: \R \to \R$ be the absolute value function: :$\forall x \in \R:  \map f x = \begin{cases} x & : x \ge 0 \\ -x & : x < 0 \end{cases}$ Then $f$ is a many-to-one relation.


Laws of Logarithms

Let $x, y, b \in \R_{>0}$ be (strictly) positive real numbers. Let $a \in \R$ be any real number such that $a > 0$ and $a \ne 1$. Let $\log_a$ denote the logarithm to base $a$. Then:


Subset of Preimage under Relation is Preimage of Subset

Let $\mathcal R \subseteq S \times T$ be a relation. Let $X \subseteq S, Y \subseteq T$. Then: :$X \subseteq \mathcal R^{-1} \sqbrk Y \iff \mathcal R \sqbrk X \subseteq Y$ In the language of direct image mappings, this can be written: :$X \subseteq \map {\mathcal R^\gets} Y \iff \map {\mathcal R^\to} X \subseteq Y$


Composition of Direct Image Mappings of Relations

Let $A, B, C$ be non-empty sets. Let $\mathcal R_1 \subseteq A \times B, \mathcal R_2 \subseteq B \times C$ be relations. Let: :${\mathcal R_1}^\to: \powerset A \to \powerset B$ and :${\mathcal R_2}^\to: \powerset B \to \powerset C$ be the direct image mappings of $\mathcal R_1$ and $\mathcal R_2$. Then: :$\paren {\mathcal R_2 \circ \mathcal R_1}^\to = {\mathcal R_2}^\to \circ {\mathcal R_1}^\to$


Subset equals Preimage of Image iff Mapping is Injection

Let $f: S \to T$ be a mapping. Let $f^{-1}$ denote the inverse of $f$. Then: :$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$ {{iff}} $f$ is an injection where: :$f \sqbrk A$ denotes the image of $A$ under $f$ :$f^{-1} \circ f$ denotes the composition of $f^{-1}$ and $f$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\forall A \in \powerset S: A = \map {\paren {f^\gets \circ f^\to} } A$ {{iff}} $f$ is an injection


Subset equals Image of Preimage iff Mapping is Surjection

Let $f: S \to T$ be a mapping. Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$. Similarly, let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$. Then: :$\forall B \in \powerset T: B = \map {\paren {f^\to \circ f^\gets} } B$ {{iff}} $f$ is a surjection.


Inverse of Injection is Many-to-One Relation

Let $f: S \to T$ be an injection. Let $f^{-1}: T \to S$ be the inverse relation of $f$. Then $f^{-1}$ is many-to-one.


Bijection iff Left and Right Cancellable

Let $f$ be a mapping. Then $f$ is a bijection {{iff}} $f$ is both left cancellable and right cancellable.


Relation Induced by Mapping is Equivalence Relation

Let $f: S \to T$ be a mapping. Let $\RR_f \subseteq S \times S$ be the relation induced by $f$: :$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$ Then $\RR_f$ is an equivalence relation.


Canonical Injection is Monomorphism

Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The canonical injections: :$\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_1: \map {\inj_1} x = \tuple {x, e_2}$ :$\inj_2: \struct {S_2, \circ_2} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_2: \map {\inj_2} x = \tuple {e_1, x}$ are monomorphisms.


Order Embedding is Injection

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets. Let $\phi: S \to T$ be an order embedding. That is: : $\forall x, y \in S: x \preceq_1 y \iff \phi \left({x}\right) \preceq_2 \phi \left({y}\right)$ Then $\phi$ is an injection.


Order Isomorphism is Surjective Order Embedding

Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets. Let $f: S \to T$ be a mapping. Then $f$ is an order isomorphism {{iff}}: :$(1): \quad f$ is a surjection :$(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map f x \preceq_2 \map f y$ That is, {{iff}} $f$ is an order embedding which is also a surjection.


Infimum of Power Set

Let $S$ be a set. Let $\mathcal P \left({S}\right)$ be the power set of $S$. Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the relational structure defined on $\mathcal P \left({S}\right)$ by the relation $\subseteq$. (From Subset Relation on Power Set is Partial Ordering, this is an ordered set.) Then the infimum of $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is the empty set $\varnothing$.


Supremum of Power Set

Let $S$ be a set. Let $\mathcal P \left({S}\right)$ be the power set of $S$. Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the relational structure defined on $\mathcal P \left({S}\right)$ by the relation $\subseteq$. (From Subset Relation on Power Set is Partial Ordering, this is an ordered set.) Then the supremum of $\left({\mathcal P \left({S}\right), \subseteq}\right)$ is the set $S$.


Diagonal Relation is Ordering and Equivalence

Let $\left({S, \Delta_S}\right)$ be a relational structure where $\Delta_S$ is the diagonal relation, defined as: :$\forall x, y \in S: \paren {x, y} \in \Delta_S \iff x = y$ Then $\Delta_S$ is the '''only''' relation on $S$ which is both an equivalence and an ordering.


Cantor-Bernstein-Schröder Theorem/Proof 3

Let $S$ and $T$ be sets, such that: :$\exists f: S \to T$ such that $f$ is an injection :$\exists g: T \to S$ such that $g$ is an injection. Then there exists a bijection from $S$ to $T$.


Cantor-Bernstein-Schröder Theorem/Proof 2

Let $S$ and $T$ be sets, such that: :$S \preccurlyeq T$, that is: $T$ dominates $S$ :$T \preccurlyeq S$, that is: $S$ dominates $T$. Then: :$S \sim T$ that is, $S$ is equivalent to $T$.


Huygens-Steiner Theorem

Let $B$ be a body of mass $M$. Let $I_0$ be the moment of inertia of $B$ about some axis $A$ through the centre of mass of $B$. Let $I$ the moment of inertia of $B$ about another axis $A'$ parallel to $A$. Then $I_0$ and $I$ are related by: :$I = I_0 + M l^2$ where $l$ is the perpendicular distance between $A$ and $A'$.


Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2

Let $\left({S, \preceq}\right)$ be an ordered set such that: : $\forall T \subseteq S: \exists a \in T: \forall x \in T: a \preceq x$ That is, such that $\preceq$ is a well-ordering by definition 1. Then $\preceq$ is a total ordering.


Choice Function Exists for Well-Orderable Union of Sets

Let $\mathbb S$ be a set of sets such that: :$\forall S \in \mathbb S: S \ne \varnothing$ that is, none of the sets in $\mathbb S$ may be empty. Let the union $\bigcup \mathbb S$ be well-orderable. Then there exists a choice function $f: \mathbb S \to \bigcup \mathbb S$ defined as: :$\forall S \in \mathbb S: \exists x \in S: f \left({S}\right) = x$ Thus, if every member of $\mathbb S$ is a well-ordered, then we can create a choice function $f$ defined as: :$\forall S \in \mathbb S: f \left({S}\right) = \inf \left({S}\right)$ True, we may be making infinitely many choices, but we have a rule for doing so.


Principle of Finite Choice

Let $I$ be a non-empty finite indexing set. Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ be an $I$-indexed family of non-empty sets. Then there exists an $I$-indexed family $\left\langle{x_i}\right\rangle_{i \mathop \in I}$ such that: :$\forall i \in I: x_i \in S_i$ That is, there exists a mapping: :$\displaystyle f: I \to \bigcup_{i \mathop \in I} S_i$ such that: :$\forall i \in I: f \left({i}\right) \in S_i$


Choice Function Exists for Set of Well-Ordered Sets

Let $\mathbb S$ be a set of sets such that: :$\forall S \in \mathbb S: S \ne \varnothing$ that is, none of the sets in $\mathbb S$ may be empty. Let every element of $\mathbb S$ be well-ordered. Then there exists a choice function $f: \mathbb S \to \bigcup \mathbb S$ satisfying: :$\forall S \in \mathbb S: \exists x \in S: f \left({S}\right) = x$


Identity of Power Set with Union

Let $S$ be a set and let $\powerset S$ be its power set. Consider the algebraic structure $\struct {\powerset S, \cup}$, where $\cup$ denotes set union. Then the empty set $\O$ serves as the identity for $\struct {\powerset S, \cup}$.


Identity of Power Set with Intersection

Let $S$ be a set and let $\powerset S$ be its power set. Consider the algebraic structure $\struct {\powerset S, \cap}$, where $\cap$ denotes set intersection. Then $S$ serves as the identity for $\struct {\powerset S, \cap}$.


Commutation with Inverse in Monoid

Let $x, y \in S$ such that $y$ is invertible. Then $x$ commutes with $y$ {{iff}} $x$ commutes with $y^{-1}$.


Commutation of Inverses in Monoid

Let $x, y \in S$ such that $x$ and $y$ are both invertible. Then $x$ commutes with $y$ {{iff}} $x^{-1}$ commutes with $y^{-1}$.


Inverse of Commuting Pair

Let $x, y \in S$ such that $x$ and $y$ are both invertible. Then $x$ commutes with $y$ {{iff}}: : $\struct {x \circ y}^{-1} = x^{-1} \circ y^{-1}$


Conjugate of Commuting Elements

Let $x, y \in S$ such that $x$ and $y$ are both invertible. Then $x \circ y \circ x^{-1} = y$ {{iff}} $x$ and $y$ commute.


Product of Commuting Elements with Inverses

Let $x, y \in S$ such that $x$ and $y$ are both invertible. Then: :$x \circ y \circ x^{-1} \circ y^{-1} = e_S = x^{-1} \circ y^{-1} \circ x \circ y$ {{iff}} $x$ and $y$ commute.


Exists Element Not in Set

Let $S$ be a set. Then $\exists x: x \notin S$. That is, for any set, there exists some element which is not in that set.


Rule of Material Implication

==== Formulation 1 ==== {{:Rule of Material Implication/Formulation 1}} ==== Formulation 2 ==== {{:Rule of Material Implication/Formulation 2}}


De Morgan's Laws (Set Theory)/Proof by Induction

Let $\mathbb T = \left\{{T_i: i \mathop \in I}\right\}$, where each $T_i$ is a set and $I$ is some finite indexing set. Then:


Union equals Intersection iff Sets are Equal

Let $S$ and $T$ be sets. Then: :$\paren {S \cup T = S} \land \paren {S \cap T = S} \iff S = T$ where: :$S \cup T$ denotes set union :$S \cap T$ denotes set intersection.


Intersection with Set Difference is Set Difference with Intersection

: $\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$


Set Difference is Right Distributive over Set Intersection

:$\paren {R \cap S} \setminus T = \paren {R \setminus T} \cap \paren {S \setminus T}$


Set Intersection Distributes over Set Difference

: $\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus \left({S \cap T}\right)$ : $R \cap \left({S \setminus T}\right) = \left({R \cap S}\right) \setminus \left({R \cap T}\right)$


Set Difference with Intersection is Difference

:$S \setminus \paren {S \cap T} = S \setminus T$


Set Difference of Intersection with Set is Empty Set

:$\left({S \cap T}\right) \setminus S = \varnothing$ :$\left({S \cap T}\right) \setminus T = \varnothing$


Set Difference Intersection with First Set is Set Difference

:$\left({S \setminus T}\right) \cap S = S \setminus T$


Set Difference Intersection with Second Set is Empty Set

:$\paren {S \setminus T} \cap T = \O$


Set Difference with Union is Set Difference

:$\left({S \cup T}\right) \setminus T = S \setminus T$


Set Difference Union First Set is First Set

:$\paren {S \setminus T} \cup S = S$


Set Difference Union Second Set is Union

:$\left({S \setminus T}\right) \cup T = S \cup T$


Set Difference with Union

:$R \setminus \paren {S \cup T} = \paren {R \cup T} \setminus \paren {S \cup T} = \paren {R \setminus S} \setminus T = \paren {R \setminus T} \setminus S$


Set Difference is Right Distributive over Union

:$\paren {R \cup S} \setminus T = \paren {R \setminus T} \cup \paren {S \setminus T}$


Set Difference with Set Difference is Union of Set Difference with Intersection

:$R \setminus \paren {S \setminus T} = \paren {R \setminus S} \cup \paren {R \cap T}$


Set Difference is Subset of Union of Differences

: $R \setminus S \subseteq \left({R \setminus T}\right) \cup \left({T \setminus S}\right)$


Ordered Sum of Tosets is Totally Ordered Set

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be tosets. Let $S + T = \left({S \cup T, \preceq}\right)$ be the ordered sum of $S$ and $T$. Then $\left({S \cup T, \preceq}\right)$ is itself a toset.


Ordered Product of Tosets is Totally Ordered Set

Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be tosets. Let $S_1 \cdot S_2 = \struct {S_1 \times S_2, \preceq}$ be the ordered product of $S_1$ and $S_2$. Then $\struct {S_1 \times S_2, \preceq}$ is itself a toset.


Restriction of Mapping is Mapping

Let $f: S \to T$ be a mapping. Let $X \subseteq S$. Let $f \restriction_X$ be the restriction of $f$ to $X$. Then $f \restriction_X: X \to T$ is a mapping: : whose domain is $X$ : whose preimage is $X$.


Hausdorff Maximal Principle

Let $\struct {\PP, \preceq}$ be a partially ordered set. Then there exists a maximal chain in $\PP$.


Logical Consequence with Union

Let $U$ be a set of propositional formulas. Let $P$ be a propositional formula. Let $U \models P$ denote that $P$ is a semantic consequence of $U$. Then: : $U \models P$ iff: : $U \cup P \models P$


Logical Consequence Union Negation

Let $U$ be a set of propositional formulas. Let $P$ be a propositional formula. Let $U \models P$ denote that $U$ is a semantic consequence $P$. Then: : $U \models P$ iff: : $U \cup \left\{{\neg P}\right\}$ has no models.


Union of Singleton

Consider the set of sets $\mathbb S$ such that $\mathbb S$ consists of just one set $S$. Then the union of $\mathbb S$ is $S$: :$\displaystyle \mathbb S = \set S \implies \bigcup \mathbb S = S$


Intersection of Singleton

Consider the set of sets $\mathbb S$ such that $\mathbb S$ consists of just one set $S$. Then the intersection of $\mathbb S$ is $S$: :$\displaystyle \mathbb S = \set S \implies \bigcap \mathbb S = S$


Union of Empty Set

Consider the set of sets $\mathbb S$ such that $\mathbb S$ is the empty set $\O$. Then the union of $\mathbb S$ is $\O$: :$\mathbb S = \O \implies \displaystyle \bigcup \mathbb S = \O$


Intersection of Empty Set

Consider the set of sets $\mathbb S$ such that $\mathbb S$ is the empty set $\O$. Then the intersection of $\mathbb S$ is $\mathbb U$: :$\mathbb S = \O \implies \displaystyle \bigcap \mathbb S = \mathbb U$ where $\mathbb U$ is the universe. A paradoxical result.


Inverse of Identity Mapping

Let $S$ be a set. Let $I_S: S \to S$ be the identity mapping on $S$. Then the inverse of $I_S$ is itself: :$\paren {I_S}^{-1} = I_S$


Factoring Mapping into Quotient and Injection

Let $f: S \to T$ be a mapping. Then $f$ can be uniquely '''factored into''' a quotient mapping, followed by an injection. Thus: :$f = h \circ q_{\RR_f}$ where: :$q_{\RR_f}: S \to S / \RR_f: \map {q_{\RR_f} } s = \eqclass s {\RR_f}$ :$h: S / \RR_f \to T: \map h {\eqclass s {\RR_f} } = \map f s$ :$\eqclass s {\RR_f}$ denotes the equivalence class of $s$ with respect to the equivalence relation $\RR$ induced on $S$ by $f$. This can be illustrated using a commutative diagram as follows: ::$\begin{xy}\xymatrix@L+2mu@+1em{  S \ar[r]^*{q_{\RR_f} }      \ar@{-->}[rd]_*{f = h \circ q_{\RR_f} } &  S / \RR_f \ar[d]^*{h} \\ &  T }\end{xy}$


Subsets in Increasing Union

Let $S_0, S_1, S_2, \ldots, S_i, \ldots$ be a nested sequence of sets, that is: :$S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ Let $S$ be the increasing union of $S_0, S_1, S_2, \ldots, S_i, \ldots$: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ Then: :$\forall s \in S: \exists k \in \N: \forall j \ge k: s \in S_j$


Mapping on Increasing Union

Let $S_0, S_1, S_2, \ldots, S_i, \ldots$ be sets such that: :$S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ that is, each set is contained in the next as a subset. Let $S$ be the increasing union of $S_0, S_1, S_2, \ldots, S_i, \ldots$: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ For each $i \in \N$, let $f_i: S_i \to T$ be a mapping such that: :$\forall j < i: f_i \restriction_{S_j} = f_j$ where $f_i \restriction_{S_j}$ denotes the restriction of $f_i$ to $S_j$. Then there is a '''unique''' mapping: :$f: S \to T$ which extends each $f_i$ to $S$.


Domain of Injection Not Larger than Codomain

Let $f: S \rightarrowtail T$ be an injection. Then: :$\left|{S}\right| \le \left|{T}\right|$ where $\left|{S}\right|$ denotes the cardinality of $S$.


Futurama Theorem

Let $A_{n - 2} \subset A_n$ be a subgroup of the alternating group on $n$ letters $A_n$. For any element $x \in A_{n - 2}$, let $x = x_1 x_2 \dots x_k$, where $x_i \in H$ is a transposition. {{explain|What is $H$?}} Then there exists $y$ which can be represented as a series of transpositions $y_1 y_2 \dots y_j \in A_n$ such that: :$(1): \quad y x = z$, where $z$ contains no transpositions from $H$ :$(2): \quad y_a \ne x_b$ for all $a, b$.


Pointwise Addition on Continuous Real-Valued Functions forms Group

Let $C$ be the set of all continuous real functions on the set of real numbers $\R$. Let $f, g \in C$. Let $f + g$ be the pointwise sum of $f$ and $g$: :$\forall x \in R: \map {\paren {f + g} } x = \map f x + \map g x$ Then $\struct {C, +}$, the algebraic structure on $C$ induced by $+$, forms a group.


General Intersection Property of Topological Space

Let $\struct {S, \tau}$ be a topological space. Let $S_1, S_2, \ldots, S_n$ be open sets of $\struct {S, \tau}$. Then: :$\displaystyle \bigcap_{i \mathop = 1}^n S_i$ is also an open set of $\struct {S, \tau}$. That is, the intersection of any finite number of open sets of a topology is also in $\tau$. Conversely, if the intersection of any finite number of open sets of a topology is also in $\tau$, then: :$(1): \quad$ The intersection of any two elements of $\tau$ is an element of $\tau$ :$(2): \quad S$ is itself an element of $\tau$.


Empty Set is Element of Topology

Let $\struct {X, \tau}$ be a topological space. Then $\O$ is an open set of $\struct {X, \tau}$.


Coarseness Relation on Topologies is Partial Ordering

Let $S$ be a set. Let $\mathbb T$ be the set of all topologies on $S$. Let $\le$ be the relation on $\mathbb T$ defined as: :$\forall \tau_1, \tau_2 \in \mathbb T: \tau_1 \le \tau_2$ {{iff}} $\tau_1$ is coarser than $\tau_2$. Then $\le$ is a partial ordering on $\mathbb T$.


Topology Defined by Closed Sets

Let $X$ be any set and let $\tau$ be a collection of subsets of $X$. Then $\tau$ is a topology on $X$ {{iff}}: :$(1): \quad$ Any intersection of arbitrarily many closed sets of $X$ under $\tau$ is a closed set of $X$ under $\tau$ :$(2): \quad$ The union of any finite number of closed sets of $X$ under $\tau$ is a closed set of $X$ under $\tau$ :$(3): \quad X$ and $\O$ are both closed sets of $X$ under $\tau$.


Complement of F-Sigma Set is G-Delta Set

Let $T = \struct {S, \tau}$ be a topological space. Let $X$ be an $F_\sigma$ set of $T$. Then its complement $S \setminus X$ is a $G_\delta$ set of $S$.


Set is Open iff Neighborhood of all its Points

Let $T = \struct {S, \tau}$ be a topological space. Let $V \subseteq S$ be a subset of $T$. Then: :$V$ is an open set of $T$ {{iff}}: :$V$ is a neighborhood of all the points in $V$.


Relationship between Limit Point Types

Let $T = \struct {X, \tau}$ be a topological space. Let $A \subseteq X$. Let: :$C$ be the set of condensation points of $A$ :$W$ be the set of $\omega$-accumulation points of $A$ :$L$ be the set of limit points of $A$ :$D$ be the set of adherent points of $A$. Then: :$C \subseteq W \subseteq L \subseteq D$ That is: :Every condensation point is an $\omega$-accumulation point :Every $\omega$-accumulation point is a limit point :Every limit point is an adherent point. In general, the inclusions do not hold in the other direction.


Limit Point of Sequence is Accumulation Point

Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$. Let $\sequence {x_n}$ be a sequence in $A$. Let $\alpha$ be a limit point of $\sequence {x_n}$. Then $\alpha$ is also an accumulation point of $\sequence {x_n}$.


Filter on Set is Proper Filter

Let $S$ be a set. Let $\mathcal P \left({S}\right)$ denote the power set of $S$. Let $\left({\mathcal P \left({S}\right), \subseteq}\right)$ be the poset defined on $\mathcal P \left({S}\right)$ by the subset relation. Let $\mathcal F$ be a filter on $S$. Then $\mathcal F$ is a proper filter on $\left({\mathcal P \left({S}\right), \subseteq}\right)$.


Peirce's Law is Equivalent to Law of Excluded Middle

Peirce's Law: :$\paren {p \implies q} \implies p \vdash p$ is logically equivalent to the Law of Excluded Middle: :$\vdash p \lor \neg p$ That is, Peirce's Law holds {{iff}} the Law of Excluded Middle holds.


Implication is Left Distributive over Conjunction

The implication operator is left distributive over the conjunction operator:


Existence of Negation Normal Form of Statement

Any propositional formula can be expressed in negation normal form (NNF).


Existence of Conjunctive Normal Form of Statement

Any propositional formula can be expressed in conjunctive normal form (CNF).


Existence of Disjunctive Normal Form of Statement

Any propositional formula can be expressed in disjunctive normal form (DNF).


Existence of Base-N Representation

{{refactor|lemmata and sub proofs}} {{tidy|Could be improved in presentational style}} Given a number $x \in \hointr 0 1$, there exists a representation of that number in a base-$p$ positional system. Specifically, there exists a sequence $\sequence {a_n}$ such that: :$0 \le a_n < p$, and :$\displaystyle \sum_{n \mathop = 1}^\infty \frac {a_n} {p^n}$ converges to $x$. Unless $\sequence {a_n}$ terminates (that is $a_n = 0$ for all sufficiently large $n$), then this representation is unique. If $\sequence {a_n}$ does terminate, then there is exactly one other sequence which satisfies the criteria of the theorem.


Symmetry Group of Equilateral Triangle is Group

The symmetry group of the equilateral triangle is a group.


Hilbert's Basis Theorem

Let $A$ be a Noetherian ring. Let $A \sqbrk x$ be the ring of polynomial forms over $A$ in the single indeterminate $x$. Then $A \sqbrk x$ is also a Noetherian ring.


Existence of Cyclic Group of Order n

Let $n \in \Z_{>0}$. Then there exists a cyclic group of order $n$ which is unique up to isomorphism.


Quotient Group is Group

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then the quotient group $G / N$ is indeed a group.


Principal Ideal is Ideal

Let $\struct {R, +, \circ}$ be a ring with unity. Let $a \in R$. Let $\ideal a$ be the principal ideal of $R$ generated by $a$. Then $\ideal a$ is an ideal of $R$. Also, if $J$ is an ideal of $R$ and $a \in J$, then $\ideal a \subseteq J$. That is, $\ideal a$ is the smallest ideal of $R$ to which $a$ belongs. {{refactor|There are a number of results that this takes in, originally taken from Whitelaw: :$(1): \quad \forall a \in R: \ideal a$ is an ideal of $R$ :$(2): \quad \forall a \in R: a \in \ideal a$ :$(3): \quad \forall a \in R:$ if $J$ is an ideal of $R$, and $a \in J$, then $\ideal a \subseteq J$. :::That is, $\ideal a$ is the smallest ideal of $R$ containing $a$. }}


Bézout's Theorem

Let $X$ and $Y$ be two plane projective curves defined over a field $F$ that do not have a common component. (This condition is true if both $X$ and $Y$ are defined by different irreducible polynomials. In particular, it holds for a pair of "generic" curves.) {{refactor|Link to a proof of the above, and (preferably) move this statement to the proof itself.}} Then the total number of intersection points of $X$ and $Y$ with coordinates in an algebraically closed field $E$ which contains $F$, counted with their multiplicities, is equal to the product of the degrees of $X$ and $Y$.


Whitney Immersion Theorem

Let $m > 1$ be a natural number. Every smooth $m$-dimensional manifold can be immersed in Euclidean $\left({2m-1}\right)$-space.


Pasch's Theorem

Let $a, b, c, d$ be points on a line. Let $\left({a, b, c}\right)$ denote that $b$ lies between $a$ and $c$. Then $\left({a, b, c}\right)$ and $\left({b, c, d}\right)$ together imply that $\left({a, b, d}\right)$. That is, if: : $b$ is between $a$ and $c$ and: : $c$ is between $b$ and $d$ then: : $b$ is between $a$ and $d$.


Forward Difference of Power

:$\Delta c^x = \paren {c - 1} c^x$ where $\Delta$ denotes the forward difference operator.


Forward Difference of Harmonic Number Function

Let $H_x$ denote the harmonic number function. Then: :$\Delta H_x = \dfrac 1 {x + 1}$ where $\Delta H_x$ denotes the forward difference operator.


Equality of Ordered Tuples

Let $a = \tuple {a_1, a_2, \ldots, a_n}$ and $b = \tuple {b_1, b_2, \ldots, b_n}$ be ordered tuples. Then: :$a = b \iff \forall i: 1 \le i \le n: a_i = b_i$ That is, for two ordered tuples to be equal, all the corresponding elements have to be equal.


Zero of Power Set with Intersection

Let $S$ be a set and let $\powerset S$ be its power set. Consider the algebraic structure $\struct {\powerset S, \cap}$, where $\cap$ denotes set intersection. Then the empty set $\O$ serves as the zero element for $\struct {\powerset S, \cap}$.


Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group

Let $\R_{>0}$ be the set of strictly positive real numbers: :$\R_{>0} = \set {x \in \R: x > 0}$ The structure $\struct {\R_{>0}, \times}$ is an uncountable abelian group.


Law of Mass Action

Let $\AA$ and $\BB$ be two chemical substances in a solution $C$ which are involved in a second-order reaction. Let $x$ grams of $\CC$ contain $a x$ grams of $\AA$ and $b x$ grams of $\BB$, where $a + b = 1$. Let there be $a A$ grams of $\AA$ and $b B$ grams of $\BB$ at time $t = t_0$, at which time $x = 0$. Then: :$x = \begin{cases} \dfrac {k A^2 a b t} {k A a b t + 1} & : A = B \\ & \\ \dfrac {A B e^{-k \paren {A - B} a b t}} {A - B e^{-k \paren {A - B} a b t}} & : A \ne B \end{cases}$ This is known as the '''law of mass action'''


Motion of Pendulum

Consider a pendulum consisting of a bob whose mass is $m$, at the end of a rod of negligible mass of length $a$. Let the bob be pulled to one side so that the rod is at an angle $\alpha$ from the vertical and then released. Let $T$ be the time period of the pendulum, that is, the time through which the bob takes to travel from one end of its path to the other, and back again. Then: :$T = 2 \sqrt {\dfrac a g} \map K k$ where: :$k = \map \sin {\dfrac \alpha 2}$ :$\map K k$ is the complete elliptic integral of the first kind.


Body under Constant Acceleration

=== $(1):$ Velocity after Time === {{:Body under Constant Acceleration/Velocity after Time}} === $(2):$ Distance after Time === {{:Body under Constant Acceleration/Distance after Time}} === $(3):$ Velocity after Distance === {{:Body under Constant Acceleration/Velocity after Distance}} where: :$\mathbf u$ is the velocity at time $t = 0$ :$\mathbf v$ is the velocity at time $t$ :$\mathbf s$ is the displacement of $B$ from its initial position at time $t$ :$\cdot$ denotes the dot product.


One-Parameter Family of Curves for First Order ODE

Every one-parameter family of curves is the general solution of some first order ordinary differential equation. Conversely, every first order ordinary differential equation has as its general solution some one-parameter family of curves.


Orthogonal Trajectories of One-Parameter Family of Curves

Every one-parameter family of curves has a unique family of orthogonal trajectories.


Brachistochrone is Cycloid

The shape of the brachistochrone is a cycloid.


Equation of Cycloid

Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$


Length of Arc of Cycloid

Let $C$ be a cycloid generated by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$ Then the length of one arc of the cycloid is $8 a$.


Goldbach's Weak Conjecture

Every odd integer greater than $7$ is the sum of three odd primes.


Goldbach Conjecture implies Weak Goldbach Conjecture

The Goldbach Conjecture: : Every even integer greater than $2$ is the sum of two primes implies Goldbach's Weak Conjecture: : Every odd integer greater than $7$ is the sum of three odd primes.


Unique Factorization Theorem

Let $\struct {D, +, \times}$ be a Euclidean domain. Then $\struct {D, +, \times}$ is a unique factorization domain.


Integral of Power

:$\displaystyle \forall n \in \R_{\ne -1}: \int_0^b x^n \rd x = \frac {b^{n + 1} } {n + 1}$


Volume of Sphere

The volume $V$ of a sphere of radius $r$ is given by: :$V = \dfrac {4 \pi r^3} 3$


Einstein's Mass-Energy Equation

The energy imparted to a body to cause that body to move causes the body to increase in mass by a value $M$ as given by the equation: :$E = M c^2$ where $c$ is the speed of light.


Dimension of Gravitational Constant

The dimension of the gravitational constant $G$ is $M^{-1} L^3 T^{-2}$.


Dimension of Spring Force Constant

The dimension of a spring force constant is $\mathsf {M T}^{-2}$.


Vieta's Formula for Pi

:$\pi = 2 \times \dfrac 2 {\sqrt 2} \times \dfrac 2 {\sqrt {2 + \sqrt 2} } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt 2} } } \times \dfrac 2 {\sqrt {2 + \sqrt {2 + \sqrt {2 + \sqrt 2 } } } } \times \cdots$


Derivative of Arc Length

Let $C$ be a curve in the cartesian plane described by the equation $y = \map f x$. Let $s$ be the length along the arc of the curve from some reference point $P$. Then the derivative of $s$ with respect to $x$ is given by: :$\dfrac {\d s} {\d x} = \sqrt {1 + \paren {\dfrac {\d y} {\d x} }^2}$


Duality Principle for Sets

Any identity in set theory which uses any or all of the operations: :Set intersection $\cap$ :Set union $\cup$ :Empty set $\O$ :Universal set $\mathbb U$ and none other, remains valid if: :$\cap$ and $\cup$ are exchanged throughout :$\O$ and $\mathbb U$ are exchanged throughout.


Cartesian Product is not Associative

Let $A, B, C$ be non-empty sets. Then: :$A \times \paren {B \times C} \ne \paren {A \times B} \times C$ where $A \times B$ is the cartesian product of $A$ and $B$.


Rational Addition is Commutative

The operation of addition on the set of rational numbers $\Q$ is commutative: :$\forall x, y \in \Q: x + y = y + x$


Rational Addition is Associative

The operation of addition on the set of rational numbers $\Q$ is associative: :$\forall x, y, z \in \Q: x + \paren {y + z} = \paren {x + y} + z$


Subtraction on Numbers is Anticommutative

The operation of subtraction on the numbers is anticommutative. That is: :$a - b = b - a \iff a = b$


Commutative Law of Addition

:$\forall x, y \in \mathbb F: x + y = y + x$ That is, the operation of addition on the standard number sets is commutative.


Integer Addition is Commutative

The operation of addition on the set of integers $\Z$ is commutative: :$\forall x, y \in \Z: x + y = y + x$


Integer Addition is Associative

The operation of addition on the set of integers $\Z$ is associative: :$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$


Natural Number Addition is Commutative

The operation of addition on the set of natural numbers $\N$ is commutative: :$\forall m, n \in \N: m + n = n + m$


Associative Law of Addition

:$\forall x, y, z \in \mathbb F: x + \left({y + z}\right) = \left({x + y}\right) + z$ That is, the operation of addition on the standard number sets is associative.


Natural Number Addition is Associative

The operation of addition on the set of natural numbers $\N$ is associative: :$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$


Rational Multiplication is Commutative

The operation of multiplication on the set of rational numbers $\Q$ is commutative: :$\forall x, y \in \Q: x \times y = y \times x$


Rational Multiplication is Associative

The operation of multiplication on the set of rational numbers $\Q$ is associative: :$\forall x, y, z \in \Q: x \times \paren {y \times z} = \paren {x \times y} \times z$


Natural Number Multiplication is Commutative

The operation of multiplication on the set of natural numbers $\N$ is commutative: :$\forall x, y \in \N: x \times y = y \times x$


Natural Number Multiplication is Associative

The operation of multiplication on the set of natural numbers $\N$ is associative: :$\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$


Natural Number Multiplication Distributes over Addition

The operation of multiplication is distributive over addition on the set of natural numbers $\N$: :$\forall x, y, z \in \N:$ ::$\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$ ::$z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$


Multiplication of Numbers Distributes over Addition

On all the number systems: :natural numbers $\N$ :integers $\Z$ :rational numbers $\Q$ :real numbers $\R$ :complex numbers $\C$ the operation of multiplication is distributive over addition: :$m \paren {n + p} = m n + m p$ :$\paren {m + n} p = m p + n p$


Rational Multiplication Distributes over Addition

The operation of multiplication on the set of rational numbers $\Q$ is distributive over addition: :$\forall x, y, z \in \Q: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$ :$\forall x, y, z \in \Q: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$


Associative Law of Multiplication

:$\forall x, y, z \in \mathbb F: x \times \paren {y \times z} = \paren {x \times y} \times z$ That is, the operation of multiplication on the standard number sets is associative.


Commutative Law of Multiplication

:$\forall x, y \in \mathbb F: x + y = y + x$ That is, the operation of multiplication on the standard number sets is commutative.


Subtraction on Numbers is Not Associative

The operation of subtraction on the numbers is not associative. That is, in general: :$a - \paren {b - c} \ne \paren {a - b} - c$


Identity Element of Multiplication on Numbers

On all the number systems: * natural numbers $\N$ * integers $\Z$ * rational numbers $\Q$ * real numbers $\R$ * complex numbers $\C$ the identity element of multiplication is one ($1$).


Identity Element of Addition on Numbers

On all the number systems: * natural numbers $\N$ * integers $\Z$ * rational numbers $\Q$ * real numbers $\R$ * complex numbers $\C$ the identity element of addition is zero ($0$).


Zero Element is Unique

Let $\struct {S, \circ}$ be an algebraic structure that has a zero element $z \in S$. Then $z$ is unique.


Group with Zero Element is Trivial

Let $\struct {G, \circ}$ be a group. Let $\struct {G, \circ}$ have a zero element. Then $\struct {G, \circ}$ is the trivial group.


Count of Binary Operations on Set

Let $S$ be a set whose cardinality is $n$. The number $N$ of different binary operations that can be applied to $S$ is given by: :$N = n^{\paren {n^2} }$


Count of Commutative Binary Operations on Set

Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different commutative binary operations that can be applied to $S$ is given by: :$N = n^{\frac {n \paren {n + 1} } 2}$


Count of Binary Operations with Fixed Identity

Let $S$ be a set whose cardinality is $n$. Let $x \in S$. The number $N$ of possible different binary operations such that $x$ is an identity element that can be applied to $S$ is given by: :$N = n^{\left({\left({n-1}\right)^2}\right)}$


Count of Commutative Binary Operations with Fixed Identity

Let $S$ be a set whose cardinality is $n$. Let $x \in S$. The number $N$ of possible different commutative binary operations such that $x$ is an identity element that can be applied to $S$ is given by: :$N = n^{\frac {n \left({n-1}\right)}2}$


Count of Binary Operations with Identity

Let $S$ be a set whose cardinality is $n$. The number $N$ of possible different binary operations which have an identity element that can be applied to $S$ is given by: :$N = n^{\paren {n - 1}^2 + 1}$


Power of Element in Subgroup

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\struct {H, \circ}$ be a subgroup of $\struct {G, \circ}$. Let $x \in H$. Then: :$\forall n \in \Z: x^n \in H$


Element to Power of Remainder

:$\forall n \in \Z: n = q k + r: 0 \le r < k \iff a^n = a^r$


Element to Power of Multiple of Order is Identity

:$\forall n \in \Z: k \divides n \iff a^n = e$


List of Elements in Finite Cyclic Group

:$\set {a^0, a^1, a^2, \ldots, a^{k - 1} }$ is a complete repetition-free list of the elements of $\gen a$


Rising Sum of Binomial Coefficients

:$\displaystyle \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$


Sum of Even Index Binomial Coefficients

:$\displaystyle \sum_{i \mathop \ge 0} \binom n {2 i} = 2^{n - 1}$


Sum of Odd Index Binomial Coefficients

:$\displaystyle \sum_{i \mathop \ge 0} \binom n {2 i + 1} = 2^{n - 1}$


Increasing Alternating Sum of Binomial Coefficients

:$\displaystyle \sum_{j \mathop = 0}^n \left({-1}\right)^{n + 1} j \binom n j = 0$


Sum of Squares of Binomial Coefficients

:$\displaystyle \sum_{i \mathop = 0}^n \binom n i^2 = \binom {2 n} n$


Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces

Let $\mathbb K$ be a subfield of $\C$. Let $V$ be a semi-inner product space over $\mathbb K$. Let $x, y$ be vectors in $V$. Then: :$\size {\innerprod x y}^2 \le \innerprod x x \innerprod y y$


Cauchy-Bunyakovsky-Schwarz Inequality/Definite Integrals

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$. Then: :$\displaystyle \paren {\int_a^b \map f t \, \map g t \rd t}^2 \le \int_a^b \paren {\map f t}^2 \rd t \int_a^b \paren {\map g t}^2 \rd t$


Cauchy-Schwarz Inequality/Complex Numbers

:$\displaystyle \paren {\sum \cmod {w_i}^2} \paren {\sum \cmod {z_i}^2} \ge \cmod {\sum w_i z_i}^2$ where all of $w_i, z_i \in \C$.


Ceiling of Floor is Floor

:$\left \lceil {\left \lfloor {x}\right \rfloor}\right \rceil = \left \lfloor {x}\right \rfloor$


Floor of Ceiling is Ceiling

: $\left \lfloor {\left \lceil {x}\right \rceil}\right \rfloor = \left \lceil {x}\right \rceil$


Ceiling of Negative equals Negative of Floor

:$\ceiling {-x} = -\floor x$


Floor of Negative equals Negative of Ceiling

:$\left \lfloor {-x}\right \rfloor = - \left \lceil {x}\right \rceil$


Integer is Coprime to 1

Every integer is coprime to $1$. That is: :$\forall n \in \Z: n \perp 1$


Integers Coprime to Zero

The only integers which are coprime to zero are $1$ and $-1$. That is: :$n \in \Z: n \perp 0 \iff n \in \set {1, -1}$ In particular, note that two integers which are coprime to each other cannot both be $0$.


Sum of Binomial Coefficients over Upper Index

{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^n \binom j m       | r = \binom {n + 1} {m + 1}       | c =  }} {{eqn | r = \dbinom 0 m + \dbinom 1 m + \dbinom 2 m + \cdots + \dbinom n m = \dbinom {n + 1} {m + 1}       | c =  }} {{end-eqn}}


Divisibility of Product of Consecutive Integers

The product of $n$ consecutive positive integers is divisible by the product of the ''first'' $n$ consecutive positive integers. That is: :$\displaystyle \forall m, n \in \Z_{>0}: \exists r \in \Z: \prod_{k \mathop = 1}^n \paren {m + k} = r \prod_{k \mathop = 1}^n k$


Equal Numbers are Congruent

:$\forall x, y, z \in \R: x = y \implies x \equiv y \pmod z$ where $x \equiv y \pmod z$ denotes congruence modulo $z$.


Complex Roots of Unity in Exponential Form

Let $n \in \Z$ be an integer such that $n > 0$. Let $z \in \C$ be a complex number such that $z^n = 1$. Then: :$U_n = \set {e^{2 i k \pi / n}: k \in \N_n}$ where $U_n$ is the set of $n$th roots of unity. That is: :$z \in \set {1, e^{2 i \pi / n}, e^{4 i \pi / n}, \ldots, e^{2 \paren {n - 1} i \pi / n} }$ Thus for every integer $n$, the number of $n$th roots of unity is $n$. Setting $\omega := e^{2 i \pi / n}$, $U_n$ can then be written as: :$U_n = \set {1, \omega, \omega^2, \ldots, \omega^{n - 1} }$


Roots of Unity under Multiplication form Cyclic Group

Let $n \in \Z$ be an integer such that $n > 0$. The $n$th complex roots of unity under the operation of multiplication form the cyclic group which is isomorphic to $C_n$.


Inverse Mapping is Unique

Let $f: S \to T$ be a mapping. If $f$ has an inverse mapping, then that inverse mapping is unique. That is, if: :$f$ and $g$ are inverse mappings of each other and :$f$ and $h$ are inverse mappings of each other then $g = h$.


Special Linear Group is Subgroup of General Linear Group

Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\SL {n, K}$ be the special linear group of order $n$ over $K$. Then $\SL {n, K}$ is a subgroup of the general linear group $\GL {n, K}$.


Determinant of Unit Matrix

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. The determinant of the unit matrix of order $n$ over $R$ is equal to $1_R$.


Determinant of Inverse Matrix

Let $K$ be a field whose zero is $0_K$ and whose unity is $1_K$. Let $\mathbf A$ be an invertible matrix of order $n$ over $K$. Then the determinant of its inverse is given by: :$\map \det {\mathbf A^{-1} } = \dfrac {1_K} {\map \det {\mathbf A} }$


Transposition is of Odd Parity

Let $S_n$ denote the set of permutations on $n$ letters. Let $\pi \in S_n$ be a transposition. Then $\pi$ is of odd parity.


Subset Product Action is Group Action

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\powerset G$ be the power set of $\struct {G, \circ}$. For any $S \in \powerset G$ and for any $g \in G$, the subset product action: :$\forall g \in G: \forall S \in \powerset G: g * S = g \circ S$ is a group action.


Subgroup of Elements whose Order Divides Integer

Let $A$ be an abelian group. Let $k \in \Z$ and $B$ be a set of the form: :$\left\{{x \in A : x^k = e}\right\}$ Then $B$ is a subgroup of $A$.


Abelian Group Factored by Prime

Let $G$ be a finite abelian group. Let $p$ be a prime. Factor the order of $G$ as: :$\order G = m p^n$ such that $p$ does not divide $m$. Then: :$G = H \times K$ where: :$H = \set {x \in G : x^{p^n} = e}$ and: :$K = \set {x \in G : x^m = e}$ Furthermore: :$\order H = p^n$


Abelian Group of Prime-power Order is Product of Cyclic Groups

Let $G$ be an abelian group of prime-power order. Let $a$ be an element of maximal order in $G$. Then $G$ can be written in the form $\gen a \times K$ for some $K \le G$.


Power of Product in Abelian Group

Let $\struct {G, \circ}$ be an abelian group. Then: :$\forall x, y \in G: \forall k \in \Z: \paren {x \circ y}^k = x^k \circ y^k$


Identity Element of Natural Number Addition is Zero

The identity element for the natural numbers under addition is zero ($0$): :$\forall n \in \N: 0 + n = n$


Simpson's Formulas

=== Cosine by Cosine === {{:Simpson's Formulas/Cosine by Cosine}} === Sine by Sine === {{:Simpson's Formulas/Sine by Sine}} === Sine by Cosine === {{:Simpson's Formulas/Sine by Cosine}} === Cosine by Sine === {{:Simpson's Formulas/Cosine by Sine}}


Prosthaphaeresis Formulas

=== Sine plus Sine === {{:Prosthaphaeresis Formulas/Sine plus Sine}} === Sine minus Sine === {{:Prosthaphaeresis Formulas/Sine minus Sine}} === Cosine plus Cosine === {{:Prosthaphaeresis Formulas/Cosine plus Cosine}} === Cosine minus Cosine === {{:Prosthaphaeresis Formulas/Cosine minus Cosine}}


Sum of Tangent and Cotangent

:$\tan x + \cot x = \sec x \csc x$


Tangent times Tangent plus Cotangent

:$\tan x \paren {\tan x + \cot x} = \sec^2 x$


Minor Trigonometric Identities

=== Sum of Tangent and Cotangent === {{:Sum of Tangent and Cotangent}} === Tangent times Tangent plus Cotangent === {{:Tangent times Tangent plus Cotangent}} === Secant Minus Cosine === {{:Secant Minus Cosine}} === Square of Tangent Minus Square of Sine === {{:Square of Tangent Minus Square of Sine}} === Difference of Fourth Powers of Cosine and Sine === {{:Difference of Fourth Powers of Cosine and Sine}} === Cosecant Minus Sine === {{:Cosecant Minus Sine}} === Cotangent Minus Tangent === {{:Cotangent Minus Tangent}} === Sum of Cosecant and Cotangent === {{:Sum of Cosecant and Cotangent}} === Sum of Squares of Secant and Cosecant === {{:Sum of Squares of Secant and Cosecant}} === Difference of Fourth Powers of Secant and Tangent === {{:Difference of Fourth Powers of Secant and Tangent}} === Reciprocal of One Plus Sine === {{:Reciprocal of One Plus Sine}} === Reciprocal of One Minus Sine === {{:Reciprocal of One Minus Sine}} === Sum of Reciprocals of One Plus and Minus Sine === {{:Sum of Reciprocals of One Plus and Minus Sine}} === Difference of Reciprocals of One Plus and Minus Sine === {{:Difference  of Reciprocals of One Plus and Minus Sine}} === Reciprocal of One Plus Cosine === {{:Reciprocal of One Plus Cosine}} === Reciprocal of One Minus Cosine === {{:Reciprocal of One Minus Cosine}} === Sum of Secant and Tangent === {{:Sum of Secant and Tangent}} === Cosine over Sum of Secant and Tangent === {{:Cosine over Sum of Secant and Tangent}} === Secant Plus One over Secant Squared === {{:Secant Plus One over Secant Squared}} === Sine Plus Cosine times Tangent Plus Cotangent === {{:Sine Plus Cosine times Tangent Plus Cotangent}} === Tangent over Secant Plus One === {{:Tangent over Secant Plus One}} === Squares of Linear Combination of Sine and Cosine === {{:Squares of Linear Combination of Sine and Cosine}} === Reciprocal of One Minus Secant === {{:Reciprocal of One Minus Secant}} === Reciprocal of One Plus Cosecant === {{:Reciprocal of One Plus Cosecant}}


Secant Minus Cosine

:$\sec x - \cos x = \sin x \tan x$


Square of Tangent Minus Square of Sine

:$\tan^2 x - \sin^2 x = \tan^2 x \ \sin^2 x$


Difference of Fourth Powers of Cosine and Sine

:$\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x$


Cosecant Minus Sine

:$\csc x - \sin x = \cos x \ \cot x$


Sum of Squares of Secant and Cosecant

:$\sec^2 x + \csc^2 x = \sec^2 x \csc^2 x$


Difference of Fourth Powers of Secant and Tangent

:$\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x$


Sum of Reciprocals of One Plus and Minus Sine

:$\dfrac 1 {1 - \sin x} + \dfrac 1 {1 + \sin x} = 2 \sec^2 x$


Difference of Reciprocals of One Plus and Minus Sine

:$\displaystyle \frac 1 {1 - \sin x} - \frac 1 {1 + \sin x} = 2 \tan x \ \sec x$


Sum of Secant and Tangent

:$\sec x + \tan x = \dfrac {1 + \sin x} {\cos x}$


Cosine over Sum of Secant and Tangent

:$\dfrac {\cos x} {\sec x + \tan x} = 1 - \sin x$


Secant Plus One over Secant Squared

:$\displaystyle \frac {\sec x + 1} {\sec^2 x} = \frac {\sin^2 x} {\sec x - 1}$


Sine Plus Cosine times Tangent Plus Cotangent

:$\paren {\sin x + \cos x} \paren {\tan x + \cot x} = \sec x + \csc x$


Tangent over Secant Plus One

:$\displaystyle \frac {\tan x} {\sec x + 1} = \frac {\sec x - 1} {\tan x}$


Squares of Linear Combination of Sine and Cosine

:$\left({a \cos x + b \sin x}\right)^2 + \left({b \cos x - a \sin x}\right)^2 = a^2 + b^2$


Reciprocal of One Minus Secant

:$\dfrac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3} = \dfrac 1 {1 - \sec x}$


Reciprocal of One Plus Cosecant

:$\dfrac {\cos^2 x + 3 \sin x - 1} {\cos^2 x + 2 \sin x + 2} = \dfrac 1 {1 + \csc x}$


Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE/General Result

Let $n \in \Z_{>0}$ be a strictly positive integer. The ordinary differential equation: :$a_n x^n \, \map {f^{\paren n} } x + \dotsb + a_1 x \, \map {f'} c + a_0 \, \map f x = 0$ can be transformed to a linear differential equation with constant coefficients by substitution $x = e^t$.


Condition for Composite Relation with Inverse to be Identity

Let $\RR \subseteq S \times T$ be a relation on $S \times T$. Let $\RR \circ \RR^{-1}$ be the composite of $\RR$ with its inverse. Let $I_T$ be the identity mapping on $T$. Then: :$\RR \circ \RR^{-1} = I_T$ {{iff}}: :$\RR$ is many-to-one and: :$\RR$ is right-total.


Set of All Relations is a Monoid

The set of all relations $\Bbb E = \left\{{\mathcal R: \mathcal R \subseteq S \times S}\right\}$ on a set $S$ forms a monoid in which the operation is composition of relations. The only invertible elements of $\Bbb E$ are permutations. If $\mathcal R$ is invertible, its inverse is $\mathcal R^{-1}$.


Inverses of Right-Total and Left-Total Relations

Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$. Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse of $\mathcal R$. Then:


Inverse Relation is Left and Right Inverse iff Bijection

Let $\RR \subseteq S \times T$ be a relation on a cartesian product $S \times T$. Let: :$I_S$ be the identity mapping on $S$ :$I_T$ be the identity mapping on $T$. Let $\RR^{-1}$ be the inverse relation of $\RR$. Then $\RR$ is a bijection {{iff}}: :$\RR^{-1} \circ \RR = I_S$ and :$\RR \circ \RR^{-1} = I_T$ where $\circ$ denotes composition of relations.


Integer Multiples Closed under Addition

Let $n \Z$ be the set of integer multiples of $n$. Then the algebraic structure $\struct {n \Z, +}$ is closed under addition.


Integer Multiples Closed under Multiplication

Let $n \Z$ be the set of integer multiples of $n$. Then the algebraic structure $\struct {n \Z, \times}$ is closed under multiplication.


Non-Zero Integers Closed under Multiplication

The set of non-zero integers is closed under multiplication.


Two-Step Subgroup Test using Subset Product

Let $G$ be a group. Let $\O\subset H \subseteq G$ be a non-empty subset of $G$. Then $H$ is a subgroup of $G$ {{iff}}: :$H H \subseteq H$ :$H^{-1} \subseteq H$ where: :$H H$ is the product of $H$ with itself :$H^{-1}$ is the inverse of $H$.


One-Step Subgroup Test using Subset Product

Let $G$ be a group. Let $\varnothing \subset H \subseteq G$ be a non-empty subset of $G$. Then $H$ is a subgroup of $G$ {{iff}}: :$H H^{-1} \subseteq H$ where: :$H^{-1}$ is the inverse of $H$ :$H H ^{-1}$ is the product of $H$ with $H^{-1}$.


Product of Subset with Union

Let $\left({G, \circ}\right)$ be an algebraic structure. Let $X, Y, Z \subseteq G$. Then: : $X \circ \left({Y \cup Z}\right) = \left({X \circ Y}\right) \cup \left({X \circ Z}\right)$ : $\left({Y \cup Z}\right) \circ X = \left({Y \circ X}\right) \cup \left({Z \circ X}\right)$ where $X \circ Y$ denotes the subset product of $X$ and $Y$.


Hilbert's Nullstellensatz

Let $k$ be an algebraically closed field. Let $n \geq 0$ be an natural number. Let $k \left[{x_1,\ldots, x_n}\right]$ be the polynomial ring in $n$ variables over $k$. Then for every ideal $J \subseteq k \left[{x_1,\ldots, x_n}\right]$, the associated ideal of its zero-locus equals its radical: :$I \left({Z \left({J}\right)}\right) = \sqrt J$


Subset of Domain is Subset of Preimage of Image

Let $f: S \to T$ be a mapping. Then: :$A \subseteq S \implies A \subseteq \paren {f^{-1} \circ f} \sqbrk A$ where: :$f \sqbrk A$ denotes the image of $A$ under $f$ :$f^{-1} \sqbrk A$ denotes the preimage of $A$ under $f$ :$f^{-1} \circ f$ denotes composition of $f^{-1}$ and $f$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\forall A \in \powerset S: A \subseteq \map {\paren {f^\gets \circ f^\to} } A$


Clairaut's Differential Equation

'''Clairaut's differential equation''' is a first order ordinary differential equation which can be put into the form: :$y = x y' + \map f {y'}$ Its general solution is: :$y = C x + \map f C$ where $C$ is a constant.


Noether Normalization Lemma

Let $k$ be a field. Let $A$ be a non-trivial finitely generated $k$-algebra. {{explain|the above link is for Definition:Non-Trivial Ring -- we need to define a Definition:Non-Trivial Algebra}} Then there exists $n \in \N$ and a finite injective morphism of $k$-algebra: :$k \sqbrk {x_1, \dotsc, x_n} \to A$ {{DefinitionWanted|Instead of using Definition:Finite Ring Homomorphism, another page is to be generated defining the module-theory version.}}


Image of Preimage under Mapping

Let $f: S \to T$ be a mapping. Then: :$B \subseteq T \implies \paren {f \circ f^{-1} } \sqbrk B = B \cap f \sqbrk S$


Isomorphism of External Direct Products

Let: :$\struct {S_1 \times S_2, \circ}$ be the external direct product of two algebraic structures $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ :$\struct {T_1 \times T_2, *}$ be the external direct product of two algebraic structures $\struct {T_1, *_1}$ and $\struct {T_2, *_2}$ :$\phi_1$ be an isomorphism from $\struct {S_1, \circ_1}$ onto $\struct {T_1, *_1}$ :$\phi_2$ be an isomorphism from $\struct {S_2, \circ_2}$ onto $\struct {T_2, *_2}$. Then the mapping $\phi_1 \times \phi_2: \struct {S_1 \times S_2, \circ} \to \struct {T_1 \times T_2, *}$ defined as: :$\map {\paren {\phi_1 \times \phi_2} } {x, y} = \tuple {\map {\phi_1} x, \map {\phi_2} y}$ is an isomorphism from $\struct {S_1 \times S_2, \circ}$ to $\struct {T_1 \times T_2, *}$.


Dual of Lattice Ordering is Lattice Ordering

Let $\struct {S, \preccurlyeq}$ be a lattice. Let $\preccurlyeq$ be the lattice ordering on $\struct {S, \preccurlyeq}$. Then its dual ordering $\succcurlyeq$ is also a lattice ordering.


Order Isomorphism on Totally Ordered Set preserves Total Ordering

Let $\left({S, \preccurlyeq_1}\right)$ and $\left({T, \preccurlyeq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preccurlyeq_1}\right) \to \left({T, \preccurlyeq_2}\right)$ be an order isomorphism. Then $\left({S, \preccurlyeq_1}\right)$ is a totally ordered set {{iff}} $\left({T, \preccurlyeq_2}\right)$ is also a totally ordered set.


Order Isomorphism on Well-Ordered Set preserves Well-Ordering

Let $\struct {S_1, \preccurlyeq_1}$ and $\struct {S_2, \preccurlyeq_2}$ be ordered sets. Let $\phi: \struct {S_1, \preccurlyeq_1} \to \struct {S_2, \preccurlyeq_2}$ be an order isomorphism. Then $\struct {S_1, \preccurlyeq_1}$ is a well-ordered set {{iff}} $\struct {S_2, \preccurlyeq_2}$ is also a well-ordered set.


Order Isomorphism on Lattice preserves Lattice Structure

Let $\left({S, \preccurlyeq_1}\right)$ and $\left({T, \preccurlyeq_2}\right)$ be ordered sets. Let $\phi: \left({S, \preccurlyeq_1}\right) \to \left({T, \preccurlyeq_2}\right)$ be an order isomorphism. Then $\left({S, \preccurlyeq_1}\right)$ is a lattice iff $\left({T, \preccurlyeq_2}\right)$ is also a lattice.


Set of Subgroups forms Complete Lattice

Let $\struct {G, \circ}$ be a group, and let $\mathbb G$ be the set of all subgroups of $G$. Then $\struct {\mathbb G, \subseteq}$ is a complete lattice.


Infimum and Supremum of Subgroups

Let $\left({G, \circ}\right)$ be a group. Let $\mathbb G$ be the set of all subgroups of $G$. Let $\left({\mathbb G, \subseteq}\right)$ be the complete lattice formed by $\mathbb G$ and $\subseteq$. Let $H, K \in \mathbb G$.


Index Laws/Sum of Indices/Monoid

Let $\left({S, \circ}\right)$ be a monoid whose identity element is $e$. For $a \in S$, let $\circ^n a = a^n$ be defined as the $n$th power of $a$: :$a^n = \begin{cases} e & : n = 0 \\ a^x \circ a & : n = x + 1 \end{cases}$ That is: :$a^n = \underbrace{a \circ a \circ \cdots \circ a}_{n \text{ copies of } a} = \circ^n \left({a}\right)$ while: :$a^0 = e$ Then: :$\forall m, n \in \N: a^{n + m} = a^n \circ a^m$


Power of Product of Commuting Elements in Semigroup equals Product of Powers

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


Sum of Sequences of Fifth and Seventh Powers

:$\displaystyle \sum_{i \mathop = 1}^n i^5 + \sum_{i \mathop = 1}^n i^7 = 2 \paren {\sum_{i \mathop = 1}^n i}^4$


Cardinality of Set of Strictly Increasing Mappings

Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be tosets. Let the cardinality of $S$ and $T$ be: :$\card S = m, \card T = n$ Then the number of strictly increasing mappings from $S$ to $T$ is: :$\dbinom n m = \dfrac {n!} {m! \ \paren {n - m}!}$. where $\dbinom n m$ is a binomial coefficient.


Construction of Inverse Completion/Equivalence Relation

The cross-relation $\boxtimes$ is an equivalence relation on $\left({S \times C, \oplus}\right)$.


Construction of Inverse Completion/Properties of Quotient Structure

=== Identity of Quotient Structure === {{:Construction of Inverse Completion/Identity of Quotient Structure}} === Invertible Elements in Quotient Structure === {{:Construction of Inverse Completion/Invertible Elements in Quotient Structure}} === Generator for Quotient Structure === {{:Construction of Inverse Completion/Generator for Quotient Structure}} === Quotient Structure is Inverse Completion === {{:Construction of Inverse Completion/Quotient Structure is Inverse Completion}}


Integers form Commutative Ring

The set of integers $\Z$ forms a commutative ring under addition and multiplication.


Symmetric Group is Group

Let $S$ be a set. Let $\map \Gamma S$ denote the set of all permutations on $S$. Then $\struct {\map \Gamma S, \circ}$, the symmetric group on $S$, forms a group.


Quotient Ring Defined by Ring Itself is Null Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\struct {R / R, +, \circ}$ be the quotient ring defined by $R$. Then $\struct {R / R, +, \circ}$ is a null ring.


Set of Subrings forms Complete Lattice

Let $\struct {K, +, \circ}$ be a ring. Let $\mathbb K$ be the set of all subrings of $K$. Then $\struct {\mathbb K, \subseteq}$ is a complete lattice.


Set of Ideals forms Complete Lattice

Let $\struct {K, +, \circ}$ be a ring. Let $\mathbb K$ be the set of all ideals of $K$. Then $\struct {\mathbb K, \subseteq}$ is a complete lattice.


Subset Product of Abelian Subgroups

Let $\left({G, \circ}\right)$ be an abelian group. Let $H_1$ and $H_2$ be subgroups of $G$. Then $H_1 \circ H_2$ is a subgroup of $G$.


Sum of Ideals is Ideal

Let $J_1$ and $J_2$ be ideals of a ring $\struct{R, +, \circ}$. Then: : $J = J_1 + J_2$ is an ideal of $R$ where $J_1 + J_2$ is as defined in subset product.


Principal Ideal from Element in Center of Ring

Let $\left({R, +, \circ}\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $b \in R$ be in the center of $R$. Then: :$\left({b}\right) = R \circ b = \left\{{x \circ b: x \in R}\right\}$ where $\left({b}\right)$ is the principal ideal generated by $b$.


External Direct Sum of Rings is Ring

Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings. Then their (external) direct product: : $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$ is a ring.


External Direct Product Distributivity

Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be ringoids. Let their external direct product be $\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$. Then the operation $\circ$ is distributive over $+$.


Ideal of External Direct Sum of Rings

Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be rings. Let  :$\displaystyle \left({R, +, \circ}\right) = \prod_{k \mathop = 1}^n \left({R_k, +_k, \circ_k}\right)$  be their direct product. For each $k \in \left[{1 \,.\,.\, n}\right]$, let: :$R\,'_k = \left\{{\left({x_1, \ldots, x_n}\right) \in R: \forall j \ne k: x_j = 0}\right\}$ Then: : $\forall k \in \left[{1 \,.\,.\, n}\right]: R\,'_k$ is an ideal of $R$.


Canonical Injection from Ideal of External Direct Sum of Rings

Let $\struct {R_1, +_1, \circ_1}, \struct {R_2, +_2, \circ_2}, \dotsc, \struct {R_n, +_n, \circ_n}$ be rings. Let $\displaystyle \struct {R, +, \circ} = \prod_{k \mathop = 1}^n \struct {R_k, +_k, \circ_k}$ be their external direct product. For each $k \in \closedint 1 n$, let: :$R'_k = \set {\tuple {x_1, \dotsc, x_n} \in R: \forall j \ne k: x_j = 0}$ Let $\inj_k: R_k \to R$ be the canonical injection on the $k$th coordinate from $R_k$ into $\struct {R, +, \circ}$. Let $\pr_k: R \to R'_k$ be the projection on the $k$th coordinate of $\struct {R, +, \circ}$ onto $R'_k$. Then: :$\inj_k: R_k \to R'_k$ is an isomorphism :Its inverse is the restriction of $\pr_k$ to $R'_k$.


Conditions for Internal Ring Direct Sum

Let $\struct {R, +, \circ}$ be a ring. Let $\sequence {\struct {S_k, +, \circ} }$ be a sequence of subrings of $R$. Then $R$ is the ring direct sum of $\sequence {S_k}_{1 \mathop \le k \mathop \le n}$ {{iff}}: :$(1): \quad R = S_1 + S_2 + \cdots + S_n$ :$(2): \quad \sequence {\struct {S_k, +} }_{1 \mathop \le k \mathop \le n}$ is a sequence of independent subgroups of $\struct {R, +}$ :$(3): \quad \forall k \in \closedint 1 n: S_k$ is an ideal of $R$.


Structure Induced by Ring Operations is Ring

Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Then $\struct {R^S, +', \circ'}$ is a ring, where $+'$ and $\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\circ$.


Division Subring Test

Let $\struct {K, +, \circ}$ be a division ring, and let $L$ be a subset of $K$. Then $\struct {L, +, \circ}$ is a division subring of $\struct {K, +, \circ}$ {{iff}} these all hold: :$(1) \quad L^* \ne \O$ :$(2) \quad \forall x, y \in L: x + \paren {-y} \in L$ :$(3) \quad \forall x, y \in L: x \circ y \in L$ :$(4) \quad x \in L^* \implies x^{-1} \in L^*$


Homomorphism from Integers into Ring with Unity

Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the characteristic of $R$ be $p$. For any $a \in R$, we define the mapping $g_a: \Z \to R$ from the integers into $R$ as: :$\forall n \in \Z: \map {g_a} n = n \cdot a$ Then $g_a$ is a group homomorphism from $\struct {\Z, +}$ to $\struct {R, +}$. Also: :$\ideal p \subseteq \map \ker {g_a}$ where: :$\map \ker {g_a}$ is the kernel of $g_a$; :$\ideal p$ is the principal ideal of $\Z$ generated by $p$. Also: :$p \divides n \implies n \cdot a = 0$ where $p \divides n$ denotes that $p$ is a divisor of $n$.


Properties of Prime Subfield

Let $F$ be a field. Let $K$ be the prime subfield of $F$. Then $K$ is isomorphic to either: :$\Q$, the field of rational numbers, or :$\Z_p$, the Ring of Integers Modulo $p$, where $p$ is prime.


Subgroup of Additive Group Modulo m is Ideal of Ring

Let $m \in \Z: m > 1$. Let $\struct {\Z_m, +_m}$ be the additive group of integers modulo $m$. Then every subgroup of $\struct {\Z_m, +_m}$ is an ideal of the ring of integers modulo $m$ $\struct {\Z_m, +_m, \times_m}$.


Bijection from Divisors to Subgroups of Cyclic Group

Let $G$ be a cyclic group of order $n$ generated by $a$. Let $S = \left\{{m \in \Z_{>0}: m \divides n}\right\}$ be the set of all divisors of $n$. Let $T$ be the set of all subgroups of $G$ Let $\phi: S \to T$ be the mapping defined as: :$\phi: m \to \left\langle{a^{n/m}}\right\rangle$ where  $\left\langle{a^{n/m}}\right\rangle$ is the subgroup generated by $a^{n/m}$. Then $\phi$ is a bijection.


Generator of Additive Group Modulo m iff Unit of Ring

Let $m \in \Z: m > 1$. Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$. Let $\struct {\Z_m, +_m, \times_m}$ be the ring of integers modulo $m$. Let $a \in \Z_m$. Then: :$a$ is a generator of $\struct {\Z_m, +_m}$  {{iff}} :$a$ is a unit of $\struct {\Z_m, +_m, \times_m}$


Subspace of Real Differentiable Functions

Let $\mathbb J$ be an open interval of the real number line $\R$. Let $\map \DD {\mathbb J}$ be the set of all differentiable real functions on $\mathbb J$. Then $\struct {\map \DD {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.


Subspace of Riemann Integrable Functions

Let $\mathbb J = \set {x \in \R: a \le x \le b}$ be a closed interval of the real number line $\R$. Let $\map \RR {\mathbb J}$ be the set of all Riemann integrable functions on $\mathbb J$. Then $\struct {\map \RR {\mathbb J}, +, \times}_\R$ is a subspace of the $\R$-vector space $\struct {\R^{\mathbb J}, +, \times}_\R$.


Basis of Vector Space of Polynomial Functions

Let $B$ be the set of all the identity functions $I^n$ on $\R^n$ where $n \in \N^*$. Then $B$ is a basis of the $\R$-vector space $\map P \R$ of all polynomial functions on $\R$.


Ordered Basis for Coordinate Plane

Let $a_1, a_2 \in \R^2$ such that $\left\{{a_1, a_2}\right\}$ forms a linearly independent set. Then $\left({a_1, a_2}\right)$ is an ordered basis for the $\R$-vector space $\R^2$. Hence the points on the plane can be uniquely identified by means of linear combinations of $a_1$ and $a_2$.


Differentiation on Polynomials is Linear Operator

Let $P \left({\R}\right)$ be the vector space of all polynomial functions on the real number line $\R$. Then the differentiation operator $D$ on $P \left({\R}\right)$ is a linear operator.


Noether's Theorem (Calculus of Variations)

Let $y_i$, $F$, $\Psi_i$, $\Phi$ be real functions. Let $x, \epsilon \in \R$. Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$ and $\mathbf \Psi = \sequence{\Psi_i}_{1 \mathop \le i \mathop \le n}$ be vectors. Let  :$\Phi = \map \Phi {x, \mathbf y, \mathbf y'; \epsilon}, \quad \Psi_i = \map {\Psi_i} {x, \mathbf y, \mathbf y'; \epsilon}$  such that: :$\map \Phi {x, \mathbf y, \mathbf y'; 0} = x, \quad \map {\Psi_i} {x, \mathbf y, \mathbf y'; 0} = y_i$ where $x, \mathbf y, \mathbf y', \epsilon$ are variables. Let: :$\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ be a functional. Let: :$X = \map \Phi {x, \mathbf y, \mathbf y'; \epsilon}, \quad \mathbf Y = \map {\mathbf \Psi} {x, \mathbf y, \mathbf y'; \epsilon}$ Suppose, $J \sqbrk {\mathbf y}$ is invariant under transformations $x \rightarrow X$ and $\mathbf y \rightarrow \mathbf Y$ for arbitrary $x_0$ and $x_1$. Then: :$\nabla_{\mathbf y'} F \cdot \boldsymbol \psi + \paren {F - \mathbf y' \cdot \nabla_{\mathbf y'} F} \phi = C$ where $C$ is a constant and: :$\map {\boldsymbol \psi} {x, \mathbf y, \mathbf y'} = \dfrac {\partial \map {\mathbf \Psi} {x, \mathbf y, \mathbf y'; \epsilon} } {\partial \epsilon} \Bigg\rvert_{\epsilon \mathop = 0}$ :$\map \phi {x, \mathbf y, \mathbf y'} = \dfrac {\partial \map \Phi {x, \mathbf y, \mathbf y'; \epsilon} } {\partial \epsilon} \Bigg\rvert_{\epsilon \mathop = 0}$


Equivalence of Definitions of Supremum of Real-Valued Function

Let $S \subseteq \R$ be a subset of the real numbers. Let $f: S \to \R$ be a real function on $S$. {{TFAE|def = Supremum of Real-Valued Function}}


Supremum does not Precede Infimum

Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$ admit both a supremum $M$ and an infimum $m$. Then $m \preceq M$.


Integer Multiples form Commutative Ring

Let $n \Z$ be the set of integer multiples of $n$. Then $\struct {n \Z, +, \times}$ is a commutative ring. Unless $n = 1$, $\struct {n \Z, +, \times}$ is not a ring with unity.


Quaternions Defined by Matrices

Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\mathcal M_\C} 2$: :$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} $ where $\C$ is the set of complex numbers. Then $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ are related to each other in the following way: {{begin-eqn}} {{eqn | l = \mathbf i \mathbf j = - \mathbf j \mathbf i       | r = \mathbf k }} {{eqn | l = \mathbf j \mathbf k = - \mathbf k \mathbf j       | r = \mathbf i }} {{eqn | l = \mathbf k \mathbf i = - \mathbf i \mathbf k       | r = \mathbf j }} {{eqn | l = \mathbf i^2 = \mathbf j^2 = \mathbf k^2 = \mathbf i \mathbf j \mathbf k       | r = -\mathbf 1 }} {{end-eqn}}


Matrix Form of Quaternion

Let $\mathbf x$ be a quaternion such that: : $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ When the quaternion basis is expressed in the form of matrices: :$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$ the general quaternion $\mathbf x$ has the form: :$\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$ or: :$\mathbf x = \begin{bmatrix} w & z \\ -\overline z & \overline w \end{bmatrix}$ where : :$w$ and $z$ are complex numbers :$\overline z$ is the complex conjugate of $z$.


Quaternion Multiplication

Let $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ be quaternions. Then their '''product''' is given by: {{begin-eqn}} {{eqn | ll=\mathbf x_1 \mathbf x_2       | l==       | o=       | r=\left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) \mathbf 1       | c= }} {{eqn | o=+       | r=\left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) \mathbf i       | c= }} {{eqn | o=+       | r=\left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) \mathbf j       | c= }} {{eqn | o=+       | r=\left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) \mathbf k       | c= }} {{end-eqn}}


Quaternion Addition forms Abelian Group

Let $\mathbb H$ be the set of quaternions. Then $\left({\mathbb H, +}\right)$, where $+$ denotes quaternion addition, is an abelian group.


Ring of Quaternions is Ring

The set $\mathbb H$ of quaternions forms a ring under the operations of addition and multiplication.


Quaternions Subring of Complex Matrix Space

The Ring of Quaternions is a subring of the matrix space $\mathcal M_\C \left({2}\right)$.


Product of Quaternion with Conjugate

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. Then their product is given by: :$\mathbf x \overline {\mathbf x} = \paren {a^2 + b^2 + c^2 + d^2} \mathbf 1 = \overline {\mathbf x} \mathbf x$


Multiplicative Inverse of Quaternion

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion such that $\mathbf x \ne \mathbf 0$. Then $\mathbf x$ has an inverse $\mathbf x^{-1}$ under the operation of quaternion multiplication: :$\mathbf x^{-1} = \lambda \overline {\mathbf x}$ where: :$\lambda = \dfrac 1 {a^2 + b^2 + c^2 + d^2} \mathbf 1$


Multiplicative Identity for Quaternions

In the set of quaternions $\mathbb H$, the element: :$\mathbf 1 + 0 \mathbf i + 0 \mathbf j + 0 \mathbf k$ serves as the identity element for quaternion multiplication. This element is written $\mathbf 1$.


Quaternions form Skew Field

The set $\H$ of quaternions forms a skew field under the operations of addition and multiplication.


Cancellation Law of Ring Product of Integral Domain

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $D^*$ denote $D \setminus \set {0_D}$, that is, $D$ without its zero. Let $a \in D^*$. Then: :$\forall x, y \in D: a \circ x = a \circ y \implies x = y$ That is, all elements of $D^*$ are cancellable for the ring product.


Gaussian Integers form Integral Domain

The ring of Gaussian integers $\struct {\Z \sqbrk i, +, \times}$ is an integral domain.


Set Intersection Not Cancellable

Let $S$ be a set and let $\mathcal P \left({S}\right)$ be the power set of $S$. Let $S_1, S_2, T \in \mathcal P \left({S}\right)$. Suppose that $S_1 \cap T = S_2 \cap T$. Then it is not necessarily the case that $S_1 = S_2$.


Sum of Triangular Matrices

Let $\mathbf A = \left[{a}\right]_{n}, \mathbf B = \left[{b}\right]_{n}$ be square matrices of order $n$. Let $\mathbf C = \mathbf A + \mathbf B$ be the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, then so is $\mathbf C$. If $\mathbf A$ and $\mathbf B$ are lower triangular matrices, then so is $\mathbf C$.


Negative of Triangular Matrix

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $-\mathbf A$ be the negative of $\mathbf A$. If $\mathbf A$ is an upper triangular matrix, then so is $-\mathbf A$. If $\mathbf A$ is a lower triangular matrix, then so is $-\mathbf A$.


Natural Numbers form Subsemiring of Integers

The semiring of natural numbers $\struct {\N, +, \times}$ forms a subsemiring of the ring of integers $\struct {\Z, +, \times}$.


Integers form Subdomain of Rationals

The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of rational numbers.


Real Numbers form Subfield of Complex Numbers

The field of real numbers $\struct {\R, +, \times}$ forms a subfield of the field of complex numbers $\struct {\C, +, \times}$.


Sum of All Ring Products is Additive Subgroup

Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +}\right)$ and $\left({T, +}\right)$ be additive subgroups of $\left({R, +, \circ}\right)$. Let $S + T$ be defined as subset product. Let $S T$ be defined as: :$\displaystyle S T = \left\{{\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \left[{1 \,.\,.\ n}\right]}\right\}$ Then both $S + T$ and $S T$ are additive subgroups of $\left({R, +, \circ}\right)$.


Sum of All Ring Products is Closed under Addition

Let $\left({R, +, \circ}\right)$ be a ring. Let $\left({S, +}\right)$ and $\left({T, +}\right)$ be additive subgroups of $\left({R, +, \circ}\right)$. Let $S T$ be defined as: :$\displaystyle S T = \left\{{\sum_{i \mathop = 1}^n s_i \circ t_i: s_1 \in S, t_i \in T, i \in \left[{1 \,.\,.\ n}\right]}\right\}$ Then $\left({S T, +}\right)$ is a closed subset of $\left({R, +}\right)$.


Complex Numbers form Subfield of Quaternions

The field of complex numbers $\left({\C, +, \times}\right)$ is isomorphic to the subfields of the quaternions $\left({\mathbb H, +, \times}\right)$ whose underlying subsets are: :$(1): \quad \mathbb H_\mathbf i = \left\{{a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k \in \mathbb H: c = d = 0}\right\}$ :$(2): \quad \mathbb H_\mathbf j = \left\{{a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k \in \mathbb H: b = d = 0}\right\}$ :$(3): \quad \mathbb H_\mathbf k = \left\{{a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k \in \mathbb H: b = c = 0}\right\}$ That is: :$(1): \quad \mathbb H_\mathbf i = \left\{{a \mathbf 1 + b \mathbf i \in \mathbb H}\right\}$ :$(2): \quad \mathbb H_\mathbf j = \left\{{a \mathbf 1 + c \mathbf j \in \mathbb H}\right\}$ :$(3): \quad \mathbb H_\mathbf k = \left\{{a \mathbf 1 + d \mathbf k \in \mathbb H}\right\}$


Ring Homomorphism whose Kernel contains Ideal

Let $R$ be a ring. Let $J$ be an ideal of $R$. Let $\nu: R \to R / J$ be the quotient epimorphism. Let $\phi: R \to S$ be a ring homomorphism such that: :$J \subseteq \map \ker \phi$ where $\map \ker \phi$ is the kernel of $\phi$. Then there exists a unique ring homomorphism $\psi: R / J \to S$ such that: :$\phi = \psi \circ \nu$ where $\circ$ denotes composition of mappings. :300px Also: :$\map \ker \psi = \map \ker \phi / J$


Decay Equation

The first order ordinary differential equation: :$\dfrac {\d y} {\d x} = k \paren {y_a - y}$ where $k \in \R: k > 0$ has the general solution: :$y = y_a + C e^{-k x}$ where $C$ is an arbitrary constant. If $y = y_0$ at $x = 0$, then: :$y = y_a + \paren {y_0 - y_a} e^{-k x}$ This differential equation is known as the '''decay equation'''.


Relation on Empty Set is Equivalence

Let $S = \varnothing$, that is, the empty set. Let $\mathcal R \subseteq S \times S$ be a relation on $S$. Then $\mathcal R$ is the null relation and is an equivalence relation.


Increasing Union of Subrings is Subring

Let $R$ be a ring. Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \ldots \subseteq S_i \subseteq \ldots$ be subrings of $R$. Then the increasing union $S$: :$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$ is a subring of $R$.


Subrings of Integers are Sets of Integer Multiples

Let $\struct {\Z, +, \times}$ be the integral domain of integers. The subrings of $\struct {\Z, +, \times}$ are the rings of integer multiples: :$\struct {n \Z, +, \times}$ where $n \in \Z: n \ge 0$. There are no other subrings of $\struct {\Z, +, \times}$ but these.


Subring of Integers is Ideal

Every subring of $\struct {\Z, +, \times}$ is an ideal of the ring $\struct {\Z, +, \times}$.


Polynomial Ring of Sequences is Ring

Let $R$ be a ring. Let $P \left[{R}\right]$ be the polynomial ring over sequences in $R$. Then $P \left[{R}\right]$ is itself a ring.


Integer Addition is Closed

The set of integers is closed under addition: :$\forall a, b \in \Z: a + b \in \Z$


Integer Addition Identity is Zero

The identity of integer addition is $0$: :$\exists 0 \in \Z: \forall a \in \Z: a + 0 = a = 0 + a$


Inverses for Integer Addition

Each element $x$ of the set of integers $\Z$ has an inverse element $-x$ under the operation of integer addition: :$\forall x \in \Z: \exists -x \in \Z: x + \paren {-x} = 0 = \paren {-x} + x$


Rational Numbers form Ring

The set of rational numbers $\Q$ forms a ring under addition and multiplication: $\struct {\Q, +, \times}$.


Real Numbers form Ring

The set of real numbers $\R$ forms a ring under addition and multiplication: $\struct {\R, +, \times}$.


Complex Numbers form Ring

The set of complex numbers $\C$ forms a ring under addition and multiplication: $\struct {\C, +, \times}$.


Complex Addition Identity is Zero

Let $\C$ be the set of complex numbers. The identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$.


Rational Addition Identity is Zero

The identity of rational number addition is $0$: :$\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$


Inverse for Complex Addition

Let $z = x + i y \in \C$ be a complex number. Let $-z = -x - i y \in \C$ be the negative of $z$. Then $-z$ is the inverse element of $z$ under the operation of complex addition: :$\forall z \in \C: \exists -z \in \C: z + \paren {-z} = 0 = \paren {-z} + z$


Equal Elements of Field of Quotients

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $\struct {K, +, \circ}$ be the field of quotients of $\struct {D, +, \circ}$. Let $x = \dfrac p q \in K$. Then: :$\forall k \in D^*: x = \dfrac {p \circ k} {q \circ k}$ where: :$D^* := D \setminus \set {0_D}$ that is, $D$ with its zero removed.


Rational Addition is Closed

The operation of addition on the set of rational numbers $\Q$ is well-defined and closed: :$\forall x, y \in \Q: x + y \in \Q$


Rational Multiplication is Closed

The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed: :$\forall x, y \in \Q: x \times y \in \Q$


Rational Multiplication Identity is One

The identity of rational number multiplication is $1$: :$\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$


Inverses for Rational Multiplication

Each element $x$ of the set of non-zero rational numbers $\Q^*$ has an inverse element $\dfrac 1 x$ under the operation of rational number multiplication: :$\displaystyle \forall x \in \Q^*: \exists \frac 1 x \in \Q^*: x \times \frac 1 x = 1 = \frac 1 x \times x$


Real Addition Identity is Zero

The identity of real number addition is $0$: :$\exists 0 \in \R: \forall x \in \R: x + 0 = x = 0 + x$


Inverses for Real Addition

Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition: :$\forall x \in \R: \exists -x \in \R: x + \paren {-x} = 0 = \paren {-x} + x$


Real Multiplication Identity is One

The identity element of real number multiplication is the real number $1$: :$\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$


Inverses for Real Multiplication

Each element $x$ of the set of non-zero real numbers $\R_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication: :$\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$


Complex Multiplication Identity is One

Let $\C_{\ne 0}$ be the set of complex numbers without zero. The identity element of $\struct {\C_{\ne 0}, \times}$ is the complex number $1 + 0 i$.